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

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 $S^1\times R^2$.