# Automorphic properties of (2, 0) theory on T6

## TL;DR

This paper studies the automorphic properties of the (2, 0) theory on six-tori, deriving transformation laws for the partition vector and a shifted momentum quantization law, revealing non-trivial constraints without a Lagrangian.

## Contribution

It derives the automorphic transformation law for the (2, 0) theory's partition vector and establishes a shifted momentum quantization law, providing new insights into the theory's structure.

## Key findings

- Derived automorphic transformation law for the partition vector.
- Established a shifted quantization law for spatial momentum.
- Connected results to gauge bundle properties and previous Yang-Mills insights.

## Abstract

We consider ADE-type (2, 0) theory on a family of flat six-tori endowed with flat Sp(4) connections coupled to the R-symmetry. Our main objects of interest are the components of the `partition vector' of the theory. These constitute an element of a certain finite dimensional vector space, carrying an irreducible representation of a discrete Heisenberg group related to the 't Hooft fluxes of the theory. Covariance under the SL_6(Z) mapping class group of a six-torus amounts to a certain automorphic transformation law for the partition vector, which we derive. Because of the absence of a Lagrangian formulation of (2, 0) theory, this transformation property is not manifest, and gives useful non-trivial constraints on the partition vector. As an application, we derive a shifted quantization law for the spatial momentum of (2, 0) theory on a space-time of the form R x T5. This quantization law is in agreement with an earlier result based on the relationship between (2, 0) theory and maximally supersymmetric Yang-Mills theory together with certain geometric facts about gauge bundles.