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arXiv:2405.17562

Angular fractals in thermal QFT

Nathan Benjamin, Jaeha Lee, Sridip Pal, David Simmons–Duffin, Yixin Xu

Published May 27, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2405.17562

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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.
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We explore the idea that the mass parameter mmm can be used as a diagnostic tool for identifying the critical points of thermal two-point functions of massive free scalar fields in odd ddd dimensions. For general values of mmm these correlators admit an expansion in thermal conformal blocks associated with symmetric traceless operators of definite scaling dimension. For each spin the higher-twist contributions can be resummed into a closed-form dressing factor of the leading-twist coefficient. At special values of mmm for which the thermal trace of the energy-momentum tensor vanishes they are identified as two-point functions of thermal CFTs. Through a dimensional uplift, valid for all mmm, we map these correlators to thermal two-point functions of higher-derivative massive theories in d+2d+2d+2 dimensions. We show that the thermal trace of the energy-momentum tensor of the d+2d+2d+2-dimensional theory is proportional to the free energy of the original ddd-dimensional theory. When that thermal trace vanishes for m≠0m\neq 0m=0 the uplifted correlators are identified as two-point functions of a higher-derivative nonunitary thermal CFTd+2_{d+2}d+2​, but they differ from the uplifted nontrivial thermal two-point functions of the original unitary CFTd_dd​.