One-Loop Partition Functions in Deformed $\mathcal{N}=4$ SYM Theory

TL;DR

This paper computes one-loop corrections to the partition functions of deformed $ ext{N}=4$ SYM theories, revealing that the Hagedorn temperature correction is deformation-independent despite the partition function's complex dependence.

Contribution

It extends the calculation of one-loop partition functions to deformed $ ext{N}=4$ SYM theories, including finite-size effects, and uncovers the invariance of the Hagedorn temperature correction.

Findings

One-loop correction to the Hagedorn temperature is deformation-independent.

Partition functions depend non-trivially on deformation parameters.

Finite-size effects are accounted for in the calculations.

Abstract

We study the thermodynamic behaviour of the real $\beta$- and $\gamma_i$-deformation of $\mathcal{N}=4$ Super Yang-Mills theory on $\mathbb{R}\times S^3$ in the planar limit. These theories were shown to be the most general asymptotically integrable supersymmetric and non-supersymmetric field-theory deformations of $\mathcal{N}=4$ Super Yang-Mills theory, respectively. We calculate the first loop correction to their partition functions using an extension of the dilatation-operator and P\'{o}lya-counting approach. In particular, we account for the one-loop finite-size effects which occur for operators of length one and two. Remarkably, we find that the $\mathcal{O}(\lambda)$ correction to the Hagedorn temperature is independent of the deformation parameters, although the partition function depends on them in a non-trivial way.