Comments on Class S(YK)

TL;DR

This paper interprets the Schur half-indices of certain $ ext{SU}(2)$ gauge theories with matter as transition amplitudes in a DSSYK-like framework, linking gauge theory indices to quantum disk partition functions.

Contribution

It provides a novel DSSYK-like interpretation of Schur half-indices for $ ext{SU}(2)$ gauge theories with matter, connecting gauge theory and quantum gravity models.

Findings

Schur half-indices are understood as transition amplitudes in a non-vacuum sector of DSSYK.

Half-indices are computed via chord diagrams with special segments related to $q$-oscillator coherent states.

The $SU(2)$ gauge theory with four flavors corresponds to a quantum disk partition function.

Abstract

We present a DSSYK-like interpretation of the Schur half-indices of $\mathcal{N}=2$ $SU(2)$ gauge theories with matter, in the presence of fundamental Wilson lines. The Schur half-indices of these theories can be understood as transition amplitudes in a non-vacuum sector of ordinary DSSYK. In the language of chord diagrams, the half-indices are obtained by summing over diagrams with special segments, which correspond to coherent states of the $q$-oscillator algebra. In addition, we show that the Schur half-index of $SU(2)$ gauge theory with $n_F=4$ fundamental half-hypermultiplets corresponds to the partition function of a particle on the quantum disk.