Modular Invariance of Conformal Field Theory on $S^1\times S^3$ and Circle Fibrations

I conjecture a high-temperature–low-temperature duality for conformal field theories defined on circle fibrations like $S^3$ and its lens space family. The duality is an exchange between the thermal circle and the fiber circle in the limit where both are small. The conjecture is motivated by the fact that ${\pi }_1(S^3/{\mathbb{Z}}_{p\to \infty })=\mathbb{Z}={\pi }_1(S^1\times S^2)$ and the Gromov-Hausdorff distance between $S^3/{\mathbb{Z}}_{p\to \infty }$ and $S^1/{\mathbb{Z}}_{p\to \infty }\times S^2$ vanishes. Several checks of the conjecture are provided: free fields, $\mathcal{N}=1$ theories in four dimensions (which shows that the Di Pietro–Komargodski supersymmetric Cardy formula and its generalizations are given exactly by a supersymmetric Casimir energy), $\mathcal{N}=4$ super Yang-Mills at strong coupling, and the six-dimensional $\mathcal{N}=(2,0)$ theory. For all examples considered, the duality is powerful enough to control the high-temperature asymptotics on the unlensed $S^3$, relating it to the Casimir energy on a highly lensed $S^3$. Such large-order quotients are more generally useful for studying quantum field theory on curved spacetimes.

Figure 1

A series of approximate equivalences motivating the conjecture in the case of a lens space partition function. More generally, we can replace $S^3/{\mathbb{Z}}_k$ with a smooth manifold ${\mathcal{M}}^{d-1}/{\mathbb{Z}}_k$. In all cases considered in this Letter, the results can be analytically continued to $p=1$ and give a correct equality between the high-temperature partition function on the unlensed manifold and the low-temperature partition function on a highly lensed manifold.