# Finite temperature $\mathcal{N} = 4$ SYM on ${\bf S}^3$

Consider the following paragraph taken from page 3 of Edward Witten's paper on Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories.

To study the theory at finite temperature on ${\bf S}^3$, we must compute the partition function on ${\bf S}^3 \times {\bf S}^1$ – with supersymmetry-breaking boundary conditions in the ${\bf S}^1$ directions. We denote the circumferences of ${\bf S}^1$ and ${\bf S}^3$ as $\beta$ and $\beta'$, respectively. By conformal invariance, only the ratio $\beta/\beta'$ matters. To study the finite temperature theory on ${\bf R}^3$, we take the large $\beta′$ limit, reducing to ${\bf R}^3 \times {\bf S}^1$.

Why do we need to compute the partition function on ${\bf S}^3 \times {\bf S}^1$ if we want to study $\mathcal{N}=4$ SYM at finite temperature on ${\bf S}^3$?

Because a standard approach to thermal QFT is to compute the expectation values in the thermal state $\exp(-\beta H)$ on a space $X$ through the Euclidean quantum field theory on a spacetime $X\times S^1$ where the time component takes values on a circle with circumference $\beta$. For an explanation of why this works, consult any standard reference on thermal field theory.