Conformal Field Theory with Periodic Time

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.

• The paper constructs a globally single-valued extension for time-ordered correlators on a Lorentzian torus with periodic time and space.
• It employs analytic continuation from Minkowski space to the torus, ensuring full SO(D,2) conformal invariance even in the presence of closed timelike curves.
• The work extends explicit two-dimensional results to higher dimensions, offering new frameworks for holography and quantization on nontrivial topologies.

Conformal Field Theory on the Lorentzian Torus: Periodic Time Extension

Introduction and Motivation

This work establishes that time-ordered correlators of unitary conformal field theories (CFTs) in Lorentzian signature admit a globally single-valued, manifestly conformally invariant extension to $S^1 \times S^1$ (the Lorentzian torus with periodic time and space), provided that the temporal and spatial circles have equal radii and for integral-spin fields. This construction generalizes previous approaches using the Minkowski plane and the Einstein cylinder (with only spatial periodicity). Unlike those settings, the Lorentzian torus, dubbed the Einstein torus (ET$^D$ for dimension $D$), allows for an explicit action of the full conformal group SO$(D,2)$, with correlation functions invariant under its identifications.

The results are established explicitly for 2D CFTs and, given the assumption that branch points in correlators occur only under pairwise null separation (proven for $D=2$), extended to general $D$. This approach addresses challenges such as possible multivaluedness arising from monodromies when operator insertions are moved along nontrivial cycles, especially those containing closed timelike curves.

Analytic Continuation from Minkowski to Torus

Two-Point Functions

The analytic extension replaces Minkowski coordinates $(t^+, t^-)$ with $(\sigma, \bar{\sigma})$ via $t^\pm = \tan \sigma^\pm$, permitting periodic identifications $$(\sigma, \bar{\sigma}) \sim (\sigma + (m + n)\pi, \bar{\sigma} + (m - n)\pi)$$. The time-ordered two-point function on ET$^2$ becomes: $$\left\langle \mathcal{O}^{,J}(\sigma_1,\bar{\sigma}_1)\,\mathcal{O}^{,J}(\sigma_2,\bar{\sigma}_2) \right\rangle_{ET_2} = \frac{1}{(i - \sin \sigma_{12} \sin \bar{\sigma}_{12})^\Delta} \left[ \left(\frac{\sin \bar{\sigma}_{12} - i \sin \sigma_{12}}{\sin \sigma_{12} - i \sin \bar{\sigma}_{12}} \right)^{J} \right]$$ where the branch structure is defined by the Feynman $i\epsilon$ prescription. Under a $2\pi$ shift of either coordinate, the correlator crosses branch cuts that yield phases, but when moving along timelike or spacelike cycles, the phases cancel pairwise, ensuring single-valuedness. This result does not extend to Wightman functions or non-periodic time orderings, confirming the necessity of the specific time-ordered construction.

Higher-Point Functions

For three- and higher-point correlators, the extension carries through by expressing cross-ratios and kinematic factors in terms of $(\sigma_k, \bar{\sigma}_k)$ via trigonometric identities, maintaining the conformal invariance and single-valuedness. The analysis for four points focuses on the matrix $P_{ij}$ controlling conformal block expansions, showing that the lack of monodromy for time-ordered products ensures it for the Einstein torus.

For $n > 4$, the correlator depends on multi-cross ratios constructed similarly, and the inductive argument based on the absence of branch cut monodromies applies, guaranteeing global single-valuedness.

Extension to Higher Dimensions

The embedding formalism for Lorentzian CFT$_D$ recasts operator insertions as points $X^A$ on the projective light cone of $\mathbb{R}^{D,2}$, with ET$^D$ given by the section $(X^{-1})^2 + (X^{0})^2 = 1$, yielding induced metric $S^{D-1} \times S^1$ with closed null geodesics. The Minkowski region is covered by two causal diamonds, conformal to flat space, and the transition to ET$^D$ requires a Weyl transformation in the relevant section.

The extension of two-point functions is given by: $$\left\langle \mathcal{O}^\Delta(x_1)\,\mathcal{O}^\Delta(x_2)\right\rangle_{ET^D} =\frac{1}{(2\cos t_{12} - 2\cos\theta_{12} + i\epsilon)^\Delta}$$ for $D=4$, generalizing naturally for arbitrary $D$. The four-point function depends on conformal cross ratios $u$ and $v$ built out of invariants $X_i\cdot X_j$ with $i\epsilon$ prescription. The function $g(u,v)$ appearing in general four-point correlators is shown to be single-valued since all possible windings of operator positions (analytic continuations along the torus cycles) yield trajectories for $(u,v)$ in the complex plane that cannot encircle branch points at $0$ or $\infty$, a consequence of the geometric and causal structure induced by the ET$^D$ identification.

The analysis holds under the strong assumption that singularities only arise under null separation (with exceptions for certain nonlocal CFTs, e.g., in $D=3$). For general $D$, counterexamples or proofs of this assumption remain open problems. For higher-point functions, similar arguments ensure single-valuedness, provided the branch structure is as hypothesized.

Physical and Mathematical Implications

The construction enables the definition of interacting quantum field theories, including string theoretic holographic CFT$_2$s, on spacetimes with closed timelike curves (CTCs), which are otherwise problematic due to issues such as causality violations and the breakdown of standard quantum field theory. The correlators remain well-defined and single-valued despite these difficulties. There are several implications:

• String theory and holography: The result opens definitions of boundary correlators for AdS$_3/\mathbb{Z}$ (with CTCs), connecting to celestial holography and timelike $T$-duality along closed timelike circles [Hull:1998vg]. These results have consequences for the quantization of theories with periodic time and for the construction and interpretation of leaf correlators in celestial holography.
• Lorentzian quantization and nontrivial topologies: The methods provide an explicit framework for quantizing field theories on spacetimes with nontrivial causal structures, including those relevant for quantum gravity, and for understanding the behavior of CFTs under nontrivial topological identifications.
• Mathematical completeness: The work identifies the Lorentzian torus as a natural spacetime supporting full conformal symmetry, potentially enabling new investigations of representation theory, modular invariants, and analytic continuations of conformal blocks.

Outlook

The single-valued extension of time-ordered correlators implies self-consistent, interacting quantum dynamics on backgrounds with closed timelike curves, potentially serving as exact laboratories for quantum information and causality in such spacetimes [Deutsch:1991nm, Hartle:1993sg, Bennett:2009rt, Lloyd:2010nt, Luminet:2021qae, Bishop:2024cqa]. Moreover, these settings may clarify roles of periodicity, causality, and analytic structure in holographic correspondences and topological quantum field theory. The results suggest future opportunities in exploring new phases of holography, the implementation of nonlocality, and the study of modular invariants in CFTs with nonstandard topology.

A critical avenue for future research lies in precisely characterizing circumstances where the extension to periodic time fails (e.g., for CFTs with non-integral spin, generalized branch cut structures, or nontrivial time delays in commutators) and in constructing explicit dynamical models on ET$^D$ with interactions, as well as connecting these frameworks to celestial CFT and quantum gravity.

Conclusion

The paper rigorously constructs a conformally invariant, globally single-valued extension of time-ordered correlation functions for unitary Lorentzian CFTs to spacetimes with periodic time and space, overcoming conventional difficulties associated with closed timelike curves. Explicit formulae in two and higher dimensions demonstrate the viability of the approach under controlled assumptions on analytic structure. These results have substantive implications for the quantization and holographic duality of quantum field theories on nontrivial topologies and suggest promising future directions in quantum gravity and nonlocal QFT (2512.09089).