A very short note on the (rational) graded Hori map

Abstract

The graded Hori map has been recently introduced by Han-Mathai in the context of T-duality as a $\mathbb{Z}$-graded transform whose homogeneous components are the Hori-Fourier transforms in twisted cohomology associated with integral multiples of a basic pair of T-dual closed 3-forms. We show how in the rational homotopy theory approximation of T-duality, such a map is naturally realized as a pull-iso-push transform, where the isomorphism part corresponds to the canonical equivalence between the left and the right gerbes associated with a T-duality configuration.

Keywords:

• Rational homotopy theory
• Hori transform
• Jacobi forms
• differential graded commutative algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 55P62
• 11F50
• 13A02

Acknowledgments

D.F. thanks NYU-AD for support on occasion of the workshop M-theory and Mathematics during which the idea of this note originated, and Hisham Sati and Urs Schreiber for discussions and comments on an early version of this note.

Notes

1 Here and below, all tensor products are over $\mathbb{K}.$