Quantum Periods for Certain Four-Dimensional Fano Manifolds

ABSTRACT

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles.

KEYWORDS:

• Fano manifolds
• $\text{four}$-folds
• mirror symmetry
• quantum period

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 14J45, 14N35
• Secondary 14J33

1. Introduction

In this paper, we take the first step toward implementing a program, laid out in [Coates et al. 12], to find and classify four-dimensional Fano manifolds using mirror symmetry. We compute quantum periods and quantum differential equations for many known four-dimensional Fano manifolds, using techniques described in [Coates et al. 16]. Our basic reference for the theory of Fano manifolds is the book by Iskovskikh–Prokhorov [Iskovskikh and Prokhorov 99]. Recall that the index of a Fano manifold X is the largest integer r such that − K_X = rH for some ample divisor H. A four-dimensional Fano manifold has index at most 5 [Shokurov 85]. Four-dimensional Fano manifolds with index r > 1 have been classified. In what follows we compute the quantum periods and quantum differential equations for all four-dimensional Fano manifolds of index r > 1, for all four-dimensional Fano toric manifolds, and for certain other four-dimensional Fano manifolds of index 1.

Highlights

We draw the reader’s attention to:

• Section 6.2.4, where new tools for computing Gromov–Witten invariants (twisted I-functions for toric complete intersections [Coates et al. 14] and an improved Quantum Lefschetz theorem [Coates 14]) make a big practical difference to the computation of quantum periods. This should be contrasted with [Coates et al. 16, Section 19], where the new techniques were not available.

• Section 6.2.11, which relies on a new construction of Szurek–Wiśniewski’s null-correlation bundle [Szurek and Wiśniewski 90] that may be of independent interest.

• The tables of regularized quantum period sequences in Appendix A.

• The numerical calculation of quantum differential operators in Section 9 and Appendix B. This suggests in particular that, for each four-dimensional Fano manifold X with Fano index r > 1, the regularized quantum differential equation of X is either extremal or of low ramification.

• Section 6.2.17 and Section B.34, which together give an example of a product such that the regularized quantum differential equation for each factor is extremal, but the regularized quantum differential equation for the product itself is not.

This paper is accompanied by fully commented source code, written in the computational algebra system Magma [Bosma et al. 97]. This will allow the reader to verify the calculations presented here, or to perform similar computations.

2. Methodology

The quantum period G_X of a Fano manifold X is a generating function (2--1) $$G_X\left(t\right)=1+\sum _{d=1}^{\infty }{c}_d{t}^{d}t\in \mathbb{C}$$(2--1) for certain genus-zero Gromov–Witten invariants c_d of X. A precise definition can be found in [Coates et al. 16, Section B], but roughly speaking c_d is the “virtual number” of degree-d rational curves C in X that pass through a given point and satisfy certain constraints on their complex structure. (The degree of a curve C here is the quantity ⟨ − K_X, C⟩.) The quantum period is discussed in detail in [Coates et al. 12, Coates et al. 16]; one property that will be important in what follows is that the regularized quantum period (2--2) $${\widehat{G}}_X\left(t\right)=1+\sum _{d=1}^{\infty }d!c_d{t}^{d}t\in \mathbb{C},\left|t\right|\ll \infty$$(2--2) satisfies a differential equation called the regularized quantum differential equation of X: (2--3) $$L_X{\widehat{G}}_X\equiv 0L_X=\sum _{m=0}^{m=N}{p}_m\left(t\right)D^m,$$(2--3) where the p_m are polynomials and $D=t\frac{d}{dt}$. It is expected that the regularized quantum differential equation for a Fano manifold X is extremal or of low ramification, as described in Section 9 below. This is a strong constraint on the Gromov–Witten invariants c_d of X.

Quantum periods for a broad class of toric complete intersections can be computed using Givental’s mirror theorem [Givental 96].

Theorem 2.1 ([Coates et al. 16, Corollary C.2]).

Let X be a toric Fano manifold and let $D_1,\dots ,D_N\in H^2(X;\mathbb{Q})$ be the cohomology classes Poincaré-dual to the torus-invariant divisors on X. The quantum period of X is $$\begin{array}{c}\hfill G_X\left(t\right)=\sum _{\begin{array}{c}\beta \in H_2(X;\mathbb{Z}):\\ ⟨\beta ,D_i⟩\ge 0\forall i\end{array}}\frac{t^{⟨\beta ,-K_X⟩}}{{\prod }_{i=1}^N⟨\beta ,D_i⟩!}.\end{array}$$

Theorem 2.2 ([Coates et al. 16, Corollary D.5]).

Let Y be a toric Fano manifold, and let $D_1,\dots ,D_N\in H^2(Y;\mathbb{Q})$ be the cohomology classes Poincaré-dual to the torus-invariant divisors on Y. Let X be the complete intersection in Y defined by a regular section of E = L₁⊕⋅⋅⋅⊕L_s where each L_i is a nef line bundle, and let ρ_i = c₁(L_i), 1 ⩽ i ⩽ s. Suppose that the class c₁(Y) − Λ is ample on Y, where Λ = c₁(L₁) + ⋅⋅⋅ + c₁(L_s). Then X is Fano, and the quantum period of X is $$\begin{array}{c}\hfill G_X\left(t\right)=e^{-ct}\sum _{\begin{array}{c}\beta \in H_2(Y;\mathbb{Z}):\\ ⟨\beta ,D_i⟩\ge 0\forall i\end{array}}{t}^{⟨\beta ,-K_Y-\Lambda ⟩}\frac{{\prod }_{j=1}^s⟨\beta ,{\rho }_j⟩!}{{\prod }_{i=1}^N⟨\beta ,D_i⟩!}\end{array}$$where c is the unique rational number such that the right-hand side has the form 1 + O(t²).

An analogous mirror theorem holds for certain complete intersections in toric Deligne–Mumford stacks, but we will need only the case where the ambient stack is a weighted projective space.

Theorem 2.3 ([Coates et al. 16, Proposition D.9]).

Let Y be the weighted projective space $\mathbb{P}(w_0,\dots ,w_n)$, let X be a smooth Fano manifold given as a complete intersection in Y defined by a section of $E=\mathcal{O}\left(d_1\right)\oplus \cdots \oplus \mathcal{O}\left(d_m\right)$, and let − k = w₀ + ⋅⋅⋅ + w_n − d₁ − ⋅⋅⋅ − d_m. Suppose that each d_i is a positive integer, that − k > 0, and that w_i divides d_j for all i, j such that 0 ⩽ i ⩽ n and 1 ⩽ j ⩽ m. Then the quantum period of X is $$\begin{array}{c}\hfill G_X\left(t\right)=e^{-ct}\sum _{d=0}^{\infty }{t}^{-kd}\frac{{\prod }_{j=1}^m\left(d{d}_j\right)!}{{\prod }_{i=1}^n\left(d{w}_i\right)!}\end{array}$$where c is the unique rational number such that the right-hand side has the form 1 + O(t²).

The quantum period of a product is the product of the quantum periods.

Theorem 2.4 ([Coates et al. 16, Corollary E.4]).

Let X and Y be smooth projective complex manifolds. Then: $$\begin{array}{c}\hfill G_{X\times Y}\left(t\right)=G_X\left(t\right)G_Y\left(t\right).\end{array}$$

As we will see below, another powerful tool for computing quantum periods is the Abelian/non-Abelian Correspondence of Ciocan-Fontanine–Kim–Sabbah [Ciocan-Fontanine et al. 08]. We now proceed to the calculation of quantum periods.

3. Four-dimensional Fano manifolds of index 5

The only example here is ${\mathbb{P}}^4$ [Kobayashi and Ochiai 73, Kollár 81, Serpico 80]. This is a toric variety. Theorem 2.1 yields $$\begin{array}{ccc}\hfill G_{{\mathbb{P}}^4}\left(t\right)& =& \sum _{d=0}^{\infty }\frac{t^{5d}}{{(d!)}^5}\hfill \\ & & \times \text{[regularized}\text{quantum}\text{period}\text{p.}\text{15,}\text{operator}\text{p.}\text{25].}\hfill \end{array}$$

4. Four-dimensional Fano manifolds of index 4

The only example here is the quadric $Q^4\subset {\mathbb{P}}^5$ [Kollár 81, Serpico 80]. This is a complete intersection in a toric variety. Theorem 2.2 yields $$\begin{array}{ccc}\hfill G_{Q^4}\left(t\right)& =& \sum _{d=0}^{\infty }\frac{\left(2d\right)!}{{(d!)}^6}{t}^{4d}\hfill \\ & & \times \text{[regularized}\text{quantum}\text{period}\text{p.}\text{15,}\text{operator}\text{p.}\text{25].}\hfill \end{array}$$

5. Four-dimensional Fano manifolds of index 3

There are six examples [Fujita 80, Fujita 81, Fujita 84, Fujita 90, Iskovskih 77, Iskovskih 79, Iskovskikh and Prokhorov 99], which are known as del Pezzo fourfolds:

• a sextic hypersurface FI⁴₁ in the weighted projective space ${\mathbb{P}}^5(1^4,2,3)$;

• a quartic hypersurface FI⁴₂ in the weighted projective space ${\mathbb{P}}^5(1^5,2)$;

• a cubic hypersurface ${\mathrm{FI}}_3^4\subset {\mathbb{P}}^5$;

• a complete intersection ${\mathrm{FI}}_4^4\subset {\mathbb{P}}^6$ of type (2H)∩(2H), where $H={\mathcal{O}}_{{\mathbb{P}}^6}\left(1\right)$;

• a complete intersection FI⁴₅⊂Gr(2, 5) of type H∩H, where H is the hyperplane bundle; and

• ${\mathrm{FI}}_6^4={\mathbb{P}}^2\times {\mathbb{P}}^2$.

The first four examples here are complete intersections in weighted projective spaces. Theorem 2.3 yields $$\begin{array}{ccc}\hfill G_{{\mathrm{FI}}_1^4}\left(t\right)& =& \sum _{d=0}^{\infty }\frac{\left(6d\right)!}{\left(3d\right)!\left(2d\right)!{(d!)}^4}{t}^{3d}\hfill \\ \hfill G_{{\mathrm{FI}}_2^4}\left(t\right)& =& \sum _{d=0}^{\infty }\frac{\left(4d\right)!}{\left(2d\right)!{(d!)}^5}{t}^{3d}\hfill \\ \hfill G_{{\mathrm{FI}}_3^4}\left(t\right)& =& \sum _{d=0}^{\infty }\frac{\left(3d\right)!}{{(d!)}^6}{t}^{3d}\hfill \\ \hfill G_{{\mathrm{FI}}_4^4}\left(t\right)& =& \sum _{d=0}^{\infty }\frac{\left(2d\right)!\left(2d\right)!}{{(d!)}^7}{t}^{3d}.\hfill \end{array}$$For FI⁴₅⊂Gr(2, 5), we use the Abelian/non-Abelian Correspondence, applying Theorem F.1 in [Coates et al. 16] with a = 2, b = c = d = e = 0. This yields $$\begin{array}{ccc}\hfill G_{{\mathrm{FI}}_5^4}\left(t\right)& =& \sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{(-1)}^{l+m}{t}^{3l+3m}\frac{(l+m)!(l+m)!}{{(l!)}^5{(m!)}^5}\hfill \\ & & \times (1-5(m-l)H_m),\hfill \end{array}$$where H_m is the mth harmonic number. For ${\mathbb{P}}^2\times {\mathbb{P}}^2$, combining Theorem 2.4 with [Coates et al. 16, Example G.2] yields $$\begin{array}{c}\hfill G_{{\mathbb{P}}^2\times {\mathbb{P}}^2}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\frac{t^{3l+3m}}{{(l!)}^3{(m!)}^3}\end{array}$$

Table

X	${\widehat{G}}_X$	LX	X	${\widehat{G}}_X$	LX
FI^41	p. 15	p. 26	FI^44	p. 15	p. 27
FI^42	p. 15	p. 26	FI^45	p. 15	p. 27
FI^43	p. 15	p. 26	${\mathbb{P}}^2\times {\mathbb{P}}^2$	p. 15	p. 27

6. Four-dimensional Fano manifolds of index 2

Consider now a four-dimensional Fano manifold with index r = 2 and Picard rank ρ.

6.1. The case ρ = 1

Four-dimensional Fano manifolds with index r = 2 and Picard rank ρ = 1 have been classified [Mukai 89, Wilson 87], [Iskovskikh and Prokhorov 99, Chapter 5]. Up to deformation, there are nine examples: the “linear unsections” of smooth three-dimensional Fano manifolds with ρ = 1, r = 1, and degree at most 144. We compute the quantum periods of these examples using the constructions in [Coates et al. 16, Sections 8–16], writing V⁴_k for a four-dimensional Fano manifold with ρ = 1, r = 2, and degree 16k.

6.1.1. V⁴₂.  [regularized quantum period p. 16, operator p. 28]

This is a sextic hypersurface in ${\mathbb{P}}^5(1^5,3)$. Proposition D.9 in [Coates et al. 16] yields $$\begin{array}{c}\hfill G_{V_2^4}\left(t\right)=\sum _{d=0}^{\infty }\frac{\left(6d\right)!}{{(d!)}^5\left(3d\right)!}{t}^{2d}.\end{array}$$

6.1.2. V⁴₄.  [regularized quantum period p. 16, operator p. 28]

This is a quartic hypersurface in ${\mathbb{P}}^5$. Theorem 2.2 yields $$\begin{array}{c}\hfill G_{V_4^4}\left(t\right)=\sum _{d=0}^{\infty }\frac{\left(4d\right)!}{{(d!)}^6}{t}^{2d}.\end{array}$$

6.1.3. V⁴₆.  [regularized quantum period p. 16, operator p. 28]

This is a complete intersection of type (2H)∩(3H) in ${\mathbb{P}}^6$, where $H={\mathcal{O}}_{{\mathbb{P}}^6}\left(1\right)$. Theorem 2.2 yields $$\begin{array}{c}\hfill G_{V_6^4}\left(t\right)=\sum _{d=0}^{\infty }\frac{\left(2d\right)!\left(3d\right)!}{{(d!)}^7}{t}^{2d}.\end{array}$$

6.1.4. V⁴₈.  [regularized quantum period p. 16, operator p. 29]

This is a complete intersection of type (2H)∩(2H)∩(2H) in ${\mathbb{P}}^7$, where $H={\mathcal{O}}_{{\mathbb{P}}^7}\left(1\right)$. Theorem 2.2 yields $$\begin{array}{c}\hfill G_{V_8^4}\left(t\right)=\sum _{d=0}^{\infty }\frac{{(\left(2d\right)!)}^3}{{(d!)}^8}{t}^{2d}.\end{array}$$

6.1.5. V⁴₁₀.  [regularized quantum period p. 16, operator p. 29]

This is a complete intersection in Gr(2, 5), cut out by a regular section of $\mathcal{O}\left(1\right)\oplus \mathcal{O}\left(2\right)$ where $\mathcal{O}\left(1\right)$ is the pullback of $\mathcal{O}\left(1\right)$ on projective space under the Plücker embedding. We apply Theorem F.1 in [Coates et al. 16] with a = b = 1 and c = d = e = 0. This yields $$\begin{array}{ccc}\hfill G_{V_{10}^4}\left(t\right)& =& \sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{(-1)}^{l+m}{t}^{2l+2m}\frac{(l+m)!(2l+2m)!}{{(l!)}^5{(m!)}^5}\hfill \\ & & \times (1-5(m-l)H_m),\hfill \end{array}$$where H_m is the mth harmonic number.

6.1.6. V⁴₁₂.  [regularized quantum period p. 16, operator p. 29]

This is the subvariety of Gr(2, 5) cut out by a regular section of $S^{★}\otimes det{S}^{★}$, where S is the universal bundle of subspaces on Gr(2, 5). We apply Theorem F.1 in [Coates et al. 16] with c = 1 and a = b = d = e = 0. This yields $$\begin{array}{ccc}\hfill G_{V_{12}^4}\left(t\right)& =& \sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{\left(-1\right)}^{l+m}{t}^{2l+2m}\frac{(2l+m)!(l+2m)!}{{(l!)}^5{(m!)}^5}\hfill \\ & & \times (1+(m-l)(H_{2l+m}+2H_{l+2m}-5H_m)).\hfill \end{array}$$

6.1.7. V⁴₁₄.  [regularized quantum period p. 16, operator p. 30]

This is a complete intersection in Gr(2, 6), cut out by a regular section of $\mathcal{O}{\left(1\right)}^{\oplus 4}$ where $\mathcal{O}\left(1\right)$ is the pullback of $\mathcal{O}\left(1\right)$ on projective space under the Plücker embedding. We apply Theorem F.1 in [Coates et al. 16] with a = 4 and b = c = d = e = 0. This yields $$\begin{array}{ccc}\hfill G_{V_{14}^4}\left(t\right)& =& \sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{(-1)}^{l+m}{t}^{2l+2m}\frac{((l+m)!{)}^4}{{(l!)}^6{(m!)}^6}\hfill \\ & & \times (1-6(m-l)H_m).\hfill \end{array}$$

6.1.8. V⁴₁₆.  [regularized quantum period p. 16, operator p. 30]

This is the subvariety of Gr(3, 6) cut out by a regular section of ${\wedge }^2{S}^{★}\oplus {(det{S}^{★})}^{\oplus 2}$, where S is the universal bundle of subspaces on Gr(3, 6). We apply Theorem F.1 in [Coates et al. 16] with a = 2, b = c = d = 0, and e = 1. This shows that the quantum period $G_{V_{16}^4}\left(t\right)$ is the coefficient of (p₂ − p₁)(p₃ − p₁)(p₃ − p₂) in the expression: $$\begin{array}{ccc}& & \sum _{l_1=0}^{\infty }\sum _{l_2=0}^{\infty }\sum _{l_3=0}^{\infty }{t}^{2l_1+2l_2+2l_3}\hfill \\ & & \times \frac{{\prod }_{k=1}^{l_1+l_2+l_3}{(p_1+p_2+p_3+k)}^2}{{\prod }_{j=1}^{j=3}{\prod }_{k=1}^{k=l_j}{(p_j+k)}^6}\prod _{1\le i<j\le 3}\prod _{k=1}^{l_i+l_j}(p_i+p_j+k)\hfill \\ & & \times \prod _{1\le i<j\le 3}(p_j-p_i+(l_j-l_i)).\hfill \end{array}$$(Since this expression is totally antisymmetric in p₁, p₂, p₃, it is divisible by (p₂ − p₁)(p₃ − p₁)(p₃ − p₂).)

