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Schur correlation functions on $S^3\times S^1$

Yiwen Pan, Wolfger Peelaers

TL;DR

The paper provides a direct path-integral derivation of the Schur index as the vacuum character of the associated chiral algebra by localizing the 4d N=2 theory on S^3×S^1 with a carefully chosen supercharge Q=Q_1+Q_2, showing the hypermultiplet sector reduces to gauged symplectic bosons on a torus while vector multiplets localize to flat connections. It then extends localization to exact correlation functions of Schur operators, including fermionic insertions, by proving a generalized localizability condition and mapping the computation to a 2d Gaussian theory on T^2 coupled to a 4d matrix integral over gauge data. The resulting framework yields explicit expressions for Schur correlators and reproduces chiral-algebra data in concrete settings, such as SU(2) N=4 SYM, where T_{2d} and J currents satisfy familiar 2d relations (e.g., T_{2d} = T_{Sug.}) and connect to the vacuum character through derivatives of the index. Together, these results strengthen the 4d–2d dictionary, provide practical localization tools for Schur correlators, and open avenues toward defects, modular properties, and deformation-quantization interpretations. $I(q;\vec{a}) = \chi_0(q;\vec{a})$ encodes the central link between the Schur limit and chiral algebra vacua, with the torus localization revealing the underlying 2d CFT structure.

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 $S^3\times S^1$, 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.

Schur correlation functions on $S^3\times S^1$

TL;DR

The paper provides a direct path-integral derivation of the Schur index as the vacuum character of the associated chiral algebra by localizing the 4d N=2 theory on S^3×S^1 with a carefully chosen supercharge Q=Q_1+Q_2, showing the hypermultiplet sector reduces to gauged symplectic bosons on a torus while vector multiplets localize to flat connections. It then extends localization to exact correlation functions of Schur operators, including fermionic insertions, by proving a generalized localizability condition and mapping the computation to a 2d Gaussian theory on T^2 coupled to a 4d matrix integral over gauge data. The resulting framework yields explicit expressions for Schur correlators and reproduces chiral-algebra data in concrete settings, such as SU(2) N=4 SYM, where T_{2d} and J currents satisfy familiar 2d relations (e.g., T_{2d} = T_{Sug.}) and connect to the vacuum character through derivatives of the index. Together, these results strengthen the 4d–2d dictionary, provide practical localization tools for Schur correlators, and open avenues toward defects, modular properties, and deformation-quantization interpretations. encodes the central link between the Schur limit and chiral algebra vacua, with the torus localization revealing the underlying 2d CFT structure.

Abstract

The Schur limit of the superconformal index of four-dimensional 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 , 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.

Paper Structure

This paper contains 22 sections, 134 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Schur index and its supergravity background
  3. The Schur limit of the superconformal index
  4. Supergravity background
  5. Killing spinors
  6. Actions and supersymmetry transformation rules
  7. Localizing the Schur index
  8. Localization locus
  9. Evaluation of the classical action and one-loop determinant
  10. Hypermultiplet one-loop determinant
  11. Exact correlation functions on S³xS¹
  12. Localizable insertions
  13. Localizing fermionic insertions
  14. Schur correlation functions on S³xS¹
  15. Fermionic propagator
  16. ...and 7 more sections

Figures (1)

  • Figure 1: Depiction of $S^3\times S^1$ and its coordinates. The three-sphere is represented as a torus-fibration over an interval.

Theorems & Definitions (1)

  • Claim 1: Localizability