On inner product in modular tensor categories. I

TL;DR

This paper investigates the inner product structure in modular tensor categories, demonstrating the unitarity of the modular group action and connecting it to Macdonald polynomial evaluations at roots of unity, revealing new identities.

Contribution

It establishes the unitarity of the modular group action on morphism spaces and links these actions to Macdonald polynomial values, providing new identities and detailed theory exposition.

Findings

Modular group acts unitarily on morphism spaces in MTCs.

In U_q sl_2, modular group action expressed via Macdonald polynomial values.

New identities for Macdonald polynomial values at roots of unity.

Abstract

In this paper we study modular tensor categories (braided rigid balanced tensor categories with additional finiteness and non-degeneracy conditions), in particular, representations of quantum groups at roots of unity. We show that the action of modular group on certain spaces of morphisms in MTC is unitary with respect to the natural inner product on these spaces. In a special case of category based on representations of the quantum group U_q sl_n at roots of unity we show that in some of these spaces of morphisms (for U_q sl_2, in all of them) the action of modular group can be written in terms of values of Macdonald's polynomials of type A at roots of unity. This gives identities for these special values, both known before (symmetry identity) and new ones. The paper contains a detailed exposition of the theory of modular categories as well as construction of modular categories from representation of quantum groups at roots of unity