6.1.9. V⁴₁₈.  [regularized quantum period p. 16, operator p. 30]

This is the subvariety of Gr(5, 7) cut out by a regular section of $(S\otimes det{S}^{★})\oplus det{S}^{★}$, where S is the universal bundle of subspaces on Gr(5, 7). We apply Theorem F.1 in [Coates et al. 16] with a = d = 1 and b = c = e = 0. This shows that the quantum period $G_{V_{18}^4}\left(t\right)$ is the coefficient of ∏_{1 ⩽ i < j ⩽ 5}(p_j − p_i) in the expression: $$\begin{array}{ccc}& & \sum _{l_1=0}^{\infty }\sum _{l_2=0}^{\infty }\sum _{l_3=0}^{\infty }\sum _{l_4=0}^{\infty }\sum _{l_5=0}^{\infty }{t}^{2\left|l\right|}\frac{{\prod }_{k=1}^{k=\left|l\right|}(p_1+p_2+\cdots +p_5+k)}{{\prod }_{j=1}^{j=5}{\prod }_{k=1}^{k=l_j}{(p_j+k)}^7}\hfill \\ & & \times \prod _{j=1}^{j=5}\prod _{k=1}^{\left|l\right|-l_j}(p_1+p_2+\cdots +p_5-p_j+k)\hfill \\ & & \times \prod _{1\le i<j\le 5}(p_j-p_i+(l_j-l_i)),\hfill \end{array}$$where |l| = l₁ + l₂ + ⋅⋅⋅ + l₅. (As above, antisymmetry implies that the long formula here is divisible by ∏_{1 ⩽ i < j ⩽ 5}(p_j − p_i).)

6.2. The case ρ > 1

Four-dimensional Fano manifolds with ρ > 1 and r = 2 have been classified by Mukai [Mukai 89, Mukai 88] and Wiśniewski [Wiśniewski 90]. There are 18 deformation families, as follows. We denote the kth such deformation family, as given in [Iskovskikh and Prokhorov 99, Table 12.7], by MW⁴_k.

6.2.1. MW⁴₁.  [regularized quantum period p. 16, operator p. 31]

This is the product ${\mathbb{P}}^1\times B_1^3$. Combining Theorem 2.4 with [Coates et al. 16, Example G.1] and [Coates et al. 16, Section 3] yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_1^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\frac{\left(6m\right)!}{{(l!)}^2{(m!)}^3\left(2m\right)!\left(3m\right)!}{t}^{2l+2m}.\end{array}$$

6.2.2. MW⁴₂.  [regularized quantum period p. 16, operator p. 32]

This is the product ${\mathbb{P}}^1\times B_2^3$. Combining Theorem 2.4 with [Coates et al. 16, Example G.1] and [Coates et al. 16, Section 4] yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_2^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\frac{\left(4m\right)!}{{(l!)}^2{(m!)}^4\left(2m\right)!}{t}^{2l+2m}.\end{array}$$

6.2.3. MW⁴₃.  [regularized quantum period p. 16, operator p. 33]

This is the product ${\mathbb{P}}^1\times B_3^3$. Combining Theorem 2.4 with [Coates et al. 16, Example G.1] and [Coates et al. 16, Section 5] yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_3^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\frac{\left(3m\right)!}{{(l!)}^2{(m!)}^5}{t}^{2l+2m}.\end{array}$$

6.2.4. MW⁴₄.  [regularized quantum period p. 16, operator p. 34]

This is a double cover of ${\mathbb{P}}^2\times {\mathbb{P}}^2$, branched over a divisor of bidegree (2, 2). Consider the toric variety F with weight data: $$\begin{array}{c}\hfill \begin{array}{cccccccc}\hfill x_0& \hfill x_1& \hfill x_2& y_0& y_1& y_2& w& \\ \multicolumn{7}{c}{130pt.5pt}& \\ \hfill 1& \hfill 1& \hfill 1& 0& 0& 0& 1& L\\ \hfill 0& \hfill 0& \hfill 0& 1& 1& 1& 1& M\end{array}\end{array}$$and $\overline{\mathrm{Amp}}F=⟨L,L+M⟩$. Let X be a member of the linear system |2L + 2M| defined by the equation w² = f_{2, 2}, where f_{2, 2} is a bihomogenous polynomial of degrees 2 in x₀, x₁, x₂ and 2 in y₀, y₁, y₂. Let $p:F⤏{\mathbb{P}}^2\times {\mathbb{P}}^2$ be the rational map which sends (contravariantly) the homogenous co-ordinate functions [x₀, x₁, x₂, y₀, y₁, y₂] on ${\mathbb{P}}_{x_0,x_1,x_2}^2\times {\mathbb{P}}_{y_0,y_1,y_2}^2$ to [x₀, x₁, x₂, y₀, y₁, y₂]. The restriction of p to X is a morphism, which exhibits X as a double cover of ${\mathbb{P}}^2\times {\mathbb{P}}^2$ branched over the locus $(f_{2,2}=0)\subset {\mathbb{P}}_{x_0,x_1,x_2}^2\times {\mathbb{P}}_{y_0,y_1,y_2}^2$. Thus X = MW⁴₄.

Recall the definition of the J-function J_X(t, z) from [Coates and Givental 07, equation 11]. Recall from [Coates 14] that there is a Lagrangian cone ${\mathcal{L}}_X\subset H^{•}(X;{\Lambda }_X)\otimes \mathbb{C}\left(\left(z^{-1}\right)\right)$ that encodes all genus-zero Gromov–Witten invariants of X, and a Lagrangian cone ${\mathcal{L}}_{\mathbf{e}}\subset H^{•}(F;{\Lambda }_F)\otimes \mathbb{C}\left(\left(z^{-1}\right)\right)\otimes \mathbb{C}\left(\lambda \right)$ that encodes all genus-zero $(\mathbf{e},2L+2M)$-twisted Gromov–Witten invariants of F. Here Λ_X and Λ_F are certain Novikov rings and $\mathbf{e}$ is the total Chern class with parameter λ (or, equivalently, $\mathbf{e}$ is the S¹-equivariant Euler class with respect to an action of S¹ described in [Coates 14]; in this case one should regard λ as the standard generator for the S¹-equivariant cohomology algebra of a point). The J-function J_X is characterized by the fact that J_X(t, − z) is the unique point on ${\mathcal{L}}_X$ of the form − z + t + O(z^{− 1}).

Let p₁, $p_2\in H^2(F;\mathbb{Q})$ denote the first Chern class of L, L + M, respectively, and let P₁, $P_2\in H^2(X;\mathbb{Q})$ denote the pullbacks of p₁, p₂ along the inclusion map i: X → F. Let Q₁, Q₂ denote the elements of the Novikov ring Λ_X that are dual, respectively, to P₁, P₂, and note that Λ_X and Λ_F are canonically isomorphic (via i_⋆). Theorem 22 in [Coates et al. 14] implies that: $$\begin{array}{ccc}\hfill I(t_1,t_2,\lambda ,z)& =& z{e}^{t_1{p}_1/z}{e}^{t_2{p}_2/z}\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\hfill \\ & & \times \frac{Q_1^l{Q}_2^m{e}^{l{t}_1}{e}^{m{t}_2}{\prod }_{k=1}^{k=2m}(\lambda +2p_2+kz)}{{\prod }_{k=1}^{k=l}{(p_1+kz)}^3{\prod }_{k=1}^{k=m}(p_2+kz)}\hfill \\ & & \times \frac{{\prod }_{k=-\infty }^{k=0}{(p_2-p_1+kz)}^3}{{\prod }_{k=-\infty }^{k=m-l}{(p_2-p_1+kz)}^3}\hfill \end{array}$$satisfies $I(t_1,t_2,\lambda ,-z)\in {\mathcal{L}}_{\mathbf{e}}$. Theorem 1.1 in [Coates 14] gives that $i^{★}{\mathcal{L}}_{\mathbf{e}}{|}_{\lambda =0}\subset {\mathcal{L}}_X$, and therefore that: $$\begin{array}{c}\hfill i^{★}I(t_1,t_2,0,-z)\in {\mathcal{L}}_X.\end{array}$$Since the hypersurface X misses the locus y₁ = y₂ = y₃ = 0 in F, we have that i^⋆(p₂ − p₁)³ = 0. Thus: $$\begin{array}{ccc}\hfill i^{★}I(t_1,t_2,0,z)& =& z{e}^{t_1{P}_1/z}{e}^{t_2{P}_2/z}\sum _{l=0}^{\infty }\sum _{m=l}^{\infty }\hfill \\ & & \times \frac{Q_1^l{Q}_2^m{e}^{l{t}_1}{e}^{m{t}_2}{\prod }_{k=1}^{k=2m}(2P_2+kz)}{{\prod }_{k=1}^{k=l}{(P_1+kz)}^3{\prod }_{k=1}^{k=m}(P_2+kz)}\hfill \\ & & \times \frac{1}{{\prod }_{k=1}^{k=m-l}{(P_2-P_1+kz)}^3}.\hfill \end{array}$$In particular, i^⋆I(t₁, t₂, 0, − z) has the form − z + t₁P₁ + t₂P₂ + O(z^{− 1}) and, from the characterization of J_X discussed above, we conclude that J_X(t₁P₁ + t₂P₂, − z) = i^⋆I(t₁, t₂, 0, − z).

To extract the quantum period G_X from the J-function J_X(t₁P₁ + t₂P₂, z), we take the component along the unit class $1\in H^{•}(X;\mathbb{Q})$, set z = 1, set t₁ = t₂ = 0, and set Q₁ = 1, Q₂ = t², obtaining: $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_4^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=l}^{\infty }\frac{\left(2m\right)!}{{(l!)}^{3}m!{((m-l)!)}^3}{t}^{2m}.\end{array}$$

6.2.5. MW⁴₅.  [regularized quantum period p. 16, operator p. 34]

This is a divisor on ${\mathbb{P}}^2\times {\mathbb{P}}^3$ of bidegree (1, 2). Theorem 2.2 yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_5^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\frac{(l+2m)!}{{(l!)}^3{(m!)}^4}{t}^{2l+2m}.\end{array}$$

6.2.6. MW⁴₆.  [regularized quantum period p. 16, operator p. 35]

This is the product ${\mathbb{P}}^1\times B_4^3$. Combining Theorem 2.4 with [Coates et al. 16, Example G.1] and [Coates et al. 16, Section 6] yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_6^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\frac{\left(2m\right)!\left(2m\right)!}{{(l!)}^2{(m!)}^6}{t}^{2l+2m}.\end{array}$$

6.2.7. MW⁴₇.  [regularized quantum period p. 16, operator p. 36]

This is a complete intersection of two divisors in ${\mathbb{P}}^3\times {\mathbb{P}}^3$, each of bidegree (1, 1). Theorem 2.2 yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_7^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\frac{(l+m)!(l+m)!}{{(l!)}^4{(m!)}^4}{t}^{2l+2m}.\end{array}$$

6.2.8. MW⁴₈.  [regularized quantum period p. 16, operator p. 36]

This is a divisor on ${\mathbb{P}}^2\times Q^3$ of bidegree (1, 1). Theorem 2.2 yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_8^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\frac{(l+m)!\left(2m\right)!}{{(l!)}^3{(m!)}^5}{t}^{2l+2m}.\end{array}$$

6.2.9. MW⁴₉.  [regularized quantum period p. 16, operator p. 37]

This is the product ${\mathbb{P}}^1\times B_5^3$. Combining Theorem 2.4 with [Coates et al. 16, Example G.1] and [Coates et al. 16, Section 7] yields $$\begin{array}{ccc}\hfill G_{{\mathrm{MW}}_9^4}\left(t\right)& =& \sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{(-1)}^{m+n}{t}^{2l+2m+2n}\hfill \\ & & \times \frac{{((m+n)!)}^3}{{(l!)}^2{(m!)}^5{(n!)}^5}(1-5(n-m)H_n).\hfill \end{array}$$

6.2.10. MW⁴₁₀.  [regularized quantum period p. 16, operator p. 38]

This is the blow-up of the quadric Q⁴ along a conic that is not contained in a plane lying in Q⁴. Consider the toric variety F with weight data: $$\begin{array}{c}\hfill \begin{array}{ccccccc}\hfill s_0& \hfill s_1& \hfill s_2& x& x_3& x_4& x_5\\ \multicolumn{7}{c}{130pt.5pt}& \\ \hfill 1& \hfill 1& \hfill 1& -1& 0& 0& 0& L\\ \hfill 0& \hfill 0& \hfill 0& 1& 1& 1& 1& M\end{array}\end{array}$$and $\overline{\mathrm{Amp}}F=⟨L,M⟩$. The morphism $F\to {\mathbb{P}}^5$ that sends (contravariantly) the homogenous co-ordinate functions [x₀, x₁, …, x₅] to [xs₀, xs₁, xs₂, x₃, x₄, x₅] blows up the plane Π = (x₀ = x₁ = x₂ = 0) in ${\mathbb{P}}^5$. Thus, a general member of |2M| on F is the blow-up of Q⁴ with center a conic on Π. In other words, a general member of |2M| on F is MW⁴₁₀. We have

• − K_F = 2L + 4M is ample, so that F is a Fano variety;

• MW⁴₁₀ ∼ 2M is ample;

• − (K_F + 2M) ∼ 2L + 2M is ample.

Theorem 2.2 yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_{10}^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=l}^{\infty }\frac{\left(2m\right)!}{{(l!)}^3(m-l)!{(m!)}^3}{t}^{2l+2m}.\end{array}$$

6.2.11. MW⁴₁₁.  [regularized quantum period p. 16, operator p. 38]

This is the projective bundle ${\mathbb{P}}_{{\mathbb{P}}^3}\left({\mathcal{E}}^{\vee }\right)$, where $\mathcal{E}\to {\mathbb{P}}^3$ is the null-correlation bundle of Szurek–Wiśniewski [Szurek and Wiśniewski 90].

Remark 6.1.

For us $\mathbb{P}\left(E\right)$ denotes the projective bundle of lines in E, whereas in Szurek–Wiśniewski and Iskovskikh–Prokhorov, $\mathbb{P}\left(E\right)$ denotes the projective bundle of one-dimensional quotients. With our conventions, if $\pi :\mathbb{P}\left(E\right)\to X$ is a projective bundle then $E^{★}={\pi }_{★}{\mathcal{O}}_{\mathbb{P}\left(E\right)}\left(1\right)$, and so a regular section $s\in \Gamma (\mathbb{P}\left(E\right),{\mathcal{O}}_{\mathbb{P}\left(E\right)}\left(1\right))$ vanishes on $\mathbb{P}\left(F^{★}\right)\subset \mathbb{P}\left(E\right)$, where the vector bundle F → X is the cokernel of $s:{\mathcal{O}}_{\mathbb{P}\left(E\right)}\to E^{★}$.

Proposition 6.2.

Let $V={\mathbb{C}}^4$, so that $\mathbb{P}\left(V\right)={\mathbb{P}}^3$. Consider the partial flag manifold Fl_{1, 2}(V) and the natural projections (6--4) $$800art/UEX{M}_{A}1{448018}_{U}F{0001}_B.eps$$(6--4) Let |L| denote the linear system defined by $\mathcal{O}\left(1\right)$ for the projective bundle p₁. Then a general element of |L| is $\mathbb{P}\left({\mathcal{E}}^{\vee }\right)$, where $\mathcal{E}\to \mathbb{P}\left(V\right)$ is the null-correlation bundle.

Proof.

The null-correlation bundle has rank 2, and so the perfect pairing $\mathcal{E}\otimes \mathcal{E}\to det\mathcal{E}$ gives canonical isomorphisms ${\mathcal{E}}^{\vee }\cong \mathcal{E}\otimes {(det\mathcal{E})}^{-1}$ and $\mathbb{P}\left({\mathcal{E}}^{\vee }\right)\cong \mathbb{P}\left(\mathcal{E}\right)$. There is an exact sequence: $$\begin{array}{c}\hfill 800art/UEX{M}_{A}1{448018}_{U}F{0002}_B.eps\end{array}$$and the map $s^{★}:T_{\mathbb{P}\left(V\right)}(-2)\to {\mathcal{O}}_{\mathbb{P}\left(V\right)}$ therein defines a section $s\in \Gamma (\mathbb{P}\left(T_{\mathbb{P}\left(V\right)}(-2)\right),{\mathcal{O}}_{\mathbb{P}\left(T_{\mathbb{P}\left(V\right)}(-2)\right)}\left(1\right))$. The construction in Remark 6.1 now exhibits $\mathbb{P}\left(\mathcal{E}\right(-1\left)\right)\cong \mathbb{P}\left(\mathcal{E}\right)$ as the locus (s = 0) in $\mathbb{P}\left(T_{\mathbb{P}\left(V\right)}(-2)\right)$. We will identify $\mathbb{P}\left(T_{\mathbb{P}\left(V\right)}(-2)\right)$ with the partial flag manifold Fl_{1, 2}(V).

For a vector bundle $\mathcal{F}\to X$ of rank 3, the perfect pairing $\mathcal{F}\otimes {\wedge }^2\mathcal{F}\to det\mathcal{F}$ gives a canonical isomorphism ${\mathcal{F}}^{★}\cong \left({\wedge }^2\mathcal{F}\right)\otimes {(det\mathcal{F})}^{-1}$. Applying this with $\mathcal{F}\to X$ equal to ${\Omega }_{\mathbb{P}\left(V\right)}\left(2\right)\to \mathbb{P}\left(V\right)$ gives $$\begin{array}{c}\hfill T_{\mathbb{P}\left(V\right)}(-2)\cong {\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right),\end{array}$$where ${\Omega }_{\mathbb{P}\left(V\right)}^2:={\wedge }^2{\Omega }_{\mathbb{P}\left(V\right)}$. We thus need to identify $\mathbb{P}\left({\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right)\right)$ with Fl_{1, 2}(V).

