Abstract
We initiate an approach to constraining conformal field theory (CFT) data at finite temperature using methods inspired by the conformal bootstrap for vacuum correlation functions. We focus on thermal one- and two-point functions of local operators on the plane. The KMS condition for thermal two-point functions is cast as a crossing equation. By studying the analyticity properties of thermal two-point functions, we derive a “thermal inversion formula” whose output is the set of thermal one-point functions for all operators appearing in a given OPE. This involves identifying a kinematic regime which is the analog of the Regge regime for four-point functions. We demonstrate the effectiveness of the inversion formula by recovering the spectrum and thermal one-point functions in mean field theory, and computing thermal one-point functions for all higher-spin currents in the critical O(N) model at leading order in 1/N. Furthermore, we develop a systematic perturbation theory for thermal data in the large spin, low-twist spectrum of any CFT. We explain how the inversion formula and KMS condition may be combined to algorithmically constrain CFTs at finite temperature. Throughout, we draw analogies to the bootstrap for vacuum four-point functions. Finally, we discuss future directions for the thermal conformal bootstrap program, emphasizing applications to various types of CFTs, including those with holographic duals.
Highlights
One of the basic operations in quantum field theory (QFT) is dimensional reduction on a circle
We initiate an approach to constraining conformal field theory (CFT) data at finite temperature using methods inspired by the conformal bootstrap for vacuum correlation functions
Modern advances in the conformal bootstrap have focused almost entirely on constraining operator product expansion (OPE) data using CFT correlation functions in flat space
Summary
One of the basic operations in quantum field theory (QFT) is dimensional reduction on a circle. Our formula shows that thermal one-point functions in conformal field theory are analytic in spin, in the same way as OPE coefficients and operator dimensions. This allows us to study the thermal data of arbitrary, stronglyinteracting CFTs. Crucially, thermal two-point functions have different OPE channels with overlapping regimes of validity. Inverting terms in one channel to the other relates thermal coefficients of operators in the theory in nontrivial ways: one-point functions determine terms in the large-spin expansion of other one-point functions These relations can be posed to formulate an analytic bootstrap problem for the thermal data. The appendices further elaborate on technical details in the main text
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