The large $N$ vector model on $S^1\times S^2$

TL;DR

This paper develops a power series method to evaluate the partition function and energy density of a massive scalar on a 2-sphere at finite temperature, deriving finite size corrections for the $O(N)$ model's stress tensor and higher spin currents.

Contribution

It introduces a novel power series approach to compute finite size effects and applies it to the large $N$, critical $O(N)$ model on $S^1\times S^2$, including the derivation of corrections to stress tensor and higher spin currents.

Findings

Finite size corrections to stress tensor expectation value obtained.

Finite size corrections to higher spin currents derived.

Corrections tend to free theory results at large spin.

Abstract

We develop a method to evaluate the partition function and energy density of a massive scalar on a 2-sphere of radius $r$ and at finite temperature $\beta$ as power series in $\frac{\beta}{r}$. Each term in the power series can be written in terms of polylogarithms. We use this result to obtain the gap equation for the large $N$, critical $O(N)$ model with a quartic interaction on $S^1\times S^2$ in the large radius expansion. Solving the gap equation perturbatively we obtain the leading finite size corrections to the expectation value of stress tensor for the $O(N)$ vector model on $S^1\times S^2$. Applying the Euclidean inversion formula on the perturbative expansion of the thermal two point function we obtain the finite size corrections to the expectation value of the higher spin currents of the critical $O(N)$ model. Finally we show that these finite size corrections of higher spin currents tend to that of the free theory at large spin as seen earlier for the model on $S^1\times R^2$.