The Plücker embedding $\mathrm{Gr}(2,V)\to \mathbb{P}\left({\wedge }^{2}V\right)$ maps a subspace W ∈ Gr(2, V) to the antisymmetric linear map L_W: V^⋆ → V, well-defined up to scale, given by $$\begin{array}{c}\hfill L_W\left(f\right)=f\left(w_1\right)w_2-f\left(w_2\right)w_1,\end{array}$$where {w₁, w₂} is a basis for W. The kernel of L_W is the annihilator W^⊥⊂V^⋆. If $f\notin W^{\perp }$ then $⟨L_W\left(f\right)⟩=kerf\cap W$; this implies in particular that rkL_W = 2. Thus, the image of the Plücker embedding consists of (the lines spanned by) antisymmetric linear maps L_W: V^⋆ → V of rank 2, and one can recover W ∈ Gr(2, V) from its image ⟨L_W⟩ by taking the annihilator of the kernel: $$\begin{array}{c}\hfill W={(ker{L}_W)}^{\perp }.\end{array}$$There is a canonical isomorphism Ann: Gr(2, V) → Gr(2, V^⋆) which maps W ∈ Gr(2, V) to W^⊥.

Recall that our goal is to identify $\mathbb{P}\left({\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right)\right)$ with Fl_{1, 2}(V). Let $q_1:\mathbb{P}\left({\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right)\right)\to \mathbb{P}\left(V\right)$ denote the projection. The Euler sequence: $$\begin{array}{c}\hfill 800art/UEX{M}_{A}1{448018}_{U}F{0003}_B.eps\end{array}$$gives, via [Hartshorne 77, II, Exercise 5.16]: $$\begin{array}{c}\hfill 800art/UEX{M}_{A}1{448018}_{U}F{0004}_B.eps\end{array}$$and thus: (6--5) $$700art/UEX{M}_{A}1{448018}_{U}F{0005}_B.eps$$(6--5) This defines a map $f:\mathbb{P}\left({\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right)\right)\to \mathbb{P}\left({\wedge }^2{V}^{★}\right)$. Consider the fiber of the sequence (6–5(6--5) $$700art/UEX{M}_{A}1{448018}_{U}F{0005}_B.eps$$(6--5) ) over $\left[v\right]\in \mathbb{P}\left(V\right)$. The map π^⋆(∧²V^⋆) → π^⋆V^⋆(1) here is given by contraction with v, and so non-zero elements of the kernel are antisymmetric linear maps V → V^⋆ of rank 2. (They are antisymmetric, hence have rank 0, 2, or 4; they are non-zero, hence are not of rank 0; and they have the non-zero element v in their kernel, hence are not of rank 4.) In particular, we see that the image of f lies in $\mathrm{Gr}(2,V^{★})\subset \mathbb{P}\left({\wedge }^2{V}^{★}\right)$. Given $\left[x\right]\in \mathbb{P}\left({\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right)\right)$, write W_[x]⊂V^⋆ for the linear subspace defined by f([x]). Suppose that $\left[x\right]\in \mathbb{P}\left({\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right)\right)$ lies over $\left[v\right]\in \mathbb{P}\left(V\right)$. Then, applying the discussion in the previous paragraph but with V there replaced by V^⋆, we see that v ∈ W^⊥_[x]. Thus, writing $q_2:\mathbb{P}\left({\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right)\right)\to \mathrm{Gr}(2,V)$ for the composition $$\begin{array}{c}\hfill 800art/UEX{M}_{A}1{448018}_{U}F{0006}_B.eps\end{array}$$we have that q₁([x])⊂q₂([x]), i.e., that the diagram: $$\begin{array}{c}\hfill 800art/UEX{M}_{A}1{448018}_{U}F{0007}_B.eps\end{array}$$coincides with the diagram (6–4(6--4) $$800art/UEX{M}_{A}1{448018}_{U}F{0001}_B.eps$$(6--4) ). This identifies $\mathbb{P}\left({\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right)\right)$ with the partial flag manifold Fl_{1, 2}(V), and exhibits $\mathbb{P}\left({\mathcal{E}}^{\vee }\right)$ as an element of the linear system |L| as claimed.

Abelianization:

To compute the quantum period, we use the Abelian/non-Abelian Correspondence of Ciocan-Fontanine–Kim–Sabbah, as in [Coates et al. 16, Section 39]. Consider the situation as in Section 3.1 of [Ciocan-Fontanine et al. 08] with:

• $X={\mathbb{C}}^{10}$, regarded as the space of pairs: $$\begin{array}{c}\hfill \left\{\right(v,w):v\in {\mathbb{C}}^2\text{is}\text{a}\text{row}\text{vector,}w\text{is}\text{a}2\times 4\text{complex}\text{matrix}\}\end{array}$$

• $G={\mathbb{C}}^{\times }\times {\mathrm{GL}}_2\left(\mathbb{C}\right)$, acting on X as: $$\begin{array}{c}\hfill (\lambda ,g):(v,w)\mapsto (\lambda v{g}^{-1},gw)\end{array}$$

• $T={\left({\mathbb{C}}^{\times }\right)}^3$, the diagonal subtorus in G;

• the group that is denoted by S in [Ciocan-Fontanine et al. 08] set equal to the trivial group;

• $\mathcal{V}$ equal to the representation of G given by the determinant of the standard representation of the second factor ${\mathrm{GL}}_2\left(\mathbb{C}\right)$.

Then X//G is the partial flag manifold $\mathrm{Fl}={\mathrm{Fl}}_{1,2}\left({\mathbb{C}}^4\right)$, whereas X//T is the toric variety with weight data: $$\begin{array}{c}\hfill \begin{array}{ccccccccccc}\hfill 1& \hfill 1& \hfill 1& 1& 0& 0& 0& 0& -1& 0& L_1\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 1& 1& 1& 0& -1& L_2\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0& 1& 1& H\end{array}\end{array}$$and $\overline{\mathrm{Amp}}=⟨L_1,L_2,H⟩$, that is, X//T is the projective bundle $\mathbb{P}\left(\mathcal{O}\right(-1,0)\oplus \mathcal{O}(0,-1\left)\right)$ over ${\mathbb{P}}^3\times {\mathbb{P}}^3$. The non-trivial element of the Weyl group $W=\mathbb{Z}/2\mathbb{Z}$ exchanges the two factors of ${\mathbb{P}}^3\times {\mathbb{P}}^3$. The representation $\mathcal{V}$ induces the line bundle ${\mathcal{V}}_G=L$ over X//G = Fl, where L was defined in the statement of Proposition 6.2, whereas the representation $\mathcal{V}$ induces the line bundle ${\mathcal{V}}_T=L_1+L_2$ over X//T.

The Abelian/non-Abelian correspondence:

Let p₁, p₂, and $p_3\in H^2(X//T;\mathbb{Q})$ denote the first Chern classes of the line bundles L₁, L₂, and H, respectively. We fix a lift of $H^{•}(X//G;\mathbb{Q})$ to $H^{•}{(X//T,\mathbb{Q})}^W$ in the sense of [Ciocan-Fontanine et al. 08, Section 3]; there are many possible choices for such a lift, and the precise choice made will be unimportant in what follows. The lift allows us to regard $H^{•}(X//G;\mathbb{Q})$ as a subspace of $H^{•}{(X//T,\mathbb{Q})}^W$, which maps isomorphically to the Weyl-anti-invariant part $H^{•}{(X//T,\mathbb{Q})}^a$ of $H^{•}(X//T,\mathbb{Q})$ via $$\begin{array}{c}\hfill 800art/UEX{M}_{A}1{448018}_{U}F{0008}_B.eps\end{array}$$We compute the quantum period of MW⁴₁₁⊂X//G by computing the J-function of Fl = X//G twisted, in the sense of [Coates and Givental 07], by the Euler class and the bundle ${\mathcal{V}}_G$, using the Abelian/non-Abelian Correspondence.

Our first step is to compute the J-function of X//T twisted by the Euler class and the bundle ${\mathcal{V}}_T$. As in [Coates et al. 16, Section D.1] and as in [Ciocan-Fontanine et al. 08], consider the bundles ${\mathcal{V}}_T$ and ${\mathcal{V}}_G$ equipped with the canonical ${\mathbb{C}}^{\times }$-action that rotates fibers and acts trivially on the base. Recall the definition of the twisted J-function $J_{\mathbf{e},{\mathcal{V}}_T}$ of X//T from [Coates et al. 16, Section D.1]. We will compute $J_{\mathbf{e},{\mathcal{V}}_T}$ using the Quantum Lefschetz theorem; $J_{\mathbf{e},{\mathcal{V}}_T}$ is the restriction to the locus τ ∈ H⁰(X//T)⊕H²(X//T) of what was denoted by $J_{{\mathcal{V}}_T}^{S\times {\mathbb{C}}^{\times }}\left(\tau \right)$ in [Ciocan-Fontanine et al. 08]. The toric variety X//T is Fano, so Theorem C.1 in [Coates et al. 16] gives $$\begin{array}{ccc}\hfill J_{X//T}\left(\tau \right)& =& e^{\tau /z}\sum _{l,m,n\ge 0}\frac{Q_1^l{Q}_2^m{Q}_3^n{e}^{l{\tau }_1}{e}^{m{\tau }_2}{e}^{m{\tau }_3}}{{\prod }_{k=1}^{k=l}{(p_1+kz)}^4{\prod }_{k=1}^{k=m}{(p_2+kz)}^4}\hfill \\ & & \times \frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_1+kz}{{\prod }_{k=-\infty }^{k=n-l}{p}_3-p_1+kz}\frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_2+kz}{{\prod }_{k=-\infty }^{k=n-m}{p}_3-p_2+kz},\hfill \end{array}$$where τ = τ₁p₁ + τ₂p₂ + τ₃p₃ and we have identified the group ring $\mathbb{Q}\left[H_2(X//T;\mathbb{Z})\right]$ with $\mathbb{Q}[Q_1,Q_2,Q_3]$ via the $\mathbb{Q}$-linear map that sends Q^β to Q^{⟨β, p₁⟩}₁Q^{⟨β, p₂⟩}₂Q^{⟨β, p₃⟩}₃. The line bundles L₁, L₂, and H are nef, and $c_1(X//T)-c_1\left({\mathcal{V}}_T\right)$ is ample, so Theorem D.3 in [Coates et al. 16] gives $$\begin{array}{ccc}\hfill J_{\mathbf{e},{\mathcal{V}}_T}\left(\tau \right)& =& e^{\tau /z}\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{Q}_1^l{Q}_2^m{Q}_3^n{e}^{l{\tau }_1}{e}^{m{\tau }_2}{e}^{m{\tau }_3}\hfill \\ & & \times \frac{{\prod }_{k=1}^{k=l+m}(\lambda +p_1+p_2+kz)}{{\prod }_{k=1}^{k=l}{(p_1+kz)}^4{\prod }_{k=1}^{k=m}{(p_2+kz)}^4}\hfill \\ & & \times \frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_1+kz}{{\prod }_{k=-\infty }^{k=n-l}{p}_3-p_1+kz}\frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_2+kz}{{\prod }_{k=-\infty }^{k=n-m}{p}_3-p_2+kz}.\hfill \end{array}$$

Consider now Fl = X//G and a point t ∈ H^•(Fl). Recall that $\mathrm{Fl}=\mathbb{P}\left(S\right)$ is the projectivization of the universal bundle S of subspaces on Gr ≔ Gr(2, 4). Let ${ϵ}_1\in H^2(\mathrm{Fl};\mathbb{Q})$ be the pullback to Fl (under the projection map p₂: Fl → Gr) of the ample generator of H²(Gr), and let ${ϵ}_2\in H^2(\mathrm{Fl};\mathbb{Q})$ be the first Chern class of ${\mathcal{O}}_{\mathbb{P}\left(S\right)}\left(1\right)$. Identify the group ring $\mathbb{Q}\left[H_2(\mathrm{Fl};\mathbb{Z})\right]$ with $\mathbb{Q}[q_1,q_2]$ via the $\mathbb{Q}$-linear map which sends Q^β to q^{⟨β, ε₁⟩}₁q^{⟨β, ε₂⟩}₂. In [Ciocan-Fontanine et al. 08, Section 6.1], the authors consider the lift ${\tilde{J}}_{{\mathcal{V}}_G}^{S\times {\mathbb{C}}^{\times }}\left(t\right)$ of their twisted J-function $J_{{\mathcal{V}}_G}^{S\times {\mathbb{C}}^{\times }}\left(t\right)$ determined by a choice of lift $H^{•}(X//G;\mathbb{Q})\to H^{•}{(X//T,\mathbb{Q})}^W$. We restrict to the locus $t\in H^0(X//G;\mathbb{Q})\oplus H^2(X//G;\mathbb{Q})$, considering the lift: $$\begin{array}{c}\hfill {\tilde{J}}_{\mathbf{e},{\mathcal{V}}_G}\left(t\right):={\tilde{J}}_{{\mathcal{V}}_G}^{S\times {\mathbb{C}}^{\times }}\left(t\right)t\in H^0(X//G;\mathbb{Q})\oplus H^2(X//G;\mathbb{Q})\end{array}$$of our twisted J-function $J_{\mathbf{e},{\mathcal{V}}_G}$ determined by our choice of lift $H^{•}(X//G;\mathbb{Q})\to H^{•}{(X//T,\mathbb{Q})}^W$. Theorems 4.1.1 and 6.1.2 in [Ciocan-Fontanine et al. 08] imply that $$\begin{array}{ccc}& & {\tilde{J}}_{\mathbf{e},{\mathcal{V}}_G}\left(\phi \left(t\right)\right)\cup (p_2-p_1)\hfill \\ & & ={\left[\left(z\frac{\partial }{\partial {\tau }_2}-z\frac{\partial }{\partial {\tau }_1}\right)J_{\mathbf{e},{\mathcal{V}}_T}\left(\tau \right)\right]}_{\tau =t,Q_1=Q_2=-q_1,Q_3=q_2}\hfill \end{array}$$for some function $\phi :H^2(X//G;\mathbb{Q})\to H^{•}(X//G;{\Lambda }_G)$. Setting t = 0 gives $$\begin{array}{ccc}& & {\tilde{J}}_{\mathbf{e},{\mathcal{V}}_G}\left(\phi \left(0\right)\right)\cup (p_2-p_1)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{(-1)}^{l+m}{q}_1^{l+m}{q}_2^n\hfill \\ & & \times \frac{{\prod }_{k=1}^{k=l+m}(\lambda +p_1+p_2+kz)}{{\prod }_{k=1}^{k=l}{(p_1+kz)}^4{\prod }_{k=1}^{k=m}{(p_2+kz)}^4}\hfill \\ & & \times \frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_1+kz}{{\prod }_{k=-\infty }^{k=n-l}{p}_3-p_1+kz}\frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_2+kz}{{\prod }_{k=-\infty }^{k=n-m}{p}_3-p_2+kz}\hfill \\ & & \times (p_2-p_1+(m-l)z).\hfill \end{array}$$For symmetry reasons, the right-hand side here is divisible by p₂ − p₁; it takes the form: $$\begin{array}{c}\hfill (p_2-p_1)(1+O\left(z^{-2}\right))\end{array}$$whereas: $$\begin{array}{ccc}& & {\tilde{J}}_{\mathbf{e},{\mathcal{V}}_G}\left(\phi \left(0\right)\right)\cup (p_2-p_1)\hfill \\ & & =(p_2-p_1)(1+\phi \left(0\right)z^{-1}+O\left(z^{-2}\right)).\hfill \end{array}$$We conclude that ϕ(0) = 0. Thus, (6--6) $$\begin{array}{ccc}& & {\tilde{J}}_{\mathbf{e},{\mathcal{V}}_G}\left(0\right)\cup (p_2-p_1)\hfill \\ & & =\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{(-1)}^{l+m}{q}_1^{l+m}{q}_2^n\hfill \\ & & \times \frac{{\prod }_{k=1}^{k=l+m}(\lambda +p_1+p_2+kz)}{{\prod }_{k=1}^{k=l}{(p_1+kz)}^4{\prod }_{k=1}^{k=m}{(p_2+kz)}^4}\hfill \\ & & \times \frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_1+kz}{{\prod }_{k=-\infty }^{k=n-l}{p}_3-p_1+kz}\frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_2+kz}{{\prod }_{k=-\infty }^{k=n-m}{p}_3-p_2+kz}\hfill \\ & & \times (p_2-p_1+(m-l)z).\hfill \end{array}$$(6--6)

To extract the quantum period G⁴_MW11 from the twisted J-function $J_{\mathbf{e},{\mathcal{V}}_G}\left(0\right)$, we proceed as in [Coates et al. 16, Example D.8]: we take the non-equivariant limit, extract the component along the unit class $1\in H^{•}(X//G;\mathbb{Q})$, set z = 1, and set Q^β = t^{⟨β, − K⟩} where K = K⁴_MW11. Thus, we consider the right-hand side of (6–6(6--6) $$\begin{array}{ccc}& & {\tilde{J}}_{\mathbf{e},{\mathcal{V}}_G}\left(0\right)\cup (p_2-p_1)\hfill \\ & & =\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{(-1)}^{l+m}{q}_1^{l+m}{q}_2^n\hfill \\ & & \times \frac{{\prod }_{k=1}^{k=l+m}(\lambda +p_1+p_2+kz)}{{\prod }_{k=1}^{k=l}{(p_1+kz)}^4{\prod }_{k=1}^{k=m}{(p_2+kz)}^4}\hfill \\ & & \times \frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_1+kz}{{\prod }_{k=-\infty }^{k=n-l}{p}_3-p_1+kz}\frac{{\prod }_{k=-\infty }^{k=0}{p}_3-p_2+kz}{{\prod }_{k=-\infty }^{k=n-m}{p}_3-p_2+kz}\hfill \\ & & \times (p_2-p_1+(m-l)z).\hfill \end{array}$$(6--6) ), take the non-equivariant limit, extract the coefficient of p₂ − p₁, set z = 1, and set q₁ = q₂ = 2t, obtaining: $$\begin{array}{ccc}\hfill G_{{\mathrm{MW}}_{11}^4}\left(t\right)& =& \sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=max(l,m)}^{\infty }{(-1)}^{l+m}{t}^{2l+2m+2n}\hfill \\ & & \times \frac{(l+m)!}{{(l!)}^4{(m!)}^4(n-l)!(n-m)!}\hfill \\ & & \times (1+(m-l)(H_{n-m}-4H_m))\hfill \\ & & +\sum _{l=0}^{\infty }\sum _{m=l+1}^{\infty }\sum _{n=l}^{m-1}{(-1)}^{l+n}{t}^{2l+2m+2n}\hfill \\ & & \times \frac{(l+m)!(m-n-1)!}{{(l!)}^4{(m!)}^4(n-l)!}(m-l).\hfill \end{array}$$

Remark 6.3.

