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Hidden Conformal Symmetry in AdS$_2\times$S$^2$ Beyond Tree Level

P. J. Heslop, A. E. Lipstein, M. Santagata

TL;DR

This work reveals a hidden four-dimensional conformal symmetry organizing SU(1,1|2) four-point correlators in AdS$_2\times$S$^2$ beyond tree level. The weight-2 one-loop sector can be recast as a four-dimensional master function arising from a one-loop bubble in AdS$_2\times$S$^2$, obtained by acting with Casimir operators on the tree-level data; the finite part aligns with a four-dimensional uplift described by a weight-3 single-valued polylog combination. Building on this, the authors propose an AdS$_2\times$S$^2$ scalar effective field theory with a derivative interaction whose bulk dynamics should reproduce all-loop four-point correlators in the two-derivative sector, suggesting a non-renormalisable but UV-completable framework that mirrors higher-dimensional organizing principles. The results illuminate how hidden conformal symmetry extends into quantum corrections and offer a tractable toy model for IIB string theory on AdS$_5\times$S$^5$ and the near-horizon physics of extremal black holes, with potential applications to quantum gravity and holographic computations of loop amplitudes.

Abstract

Correlators of a certain one-dimensional superconformal field theory dual to hypermultiplets in AdS$_2\times$S$^2$ exhibit a hidden four-dimensional conformal symmetry which allows one to repackage all tree-level 4-point correlators into a single four-dimensional object corresponding to a contact diagram arising from a massless $φ^4$ theory in AdS$_2\times$S$^2$. This theory serves as a toy model for IIB string theory in AdS$_5\times$S$^5$ and is interesting in its own right because AdS$_2\times$S$^2$ describes the near-horizon limit of extremal black holes in four dimensions. We argue that after acting with an $SU(1,1)\times SU(2)$ Casimir, all one-loop correlators can similarly be encoded by a four-dimensional function which arises from a one-loop scalar bubble diagram in AdS$_2\times$S$^2$, explaining how the hidden conformal symmetry extends beyond tree level. Finally, we conjecture a scalar effective field theory with a derivative interaction in AdS$_2\times$S$^2$ whose Witten diagrams should directly reproduce 4-point correlators to all loops without acting with Casimirs.

Hidden Conformal Symmetry in AdS$_2\times$S$^2$ Beyond Tree Level

TL;DR

This work reveals a hidden four-dimensional conformal symmetry organizing SU(1,1|2) four-point correlators in AdSS beyond tree level. The weight-2 one-loop sector can be recast as a four-dimensional master function arising from a one-loop bubble in AdSS, obtained by acting with Casimir operators on the tree-level data; the finite part aligns with a four-dimensional uplift described by a weight-3 single-valued polylog combination. Building on this, the authors propose an AdSS scalar effective field theory with a derivative interaction whose bulk dynamics should reproduce all-loop four-point correlators in the two-derivative sector, suggesting a non-renormalisable but UV-completable framework that mirrors higher-dimensional organizing principles. The results illuminate how hidden conformal symmetry extends into quantum corrections and offer a tractable toy model for IIB string theory on AdSS and the near-horizon physics of extremal black holes, with potential applications to quantum gravity and holographic computations of loop amplitudes.

Abstract

Correlators of a certain one-dimensional superconformal field theory dual to hypermultiplets in AdSS exhibit a hidden four-dimensional conformal symmetry which allows one to repackage all tree-level 4-point correlators into a single four-dimensional object corresponding to a contact diagram arising from a massless theory in AdSS. This theory serves as a toy model for IIB string theory in AdSS and is interesting in its own right because AdSS describes the near-horizon limit of extremal black holes in four dimensions. We argue that after acting with an Casimir, all one-loop correlators can similarly be encoded by a four-dimensional function which arises from a one-loop scalar bubble diagram in AdSS, explaining how the hidden conformal symmetry extends beyond tree level. Finally, we conjecture a scalar effective field theory with a derivative interaction in AdSS whose Witten diagrams should directly reproduce 4-point correlators to all loops without acting with Casimirs.
Paper Structure (10 sections, 98 equations)

This paper contains 10 sections, 98 equations.

Table of Contents

  1. Introduction
  2. basics of $SU(1,1|2)$ correlators
  3. One-loop correlators
  4. Generalised Witten diagrams
  5. An all-loops conjecture
  6. Conclusion
  7. Internal coordinate conventions
  8. The functions $f^{(3)}$ and $\mathfrak{f}_s$
  9. Sphere integration
  10. AdS Integral