Surface defects, the superconformal index and q-deformed Yang-Mills

TL;DR

This work extends the 4d ${\cal N}=2$ superconformal index in the presence of surface defects from symmetric $SU(N)$ representations to arbitrary irreducible representations by constructing and composing generalized difference operators $\tilde{G}_{\mathcal R}$. In the Schur limit, these operators are self-adjoint, commuting, and have Schur polynomials as eigenfunctions with eigenvalues given by ratios of modular $S$-matrices, reflecting a Verlinde-like structure. The authors connect these 4d defects to defect punctures in $q$-deformed YM, showing that acting with $\tilde{G}_{\mathcal R}$ on the index corresponds to inserting the gauge-invariant operator $\mathcal O_{\mathcal R}=\chi_{\mathcal R}(e^{i\phi})$ in the 2d theory, and they derive explicit OPEs governed by Littlewood–Richardson coefficients. This establishes a detailed bridge between 4d surface defects and 2d topological gauge theory, enriching the index–$q$YM dictionary and suggesting paths to generalizations beyond the Schur limit.

Abstract

Recently a prescription to compute the four-dimensional N = 2 superconformal index in the presence of certain BPS surface defects has been given. These surface defects are labelled by symmetric representations of SU(N). In the present paper we give a prescription to compute the superconformal index in the presence of surface defects labelled by arbitrary representations of SU(N). Furthermore, we extend the dictionary between the N = 2 superconformal Schur-index and correlators of q-deformed Yang-Mills to incorporate such surface defects.

Figures (4)

• Figure 1: The SCFT $\mathcal{T}_{\mathrm{IR}}$, which is assumed to have only full punctures in this paper, is coupled to the SCFT associated to a sphere with three punctures (two maximal punctures and one minimal puncture, black dot in the picture). The resulting SCFT is called $\mathcal{T}_{\mathrm{UV}}$ and corresponds to the degeneration limit shown in the figure.
• Figure 2: Partition functions for the sphere with one, two and three holes/punctures respectively, for q-deformed YM in the zero area limit.
• Figure 3: Disk amplitude in the presence of a surface puncture.
• Figure 4: The index without surface defects corresponds to a correlator of q-YM. Inserting a surface defect/acting with a difference operator in the 4d theory corresponds to inserting a defect puncture on the 2d side.