2d Index and Surface operators

TL;DR

The paper develops a comprehensive framework to compute and study the 2d $(2,2)$ superconformal index (flavored elliptic genus) for both abelian and non-abelian gauge theories via unitary matrix integrals. It analyzes Landau-Ginzburg, orbifold, and CY-LG structures, establishing flop invariance and CY-LG correspondences, and proves Seiberg-type dualities in 2d through explicit index identities and $q$-difference equations. In a second major thrust, it couples 2d theories to 4d ${\cal N}=2$ theories to realize half-BPS surface operators labeled by Levi types, computes the coupled index, and demonstrates invariance under generalized S-duality, including checks via 2d dualities and vortex-string constructions. The results yield explicit index formulas for a wide class of surface defects and illuminate deep duality structures across 2d and 4d theories, with potential implications for class S theories and related holographic/brane constructions.

Abstract

In this paper we compute the superconformal index of 2d (2,2) supersymmetric gauge theories. The 2d superconformal index, a.k.a. flavored elliptic genus, is computed by a unitary matrix integral much like the matrix integral that computes 4d superconformal index. We compute the 2d index explicitly for a number of examples. In the case of abelian gauge theories we see that the index is invariant under flop transition and CY-LG correspondence. The index also provides a powerful check of the Seiberg-type duality for non-abelian gauge theories discovered by Hori and Tong. In the later half of the paper, we study half-BPS surface operators in N=2 superconformal gauge theories. They are engineered by coupling the 2d (2,2) supersymmetric gauge theory living on the support of the surface operator to the 4d N=2 theory, so that different realizations of the same surface operator with a given Levi type are related by a 2d analogue of the Seiberg duality. The index of this coupled system is computed by using the tools developed in the first half of the paper. The superconformal index in the presence of surface defect is expected to be invariant under generalized S-duality. We demonstrate that it is indeed the case. In doing so the Seiberg-type duality of the 2d theory plays an important role.

Figures (11)

• Figure 1: The pole structure of the integrand. The residues at poles $z=a^{-1}$ and $z=b$ sum to zero. The arrows indicate canceling contributions to the integral along the unit circle.
• Figure 2: $SU(2)$${\cal N}=4$ SYM coupled to the $U(1)$ gauge theory living on the support of the surface operator. The $4d$ field content is denoted in terms of ${\cal N}=2$ multiplets while the $2d$ field content is denoted in terms of $(2,2)$ multiplets.
• Figure 3: The theory ${\mathcal{T}}_{2d}$ is depicted in the red dotted box (on the right hand side). We have used $(2,2)$ multiplets to denote the field content of this $(4,4)$ theory. The $SU(N)$ flavor symmetry of ${\mathcal{T}}_{2d}$ is coupled to the $4d$${\cal N}=4$ SYM (in blue dotted box) as shown.
• Figure 4: The brane set up realizing 2d/4d coupling. In this example, we have chosen $n=5$.
• Figure 5: A convenient quiver representation of the ${\mathcal{N}}=2$ SCQCD.