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Conformal Field Theory with Periodic Time

Walker Melton, Andrew Strominger

TL;DR

The authors show that time-ordered correlators of a unitary CFT_2 in Minkowski space admit a single-valued, conformally invariant extension to the Lorentzian equal-radius torus ET^2, and they extend this construction to Lorentzian CFT_D on ET^D under a branch-cut assumption that all singularities arise only at null separations. In 2D, they provide explicit ET^2 expressions for two-, three-, and four-point functions, proving single-valuedness and no monodromy on ET^2; for higher points the cross-ratio structure remains single-valued as well. The extension is then formulated in the embedding formalism, showing how ET^D emerges as a global section with X^2=0 and a topology S^{D-1}×S^1, and arguing that four-point (and higher) functions remain single-valued on ET^D given the null-separation branch-cut assumption. The work has potential implications for holography and celestial holography, offering a consistent framework for QFT on spacetimes with closed timelike curves and informing timelike dualities in AdS_3/ℤ.

Abstract

It is shown that time-ordered correlation functions of a unitary CFT$_2$ in 2D Minkowski space admit a single-valued, conformally-invariant extension to the Lorentzian signature torus provided that the $S^1\times S^1$ spatial and temporal radii are equal. The result extends to Lorentzian CFT$_D$ on equal-radii $S^{D-1}\times S^1$ under the assumption that branch cuts occur only when a pair of operator insertions are null separated.

Conformal Field Theory with Periodic Time

TL;DR

The authors show that time-ordered correlators of a unitary CFT_2 in Minkowski space admit a single-valued, conformally invariant extension to the Lorentzian equal-radius torus ET^2, and they extend this construction to Lorentzian CFT_D on ET^D under a branch-cut assumption that all singularities arise only at null separations. In 2D, they provide explicit ET^2 expressions for two-, three-, and four-point functions, proving single-valuedness and no monodromy on ET^2; for higher points the cross-ratio structure remains single-valued as well. The extension is then formulated in the embedding formalism, showing how ET^D emerges as a global section with X^2=0 and a topology S^{D-1}×S^1, and arguing that four-point (and higher) functions remain single-valued on ET^D given the null-separation branch-cut assumption. The work has potential implications for holography and celestial holography, offering a consistent framework for QFT on spacetimes with closed timelike curves and informing timelike dualities in AdS_3/ℤ.

Abstract

It is shown that time-ordered correlation functions of a unitary CFT in 2D Minkowski space admit a single-valued, conformally-invariant extension to the Lorentzian signature torus provided that the spatial and temporal radii are equal. The result extends to Lorentzian CFT on equal-radii under the assumption that branch cuts occur only when a pair of operator insertions are null separated.

Paper Structure

This paper contains 11 sections, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two points
  3. Time-ordered correlator
  4. The cylinder
  5. Three points
  6. Four points
  7. Beyond 4 points
  8. Higher dimensions
  9. The embedding formalism
  10. Conformal correlators on the Einstein torus
  11. Discussion

Figures (2)

  • Figure 1: A fundamental domain in $(\sigma,\bar{\sigma})$ for the Einstein torus ET$^2$ . The two diamonds M$^2_I$ and M$^2_{II}$ are each Weyl-equivalent to 2D Minkowski space M$^2$ .
  • Figure 2: Trajectories in the complex $u_R$ plane. No matter the order in which the operator 1 moves past the lightcones of operators 2 and 3, $u$ never circles the origin with this $i\epsilon$ prescription. The curve always passes the real axis at $u_R = 1$. The colored dots show the location of $u_R$ each time $\mathcal{O}_1$ moves across $\mathcal{O}_2$'s or $\mathcal{O}_3$'s lightcone.