Proofs of some conjectures of Okazaki and Smith on line defect half-indices of ${\rm SU}(N)$ Chern-Simons theories

TL;DR

This paper proves conjectures relating matrix integrals and q-series as Wilson line defect half-indices in 3d SU(N) Chern-Simons theories, extending formulas and confirming specific cases.

Contribution

It provides explicit formulas for half indices of SU(N) Chern-Simons theories, proving conjectures and extending results to include additional parameters and charges.

Findings

Proved three conjectures of Okazaki and Smith.

Extended formulas to include an extra parameter.

Confirmed the SU(3)_{-4} conjecture.

Abstract

Okazaki and Smith discovered many elegant formulas expressing some matrix integrals as some celebrated $q$-series such as the Rogers--Ramanujan functions or Jacobi theta functions. These integrals arise as Wilson line defect half-indices of 3d $\mathcal{N}=2$ supersymmetric ${\rm SU}(N)$ Chern-Simons theories. We evaluate them by carefully calculating the constant terms of some infinite products. Along the way we use some crucial facts about antisymmetric multivariate formal Laurent series. Consequently, we prove three general conjectures of Okazaki and Smith which provide explicit formulas for half indices of the ${\rm SU}(N)_{-N-k}$ ($k=0,1/2,1$) Chern-Simons theories. During the process, we extend these ${\rm SU}(N)$ formulas to include one additional parameter. Furthermore, we generalize the ${\rm SU}(N)_{-N-1/2}$ and ${\rm SU}(N)_{-N-1}$ conjectures by calculating the corresponding half-indices of Wilson lines of arbitrary charge. As a special instance of our generalizations, we also confirm the ${\rm SU}(3)_{-4}$ conjecture of Okazaki and Smith.