Abstract
The Schur limit of the superconformal index of four-dimensional mathcal{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 S3 × S1, 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.
Highlights
Are represented harmonically by a specific subset of supersymmetrically protected local operators
We 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
It is clear that the space of Schur operators defines the space of states of the vertex operator algebra, and it does not come as a surprise that the Schur limit of the superconformal index, I(q; a), which counts the former with signs, equals the vacuum character χ0(q; a) of the chiral algebra [1]: χ0(q; a)
Summary
R is the U(1)r r-symmetry charge, and M and M⊥ are generators of the SO(2) rotational groups of two orthogonal planes in R4.9 Equivalently, one may characterize states contributing to the index as those annihilated by the two Poincare supercharges Q1−, Q2− ̇ and their Hermitian conjugates. They define harmonic representatives of cohomology classes of the nilpotent supercharge Q1 = Q1− + S2− ̇ or, equivalently, the charge Q2 = S1− − Q2− ̇ .10. The space of (cohomology classes of) states contributing to the Schur index is isomorphic to the space of states of said chiral algebra and the Schur index equals the (graded) vacuum character or torus partition function of the associated chiral algebra. The first goal of this paper is to elucidate this latter equality directly from the path integral
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