Schur correlation functions on S3 × S1

Abstract

The Schur limit of the superconformal index of four-dimensional N = 2 superconformal field theories has been shown to equal the supercharacter of the vacuum module of their associated chiral algebra. Applying localization techniques to the theory suitably put on S 3 ×S 1 , we obtain a direct derivation of this fact. We also show that the localization computation can be extended to calculate correlation functions of a subset of local operators, namely of the so-called Schur operators. Such correlators correspond to insertions of chiral algebra fields in the trace-formula computing the supercharacter. As a by-product of our analysis, we show that the standard lore in the localization literature stating that only off-shell supersymmetrically closed observables are amenable to localization, is incomplete, and we demonstrate how insertions of fermionic operators can be incorporated in the computation.

Sphere correlation functions and Verma modules

“…In this section we briefly review standard 3d N = 4 gauge theories and present our conventions. The basic data entering the definition of a renormalizable 3d N = 4 gauge 3 The Schur index can be also decorated by inserting certain BPS local operators, the so-called Schur operators [53]. Also see [54] for the inclusion of surface defects.…”

Testing Macdonald index as a refined character of chiral algebra

“…(A.33)The constant terms in the above expansion can be found q; q) n (q; q) m (q; q) N −n (q; q) N −m t 2N −n−m 36] that the above express is equal to (t; q)(tq; q) (q; q)(t 2 ; q) . (A 35).…”

Topological correlators and surface defects from equivariant cohomology

“…A duality between N = 2 theories on S 4 and a symplectic boson system on S 2 was identified in [12]. More recently, a correspondence between a β-γ system on the two-torus and N = 2 theories on S 3 × S 1 was also established [13,14]. The link between three and four dimensions was explored in [15][16][17].…”

“…The link between three and four dimensions was explored in [15][16][17]. Notably, for four-dimensional theories it has not yet been possible to deviate from the conformal point since the dualities explored in [12][13][14] make explicit use of features exclusive to superconformal field theories, such as non-trivial U(1) r R-symmetry background fields.…”