Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the four-point function of the stress-energy tensors in a CFT

Anatoly Dymarsky

TL;DR

This work quantifies how Ward Identities constrain the four-point functions of stress-energy tensors and conserved currents in a CFT by counting the remaining functional degrees of freedom. It reveals a precise match between these unrestricted DOF and the functional degrees of freedom governing 2→2 scattering amplitudes of corresponding higher-dimensional dual particles, supporting a deep link between CFT correlators and scattering amplitudes. The authors develop a coordinate-space framework to isolate the unrestricted DOF, propose a bootstrap formulation restricted to these DOF, and discuss two practical strategies—coordinate-space and momentum-space approaches—for solving the Ward Identities. They also illuminate a parallel between solving constraints in coordinate and momentum space and examine degeneracies in small dimensions. Overall, the paper suggests a pathway to a more efficient bootstrap program for spinning operators and hints at a broader duality between CFT data and higher-dimensional scattering.)

Abstract

We discuss to what extent the full set of Ward Identities constrain the four-point function of the stress-energy tensors or conserved currents in a conformal field theory. We calculate the number of kinematically unrestricted functional degrees of freedom governing the corresponding correlators and find that it matches the number of functional degrees of freedom governing scattering amplitudes of some "dual" massless particles in the auxiliary Minkowski space. We also formulate the conformal bootstrap constraints for the correlators in question in terms of only unrestricted degrees of freedom. As a by-product we find interesting parallels between solving Ward Identities in coordinate and momentum space.

On the four-point function of the stress-energy tensors in a CFT

TL;DR

This work quantifies how Ward Identities constrain the four-point functions of stress-energy tensors and conserved currents in a CFT by counting the remaining functional degrees of freedom. It reveals a precise match between these unrestricted DOF and the functional degrees of freedom governing 2→2 scattering amplitudes of corresponding higher-dimensional dual particles, supporting a deep link between CFT correlators and scattering amplitudes. The authors develop a coordinate-space framework to isolate the unrestricted DOF, propose a bootstrap formulation restricted to these DOF, and discuss two practical strategies—coordinate-space and momentum-space approaches—for solving the Ward Identities. They also illuminate a parallel between solving constraints in coordinate and momentum space and examine degeneracies in small dimensions. Overall, the paper suggests a pathway to a more efficient bootstrap program for spinning operators and hints at a broader duality between CFT data and higher-dimensional scattering.)

Abstract

We discuss to what extent the full set of Ward Identities constrain the four-point function of the stress-energy tensors or conserved currents in a conformal field theory. We calculate the number of kinematically unrestricted functional degrees of freedom governing the corresponding correlators and find that it matches the number of functional degrees of freedom governing scattering amplitudes of some "dual" massless particles in the auxiliary Minkowski space. We also formulate the conformal bootstrap constraints for the correlators in question in terms of only unrestricted degrees of freedom. As a by-product we find interesting parallels between solving Ward Identities in coordinate and momentum space.

Paper Structure

This paper contains 16 sections, 42 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Imposing Conservation in the Coordinate Space
  3. Conservation Constraints for Conserved Currents
  4. Permutation symmetry I
  5. Number of Unrestricted Functional Degrees of Freedom
  6. Permutation symmetry II
  7. Conformal Bootstrap for Unrestricted D.O.F.
  8. Small $\rm d$ and Degenerate Tensors
  9. Four Point Function of the Stress-Energy Tensors
  10. CFT Correlators and Scattering Amplitudes
  11. Two Easy Ways to Solve the Conservation Constraints
  12. Solving All W.I.'s Automatically
  13. Solving Ward Identities in the Momentum Space
  14. Conclusions
  15. Degenerate Tensors in Various Dimensions
  16. ...and 1 more sections

Figures (2)

  • Figure 1: (a) The $(u,v)$-plane with physically accessible area of cross-ratios highlighted in color. (b) The same area after a change of variables $(u,v)\rightarrow \omega$.
  • Figure 2: Parallel between solving W.I.'s in the coordinate and momentum space.