We study N = 2 supersymmetric gauge theories on RP 2 × S 1 and compute the superconformal index by using the localization technique. We consider not only the round real projective plane RP 2 but also the squashed real projective plane RP 2 b which turns back to RP 2 by taking a squashing parameter b as 1. In addition, we find that the result is independent of the squashing parameter b. We apply our new superconformal index to check the simplest case of 3d mirror symmetry, i.e. the equivalence between the N = 2 SQED and the XYZ model on RP 2 × S 1 . We prove it by using a mathematical formula called the q-binomial theorem. We also comment on the N = 4 version of mirror symmetry, mirror symmetry via generalized indices, and possibilities of generalizations from mathematical viewpoints.

We study AdS 7 /CFT 6 correspondence between M-theory on AdS 7 ×S 4 and the 6D N = (2, 0) superconformal field theory. In particular we focus on Wilson surfaces.We use the conjecture that the (2,0) theory compactified on S 1 is equivalent to

Abstract:We show the refinement of the prescription for the geometric transition in the refined topological string theory and, as its application, discuss a possibility to describe qq-characters from the string theory point of view. Though the suggested way to operate the refined geometric transition has passed through several checks, it is additionally found in this paper that the presence of the preferred direction brings a nontrivial effect. We provide the modified formula involving this point. We then apply our prescription of the refined geometric transition to proposing the stringy description of doubly quantized Seiberg-Witten curves called qq-characters in certain cases.

We compute the exact partition function on the branched two-sphere by the localization technique. It is found that it does not depend on a branching parameter q, which means that supersymmetric Rényi entropy defined by utilizing it is equivalent to the usual entanglement entropy. We also provide the interpretation of the conical singularities on the branched sphere as defects sit on the poles of the nonsingular two-sphere.

It has been found that surface operators have a significant role in AGT relation. This duality is an outstanding consequence of M-theory, but it is actually encoded into the brane web for which the topological string can work. From this viewpoint, the surface defect in AGT relation is geometrically engineered as a toric brane realization. Also, there is a class of the brane configuration in M-theory called M-strings which can be translated into the language of the topological string. In this work, we propose a new M-string configuration which can realize AGT relation in the presence of the surface defect by utilizing the geometric transition in the refined topological string.