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arXiv:hep-th/0403191

FINITE TEMPERATURE CORRELATION FUNCTIONS AND MODULAR FORMS IN A GLOBALLY CONFORMAL INVARIANT QFT

Nikolay M. Nikolov, Ivan T. Todorov

Published Jan 1, 2003• 2 citations
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Abstract

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

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In this paper, we revisit the claim that many partition functions are invariant under reflecting temperatures to negative values (T-reflection). The goal of this paper is to demarcate which partition functions should be invariant under T-reflection, and why. Our main claim is that finite-temperature path integrals for quantum field theories (QFTs) should be T-reflection invariant. Because multi-particle partition functions are equal to Euclidean path integrals for QFTs, we expect them to be T-reflection invariant. Single-particle partition functions though are often not invariant under T-reflection. Several exactly solvable systems are non-invariant under naive T-reflection, but are likely invariant under an extended T-reflection. We give example systems that are T-reflection invariant but are (1) non-unitary, (2) chiral, (3) interacting, (4) non-supersymmetric, or (5) non-conformal, and (6) argue that T-reflection is unrelated to time-reversal. Finally, we study the interplay between T-reflection and perturbation theory in the anharmonic harmonic oscillator in quantum mechanics and in Yang-Mills in four-dimensions. This is the first in a series of papers on temperature-reflections.

arXiv:1711.07536•Nov 20, 2017•5 citations
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
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).
Any two dimensional quantum field theory that can be consistently defined on a torus is invariant under modular transformations. In this paper we study families of quantum field theories labeled by a dimensionful parameter t, that have the additional property that the energy of a state at finite t is a function only of t and of the energy and momentum of the corresponding state at t = 0, where the theory becomes conformal. We show that under this requirement, the partition sum of the theory at t = 0 uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in t, to be that of a TTˉT\bar{T}TTˉ deformed CFT. Non-perturbatively, we find that for one sign of t (for which the energies are real) the partition sum is uniquely determined, while for the other sign we find non-perturbative ambiguities. We characterize these ambiguities and comment on their possible relations to holography.
Modular invariance and uniqueness of TT‾ T\overline{T} TT deformed CFT
Abstract Any two dimensional quantum field theory that can be consistently defined on a torus is invariant under modular transformations. In this paper we study families of quantum field theories labeled by a dimensionful parameter t, that have the additional property that the energy of a state at finite t is a function only of t and of the energy and momentum of the corresponding state at t = 0, where the theory becomes conformal. We show that under this requirement, the partition sum of the theory at t = 0 uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in t, to be that of a TT‾ T\overline{T} TT T T ¯ deformed CFT. Non-perturbatively, we find that for one sign of t (for which the energies are real) the partition sum is uniquely determined, while for the other sign we find non-perturbative ambiguities. We characterize these ambiguities and comment on their possible relations to holography.
Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.
In this paper, we revisit the claim that many partition functions are invariant under reflecting temperatures to negative values (T-reflection). The goal of this paper is to demarcate which partition functions should be invariant under T-reflection, and why. Our main claim is that finite-temperature path integrals for quantum field theories (QFTs) should be T-reflection invariant. Because multi-particle partition functions are equal to Euclidean path integrals for QFTs, we expect them to be T-reflection invariant. Single-particle partition functions though are often not invariant under T-reflection. Several exactly solvable systems are non-invariant under naive T-reflection, but are likely invariant under an extended T-reflection. We give example systems that are T-reflection invariant but are (1) non-unitary, (2) chiral, (3) interacting, (4) non-supersymmetric, or (5) non-conformal, and (6) argue that T-reflection is unrelated to time-reversal. Finally, we study the interplay between T-reflection and perturbation theory in the anharmonic harmonic oscillator in quantum mechanics and in Yang-Mills in four-dimensions. This is the first in a series of papers on temperature-reflections.