Quantum Flat Connections, KZ equations, and Integrability

TL;DR

This paper explores the quantization of flat connections in supersymmetric Yang-Mills theories, revealing their integrability and connection to KZ and BPZ equations in specific cases.

Contribution

It explicitly describes the quantum flat connections for Argyres-Douglas theories and links them to irregular KZ and BPZ equations, advancing understanding of their integrability.

Findings

Quantum flat connection for $sl_2$ valued in $gl_2$(A) is explicitly described.

Quantum connection is equivalent to irregular KZ connections.

KZ equations can be transformed into BPZ equations via gauge transformation.

Abstract

N=2 supyersymmetric Yang-Mills theories are described in terms of a Hitchin system over a Riemann surface C. Focusing on strongly coupled Argyres-Douglas theories, we show that the corresponding flat bundle over C can be quantized such that the resulting quantum flat connection is integrable. For $sl_2$, the quantum connection takes values in $gl_2$(A) where A is an associative algebra which we explicitly describe for the cases of Painlev\'e I, II and IV. Moreover, we find that the quantum connection is equivalent to irregular versions of Knizhnik-Zamolodchikov (KZ) connections. Utilizing a suitable gauge transformation, one can show that the corresponding KZ equations give rise to BPZ equations.