Quantum Kramers-Wannier Duality And Its Topology

TL;DR

This paper extends Kramers-Wannier duality to quantum groups on surfaces, linking it to quantum cohomologies and topological invariants, inspired by string theory's Poisson-Lie T-duality.

Contribution

It introduces a quantum group-based generalization of Kramers-Wannier duality applicable to surfaces with boundaries, connecting it to quantum cohomologies and 3-fold invariants.

Findings

Defined cohomologies with quantum coefficients for surfaces.

Established functorial properties under surface glueing.

Linked quantum cohomologies to q-invariants of 3-folds.

Abstract

We show for any oriented surface, possibly with a boundary, how to generalize Kramers-Wannier duality to the world of quantum groups. The generalization is motivated by quantization of Poisson-Lie T-duality from the string theory. Cohomologies with quantum coefficients are defined for surfaces and their meaning is revealed. They are functorial with respect to some glueing operations and connected with q-invariants of 3-folds.