The quantum period of $\mathbb{P}\left({\Omega }_{\mathbb{P}\left(V\right)}^2\left(2\right)\right)$ can also be computed using Strangeway’s reconstruction theorem for the quantum cohomology of Fano bundles [Strangeway 15, Theorem 1]. Thus, the quantum period of MW⁴₁₁ can be derived from this result together with the Quantum Lefschetz theorem. The Gromov–Witten invariants required as input to the reconstruction theorem can be computed via [Strangeway 15, Lemma 1], using Schubert calculus on Gr(2, 4) and intersection numbers in ${\mathbb{P}}^3$.

6.2.12. MW⁴₁₂.  [regularized quantum period p. 16, operator p. 39]

This is the blow-up of the quadric Q⁴ along a line. Consider the toric variety F with weight data: $$\begin{array}{c}\hfill \begin{array}{ccccccc}\hfill s_0& \hfill s_1& \hfill s_2& s_3& x& x_4& x_5\\ \multicolumn{7}{c}{130pt.5pt}& \\ \hfill 1& \hfill 1& \hfill 1& 1& -1& 0& 0& L\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 1& 1& M\end{array}\end{array}$$and $\overline{\mathrm{Amp}}F=⟨L,M⟩$. The morphism $F\to {\mathbb{P}}^5$ that sends (contravariantly) the homogenous co-ordinate functions [x₀, x₁, …, x₅] to [xs₀, xs₁, xs₂, xs₃, x₄, x₅] blows up the line (x₀ = x₁ = x₂ = x₃ = 0) in ${\mathbb{P}}^5$, and MW⁴₁₂ is the proper transform of a quadric containing this line. Thus MW⁴₁₂ is a member of |L + M| in the toric variety F. We have:

• − K_F = 3L + 3M is ample, so that F is a Fano variety;

• MW⁴₁₂ ∼ L + M is ample;

• − (K_F + L + M) ∼ 2L + 2M is ample.

Theorem 2.2 yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_{12}^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=l}^{\infty }\frac{(l+m)!}{{(l!)}^4(m-l)!{(m!)}^2}{t}^{2l+2m}.\end{array}$$

6.2.13. MW⁴₁₃.  [regularized quantum period p. 16, operator p. 40]

This is the projective bundle ${\mathbb{P}}_{Q^3}(\mathcal{O}\left(1\right)\oplus \mathcal{O})$ or, equivalently, a member of |2L| in the toric variety F with weight data: $$\begin{array}{c}\hfill \begin{array}{ccccccc}\hfill x_0& \hfill x_1& \hfill x_2& x_3& x_4& u& v\\ \multicolumn{7}{c}{130pt.5pt}& \\ \hfill 1& \hfill 1& \hfill 1& 1& 1& 0& -1& L\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1& 1& M\end{array}\end{array}$$and $\overline{\mathrm{Amp}}F=⟨L,M⟩$. We have

• − K_F = 4L + 2M is ample, that is, F is a Fano variety;

• MW⁴₁₃ ∼ 2L is nef;

• − (K_F + 2L) ∼ 2L + 2M is ample.

The projection [x₀: x₁: x₂: x₃: x₄: x₅: u: v]↦[x₀: x₁: x₂: x₃: x₄: x₅] exhibits F as the scroll ${\mathbb{P}}_{{\mathbb{P}}^4}(\mathcal{O}\left(1\right)\oplus \mathcal{O})$ over ${\mathbb{P}}^4$, and passing to a member of |2L| restricts this scroll to $Q^3\subset {\mathbb{P}}^4$. Theorem 2.2 yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_{13}^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=l}^{\infty }\frac{\left(2l\right)!}{{(l!)}^{5}m!(m-l)!}{t}^{2l+2m.}\end{array}$$

6.2.14. MW⁴₁₄.  [regularized quantum period p. 16, operator p. 40]

This is the product ${\mathbb{P}}^1\times {\mathbb{P}}^3$. Combining Theorem 2.4 with [Coates et al. 16, Example G.1] and [Coates et al. 16, Section 1] yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_{14}^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\frac{t^{2l+4m}}{{(l!)}^2{(m!)}^4}.\end{array}$$

6.2.15. MW⁴₁₅.  [regularized quantum period p. 16, operator p. 41]

This is the projective bundle ${\mathbb{P}}_{{\mathbb{P}}^3}(\mathcal{O}\left(1\right)\oplus \mathcal{O}(-1))$, or in other words, the toric variety with weight data: $$\begin{array}{c}\hfill \begin{array}{cccccc}\hfill x_0& \hfill x_1& \hfill x_2& x_3& u& v\\ \multicolumn{6}{c}{110pt.5pt}& \\ \hfill 1& \hfill 1& \hfill 1& 1& 0& -2& L\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 1& M\end{array}\end{array}$$and $\overline{\mathrm{Amp}}F=⟨L,M⟩$. Theorem 2.1 yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_{15}^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=2l}^{\infty }\frac{t^{2l+2m}}{{(l!)}^{4}m!(m-2l)!}.\end{array}$$

6.2.16. MW⁴₁₆.  [regularized quantum period p. 16, operator p. 42]

This is the product ${\mathbb{P}}^1\times W^3$, where $W^3\subset {\mathbb{P}}^2\times {\mathbb{P}}^2$ is a divisor of bidegree (1, 1). Theorem 2.2 yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_{16}^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\frac{(m+n)!}{{(l!)}^2{(m!)}^3{(n!)}^3}{t}^{2l+2m+2n}.\end{array}$$

6.2.17. MW⁴₁₇.  [regularized quantum period p. 16, operator p. 43]

This is the product ${\mathbb{P}}^1\times B_7^3$, where B³₇ is the blow-up of ${\mathbb{P}}^3$ at a point. Note that B³₇ is the projective bundle ${\mathbb{P}}_{{\mathbb{P}}^2}(\mathcal{O}\oplus \mathcal{O}(-1))$. It follows that MW⁴₁₇ is the toric variety with weight data: $$\begin{array}{c}\hfill \begin{array}{ccccccc}\hfill x_0& \hfill x_1& \hfill y_0& y_1& y_2& u& v\\ \multicolumn{7}{c}{130pt.5pt}& \\ \hfill 1& \hfill 1& \hfill 0& 0& 0& 0& 0& L\\ \hfill 0& \hfill 0& \hfill 1& 1& 1& 0& -1& M\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1& 1& N\end{array}\end{array}$$and $\overline{\mathrm{Amp}}F=⟨L,M,N⟩$. Theorem 2.1 yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_{17}^4}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=m}^{\infty }\frac{t^{2l+2m+2n}}{{(l!)}^2{(m!)}^{3}n!(n-m)!}.\end{array}$$

6.2.18. MW⁴₁₈.  [regularized quantum period p. 16, operator p. 44]

This is the product ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$. Combining Theorem 2.4 with [Coates et al. 16, Example G.1] yields $$\begin{array}{c}\hfill G_{{\mathrm{MW}}_{18}^4}\left(t\right)=\sum _{k=0}^{\infty }\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\frac{t^{2k+2l+2m+2n}}{{(k!)}^2{(l!)}^2{(m!)}^2{(n!)}^2}.\end{array}$$

7. Four-dimensional Fano toric manifolds

Four-dimensional Fano toric manifolds were classified by Batyrev [Batyrev 99] and Sato [Sato 00]. Øbro classified Fano toric manifolds in dimensions 2–8 [Øbro 07] and, to standardize notation, we will write ${\text{BØS}}_k^4$ for the kth four-dimensional Fano toric manifold in Øbro’s list. ${\text{BØS}}_k^4$ is the (23 + k)th Fano toric manifold in the Graded Ring Database [Brown and Kasprzyk], as the list there is the concatenation of Øbro’s lists in dimensions 2–8. We can compute the quantum periods of the ${\text{BØS}}_k^4$ using Theorem 2.1; the first few Taylor coefficients of their regularized quantum periods can be found in the tables in the Appendix.

8. Product manifolds and other index 1 examples

Quantum periods for one-, two- and three-dimensional Fano manifolds were computed in [Coates et al. 16]. Combining these results with Theorem 2.4 allows us to compute the quantum period of any four-dimensional Fano manifold that is a product of lower-dimensional manifolds. Many of these examples have Fano index r = 1.

In his thesis [Strangeway 14], Strangeway determined the quantum periods of two four-dimensional Fano manifolds of index r = 1 that have not yet been discussed. These manifolds arise as complete intersections in the nine-dimensional projective bundle $F=\mathbb{P}\left({\Omega }_{{\mathbb{P}}^4}^2\left(2\right)\right)$. Let $\pi :F\to {\mathbb{P}}^4$ denote the canonical projection, let p ∈ H²(F) be the first Chern class of ${\pi }^{★}{\mathcal{O}}_{{\mathbb{P}}^4}\left(1\right)$, and let ξ ∈ H²(F) be the first Chern class of the tautological bundle ${\mathcal{O}}_F\left(1\right)$. The manifold F is Fano of Picard rank 2, with nef cone generated by {ξ, p} and − K_F = 6ξ + 2p. Let: $$\begin{array}{ccc}& & {\mathrm{Str}}_1\subset F\text{denote}\text{a}\text{complete}\text{intersection}\text{of}\text{five}\text{divisors}\hfill \\ & & \mathrm{of}\mathrm{type}\xi \hfill \\ & & {\mathrm{Str}}_2\subset F\text{denote}\text{a}\text{complete}\text{intersection}\text{of}\text{four}\text{divisors}\hfill \\ & & \mathrm{of}\mathrm{type}\xi \mathrm{and}\mathrm{a}\mathrm{divisor}\mathrm{of}\mathrm{type}p\hfill \end{array}$$We consider also: $$\begin{array}{ccc}& & {\mathrm{Str}}_3\subset F\text{,}\text{a}\text{complete}\text{intersection}\text{of}\text{four}\text{divisors}\text{of}\hfill \\ & & \mathrm{type}\xi \mathrm{and}\mathrm{a}\mathrm{divisor}\mathrm{of}\mathrm{type}\xi +p\hfill \end{array}$$which was unaccountably omitted from [Strangeway 14].

The manifolds Str_k, k ∈ {1, 2, 3}, each have Picard rank two. To see this, observe that the ambient manifold F is the blow-up of ${\mathbb{P}}^9$ along Gr(2, 5), where $\mathrm{Gr}(2,5)\to {\mathbb{P}}^9$ is the Plücker embedding [Strangeway 15]; the blow-up $F\to {\mathbb{P}}^9$ and the projection $\pi :F\to {\mathbb{P}}^4$ are the extremal contractions corresponding to the two extremal rays in $\overline{\mathrm{NE}}\left(F\right)$. Thus, Str₁ is the blow-up of ${\mathbb{P}}^4$ along an elliptic curve $E_5\subset {\mathbb{P}}^4$ of degree 5. Consider the five-dimensional Fano manifold F₅ given by the complete intersection of four divisors of type ξ in F. Then F₅ is the blow-up of ${\mathbb{P}}^5$ along a del Pezzo surface S₅ of degree 5; in particular, F₅ has Picard rank two. Str₃ is an ample divisor (of type ξ + p) in F₅, so the Picard rank of Str₃ is also two. The manifold Str₂ is a divisor in F₅ of type p, and F₅ arises as the closure of the graph of the map ${\mathbb{P}}^5\to {\mathbb{P}}^4$ given by the five-dimensional linear system of quadrics passing through S₅. This exhibits Str₂ as the blow-up of a smooth four-dimensional quadric Q⁴ along S₅, which implies that the Picard rank of Str₂ is two.

We can compute the quantum periods of Str_k, k ∈ {1, 2, 3}, by observing that a complete intersection in F of five divisors of type ξ and one divisor of type p is a three-dimensional Fano manifold ${\mathrm{MM}}_{2\text{--}17}^3$, “unsectioning” to compute the quantum period for F, and then applying the quantum Lefschetz theorem to compute the quantum periods for Str₁, Str₂, and Str₃. Recall the definition of the J-function J_X(t, z) from [Coates and Givental 07, equation 11]. The identity component of the J-function of ${\mathrm{MM}}_{2\text{--}17}^3$ is (8--7) $$\begin{array}{ccc}& & e^{-q_1-q_2}\sum _{l_1,l_2,l_3\ge 0}{(-q_1)}^{l_1+l_2}{q}_2^{l_3}\hfill \\ & & \times \frac{(l_1+l_2)!(l_1+l_3)!(l_2+l_3)!(l_1+l_2+l_3)!}{{(l_1!)}^4{(l_2!)}^4{(l_3!)}^4{z}^{l_1+l_2+l_3}}\hfill \\ & & \times (1+(l_2-l_1)(H_{l_2+l_3}-4H_{l_2})),\hfill \end{array}$$(8--7) where q₁, q₂ are generators of the Novikov ring for ${\mathrm{MM}}_{2\text{--}17}^3$ dual, respectively, to ξ and p; see [Coates et al. 16, Section 34]. The identity component of the J-function of F takes the form: $$\begin{array}{c}\hfill \sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{c}_{l,m}{z}^{-6l-2m}{q}_1^l{q}_2^m\end{array}$$for some coefficients $c_{l,m}\in \mathbb{Q}$. The Quantum Lefschetz theorem implies (see [Coates et al. 16, Section D.1]) that the identity component of the J-function of ${\mathrm{MM}}_{2\text{--}17}^3$ is equal to (8--8) $$e^{-c_{1,0}{q}_1-c_{0,1}{q}_2}\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{(l!)}^{5}m!c_{l,m}{z}^{-l-m}{q}_1^l{q}_2^m$$(8--8) and it is known that c_{1, 0} = 1 and c_{0, 1} = 0 [Strangeway 15, Section 5.1]. Equating (8–7(8--7) $$\begin{array}{ccc}& & e^{-q_1-q_2}\sum _{l_1,l_2,l_3\ge 0}{(-q_1)}^{l_1+l_2}{q}_2^{l_3}\hfill \\ & & \times \frac{(l_1+l_2)!(l_1+l_3)!(l_2+l_3)!(l_1+l_2+l_3)!}{{(l_1!)}^4{(l_2!)}^4{(l_3!)}^4{z}^{l_1+l_2+l_3}}\hfill \\ & & \times (1+(l_2-l_1)(H_{l_2+l_3}-4H_{l_2})),\hfill \end{array}$$(8--7) ) and (8–8(8--8) $$e^{-c_{1,0}{q}_1-c_{0,1}{q}_2}\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{(l!)}^{5}m!c_{l,m}{z}^{-l-m}{q}_1^l{q}_2^m$$(8--8) ) determines the c_{l, m}: $$\begin{array}{ccc}\hfill c_{l,m}& =& \sum _{i=0}^l\sum _{j=0}^m{(-1)}^{j+l}\hfill \\ & & \times \frac{(m+l-i-j)!(i+m-j)!(m+l-j)!}{{((l-i)!)}^4{(i!)}^4{((m-j)!)}^{4}j!m!{(l!)}^4}\hfill \\ & & \times (1+(2i-l)(H_{i+m-j}-4H_i)).\hfill \end{array}$$The Quantum Lefschetz theorem now gives that $$\begin{array}{ccc}& & G_{{\mathrm{Str}}_1}\left(t\right)=e^{-t}\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{(l!)}^5{c}_{l,m}{t}^{l+2m}\hfill \\ & & G_{{\mathrm{Str}}_2}\left(t\right)=\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{(l!)}^{4}m!c_{l,m}{t}^{2l+m}\hfill \\ & & G_{{\mathrm{Str}}_3}\left(t\right)=e^{-t}\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{(l!)}^4(l+m)!c_{l,m}{t}^{l+m}.\hfill \end{array}$$

9. Numerical calculations of quantum differential operators

As discussed in Section 2, the regularized quantum period ${\widehat{G}}_X\left(t\right)$ of a Fano manifold X satisfies a differential equation: (9--9) $$\begin{array}{c}\hfill L_X{\widehat{G}}_X\equiv 0L_X=\sum _{k=0}^{k=N}{p}_k\left(t\right)D^k\end{array}$$(9--9) called the regularized quantum differential equation. Here the p_m are polynomials and $D=t\frac{d}{dt}$. The regularized quantum differential equation for X coincides with the (unregularized) quantum differential equation for an anticanonical Calabi–Yau manifold Y⊂X; the study of the regularized quantum period from this point of view was pioneered by Batyrev–Ciocan-Fontanine–Kim–van Straten [Batyrev et al. 98, Batyrev et al. 00]. The differential equation (9–9(9--9) $$\begin{array}{c}\hfill L_X{\widehat{G}}_X\equiv 0L_X=\sum _{k=0}^{k=N}{p}_k\left(t\right)D^k\end{array}$$(9--9) ) is expected to be Fuchsian, and the local system of solutions to L_Xf ≡ 0 is expected to be of low ramification in the following sense.

Definition 9.1 ([Coates et al. 12]).

