Modular invariance on $S^1 \times S^3$ and circle fibrations

TL;DR

The paper posits a high-temperature/low-temperature duality for conformal field theories on circle fibrations such as $S^3$ and lens spaces, asserting an emergent $SL(2,\mathbb{Z})$-type invariance that exchanges the thermal circle with the fiber circle in the limit of infinite lensing. This conjecture is tested across multiple dimensions and theories, including free scalars, $\mathcal{N}=1$ SCFTs, $\mathcal{N}=4$ SYM at both weak and strong coupling, and the $6$D $\mathcal{N}=(2,0)$ theory, by relating high-temperature partition functions on unlensed geometries to low-temperature results on highly lensed geometries, with the vacuum (Casimir) energy governing the limit. In each case, the leading large-lensing terms exhibit cycle-swapping invariance, though subleading terms can spoil exact modularity; in SUSY theories, the vacuum energy becomes scheme-independent, enabling a cleaner modular-like relationship and hints at Cardy-like formulas in higher dimensions. The work also outlines extensions to non-free theories, orbifolds, and correlation functions, and discusses potential applications to topological volume independence and counting problems via the lens-space family, suggesting broad implications for modular structure in higher-dimensional QFTs.

Abstract

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 $π_1(S^3/\mathbb{Z}_{p\rightarrow \infty})=\mathbb{Z}=π_1(S^1\times S^2)$ and the Gromov-Hausdorff distance between $S^3/\mathbb{Z}_{p\rightarrow \infty}$ and $S^1/\mathbb{Z}_{p\rightarrow \infty} \times S^2$ 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 $S^3$.

Figures (1)

• 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 paper, 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.