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ELLIPTIC THERMAL CORRELATION FUNCTIONS AND MODULAR FORMS IN A GLOBALLY CONFORMAL INVARIANT QFT

Nikolay M. Nikolov, Ivan T. Todorov

Published Jan 1, 2008• 18 citations
OpenAlex

Abstract

Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space–time dimensions implies the Huygens ’ principle, and hence, rationality of correlation functions of observable fields [23]. The conformal Hamiltonian H has discrete spectrum assumed to be finitely degenerate. Thermal expectation values of field products on compactified Minkowski space are then represented as linear combinations of basic (doubly periodic) elliptic functions in the conformal time variables (of periods 1 and τ) whose coefficients are, in general, formal power series in q 1 2 = eiπτ involving spherical functions of the “space–like ” fields ’ arguments. As a corollary, if the resulting expansions converge to meromorphic functions, then the finite temperature correlation functions are elliptic. Thermal 2–point functions of free fields are computed which display these features. We also study modular transformation properties of Gibbs energy mean values with respect to the (complex) inverse temperature τ (Im τ = β> 0).

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Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space–time dimensions implies the Huygens ’ principle, and hence, rationality of correlation functions of observable fields [23]. The conformal Hamiltonian H has discrete spectrum assumed to be finitely degenerate. Thermal expectation values of field products on compactified Minkowski space are then represented as linear combinations of basic (doubly periodic) elliptic functions in the conformal time variables (of periods 1 and τ) whose coefficients are, in general, formal power series in q 1 2 = eiπτ involving spherical functions of the “space–like ” fields ’ arguments. As a corollary, if the resulting expansions converge to meromorphic functions, then the finite temperature correlation functions are elliptic. Thermal 2–point functions of free fields are computed which display these features. We also study modular transformation properties of Gibbs energy mean values with respect to the (complex) inverse temperature τ (Im τ = β> 0).
Abstract Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space–time dimensions implies the Huygens’ principle, and hence, rationality of correlation functions of observable fields [16]. The conformal hamiltonian H ( has discrete spectrum and we assume that the partition function tr D qH) , q = e2πi τ, Im τ> 0 (|q | < 1) as well as the finite temperature expectation values of the field products are well defined in the finite energy space D (an assumption that is verified for free fields). We then demonstrate that the finite temperature expectation values are expressed by (doubly periodic) elliptic functions in appropriate coordinates. We compute examples of 2-point functions of free fields and study the modular transformation properties of the mean value of the energy in an equilibrium state with respect to
In this paper, we present robust evidence that general finite temperature quantum field theory (QFT) path integrals are invariant under reflecting temperatures to negative values (T-reflection), up to a possible anomaly phase. Our main focus is on two-dimensional conformal field theories (2d CFTs) on the two-torus. Modular invariance for 2d CFT path integrals follows from demanding invariance under redundant encodings of the two-torus shape in the path integral. We emphasize that identical logic implies 2d CFTs are invariant under T-reflection, up to phases. We compute T-reflection anomaly phases for certain 2d CFT path integrals via a continuation, and via an extension of modular forms from the upper half-plane to the double half-plane. Crucially, they perfectly agree. Requiring QFT path integrals to be invariant under redundant encodings of the spacetime geometry implies (i) that 2d CFTs are both modular and T-reflection invariant and (ii) that general QFT path integrals are invariant under T-reflection. This quite board argument suggests T-reflection phases may indicate previously unnoticed anomalies and consistency conditions for general QFT.
A bstract We study modular properties of conformal field theories perturbed by holomorphic fields. We prove an asymptotic formula for the modular S-transform of a generalized partition function that includes zero modes of higher spin holomorphic currents. The derivation makes use of general properties of torus correlation functions, in particular the Zhu recursion relation. The asymptotic expansion of the modular transformed partition function takes a universal form that is determined iteratively by the second order pole coefficients in the operator product expansion of the holomorphic currents. This proves and generalizes a conjecture regarding the modular transformation properties of generalized Gibbs ensembles.
We review the physical meaning of modular invariance for unitary conformal quantum field theories in d = 2 . For quantum field theory models, while T invariance is necessary for locality, S invariance is not mandatory. S invariance is a form of completeness of the theory that has a precise meaning as Haag duality for arbitrary multi-interval regions. We present a mathematical proof as well as derive this result from a physical standpoint using Renyi entropies and the replica trick. For rational conformal field theories (CFTs), the failure of modular invariance or Haag duality can be measured by an index, related to the quantum dimensions of the model. We show how to compute this index from the modular transformation matrices. The index also appears in a limit of the Renyi mutual information. Cases of infinite index are briefly discussed. Part of the argument can be extended to higher dimensions, where the lack of completeness can also be diagnosed using the CFT data through the thermal partition function and measured by an index. Published by the American Physical Society 2024
A bstract We show that thermal effective field theory controls the long-distance expansion of the partition function of a d -dimensional QFT, with an insertion of any finite-order spatial isometry. Consequently, the thermal partition function on a sphere displays a fractal-like structure as a function of angular twist, reminiscent of the behavior of a modular form near the real line. As an example application, we find that for CFTs, the effective free energy of even-spin minus odd-spin operators at high temperature is smaller than the usual free energy by a factor of 1/2 d . Near certain rational angles, the partition function receives subleading contributions from “Kaluza-Klein vortex defects” in the thermal EFT, which we classify. We illustrate our results with examples in free and holographic theories, and also discuss nonperturbative corrections from worldline instantons.
We show that thermal effective field theory controls the long-distance expansion of the partition function of a ddd-dimensional QFT, with an insertion of any finite-order spatial isometry. Consequently, the thermal partition function on a sphere displays a fractal-like structure as a function of angular twist, reminiscent of the behavior of a modular form near the real line. As an example application, we find that for CFTs, the effective free energy of even-spin minus odd-spin operators at high temperature is smaller than the usual free energy by a factor of 1/2d1/2^d1/2d. Near certain rational angles, the partition function receives subleading contributions from "Kaluza-Klein vortex defects" in the thermal EFT, which we classify. We illustrate our results with examples in free and holographic theories, and also discuss nonperturbative corrections from worldline instantons.