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.
Highlights
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
A quantity closely related to the 2d superconformal index, the elliptic genus, of (2, 2) theories has been widely studied in the literature
Summary
We start our investigation of the superconformal index of the (2, 2) gauge theories. As Imζ → −∞, the moduli space is given by Calabi Yau X which is generically different from X In this limit the gauge theory has the description in terms of nonlinear sigma model on X. The poles in the fundamental domain of a given fugacity are contributed by the chiral multiplets that are charged positively as well as negatively under the corresponding Cartan. This procedure is repeated for all the Cartan generators This residue prescription is reminiscent of the contour integration that computes the S2 partition function of the (2, 2) theories. For positive (negative) values of the FI parameter the contour integration picks out the residues at the poles coming from positively (negatively) charged chiral multiplets. We will not be doing that in this paper as that will lead us far from the main point of this paper
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