Let $S\subset {\mathbb{P}}^1$ a finite set, and $\mathbb{V}\to {\mathbb{P}}^1\setminus S$ a local system. Fix a basepoint $x\in {\mathbb{P}}^1\setminus S$. For s ∈ S, choose a small loop that winds once anticlockwise around s and connect it to x via a path, thereby making a loop γ_s about s based at x. Let $T_s:{\mathbb{V}}_x\to {\mathbb{V}}_x$ denote the monodromy of $\mathbb{V}$ along γ_s. The ramification of $\mathbb{V}$ is $$\begin{array}{c}\hfill \mathrm{rf}\left(\mathbb{V}\right):=\sum _{s\in S}dim({\mathbb{V}}_x/{{\mathbb{V}}_x}^{T_s}).\end{array}$$The ramification defect of $\mathbb{V}$ is the quantity $\mathrm{rf}\left(\mathbb{V}\right)-2\mathrm{rk}\left(\mathbb{V}\right)$. Non-trivial irreducible local systems $\mathbb{V}\to {\mathbb{P}}^1\setminus S$ have non-negative ramification defect; this gives a lower bound for the ramification of $\mathbb{V}$. A local system of ramification defect zero is called extremal.

Definition 9.2.

The ramification (respectively, ramification defect) of a differential operator L_X is the ramification (respectively, ramification defect) of the local system of solutions L_Xf ≡ 0.

Definition 9.3.

The quantum differential operator for a Fano manifold X is the operator $L_X\in \mathbb{Q}\left[t\right]⟨D⟩$ such that $L_X{\widehat{G}}_X\equiv 0$ which is of lowest order in D and, among all such operators of this order, is of lowest degree in t. (This defines L_X only up to an overall scalar factor, but this suffices for our purposes.)

Suppose that each of the polynomials p₀, …, p_N are of degree at most r, and write: $$\begin{array}{c}\hfill L_X=\sum _{k=0}^{k=N}{p}_k\left(t\right)D^k{p}_k\left(t\right)=\sum _{l=0}^r{a}_{kl}{t}^l.\end{array}$$The differential equation $L_X{\widehat{G}}_X\equiv 0$ gives a system of linear equations for the coefficients a_kl which, given sufficiently many terms of the Taylor expansion of ${\widehat{G}}_X$, becomes over-determined. Given a priori bounds on N and r, therefore, we could compute the quantum differential operator L_X by calculating sufficiently many terms in the Taylor expansion. In general, we do not have such bounds, but nonetheless by ensuring the linear system for (a_kl) is highly over-determined we can be reasonably confident that the operator L_X which we compute is correct. In addition, since L_X is expected to correspond under mirror symmetry to a Picard–Fuchs differential equation for the mirror family, L_X is expected to be of Fuchsian type. This is an extremely delicate condition on the coefficients (a_kl), and it can be checked by exact computation.

We computed candidate quantum differential operators L_X for all four-dimensional Fano manifolds of Fano index r > 1, and checked the Fuchsian condition in each case. The operators L_X, together with their ramification defects and the log-monodromy data {log T_s: s ∈ S} in Jordan normal form, can be found in Appendix B. In 24 cases, the local system of solutions to the regularized quantum differential equation is extremal, and in the remaining 11 cases it is of ramification defect 1.

To compute the ramification of L_X, we follow Kedlaya [Kedlaya 10, Section 7.3]. This involves only linear algebra over a splitting field for p_N(t)—recall that every singular point of L_X occurs at a root of p_N(t)—and thus can be implemented using exact (not numerical) computer algebra. For this we use Steel’s symbolic implementation of $\bar{\mathbb{Q}}$ in the computational algebra system Magma [Bosma et al. 97, Steel 10].

The situation in lower dimensions

The classification of three-dimensional Fano manifolds is known [Iskovskih 77, Iskovskih 78, Iskovskih 79, Mori and Mukai 82, Mori and Mukai 83, Mori and Mukai 86, Mori and Mukai 03, Mori and Mukai 04], and the quantum periods of all three-dimensional Fano manifolds have been computed [Coates et al. 16]. In each case, the regularized quantum differential equation (9–9(9--9) $$\begin{array}{c}\hfill L_X{\widehat{G}}_X\equiv 0L_X=\sum _{k=0}^{k=N}{p}_k\left(t\right)D^k\end{array}$$(9--9) ) is extremal, and coincides with the Picard–Fuchs differential equation associated with a Laurent polynomial $f:{\left({\mathbb{C}}^{\times }\right)}^3\to \mathbb{C}$ in three variables.¹ This latter phenomenon is a manifestation of Mirror Symmetry, and when it occurs then we say that the Laurent polynomial f is a mirror to the corresponding Fano manifold X. The classification of two-dimensional Fano manifolds, which are called del Pezzo surfaces, is classical, and it was proved in [Akhtar et al. 16] that del Pezzo surfaces correspond under Mirror Symmetry to a distinguished family of Laurent polynomials in two variables called maximally mutable Laurent polynomials. This mirror correspondence extends to two-dimensional Fano orbifolds too [Akhtar et al. 16, Oneto and Petracci, Cavey and Prince 17], and this inspired the classification of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities by Corti–Heuberger [Corti and Heuberger 17, Kasprzyk et al. 17]. Kasprzyk and Tveiten have defined what it means for a Laurent polynomial in any number of variables to be maximally mutable [Kasprzyk and Tveiten], and each of the three-dimensional Fano manifolds corresponds under mirror symmetry to a maximally mutable Laurent polynomial in three variables. It would be very interesting to find out whether the correspondence between maximally mutable Laurent polynomials and Fano manifolds (or, more precisely, Fano varieties with an appropriate class of mild singularities) persists to higher dimensions. Furthermore, Golyshev has observed a connection, which holds in low dimensions, between the ramification defect of the regularized quantum differential operator L_X of a Fano manifold X and the dimension of the primitive part of the middle-dimensional cohomology of X. The regularized quantum differential operators for three-dimensional Fano manifolds are all extremal, which is consistent with the fact that this primitive part automatically vanishes in odd dimensions. It will be interesting to see how much of this picture persists beyond dimension 3.

Source code

This paper is accompanied by full source code, written in Magma. See the included file README.txt for usage instructions. The source code, but not the text of this paper, is released under a Creative Commons CC0 license [CCO]: see the included file COPYING.txt for details. If you make use of the source code in an academic or commercial context, please acknowledge this by including a reference or citation to this paper.

Acknowledgments

We thank Alessio Corti for a number of very useful conversations.

Additional information

Funding

This research was supported by a Royal Society University Research Fellowship (TC); ERC Starting Investigator Grant number 240123; the Leverhulme Trust; grant MK-1297.2014.1; AG Laboratory NRU-HSE, RF government grant ag. 11.G34.31.0023; Grant of Leading Scientific Schools (N.Sh. 2998.2014.1); EPSRC grant EP/I008128/1, and EPSRC Fellowship EP/N022513/1.

Notes

1 The proof of extremality involves applying the generalized Griffiths–Dwork algorithm of Lairez [Lairez 16] to f.

Appendix A. Regularized quantum period sequences

In this appendix, we record the description, degree, and Picard rank ρ_X for each of the four-dimensional Fano manifolds X considered in this paper, together with the first few Taylor coefficients α_d of the regularized quantum period: $$\begin{array}{c}\hfill {\widehat{G}}_X\left(t\right)=\sum _{d=0}^{\infty }{\alpha }_d{t}^d.\end{array}$$The tables are divided by Fano index r. We include only coefficients α_d with $d\equiv 0modr$, since coefficients α_d with $d\not\equiv 0modr$ are zero. Notation is as follows:

• ${\mathbb{P}}^n$ denotes n-dimensional complex projective space;

• Qⁿ denotes a quadric hypersurface in ${\mathbb{P}}^{n+1}$;

• FI⁴_k is as in Section 5 above;

• V⁴_k is as in Section 6.1 above;

• MW⁴_k is as in Section 6.2 above;

• BØS⁴_k is as in Section 7 above;

• Str_k is as in Section 8 above;

• S²_k denotes the del Pezzo surface of degree k;

• F₁ denotes the Hirzebruch surface $\mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^1}(-1)\oplus {\mathcal{O}}_{{\mathbb{P}}^1})$;

• V³_k denotes the three-dimensional Fano manifold of Picard rank 1, Fano index 1, and degree k;

• B³_k denotes the three-dimensional Fano manifold of Picard rank 1, Fano index 2, and degree 8k;

• ${\mathrm{MM}}_{\rho \text{--}k}^3$ denotes the kth entry in the Mori–Mukai list of three-dimensional Fano manifolds of Picard rank ρ [Mori and Mukai 82, Mori and Mukai 83, Mori and Mukai 86, Mori and Mukai 03, Mori and Mukai 04]. We use the ordering as in [Coates et al. 16], which agrees with the original papers of Mori–Mukai except when ρ = 4.

We prefer^{Table A.1, Table A.2, Table A.3, Table A.4, Table A.5}to express manifolds as products of lower-dimensional manifolds where possible, so, for example, ${\text{BØS}}_{122}^4$ is the product ${\mathbb{P}}^1\times {\mathbb{P}}^3$, but we refer to this space as ${\mathbb{P}}^1\times {\mathbb{P}}^3$ rather than ${\text{BØS}}_{122}^4$. The tables for Fano index r with r ∈ {2, 3, 4, 5} are complete. The table for r = 1 is very far from complete.

Table A.1. Four-dimensional Fano manifolds with Fano index r = 5.

X	( − KX)^4	ρX	α0	α5	α10	α15	α20
${\mathbb{P}}^4$	625	1	1	120	113400	168168000	305540235000

Table A.2. Four-dimensional Fano manifolds with Fano index r = 4.

X	( − KX)^4	ρX	α0	α4	α8	α12	α16
Q^4	512	1	1	48	15120	7392000	4414410000

Table A.3. Four-dimensional Fano manifolds with Fano index r = 3.

X	( − KX)^4	ρX	α0	α3	α6	α9	α12
${\mathbb{P}}^2\times {\mathbb{P}}^2$	486	2	1	12	900	94080	11988900
FI^45	405	1	1	18	1710	246960	43347150
FI^44	324	1	1	24	3240	672000	169785000
FI^43	243	1	1	36	8100	2822400	1200622500
FI^42	162	1	1	72	37800	31046400	31216185000
FI^41	81	1	1	360	1247400	6861254400	46381007673000

Table A.4. Four-dimensional Fano manifolds with Fano index r = 2.

X	( − KX)^4	ρX	α0	α2	α4	α6	α8
MW^415	640	2	1	2	6	380	6790
${\mathbb{P}}^1\times {\mathbb{P}}^3$	512	2	1	2	30	740	12670
MW^413	480	2	1	2	54	740	21910
MW^412	416	2	1	2	54	1100	28630
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}35}^3$	448	3	1	4	60	1480	41020
MW^411	384	2	1	4	84	2200	70420
MW^410	352	2	1	4	84	2560	87220
MW^47	320	2	1	4	108	3280	126700
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}32}^3$	384	3	1	6	114	3300	114450
MW^48	320	2	1	6	138	4740	194250
V^418	288	1	1	6	162	6180	284130
MW^45	256	2	1	6	186	7980	410970
${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$	384	4	1	8	168	5120	190120
${\mathbb{P}}^1\times B_5^3$	320	2	1	8	192	6920	303520
V^416	256	1	1	8	240	10880	597520
V^414	224	1	1	8	288	15200	968800
MW^44	192	2	1	8	360	22400	1695400
${\mathbb{P}}^1\times B_4^3$	256	2	1	10	318	15220	886270
V^412	192	1	1	10	438	28900	2310070
V^410	160	1	1	12	684	58800	6129900
${\mathbb{P}}^1\times B_3^3$	192	2	1	14	690	50900	4540690
V^48	128	1	1	16	1296	160000	24010000
V^46	96	1	1	24	3240	672000	169785000
${\mathbb{P}}^1\times B_2^3$	128	2	1	26	2814	447380	84832510
V^44	64	1	1	48	15120	7392000	4414410000
${\mathbb{P}}^1\times B_1^3$	64	2	1	122	84606	84187220	98308169470
V^42	32	1	1	240	498960	1633632000	6558930378000

Table A.5. Certain four-dimensional Fano manifolds with Fano index r = 1.

