Abstract
Abstract Motivated by problems arising in the study of N=2 supersymmetric gauge theories we introduce and study irregular singularities in two-dimensional conformal field theory, here Liouville theory. Irregular singularities are associated to representations of the Virasoro algebra in which a subset of the annihilation part of the algebra act diagonally. In this paper we define natural bases for the space of conformal blocks in the presence of irregular singularities, describe how to calculate their series expansions, and how such conformal blocks can be constructed by some delicate limiting procedure from ordinary conformal blocks. This leads us to a proposal for the structure functions appearing in the decomposition of physical correlation functions with irregular singularities into conformal blocks. Taken together, we get a precise prediction for the partition functions of some Argyres-Douglas type theories on S 4.
Highlights
In this paper we propose a precise answer to a natural question which arises in the study of four-dimensional N = 2 gauge theories: How can we define and compute the partition function of an Argyres-Douglas theory [2, 3] on S4? The answer to this question is suggested by the recent observation [1] that the S4 partition function [26] of a certain class of SU(2) gauge theories coincides with Liouville theory correlation functions
In this paper we have initiated the study of Virasoro conformal blocks and Liouville theory correlation functions in the presence of irregular singularities
Regular BPZ conformal blocks are usually defined through the sewing construction, which provides a convergent power series expansion around the corners of the complex structure moduli space where the Riemann surface degenerates
Summary
In this paper we propose a precise answer to a natural question which arises in the study of four-dimensional N = 2 gauge theories: How can we define and compute the partition function of an Argyres-Douglas theory [2, 3] on S4? In this paper we are going to exploit the observation that the limiting procedure which defines Argyres-Douglas theories from SU(2) gauge theories [16] has a simple interpretation as a collision limit in Liouville theory. The existence of a well-defined collision limit for Liouville theory correlation functions is far from obvious from a two-dimensional perspective. While this paper was being written, reference [7] appeared which has partial overlap with the discussion in sections 2 and 7
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