[Paper Review] Temperature-reflection II: Modular Invariance and T-reflection

David A. McGady|arXiv (Cornell University)|Jun 26, 2018

Black Holes and Theoretical Physics40 references3 citations

TL;DR

This paper provides strong evidence that finite-temperature quantum field theory path integrals are invariant under T-reflection—reflecting inverse temperature β to −β—up to a phase, with this invariance rooted in the redundancy of encoding the thermal circle geometry. For 2D CFTs on the torus, modular invariance and T-reflection invariance are shown to be equivalent, and the T-reflection anomaly phase is computed consistently via analytic continuation and modular forms on the double half-plane.

ABSTRACT

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.

Motivation & Objective

• To establish that finite-temperature QFT path integrals are invariant under T-reflection (β → −β), up to a phase, by leveraging geometric redundancy in the thermal circle.
• To demonstrate that T-reflection invariance in 2D CFTs is equivalent to modular invariance, using the torus compactification and lattice identification.
• To compute the T-reflection anomaly phase via analytic continuation and via extension of modular forms to the double half-plane, ensuring consistency.
• To explore the physical and mathematical implications of T-reflection, including potential links to anomalies, SPT phases, and Casimir energies.
• To propose that T-reflection may signal previously unnoticed quantum anomalies and consistency conditions in general QFTs.

Proposed method

• Formalizing the path integral on a thermal circle S¹_β as a function of the lattice Λ(β) = βℤ, which is invariant under β → −β, implying Z(−β) = Z(β) up to a phase.
• Using modular invariance of 2D CFTs on the torus T² = ℂ/Λ(τ) to derive T-reflection invariance via the modular group SL(2,ℤ) and its action on the complex structure τ.
• Computing the T-reflection phase γ_R via analytic continuation of modular forms from the upper half-plane to the double half-plane.
• Extending modular forms (e.g., Eisenstein series, Dedekind eta-function) to the double half-plane and showing agreement between continuation and modular extension methods.
• Applying the R-transformation (S-duality) to compute anomaly phases and linking them to homomorphisms of GL₂(ℤ).
• Analyzing specific CFT examples (e.g., free boson, N=4 SYM) to compute explicit T-reflection phases and verify consistency with modular properties.

Experimental results

Research questions

• RQ1 Is the finite-temperature path integral of a QFT invariant under T-reflection (β → −β), and if not, what is the associated anomaly phase?
• RQ2 How is T-reflection invariance related to modular invariance in 2D CFTs on the torus?
• RQ3 Can the T-reflection anomaly phase be computed consistently via analytic continuation and via extension of modular forms to the double half-plane?
• RQ4 What are the physical implications of T-reflection invariance for anomalies, SPT phases, and Casimir energies?
• RQ5 Does T-reflection invariance impose new consistency conditions on general QFTs, particularly in relation to vacuum energy and the cosmological constant?

Key findings

• T-reflection invariance of QFT path integrals is supported by the geometric redundancy of encoding the thermal circle, with Z(−β) = Z(β) up to a phase e^{iΓ_R}.
• For 2D CFTs on the torus, T-reflection invariance is equivalent to modular invariance under SL(2,ℤ), with the anomaly phase arising from the modular S-transformation.
• The T-reflection anomaly phase Γ_R is computed consistently via two methods: analytic continuation of modular forms and extension to the double half-plane, yielding identical results.
• The anomaly phase for the free boson and N=4 SYM is found to be e^{iΓ_R} = i^{2k}, matching the modular weight of the partition function.
• The Vafa-Witten partition function Z(τ) = 1/Δ(τ) for N=4 SYM on S³×S¹ is invariant under T-reflection, suggesting deep connections to modular and Jacobi forms.
• Perturbative corrections in N=4 SYM on S³×S¹ exhibit β → −β invariance, with leading-order terms matching those of a perturbed harmonic oscillator, indicating robustness of T-reflection symmetry.