X	( − KX)^4	ρX	α0	α1	α2	α3	α4	α5	α6	α7
${\text{BØS}}_{115}^4$	512	2	1	0	0	0	24	120	0	0
${\text{BØS}}_{21}^4$	594	2	1	0	0	6	0	0	90	1260
${\text{BØS}}_{118}^4$	513	2	1	0	0	6	0	120	90	0
${\text{BØS}}_{17}^4$	450	3	1	0	0	6	0	120	90	1260
${\text{BØS}}_{47}^4$	513	2	1	0	0	6	24	0	90	2520
${\text{BØS}}_{94}^4$	459	3	1	0	0	6	24	120	90	1260
${\text{BØS}}_{37}^4$	417	3	1	0	0	6	24	120	90	2520
${\text{BØS}}_{74}^4$	486	3	1	0	0	6	48	0	90	2520
${\text{BØS}}_{86}^4$	405	3	1	0	0	6	48	120	90	2520
${\text{BØS}}_{114}^4$	401	3	1	0	0	12	0	120	900	0
${\text{BØS}}_{46}^4$	406	3	1	0	0	12	24	0	900	3780
${\text{BØS}}_{87}^4$	364	4	1	0	0	12	24	120	900	3780
${\text{BØS}}_{32}^4$	322	4	1	0	0	12	24	240	900	5040
${\text{BØS}}_{30}^4$	327	4	1	0	0	12	48	120	900	7560
${\text{BØS}}_{31}^4$	249	5	1	0	0	18	72	360	2430	18900
Str1	225	2	1	0	0	30	120	240	5850	50400
${\text{BØS}}_2^4$	800	2	1	0	2	0	6	0	20	840
${\text{BØS}}_1^4$	605	3	1	0	2	0	6	0	380	840
${\text{BØS}}_{12}^4$	560	3	1	0	2	0	6	60	380	840
${\text{BØS}}_{121}^4$	544	2	1	0	2	0	6	120	20	2520
${\text{BØS}}_{105}^4$	489	3	1	0	2	0	6	120	380	2520
${\text{BØS}}_{18}^4$	529	3	1	0	2	0	30	60	380	840
${\text{BØS}}_{10}^4$	496	4	1	0	2	0	30	60	740	840
${\text{BØS}}_{109}^4$	464	3	1	0	2	0	30	120	380	2520
${\text{BØS}}_{104}^4$	431	3	1	0	2	0	30	120	740	2520
${\text{BØS}}_{15}^4$	433	4	1	0	2	0	30	180	380	3360
${\text{BØS}}_{11}^4$	415	4	1	0	2	0	30	180	740	3360
${\text{BØS}}_8^4$	576	3	1	0	2	6	6	60	110	1680
${\text{BØS}}_{26}^4$	560	3	1	0	2	6	6	60	470	420
${\text{BØS}}_7^4$	592	3	1	0	2	6	6	120	110	1260
${\text{BØS}}_{20}^4$	400	3	1	0	2	6	6	120	830	2520
${\text{BØS}}_{111}^4$	480	3	1	0	2	6	6	180	110	2940
${\text{BØS}}_{24}^4$	442	4	1	0	2	6	6	180	470	2940
${\text{BØS}}_{106}^4$	496	3	1	0	2	6	30	60	470	2940
${\text{BØS}}_{45}^4$	432	3	1	0	2	6	30	60	830	2940
${\text{BØS}}_{41}^4$	433	3	1	0	2	6	30	120	470	3780
${\text{BØS}}_6^4$	463	4	1	0	2	6	30	120	470	3780
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}33}^3$	432	3	1	0	2	6	30	120	830	2520
${\text{BØS}}_{82}^4$	432	4	1	0	2	6	30	180	470	4200
${\text{BØS}}_{113}^4$	400	3	1	0	2	6	30	180	470	5460
${\text{BØS}}_{92}^4$	384	4	1	0	2	6	30	180	830	5460
${\text{BØS}}_{70}^4$	411	4	1	0	2	6	30	240	470	5040
${\text{BØS}}_{16}^4$	337	4	1	0	2	6	30	240	1190	7560
${\text{BØS}}_{52}^4$	464	4	1	0	2	6	54	60	830	2940
${\text{BØS}}_{71}^4$	390	4	1	0	2	6	54	120	1190	3780
${\text{BØS}}_{91}^4$	384	4	1	0	2	6	54	180	830	5460
${\text{BØS}}_{13}^4$	368	4	1	0	2	6	54	180	830	5880
${\text{BØS}}_{81}^4$	357	4	1	0	2	6	54	240	1190	6300
${\mathbb{P}}^2\times F_1$	432	3	1	0	2	12	6	180	920	1680
${\mathbb{P}}^1\times Q^3$	432	2	1	0	2	12	6	240	560	2520
${\text{BØS}}_{27}^4$	417	4	1	0	2	12	6	240	560	3360
${\text{BØS}}_{60}^4$	448	4	1	0	2	12	30	120	920	4620
${\text{BØS}}_{88}^4$	389	4	1	0	2	12	30	180	1280	5460
${\text{BØS}}_{35}^4$	369	4	1	0	2	12	30	180	1280	5460
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}30}^3$	368	3	1	0	2	12	30	240	1280	5040
${\text{BØS}}_{93}^4$	347	4	1	0	2	12	30	300	1280	7980
${\text{BØS}}_{85}^4$	352	4	1	0	2	12	54	240	1280	9660
${\text{BØS}}_{42}^4$	326	4	1	0	2	12	54	240	1640	10080
${\text{BØS}}_{51}^4$	480	4	1	0	2	18	6	180	1370	1260
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}28}^3$	320	3	1	0	2	18	30	360	2090	7560
${\text{BØS}}_{73}^4$	352	4	1	0	2	18	54	180	2090	11340
Str2	240	2	1	0	2	30	54	600	6590	26040
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}36}^3$	496	3	1	0	4	0	36	60	400	3360
${\text{BØS}}_{43}^4$	464	3	1	0	4	0	36	120	400	5040
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}29}^3$	400	4	1	0	4	0	60	60	1480	3360
${\text{BØS}}_{36}^4$	384	4	1	0	4	0	60	120	1480	5040
${\text{BØS}}_3^4$	558	4	1	0	4	6	36	120	490	3360
${\text{BØS}}_{22}^4$	505	4	1	0	4	6	36	120	850	2100
${\text{BØS}}_5^4$	478	4	1	0	4	6	36	180	490	5460
${\text{BØS}}_9^4$	382	4	1	0	4	6	36	180	1210	6720
${\text{BØS}}_{95}^4$	447	4	1	0	4	6	36	240	490	7140
${\mathbb{P}}^2\times {\mathbb{P}}^1\times {\mathbb{P}}^1$	432	3	1	0	4	6	36	240	490	7560
${\text{BØS}}_{25}^4$	409	4	1	0	4	6	36	240	850	7140
${\text{BØS}}_{100}^4$	415	4	1	0	4	6	60	120	1570	4620
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}30}^3$	400	4	1	0	4	6	60	180	1570	5460
${\text{BØS}}_{34}^4$	369	4	1	0	4	6	60	180	1570	6720
${\text{BØS}}_{56}^4$	405	5	1	0	4	6	60	240	1210	8400
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}26}^3$	368	4	1	0	4	6	60	240	1570	8820
${\text{BØS}}_{102}^4$	367	4	1	0	4	6	60	240	1570	9660
${\text{BØS}}_{44}^4$	351	4	1	0	4	6	60	240	1930	9660
${\text{BØS}}_{48}^4$	442	5	1	0	4	6	84	120	1930	4620
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}22}^3$	320	4	1	0	4	6	84	300	2650	13440
${\text{BØS}}_{29}^4$	310	5	1	0	4	6	84	360	2650	15120
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}31}^3$	416	4	1	0	4	12	36	360	940	8400
F1 × F1	384	4	1	0	4	12	36	360	1300	8400
$S_7^2\times {\mathbb{P}}^2$	378	4	1	0	4	12	36	360	1300	9660
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}31}^3$	368	3	1	0	4	12	36	420	940	11760
${\text{BØS}}_{54}^4$	405	5	1	0	4	12	60	300	1660	10080
${\text{BØS}}_{58}^4$	373	5	1	0	4	12	60	300	2020	10080
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}25}^3$	352	4	1	0	4	12	60	360	2020	10920
${\text{BØS}}_{66}^4$	332	5	1	0	4	12	60	360	2020	13440
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}23}^3$	336	4	1	0	4	12	60	420	2020	14280
${\text{BØS}}_{28}^4$	321	5	1	0	4	12	60	420	2020	16380
${\text{BØS}}_{65}^4$	331	5	1	0	4	12	84	420	2380	17640
${\text{BØS}}_{80}^4$	325	5	1	0	4	12	84	420	2740	17640
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}19}^3$	304	4	1	0	4	12	84	480	3100	20160
${\text{BØS}}_{50}^4$	394	5	1	0	4	18	36	480	1750	10500
${\text{BØS}}_{68}^4$	363	5	1	0	4	18	36	480	2110	10500
${\text{BØS}}_{59}^4$	341	5	1	0	4	18	60	480	2830	15540
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}27}^3$	304	3	1	0	4	18	60	600	2830	19740
${\text{BØS}}_{53}^4$	330	5	1	0	4	18	84	480	3190	20580
${\text{BØS}}_{69}^4$	310	5	1	0	4	18	84	480	3550	20580
${\text{BØS}}_{84}^4$	299	5	1	0	4	18	84	600	3550	25620
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}14}^3$	256	4	1	0	4	18	132	780	6070	42420
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}9}^3$	208	4	1	0	4	36	228	1560	15340	122640
${\text{BØS}}_{38}^4$	385	4	1	0	6	0	90	120	1860	7560
${\mathbb{P}}^1\times {\mathbb{P}}^1\times F_1$	384	4	1	0	6	6	90	300	1950	13020
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}13}^3$	368	5	1	0	6	6	114	240	3390	9660
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}24}^3$	336	4	1	0	6	6	114	300	3390	14280
${\text{BØS}}_{33}^4$	305	5	1	0	6	6	114	360	3750	18480
${\text{BØS}}_4^4$	364	5	1	0	6	12	90	420	2760	17220
${\text{BØS}}_{23}^4$	354	5	1	0	6	12	90	480	2760	20160
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}12}^3$	352	5	1	0	6	12	90	540	2400	21420
S^27 × F1	336	5	1	0	6	12	90	540	2760	21420
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}29}^3$	320	3	1	0	6	12	90	600	2400	26040
${\text{BØS}}_{96}^4$	334	5	1	0	6	12	114	480	3840	22680
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}10}^3$	320	5	1	0	6	12	114	540	3840	23940
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}20}^3$	304	4	1	0	6	12	114	600	3840	28560
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}17}^3$	288	4	1	0	6	12	138	600	5280	31080
$S_6^2\times {\mathbb{P}}^2$	324	5	1	0	6	18	90	720	3570	28980
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}18}^3$	288	4	1	0	6	18	114	840	4650	38220
${\text{BØS}}_{57}^4$	298	6	1	0	6	18	138	780	5730	39480
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}16}^3$	272	4	1	0	6	18	138	900	6090	46620
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}25}^3$	256	3	1	0	6	24	114	1200	5820	57120
${\text{BØS}}_{49}^4$	308	6	1	0	6	24	138	960	6180	46200
${\text{BØS}}_{55}^4$	298	6	1	0	6	24	138	960	6540	46200
${\text{BØS}}_{63}^4$	278	6	1	0	6	24	138	1080	6540	53760
${\text{BØS}}_{64}^4$	268	6	1	0	6	24	162	960	7980	53760
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}24}^3$	240	3	1	0	6	24	186	1260	10140	78120
${\text{BØS}}_{39}^4$	307	5	1	0	8	0	168	120	5120	10080
$S_7^2\times {\mathbb{P}}^1\times {\mathbb{P}}^1$	336	5	1	0	8	6	168	360	5210	19740
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}21}^3$	304	4	1	0	8	6	192	360	7010	21000
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}9}^3$	304	5	1	0	8	12	168	720	5660	39480
S^27 × S^27	294	6	1	0	8	12	168	720	6020	39480
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}8}^3$	288	5	1	0	8	12	192	720	7460	42000
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}26}^3$	272	3	1	0	8	12	192	780	7460	47880
S^26 × F1	288	6	1	0	8	18	168	1020	6830	54600
${\mathbb{P}}^1\times {\mathrm{MM}}_{5\text{--}2}^3$	288	6	1	0	8	18	192	1020	7910	57120
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}7}^3$	272	5	1	0	8	18	192	1080	8270	63000
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}15}^3$	256	4	1	0	8	18	216	1140	10070	72660
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}5}^3$	256	5	1	0	8	24	216	1440	10880	89040
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}22}^3$	240	3	1	0	8	24	216	1560	11240	100800
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}13}^3$	240	4	1	0	8	24	240	1560	13040	105840
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}11}^3$	224	4	1	0	8	30	264	1980	16370	142800
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}18}^3$	192	3	1	0	8	48	360	3360	31040	295680
${\text{BØS}}_{40}^4$	230	6	1	0	10	0	270	240	10900	25200
$S_6^2\times {\mathbb{P}}^1\times {\mathbb{P}}^1$	288	6	1	0	10	12	270	840	11080	55440
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}23}^3$	240	3	1	0	10	12	318	960	15760	74760
S^26 × S^27	252	7	1	0	10	18	270	1320	12610	91560
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}6}^3$	256	5	1	0	10	18	294	1320	14050	94080
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}4}^3$	240	5	1	0	10	24	318	1800	17380	135240
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}21}^3$	224	3	1	0	10	24	342	1920	19900	154560
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}12}^3$	224	4	1	0	10	30	342	2340	21070	186060
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}19}^3$	208	3	1	0	10	30	342	2520	21430	208740
${\mathbb{P}}^2\times S_5^2$	270	6	1	0	10	36	270	2160	15040	134400
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}20}^3$	208	3	1	0	10	36	390	2940	27640	255360
S^26 × S^26	216	8	1	0	12	24	396	2160	23160	186480
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}3}^3$	224	5	1	0	12	24	444	2160	26760	191520
F1 × S^25	240	7	1	0	12	36	396	2820	24060	219240
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}10}^3$	208	4	1	0	12	36	492	3360	35220	319200
${\mathbb{P}}^1\times {\mathrm{MM}}_{5\text{--}1}^3$	224	6	1	0	12	42	468	3480	32430	300300
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}17}^3$	192	3	1	0	12	42	540	4140	43230	423360
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}7}^3$	192	4	1	0	12	48	564	4680	48000	486360
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}16}^3$	176	3	1	0	12	60	636	6120	63300	693000
${\mathbb{P}}^1\times {\mathbb{P}}^1\times S_5^2$	240	7	1	0	14	30	546	2760	33350	246540
S^27 × S^25	210	8	1	0	14	36	546	3480	37040	330540
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}15}^3$	176	3	1	0	14	36	714	4320	59720	519120
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}2}^3$	208	5	1	0	14	42	618	4200	46490	425880
${\mathbb{P}}^1\times {\mathrm{MM}}_{4\text{--}1}^3$	192	5	1	0	14	48	690	5280	59540	594720
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}8}^3$	192	4	1	0	14	54	690	5700	61070	631260
${\mathbb{P}}^1\times V_{22}^3$	176	2	1	0	14	60	786	6960	78800	859320
S^26 × S^25	180	9	1	0	16	42	720	4920	58390	567840
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}6}^3$	176	4	1	0	16	66	936	8280	97630	1086540
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}12}^3$	160	3	1	0	16	72	1056	9840	122920	1428000
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}13}^3$	160	3	1	0	16	84	1104	11400	137860	1685040
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}11}^3$	144	3	1	0	16	108	1248	15600	188260	2538480
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}14}^3$	160	3	1	0	18	90	1302	13260	168570	2089080
S^25 × S^25	150	10	1	0	20	60	1140	9120	121700	1377600
$S_4^2\times {\mathbb{P}}^2$	216	7	1	0	20	102	1188	11760	123050	1391880
${\mathbb{P}}^1\times V_{18}^3$	144	2	1	0	20	120	1788	20760	285680	3926160
Str3	86	2	1	0	20	156	2700	41040	697700	12503400
S^24 × F1	192	8	1	0	22	102	1434	13740	160510	1881180
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}3}^3$	144	4	1	0	22	132	2058	24360	345280	4867800
$S_4^2\times {\mathbb{P}}^1\times {\mathbb{P}}^1$	192	8	1	0	24	96	1704	14400	193920	2150400
S^24 × S^27	168	9	1	0	24	102	1704	15720	205530	2452380
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}5}^3$	160	4	1	0	24	126	1992	21300	290130	3813600
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}9}^3$	128	3	1	0	24	174	2784	37680	578490	9059820
S^24 × S^26	144	10	1	0	26	108	1998	19080	270440	3435600
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}4}^3$	144	4	1	0	26	156	2574	31080	457640	6657840
${\mathbb{P}}^1\times V_{16}^3$	128	2	1	0	26	192	3198	44160	700820	11249280
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}8}^3$	112	3	1	0	28	216	3900	58800	984520	17334240
S^24 × S^25	120	11	1	0	30	126	2658	27720	439590	6247500
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}10}^3$	128	3	1	0	30	216	3858	54000	891660	14726880
${\mathbb{P}}^1\times V_{14}^3$	112	2	1	0	34	312	5910	97920	1820140	34520640
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}7}^3$	112	3	1	0	38	348	6954	117840	2268560	44336040
S^24 × S^24	96	12	1	0	40	192	4776	59520	1120000	19138560
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}6}^3$	96	3	1	0	46	528	11826	238560	5341780	122340960
${\mathbb{P}}^1\times V_{12}^3$	96	2	1	0	50	600	13758	288480	6659420	157802400
$S_3^2\times {\mathbb{P}}^2$	162	8	1	0	54	498	9882	162000	2938770	54057780
S^23 × F1	144	9	1	0	56	498	10536	171900	3240110	60897480
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}1}^3$	96	4	1	0	56	672	16296	350400	8393600	205470720
$S_3^2\times {\mathbb{P}}^1\times {\mathbb{P}}^1$	144	9	1	0	58	492	11214	178440	3502120	65938320
S^23 × S^27	126	10	1	0	58	498	11214	181800	3561250	68151720
S^23 × S^26	108	11	1	0	60	504	11916	195120	3962040	78104880
${\mathbb{P}}^1\times {\mathrm{MM}}_{3\text{--}2}^3$	112	4	1	0	60	600	13884	259440	5613000	122354400
S^23 × S^25	90	12	1	0	64	522	13392	225720	4887190	102194400
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}5}^3$	96	3	1	0	68	816	21012	465960	11662880	297392760
S^23 × S^24	72	13	1	0	74	588	17550	319560	7862600	185440080
${\mathbb{P}}^1\times V_{10}^3$	80	2	1	0	80	1320	38688	1078320	32604200	1016215200
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}4}^3$	80	3	1	0	92	1518	47172	1357680	42774050	1385508600
S^23 × S^23	54	14	1	0	108	984	37260	848880	26609400	804368880
${\mathbb{P}}^1\times V_8^3$	64	2	1	0	154	3840	159486	6504960	284808340	12889551360
$S_2^2\times {\mathbb{P}}^2$	108	9	1	0	276	6822	314532	12870000	570227370	25599296520
S^22 × F1	96	10	1	0	278	6822	317850	13006380	579688190	26140920540
$S_2^2\times {\mathbb{P}}^1\times {\mathbb{P}}^1$	96	10	1	0	280	6816	321192	13126080	588430720	26621521920
S^22 × S^27	84	11	1	0	280	6822	321192	13142760	589248730	26688271260
S^22 × S^26	72	12	1	0	282	6828	324558	13295880	599727720	27308448720
S^22 × S^25	60	13	1	0	286	6846	331362	13619400	621807910	28636256460
S^22 × S^24	48	14	1	0	296	6912	348840	14492160	681885440	32334274560
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}3}^3$	64	3	1	0	302	8472	442194	21352560	1128405740	61403700960
S^22 × S^23	36	15	1	0	330	7308	413838	18050760	935040840	48854892240
${\mathbb{P}}^1\times V_6^3$	48	2	1	0	398	17616	1221810	85572960	6386359700	493612489440
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}2}^3$	48	3	1	0	472	21216	1568424	115141440	9050108800	736102993920
S^22 × S^22	24	16	1	0	552	13632	1086120	63331200	4672300800	350133073920
${\mathbb{P}}^1\times V_4^3$	32	2	1	0	1946	215808	35318526	5981882880	1074550170260	200205416839680
$S_1^2\times {\mathbb{P}}^2$	54	10	1	0	10260	2021286	618874020	184451042160	57876574021290	18570362883899400
S^21 × F1	48	11	1	0	10262	2021286	618997146	184491467820	57895141165310	18578110239211740
$S_1^2\times {\mathbb{P}}^1\times {\mathbb{P}}^1$	48	11	1	0	10264	2021280	619120296	184531277760	57913469449600	18585729302999040
S^21 × S^27	42	12	1	0	10264	2021286	619120296	184531893480	57913712003290	18585859292400540
S^21 × S^26	36	13	1	0	10266	2021292	619243470	184572934920	57932529089640	18593740045797840
S^21 × S^25	30	14	1	0	10270	2021310	619489890	184655634120	57970416902950	18609636764945100
S^21 × S^24	24	15	1	0	10280	2021376	620106408	184864542720	58066057475840	18649867837440000
S^21 × S^23	18	16	1	0	10314	2021772	622208142	185592555720	58399032538440	18790790073224400
${\mathbb{P}}^1\times {\mathrm{MM}}_{2\text{--}1}^3$	32	3	1	0	10382	2082840	650724306	199392674160	64624270834220	21530238491351520
S^21 × S^22	12	17	1	0	10536	2028096	636179112	190741334400	60762986684160	19811992617768960
S^21 × S^21	6	18	1	0	20520	4042560	1869353640	783667509120	387953543059200	204188081194137600
${\mathbb{P}}^1\times V_2^3$	16	2	1	0	68762	55200000	61055606526	71592493125120	88810659628444820	114429017109750013440

It appears from Table A.5 as if the regularized quantum period might coincide for the pairs $\{{\text{BØS}}_6^4,{\text{BØS}}_{41}^4\}$ and $\{{\text{BØS}}_{35}^4,{\text{BØS}}_{88}^4\}$. This is not the case. The coefficients α₈, α₉ in these cases are:

Table

X	α8	α9
${\text{BØS}}_6^4$	14350	87360
${\text{BØS}}_{35}^4$	32830	227640
${\text{BØS}}_{41}^4$	10990	102480
${\text{BØS}}_{88}^4$	32830	212520

Thus 10 terms of the Taylor expansion of the regularized quantum period suffice to distinguish all of the four-dimensional Fano manifolds considered in this paper.

Appendix B. Quantum differential operators for four-dimensional Fano manifolds of index r > 1: Numerical results

In this appendix, we record the quantum differential operators for all four-dimensional Fano manifolds of Fano index r > 1. These were computed numerically, as described in Section 9, from 500 terms of the Taylor expansion of the quantum period. They pass a number of strong consistency checks, and so we are reasonably confident that they are correct, but this has not been rigorously proven. We record also the local log-monodromies and ramification defect for the quantum local system, that is, for the local system of solutions to the regularized quantum differential equation. These are derived using exact computer algebra from the (numerically computed) operators L_X, as described in Section 9.

B1. ${\mathbb{P}}^4$.  [description p. 2, regularized quantum period p. 15]

The quantum differential operator is $$\begin{array}{ccc}& & (5t-1)(625t^4+125t^3+25t^2+5t+1)D^4+31250t^5{D}^3\hfill \\ & & +109375t^5{D}^2+156250t^{5}D+75000t^5.\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{5}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\begin{array}{c}\text{at}\text{the}\text{roots}\text{of}625t^4+125t^3\hfill \\ +25t^2+5t+1=0.\hfill \end{array}\hfill \end{array}$$The operator L_X is extremal.

