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Integral constraints for $\mathcal{N}=4$ super-Yang-Mills from a squashed sphere

Shai M. Chester, Ross Dempsey, Debaditya Pramanik, Silviu S. Pufu

TL;DR

The paper demonstrates that three new integral constraints on four-point correlators of the N=4 stress-tensor multiplet, derived from squashed S^4 deformations, are not independent but follow from the two previously known constraints via N=4 superconformal Ward identities. By combining supersymmetric localization inputs with Ward identities, the authors express all relevant integrated correlators in terms of the reduced correlator T(u,v) and standard conformal D- and overline{D}-functions, using a robust toolkit that includes 6d embeddings and SO(5) averaging. The main technical advance lies in constructing finite, crossing-invariant, and regulator-friendly Ward-identity expressions for both non-squashed and squashed deformations, and in demonstrating that the squashing-induced constraints are consequences of established Ward identities. The results have implications for conformal bootstrap studies of N=4 SCFTs and suggest a pathway to derive analogous independent constraints in other theories with less-than-maximal supersymmetry, potentially improving bootstrap bounds and cross-checks. Overall, the work strengthens the bridge between localization-derived data and the conformal bootstrap program by showing squashing yields no new independent integral constraints beyond the known ones in N=4 SYM.

Abstract

Supersymmetric localization and Ward identities have been used in the past several years to derive two integral constraints on the four-point function of the stress-tensor multiplet in $\mathcal{N} = 4$ super-Yang-Mills theory. These constraints are powerful tools for studying the theory, especially when used in tandem with analytic and/or numerical bootstrap techniques. In this paper, we consider three additional integral constraints that can be derived starting from the $\mathcal{N} = 4$ super-Yang-Mills theory placed on a squashed four-sphere. These constraints are technically much more challenging to derive than the ones in the literature, and much of the paper is devoted to developing techniques to make this computation tractable. Our end result is that these three constraints are implied by the two constraints already appearing in the literature.

Integral constraints for $\mathcal{N}=4$ super-Yang-Mills from a squashed sphere

TL;DR

The paper demonstrates that three new integral constraints on four-point correlators of the N=4 stress-tensor multiplet, derived from squashed S^4 deformations, are not independent but follow from the two previously known constraints via N=4 superconformal Ward identities. By combining supersymmetric localization inputs with Ward identities, the authors express all relevant integrated correlators in terms of the reduced correlator T(u,v) and standard conformal D- and overline{D}-functions, using a robust toolkit that includes 6d embeddings and SO(5) averaging. The main technical advance lies in constructing finite, crossing-invariant, and regulator-friendly Ward-identity expressions for both non-squashed and squashed deformations, and in demonstrating that the squashing-induced constraints are consequences of established Ward identities. The results have implications for conformal bootstrap studies of N=4 SCFTs and suggest a pathway to derive analogous independent constraints in other theories with less-than-maximal supersymmetry, potentially improving bootstrap bounds and cross-checks. Overall, the work strengthens the bridge between localization-derived data and the conformal bootstrap program by showing squashing yields no new independent integral constraints beyond the known ones in N=4 SYM.

Abstract

Supersymmetric localization and Ward identities have been used in the past several years to derive two integral constraints on the four-point function of the stress-tensor multiplet in super-Yang-Mills theory. These constraints are powerful tools for studying the theory, especially when used in tandem with analytic and/or numerical bootstrap techniques. In this paper, we consider three additional integral constraints that can be derived starting from the super-Yang-Mills theory placed on a squashed four-sphere. These constraints are technically much more challenging to derive than the ones in the literature, and much of the paper is devoted to developing techniques to make this computation tractable. Our end result is that these three constraints are implied by the two constraints already appearing in the literature.

Paper Structure

This paper contains 47 sections, 285 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction and Summary
  2. Deformations of N = 2 SCFTs on S4̂
  3. Deformation by chiral and anti-chiral operators
  4. Mass deformation
  5. Squashing deformation
  6. Stress-tensor correlation functions in N = 4 SCFTs
  7. Computing N = 4 stress-tensor correlation functions
  8. N = 2 SCFT operators embedded in N = 4 stress-tensor multiplet
  9. Revisiting non-squashed integral constraints
  10. Mass² tau² derivative
  11. Mass⁴ derivative
  12. Integration by parts
  13. Converting to $D$-functions in 6d
  14. Integral constraints with squashing
  15. b² tau² derivative
  16. ...and 32 more sections

Figures (3)

  • Figure 1: Operators in the flavor multiplet of $\mathcal{N}=2$ SCFT. The $\mathfrak{u}(1)_R$ charge of the operators are listed below.
  • Figure 2: Operators appearing in the stress-tensor multiplet of $\mathcal{N}=2$ SCFTs. The $\mathfrak{u}(1)_R$ charges are listed for operators in each column are listed at the bottom.
  • Figure 3: Conformal primary operators in the stress-tensor multiplet of an $\mathcal{N}=4$ SCFT. The lines joining operators indicate which operators are related to each other by a SUSY transformation.