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

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 ${\pi }_1(S^3/{\mathbb{Z}}_{p\to \infty })=\mathbb{Z}={\pi }_1(S^1\times S^2)$ 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 $S^3$.

12 pages

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Cited by in corpus (31)

• Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS${}_5$ black holes
• The asymptotic growth of states of the 4d N=1 superconformal index
• Global symmetry, Euclidean gravity, and the black hole information problem
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• Supersymmetric phases of 4d N=4 SYM at large N
• Constraints on Flavored 2d CFT Partition Functions
• Universality of squashed-sphere partition functions
• OPE statistics from higher-point crossing
• Universal Asymptotics for High Energy CFT Data
• Thermal Conformal Blocks
• $SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index
• Parity and the modular bootstrap
• Localization of 4d $\mathcal{N}=1$ theories on ${\mathbb{D}}^2\times {\mathbb{T}}^2$
• EFT Asymptotics: the Growth of Operator Degeneracy
• Detachable circles and temperature-inversion dualities for CFT${}_d$
• Bose-Fermi cancellations without supersymmetry
• Quantum Thermodynamics of an M2-M5 Brane System
• Comments on a state-operator correspondence for the torus
• Modularity of supersymmetric partition functions
• Phase Transitions and Stability of Eguchi-Hanson-AdS Solitons
• Light-state Dominance from the Conformal Bootstrap
• Notes on massless scalar field partition functions, modular invariance and Eisenstein series
• Conformal Quantum Mechanics, Holomorphic Factorisation, and Ultra-Spinning Black Holes
• From Exact Results to Gauge Dynamics on ${\mathbb{R}}^3\times S^1$
• Sparseness bounds on local operators in holographic $CF{T}_d$
• Higher-Curvature Gravity, Black Holes and Holography
• Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs
• Modular properties of massive scalar partition functions
• Superconformal indices on $S^1\times (S^5/Z_p)$
• The thermal representation of conformal ladder integrals
• Quantum Field Theory on spherical space forms