B2. Q⁴.  [description p. 2, regularized quantum period p. 15]

The quantum differential operator is $$\begin{array}{ccc}& & (32t^2-1)(32t^2+1)D^4+8192t^4{D}^3+23552t^4{D}^2\hfill \\ & & +28672t^{4}D+12288t^4.\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}32t^2-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}32t^2+1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B3. FI⁴₁.  [description p. 3, regularized quantum period p. 15]

The quantum differential operator is $$\begin{array}{ccc}& & (11664t^3-1)D^4+69984t^3{D}^3+142884t^3{D}^2\hfill \\ & & +113724t^{3}D+29160t^3.\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}11664t^3-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill \frac{1}{2}& 0\\ \hfill 0& \hfill 0& \hfill 0& \frac{1}{2}\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B4. FI⁴₂.  [description p. 3, regularized quantum period p. 15]

The quantum differential operator is $$\begin{array}{ccc}& & (12t-1)(144t^2+12t+1)D^4+10368t^3{D}^3\hfill \\ & & +21924t^3{D}^2+19116t^{3}D+5832t^3.\hfill \end{array}$$

The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{12}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}144t^2+12t+1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill \frac{3}{4}& 0\\ \hfill 0& \hfill 0& \hfill 0& \frac{1}{4}\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B5. FI⁴₃.  [description p. 3, regularized quantum period p. 15]

The quantum differential operator is $$\begin{array}{ccc}& & (9t-1)(81t^2+9t+1)D^4+4374t^3{D}^3+9477t^3{D}^2\hfill \\ & & +8748t^{3}D+2916t^3.\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{9}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}81t^2+9t+1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B6. FI⁴₄.  [description p. 3, regularized quantum period p. 15]

The quantum differential operator is $$\begin{array}{ccc}& & (432t^3-1)D^4+2592t^3{D}^3+5724t^3{D}^2\hfill \\ & & +5508t^{3}D+1944t^3.\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}432t^3-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill \frac{1}{2}& 1\\ \hfill 0& \hfill 0& \hfill 0& \frac{1}{2}\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B7. FI⁴₅.  [description p. 3, regularized quantum period p. 15]

The quantum differential operator is $$\begin{array}{ccc}& & (729t^6+297t^3-1)D^4+162t^3(54t^3+11)D^3\hfill \\ & & +27t^3(1323t^3+148)D^2+81t^3(702t^3+49)D\hfill \\ & & +1458t^3(20t^3+1).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}729t^6+297t^3-1=0.\hfill \end{array}$$The ramification defect of L_X is 1.

B8. ${\mathbb{P}}^2\times {\mathbb{P}}^2$.  [description p. 3, regularized quantum period p. 15]

The quantum differential operator is $$\begin{array}{ccc}& & (3t+1)(6t-1)(9t^2-3t+1)(36t^2+6t+1)D^4\hfill \\ & & +162t^3(432t^3+7)D^3+27t^3(10584t^3+95)D^2\hfill \\ & & +1296t^3(351t^3+2)D+972t^3(240t^3+1).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{6}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{3}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}9t^2-3t+1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}36t^2+6t+1=0.\hfill \end{array}$$The ramification defect of L_X is 1.

B9. V⁴₂.  [description p. 3, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (6912t^2-1)D^4+27648t^2{D}^3+38400t^2{D}^2\hfill \\ & & +21504t^{2}D+3840t^2.\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}6912t^2-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill \frac{2}{3}& 0\\ \hfill 0& \hfill 0& \hfill 0& \frac{1}{3}\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B10. V⁴₄.  [description p. 3, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (32t-1)(32t+1)D^4+4096t^2{D}^3+5888t^2{D}^2\hfill \\ & & +3584t^{2}D+768t^2.\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{32}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{32}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill \frac{1}{2}& 0\\ \hfill 0& \hfill 0& \hfill 0& \frac{1}{2}\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B11. V⁴₆.  [description p. 4, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (432t^2-1)D^4+1728t^2{D}^3+2544t^2{D}^2\hfill \\ & & +1632t^{2}D+384t^2.\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}432t^2-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill \frac{2}{3}& 0\\ \hfill 0& \hfill 0& \hfill 0& \frac{1}{3}\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B12. V⁴₈.  [description p. 4, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (16t-1)(16t+1)D^4+1024t^2{D}^3+1536t^2{D}^2\hfill \\ & & +1024t^{2}D+256t^2.\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{16}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{16}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B13. V⁴₁₀.  [description p. 4, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (256t^4+176t^2-1)D^4+64t^2(32t^2+11)D^3\hfill \\ & & +16t^2(352t^2+67)D^2+32t^2(192t^2+23)D\hfill \\ & & +192t^2(12t^2+1).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}256t^4+176t^2-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The ramification defect of L_X is 1.

B14. V⁴₁₂.  [description p. 4, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (4t^2-12t+1)(4t^2+12t+1)D^4+32t^2(4t^2-17)D^3\hfill \\ & & +8t^2(46t^2-105)D^2+16t^2(28t^2-37)D\hfill \\ & & +32t^2(6t^2-5).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}4t^2-12t+1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}4t^2+12t+1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B15. V⁴₁₄.  [description p. 4, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (4t^2+1)(108t^2-1)D^4+32t^2(108t^2+13)D^3\hfill \\ & & +24t^2(406t^2+27)D^2+16t^2(708t^2+29)D\hfill \\ & & +128t^2(36t^2+1).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}108t^2-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}4t^2+1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill \frac{2}{3}& 0\\ \hfill 0& \hfill 0& \hfill 0& \frac{1}{3}\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The ramification defect of L_X is 1.

B16. V⁴₁₆.  [description p. 4, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (16t^2-8t-1)(16t^2+8t-1)D^4+128t^2(16t^2-3)D^3\hfill \\ & & +32t^2(184t^2-19)D^2\hfill \\ & & +448t^2(4t-1)(4t+1)D+128t^2(24t^2-1).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \end{array}$$

$$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}16t^2-8t-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}16t^2+8t-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B17. V⁴₁₈.  [description p. 4, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (432t^4+72t^2-1)D^4+288t^2(12t^2+1)D^3\hfill \\ & & +24t^2(414t^2+19)D^2+336t^2(36t^2+1)D\hfill \\ & & +96t^2(54t^2+1).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}432t^4+72t^2-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.

B18. MW⁴₁.  [description p. 5, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & {(2t-1)}^2{(2t+1)}^2(1724t^2-4t-1)(1724t^2+4t-1)\hfill \\ & & (1384128950480t^6-34997928616t^4-263676995t^2+9409)D^6\hfill \\ & & +2(2t-1)(2t+1)(181010493290961157120t^{12}-17468834144875533568t^{10}+374429340495784832t^8\hfill \\ & & +3959280486757728t^6-1227737299988t^4-725619617t^2+18818)D^5\hfill \\ & & +4\left(3126544884116601804800t^{14}-725557954486979610624t^{12}+31166631689741025792t^{10}\right.\hfill \\ & & -\left.440963660134839040t^8-7399870298607304t^6-348759582360t^4-504354223t^2+9409\right)D^4\hfill \\ & & +8t^2(6746754749935824947200t^{12}-1095289161198143939072t^{10}+33852557194447324800t^8\hfill \\ & & -138056179574882528t^6-6267098983057824t^4-620133376448t^2-10735669)D^3\hfill \\ & & +32t^2(3803277296533945221760t^{12}-460852243532660846400t^{10}+13408867107650109352t^8\hfill \\ & & +72880113460188392t^6-1247120973283936t^4-144892007990t^2-573949)D^2\hfill \\ & & +256t^4(524004808703094940640t^{10}-51043969670376668752t^8+1573801437077923102t^6\hfill \\ & & +16338545311012128t^4-54334824441981t^2-5942952755)D\hfill \\ & & +1536t^4(35996404915816139200t^{10}-3025511420019920960t^8\hfill \\ & & +99559010182515260t^6+1234528802429310t^4-1125770982819t^2-89024854)\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& \frac{5}{6}& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& \frac{1}{6}\end{array}\right)\text{at}{\displaystyle t=\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& \frac{5}{6}& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& \frac{1}{6}\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}1724t^2-4t-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}1724t^2+4t-1=0.\hfill \end{array}$$The operator L_X is extremal.

B19. MW⁴₂.  [description p. 5, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & {(2t-1)}^2{(2t+1)}^2(14t-1)(14t+1)(18t-1)(18t+1)(7544656t^6-3112t^4-6667t^2+1)D^6\hfill \\ & & +2(2t-1)(2t+1)\hfill \\ & & (21081096723456t^{12}-1556683668736t^{10}-25006224512t^8+2282941792t^6-4014452t^4-18937t^2+2)D^5\hfill \\ & & +4\left(364128034314240t^{14}-74616093755392t^{12}+1020882362880t^{10}\right.\hfill \\ & & +\left.104092887296t^8-4150928136t^6-1064664t^4-12743t^2+1\right)D^4\hfill \\ & & +8t^2(785749968783360t^{12}-103037075309056t^{10}-578957540736t^8\hfill \\ & & +167061307168t^6-3800819360t^4-2728128t^2-205)D^3\hfill \\ & & +32t^2(442942589109888t^{12}-38313217780544t^{10}-591376611224t^8\hfill \\ & & +79457857384t^6-864235264t^4-697078t^2-13)D^2\hfill \\ & & +256t^4(61027379435232t^{10}-3662113129808t^8-80739092050t^6+9991256448t^4-46797421t^2-33179)D\hfill \\ & & +7680t^4(838452710592t^{10}-37503518528t^8-1048500436t^6+130762470t^4-249799t^2-110)\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{14}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{18}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{18}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{14}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{2}}\text{.}\hfill \end{array}$$The operator L_X is extremal.

B20. MW⁴₃.  [description p. 5, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & {(2t-1)}^2{(2t+1)}^2(104t^2-4t-1)(104t^2+4t-1)(124883200t^6+3445552t^4-190621t^2+50)D^6\hfill \\ & & +2(2t-1)(2t+1)(59432414412800t^{12}-3046657163264t^{10}-307452667136t^8\hfill \\ & & +12555781056t^6-36422116t^4-548263t^2+100)D^5\hfill \\ & & +4\left(1026559885312000t^{14}-181195540611072t^{12}-5355282845184t^{10}+794966941312t^8\right.\hfill \\ & & \left.-20516326820t^6-1544700t^4-364427t^2+50\right)D^4\hfill \\ & & +8t^2(2215208173568000t^{12}-216948202172416t^{10}-15687219459072t^8\hfill \\ & & +1005364344896t^6-20392735992t^4-36932632t^2-5075)D^3\hfill \\ & & +64t^2(624378035507200t^{12}-30899527376640t^{10}-4029754891664t^8\hfill \\ & & +205143060452t^6-2542262335t^4-5022410t^2-175)D^2\hfill \\ & & +256t^4(172050086041600t^{10}-3462364820096t^8\hfill \\ & & -968115899896t^6+47585432988t^4-312087909t^2-517438)D\hfill \\ & & +1536t^4(11818946048000t^{10}-33971342080t^8-59641497680t^6+2994403590t^4-9592347t^2-9140)\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& \frac{5}{6}& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& \frac{1}{6}\end{array}\right)\text{at}{\displaystyle t=\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& \frac{5}{6}& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& \frac{1}{6}\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}104t^2-4t-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}104t^2+4t-1=0\text{.}\hfill \end{array}$$The operator L_X is extremal.

B21. MW⁴₄.  [description p. 5, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{c}\hfill (16t^2+1)(128t^2-1)D^4+64t^2(256t^2+7)D^3+16t^2(2816t^2+43)D^2+96t^2(512t^2+5)D+128t^2(144t^2+1).\end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}128t^2-1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}16t^2+1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The ramification defect of L_X is 1.

B22. MW⁴₅.  [description p. 6, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (32t^4-144t^3+40t^2-12t+1)(32t^4+144t^3+40t^2+12t+1)\hfill \\ & & (1379024896t^8-181690112t^6+32203856t^4+160775t^2+136)D^6\hfill \\ & & +2\left(14121214935040t^{16}-152437496479744t^{14}+18570092085248t^{12}-3527276827648t^{10}\right.\hfill \\ & & -\left.277037824256t^8-556812800t^6-140567296t^4-499733t^2-272\right)D^5\hfill \\ & & +4\left(54366677499904t^{16}-340827602288640t^{14}+64054093561856t^{12}-13315483720192t^{10}\right.\hfill \\ & & \left.-406200526464t^8-986296160t^6+122964452t^4+370629t^2+136\right)D^4\hfill \\ & & +8t^2(103084869025792t^{14}-369261049675776t^{12}+91720177627136t^{10}-21127821675520t^8\hfill \\ & & -305989253824t^6-1259677440t^4-1614768t^2-5831)D^3\hfill \\ & & +128t^2(12720125640704t^{14}-26591028115456t^{12}+8287153333632t^{10}\hfill \\ & & -2016286904136t^8-15422020074t^6-337096385t^4-533452t^2-102)D^2\hfill \\ & & +256t^4(6200095932416t^{12}-7989995554816t^{10}+3020985441920t^8\hfill \\ & & -739274260600t^6-4109085596t^4-179641501t^2-331640)D\hfill \\ & & +1536t^4(386126970880t^{12}-333456404480t^{10}+147420386560t^8\hfill \\ & & -34919816144t^6-204213358t^4-10473275t^2-20944).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}32t^4-144t^3+40t^2-12t+1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}32t^4+144t^3+40t^2+12t+1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The ramification defect of L_X is 1.

B23. MW⁴₆.  [description p. 6, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & {(2t-1)}^2{(2t+1)}^2(6t-1)(6t+1)(10t-1)(10t+1)(1433040t^6+80728t^4-2579t^2+1)D^6\hfill \\ & & +2(2t-1)(2t+1)(226993536000t^{12}-6984049920t^{10}\hfill \\ & & -2174297216t^8+67707232t^6-106452t^4-7441t^2+2)D^5\hfill \\ & & +4\left(3920797440000t^{14}-584516290560t^{12}-52264472064t^{10}\right.\hfill \\ & & \left.+4812821760t^8-86750536t^6+175464t^4-4911t^2+1\right)D^4\hfill \\ & & +8t^2(8460668160000t^{12}-550314631680t^{10}-112753881472t^8\hfill \\ & & +5245181728t^6-97715360t^4-329920t^2-69)D^3\hfill \\ & & +32t^2(4769443728000t^{12}-59034734400t^{10}-53369502424t^8\hfill \\ & & +1881440232t^6-26802560t^4-94678t^2-5)D^2\hfill \\ & & +768t^4(219040164000t^{10}+4402781840t^8-2035415446t^6+67181120t^4-620103t^2-1737)D\hfill \\ & & +4608t^4(15046920000t^{10}+593723200t^8-120630700t^6+4065994t^4-22137t^2-34).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& \frac{1}{2}& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& \frac{1}{2}\end{array}\right)\text{at}{\displaystyle t=\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{6}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{10}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{10}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{6}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& \frac{1}{2}& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& \frac{1}{2}\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{2}}\text{.}\hfill \end{array}$$The operator L_X is extremal.

B24. MW⁴₇.  [description p. 6, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{c}\hfill (8t-1)(8t+1)(16t^2+1)D^4+64t^2(128t^2+3)D^3+16t^2(1456t^2+19)D^2+32t^2(864t^2+7)D+64t^2(180t^2+1).\end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{8}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{8}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}16t^2+1=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill \frac{1}{2}& 0\\ \hfill 0& \hfill 0& \hfill 0& \frac{1}{2}\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The ramification defect of L_X is 1.

B25. MW⁴₈.  [description p. 6, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (2t-1)(2t+1)(8000t^6+528t^4+60t^2-1)(64404500t^8-1791160t^6+729408t^4+2144t^2+5)D^6\hfill \\ & & +4\left(10304720000000t^{16}-1452885248000t^{14}+158410065280t^{12}\right.\hfill \\ & & \left.-11837396096t^{10}-726357052t^8+2787020t^6-1531744t^4-3536t^2-5\right)D^5\hfill \\ & & +4\left(79346344000000t^{16}-7715142848000t^{14}+1509213811040t^{12}\right.\hfill \\ & & \left.-96608493568t^{10}-1083561820t^8+48291628t^6+2808508t^4+5602t^2+5\right)D^4\hfill \\ & & +32t^2(37612228000000t^{14}-2748389976000t^{12}+827208001440t^{10}\hfill \\ & & -40569448408t^8-339211968t^6+5857560t^4-32366t^2-51)D^3\hfill \\ & & +32t^2(74258388500000t^{14}-4572988268000t^{12}+1872888803480t^{10}\hfill \\ & & -62657669272t^8-239219588t^6-13755238t^4-70290t^2-15)D^2\hfill \\ & & +512t^4(4524416125000t^{12}-258077892500t^{10}+128903007200t^8\hfill \\ & & -2846763420t^6-5734916t^4-1401952t^2-4225)D\hfill \\ & & +3072t^4(281769687500t^{12}-15710668750t^{10}+8827474000t^8-131446365t^6-200152t^4-95918t^2-245).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}8000t^6+528t^4+60t^2-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The ramification defect of L_X is 1.

B26. MW⁴₉.  [description p. 6, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (176t^4-144t^3+20t^2+8t-1)(176t^4+144t^3+20t^2-8t-1)(2109888t^6+174528t^4-2941t^2+2)D^6\hfill \\ & & +2\left(718914797568t^{14}-137405030400t^{12}-5419619584t^{10}\right.\hfill \\ & & \left.+2224774528t^8-45409520t^6-364024t^4+8407t^2-4\right)D^5\hfill \\ & & +4\left(3104404807680t^{14}-245927227392t^{12}-51116043392t^{10}\right.\hfill \\ & & \left.+3834427616t^8-24121292t^6+553996t^4-5471t^2+2\right)D^4\hfill \\ & & +8t^2(6698978795520t^{12}-43313504256t^{10}-101803764992t^8\hfill \\ & & +3311154624t^6-46433864t^4-300504t^2-107)D^3\hfill \\ & & +64t^2(1888172529408t^{12}+75804905088t^{10}-23177683476t^8+452012856t^6-8043103t^4-46340t^2-4)D^2\hfill \\ & & +4608t^4(28905231168t^{10}+2010341344t^8-284249759t^6+4423780t^4-79284t^2-316)D\hfill \\ & & +55296t^4(992819520t^{10}+86769760t^8-8133271t^6+123438t^4-1829t^2-4).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}176t^4-144t^3+20t^2+8t-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}176t^4+144t^3+20t^2-8t-1=0.\hfill \end{array}$$The operator L_X is extremal.

B27. MW⁴₁₀.  [description p. 6, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (4t^2+1)(16384t^6+512t^4+44t^2-1)(1473757184t^8+13746176t^6+7592448t^4+46808t^2+55)D^6\hfill \\ & & +4\left(482920754053120t^{16}+86447658893312t^{14}+4886210281472t^{12}+534951231488t^{10}\right.\hfill \\ & & \left.+18501230592t^8+119944000t^6+16234576t^4+72412t^2+55\right)D^5\hfill \\ & & +4\left(3718489806209024t^{16}+396693178679296t^{14}+31496439267328t^{12}+3577124749312t^{10}\right.\hfill \\ & & \left.+37524877312t^8-202496928t^6-28001956t^4-105906t^2-55\right)D^4\hfill \\ & & +16t^2(3525321504587776t^{14}+219939516448768t^{12}+31919813689344t^{10}\hfill \\ & & +2870602727424t^8+30098031616t^6+29514768t^4+38264t^2+755)D^3\hfill \\ & & +64t^2(1740023841947648t^{14}+63218454626304t^{12}+17856939016192t^{10}\hfill \\ & & +1163732019200t^8+10148827776t^6+77992960t^4+145739t^2+55)D^2\hfill \\ & & +256t^4(424064787152896t^{12}+8915881033728t^{10}+4878831042560t^8\hfill \\ & & +233990370304t^6+2025100864t^4+24310608t^2+53911)D\hfill \\ & & +30720t^4(1320486436864t^{12}+15835594752t^{10}+16557547520t^8+617585152t^6+5801632t^4+78192t^2+187).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}4t^2+1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}16384t^6+512t^4+44t^2-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The ramification defect of L_X is 1.

B28. MW⁴₁₁.  [description p. 7, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (176t^4-96t^3+8t^2-4t-1)(176t^4+96t^3+8t^2+4t-1)(46963840t^6-6320080t^4+10817t^2-60)D^6\hfill \\ & & +2\left(16002270986240t^{14}-4453229608960t^{12}+179162526976t^{10}+27549781760t^8\right.\hfill \\ & & \left.-288092960t^6+25314336t^4-28611t^2+120\right)D^5\hfill \\ & & +4\left(69100715622400t^{14}-16714357862400t^{12}+879319798400t^{10}\right.\hfill \\ & & \left.+38172435520t^8-54159304t^6-23633284t^4+6819t^2-60\right)D^4\hfill \\ & & +8t^2(149112070553600t^{12}-33265162726400t^{10}+1526941953280t^8\hfill \\ & & +17140032640t^6-188273072t^4+3160464t^2+1615)D^3\hfill \\ & & +128t^2(21014345918720t^{12}-4511525020640t^{10}+157602014826t^8\hfill \\ & & -362882280t^6+22427697t^4+282960t^2+30)D^2\hfill \\ & & +512t^4(5790594508160t^{10}-1225795140240t^8+31283552519t^6-265518840t^4+10109387t^2+49308)D\hfill \\ & & +15360t^4(79556744960t^{10}-16784238240t^8+319629734t^6-3585072t^4+146945t^2+456).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}176t^4-96t^3+8t^2-4t-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}176t^4+96t^3+8t^2+4t-1=0.\hfill \end{array}$$The operator L_X is extremal.

B29. MW⁴₁₂.  [description p. 10, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (16t^4+44t^2-1)(432t^4+36t^2+1)(314924112t^8-117964512t^6+14238144t^4+164850t^2+221)D^6\hfill \\ & & +4\left(10883777310720t^{16}+14017882540032t^{14}-7127443839744t^{12}\right.\hfill \\ & & \left.+913572411264t^{10}+49215528432t^8+208091040t^6+29910072t^4+249043t^2+221\right)D^5\hfill \\ & & +4\left(83805085292544t^{16}+42638961696768t^{14}-38573795723328t^{12}+6607519967520t^{10}\right.\hfill \\ & & \left.+197257285008t^8+2651761872t^6-52981980t^4-386208t^2-221\right)D^4\hfill \\ & & +16t^2(79451574368256t^{14}+656487804672t^{12}-24209536131072t^{10}\hfill \\ & & +5352527374416t^8+119397810240t^6+1063433832t^4+1841528t^2+1513)D^3\hfill \\ & & +32t^2(78431220245376t^{14}-24146078481216t^{12}-14886119891520t^{10}\hfill \\ & & +4353594087744t^8+89900843112t^6+822085824t^4+1734954t^2+221)D^2\hfill \\ & & +2304t^4(1061924105664t^{12}-538846899984t^{10}-114223084416t^8\hfill \\ & & +48172072716t^6+1084223724t^4+9219514t^2+22737)D\hfill \\ & & +27648t^4(33067031760t^{12}-20911507560t^{10}-1683208512t^8+1243973394t^6+32605206t^4+248744t^2+663).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}432t^4+36t^2+1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}16t^4+44t^2-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The ramification defect of L_X is 1.

B30. MW⁴₁₃.  [description p. 10, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (2t-1)(2t+1)(136t^3-20t^2+6t-1)(136t^3+20t^2+6t+1)(66683996t^6-1058780t^4-19394t^2+15)D^6\hfill \\ & & +4\left(27134518180352t^{14}-3636686279168t^{12}-76696829376t^{10}\right.\hfill \\ & & \left.+3834603264t^8-107218310t^6+2061400t^4+29091t^2-15\right)D^5\hfill \\ & & +4\left(234343566103040t^{14}-21804386465792t^{12}+102596245216t^{10}\right.\hfill \\ & & \left.+15956069184t^8+542705432t^6-1694520t^4-36762t^2+15\right)D^4\hfill \\ & & +32t^2(126422186976640t^{12}-8667906294144t^{10}+129634375864t^8\hfill \\ & & +2690207848t^6+17651600t^4-204240t^2-49)D^3\hfill \\ & & +32t^2(285066614292448t^{12}-15534154687040t^{10}+251203935368t^8\hfill \\ & & -291393828t^6-1136928t^4-258660t^2-15)D^2\hfill \\ & & +128t^4(78551346664144t^{10}-3663416463632t^8+48257529676t^6-820898270t^4-4423658t^2-37545)D\hfill \\ & & +3072t^4(1349017239080t^{10}-57143052340t^8+533807468t^6-19445050t^4-103234t^2-345).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}136t^3-20t^2+6t-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}136t^3+20t^2+6t+1=0.\hfill \end{array}$$The operator L_X is extremal.

B31. MW⁴₁₄.  [description p. 10, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (2t-1)(2t+1)(6t-1)(6t+1)(20t^2-4t+1)(20t^2+4t+1)(127920t^6-45016t^4-293t^2-1)D^6\hfill \\ & & +2\left(81050112000t^{14}-42347458560t^{12}+4138403072t^{10}\right.\hfill \\ & & \left.+109225472t^8+95280t^6+187248t^4+911t^2+2\right)D^5\hfill \\ & & +4\left(349989120000t^{14}-177028331520t^{12}+12857605632t^{10}\right.\hfill \\ & & \left.+227619584t^8+1578760t^6-171496t^4-769t^2-1\right)D^4\hfill \\ & & +8t^2(755239680000t^{12}-383806594560t^{10}+19129972096t^8\hfill \\ & & +278237408t^6+1853984t^4+10304t^2+13)D^3\hfill \\ & & +32t^2(425743344000t^{12}-221217604800t^{10}+7159648792t^8+109530520t^6+1224240t^4+4582t^2+1)D^2\hfill \\ & & +768t^4(19552572000t^{10}-10422296720t^8+211812198t^6+4163952t^4+55827t^2+173)D\hfill \\ & & +23040t^4(268632000t^{10}-146189120t^8+1864348t^6+52366t^4+729t^2+2).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{6}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{6}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{2}}\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}20t^2-4t+1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}20t^2+4t+1=0.\hfill \end{array}$$The operator L_X is extremal.

B32. MW⁴₁₅.  [description p. 10, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (416t^4-144t^3+8t^2+4t-1)(416t^4+144t^3+8t^2-4t-1)(119179008t^6+10942640t^4+192779t^2+980)D^6\hfill \\ & & +2\left(226871066492928t^{14}+10977991065600t^{12}-661584438272t^{10}\right.\hfill \\ & & \left.-1433003264t^8-363743360t^6-47681568t^4-641057t^2-1960\right)D^5\hfill \\ & & +4\left(979670514401280t^{14}+74598187089920t^{12}-1635236114944t^{10}\right.\hfill \\ & & \left.-46841533568t^8+1268915136t^6+55009348t^4+503977t^2+980\right)D^4\hfill \\ & & +8t^2(2114025846865920t^{12}+203938713538560t^{10}-1101057117184t^8\hfill \\ & & -117410965440t^6-475968000t^4+3889200t^2-12299)D^3\hfill \\ & & +128t^2(297929404790784t^{12}+33127871570560t^{10}+111493204952t^8\hfill \\ & & -15417741170t^6-129552291t^4+148450t^2-245)D^2\hfill \\ & & +256t^4(164191489173504t^{10}+19958501546880t^8+167108388648t^6-7614297740t^4-82910507t^2-24740)D\hfill \\ & & +1536t^4(11279101317120t^{10}+1447829510400t^8+16271407536t^6-474232070t^4-5907869t^2-4760).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}416t^4-144t^3+8t^2+4t-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}416t^4+144t^3+8t^2-4t-1=0.\hfill \end{array}$$The operator L_X is extremal.

B33. MW⁴₁₆.  [description p. 11, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (8t^2-4t+1)(8t^2+4t+1)(28t^2-4t-1)(28t^2+4t-1)(1245440t^6+159472t^4-353t^2+2)D^6\hfill \\ & & +2\left(687403171840t^{14}+55847092224t^{12}-2940634112t^{10}\right.\hfill \\ & & \left.+646645248t^8-7816080t^6-602296t^4+771t^2-4\right)D^5\hfill \\ & & +4\left(2968331878400t^{14}+342433251328t^{12}-16569335296t^{10}\right.\hfill \\ & & \left.+962246144t^8+20691276t^6+609636t^4-327t^2+2\right)D^4\hfill \\ & & +8t^2(6405347737600t^{12}+905030160384t^{10}-30840349696t^8+529133120t^6+4111656t^4-69016t^2-79)D^3\hfill \\ & & +64t^2(1805409751040t^{12}+290148947200t^{10}-6893123664t^8+13678020t^6-1333787t^4-13834t^2-3)D^2\hfill \\ & & +256t^4(497488517120t^{10}+86925519744t^8-1508603672t^6-6132868t^4-554137t^2-2494)D\hfill \\ & & +1536t^4(34174873600t^{10}+6291237120t^8-85724560t^6-530810t^4-38831t^2-116).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}8t^2-4t+1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}28t^2-4t-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}28t^2+4t-1=0\hfill \\ & & \left(\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}8t^2+4t+1=0.\hfill \end{array}$$The operator L_X is extremal.

B34. MW⁴₁₇.  [description p. 11, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (2752t^6-1152t^5+224t^4+96t^3-52t^2+12t-1)(2752t^6+1152t^5+224t^4-96t^3-52t^2-12t-1)\hfill \\ & & (4516691026120601600t^{18}+10894175535784019520t^{16}+161057014788668272t^{14}+223186423846901825t^{12}\hfill \\ & & +1489656860655194t^{10}-37076036387883t^8-944190030122t^6+1145046509t^4+34077463t^2+3136)D^8\hfill \\ & & +2\left(718350728614858094700134400t^{30}+1808773177510341584824565760t^{28}\right.\hfill \\ & & +11029946474586078383243264t^{26}+38551285621538701581713408t^{24}-573876948723528931270656t^{22}\hfill \\ & & -55410907815277159905280t^{20}-10868619181236655325696t^{18}-205717988100675369984t^{16}\hfill \\ & & +54690133059712073568t^{14}-2288987306843185052t^{12}-22309755134355486t^{10}\hfill \\ & & \left.+253350406694334t^8+8580745232122t^6-7798176720t^4-238542241t^2-18816\right)D^7\hfill \\ & & +4\left(6234258109050375607576166400t^{30}+16563148489700147386089144320t^{28}\right.\hfill \\ & & +153884025675978814793531392t^{26}+396376397592542781690715136t^{24}-244086311073605059692544t^{22}\hfill \\ & & -293272503356977770941696t^{20}-34041797107502106577312t^{18}+652775043341067168068t^{16}\hfill \\ & & -44917328848849757800t^{14}+11574061561240458564t^{12}+61805608644992588t^{10}\hfill \\ & & \left.-1336778298658238t^8-25795200795638t^6+35399715688t^4+581693399t^2+40768\right)D^6\hfill \\ & & +8\left(29067549125736936474830438400t^{30}+81851870301201769764088381440t^{28}\right.\hfill \\ & & +939457832323447863369383936t^{26}+2148139133199504182578666496t^{24}+5371023638579317537625088t^{22}\hfill \\ & & -957869720387013465562880t^{20}-5020278555805201268384t^{18}-1791882536502223335404t^{16}\hfill \\ & & -85650713028232429760t^{14}-20843283163258659946t^{12}-101280721448407430t^{10}\hfill \\ & & \left.+1741410204251200t^8+30185303650830t^6-49184868070t^4-583650151t^2-37632\right)D^5\hfill \\ & & +16\left(79230237058744125391346073600t^{30}+236991057245518887863871979520t^{28}\right.\hfill \\ & & +3064756009927980485218619392t^{26}+6725558154012574042351982336t^{24}+25714515380315969257756928t^{22}\hfill \\ & & -2214587183336657609276512t^{20}+14905942833336792786992t^{18}+2425296396558205778092t^{16}\hfill \\ & & +70030906648265241400t^{14}+13039662696372133523t^{12}+68594134702584874t^{10}\hfill \\ & & \left.-642583292426727t^8-12336212241784t^6+23535298811t^4+206616410t^2+12544\right)D^4\hfill \\ & & +64t^2(64244286367595768799677644800t^{28}+203835202251539030078676664320t^{26}\hfill \\ & & +2826396757245234563525490688t^{24}+6176058394659175094246477056t^{22}+30060979005675148672022400t^{20}\hfill \\ & & -1750848106552043666387424t^{18}-31680726103451488064224t^{16}+328919199593772520366t^{14}\hfill \\ & & +33052797280494574660t^{12}-311682535251628245t^{10}-3943292702734802t^8\hfill \\ & & -5555362326911t^6+105428531260t^4-716272453t^2-28448)D^3\hfill \\ & & +512t^2(15009134071247813042455347200t^{28}+50222389883894861028705538560t^{26}\hfill \\ & & +725335071957348016538310016t^{24}+1603230224901344022068592328t^{22}+9154893897382658634810352t^{20}\hfill \\ & & -428192374996045407448412t^{18}-15519523249001954206792t^{16}+52781806444610795042t^{14}\hfill \\ & & -1276462080564820298t^{12}-107386844952242335t^{10}-792211501649686t^8\hfill \\ & & -5580848553730t^6+16358540366t^4-35218463t^2-1568)D^2\hfill \\ & & +6144t^6(1216860211646830162557747200t^{24}+4254138344852923650551665920t^{22}\hfill \\ & & +62879395621367262268113088t^{20}+141203555473546515828319348t^{18}+892698338486015205162136t^{16}\hfill \\ & & -36871341336439978899176t^{14}-1637218553379789670928t^{12}+2207989224928555682t^{10}\hfill \\ & & -562698415042789318t^8-9517978064339709t^6-42261757561740t^4-328350172920t^2+368309382)D\hfill \\ & & +15482880t^6(183729957560533831884800t^{24}+663265247268250820234880t^{22}\hfill \\ & & +9929341713271041302112t^{20}+22607862359961003907882t^{18}+151963033359678362364t^{16}\hfill \\ & & -5846359991941694249t^{14}-276152453516510822t^{12}+64100253358393t^{10}\hfill \\ & & -116439515097332t^8-1405235953421t^6-3886412020t^4-21911230t^2-41272).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccccccc}\hfill 0& \hfill 1& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 1& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}2752t^6-1152t^5+224t^4+96t^3-52t^2+12t-1=0\hfill \\ & & \left(\begin{array}{cccccccc}\hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0& 0& 0& 0& 0\end{array}\right)\text{at}\text{the}\text{roots}\text{of}2752t^6+1152t^5+224t^4-96t^3-52t^2-12t-1=0.\hfill \end{array}$$The ramification defect of L_X is 1.

B35. MW⁴₁₈.  [description p. 11, regularized quantum period p. 16]

The quantum differential operator is $$\begin{array}{ccc}& & (4t-1)(4t+1)(8t-1)(8t+1)D^4+64t^2(128t^2-5)D^3+16t^2(1472t^2-33)D^2\hfill \\ & & +32t^2(896t^2-13)D+128t^2(96t^2-1).\hfill \end{array}$$The local log-monodromies for the quantum local system: $$\begin{array}{ccc}& & \left(\begin{array}{cccc}\hfill 0& \hfill 1& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 1& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=0\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{4}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=\frac{1}{8}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{8}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}{\displaystyle t=-\frac{1}{4}}\hfill \\ & & \left(\begin{array}{cccc}\hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\\ \hfill 0& \hfill 0& \hfill 0& 0\end{array}\right)\text{at}t=\infty .\hfill \end{array}$$The operator L_X is extremal.