We apply the operator product expansion inversion formula to thermal two-point functions of bosonic and fermionic conformal field theories in general odd dimensions. This allows us to analyze in detail the operator spectrum of these theories. We find that nontrivial thermal conformal field theories arise when the thermal mass satisfies an algebraic transcendental equation that ensures the absence of an infinite set of operators from the spectrum. The solutions of these gap equations for general odd dimensions are in general complex numbers and follow a particular pattern. We argue that this pattern unveils the large-N vacuum structure of the corresponding theories at zero temperature.
A bstract In this work we address partial wave decompositions of thermal one-point functions in conformal field theories on S 1 × S d− 1 . With the help of Casimir differential equations we develop efficient algorithms to compute the relevant conformal blocks for an external field of arbitrary spin and with any spin exchange along the thermal circle, at least in three dimensions. This is achieved by identifying solutions to the Casimir equations with a special class of spherical functions in the harmonic analysis of the conformal group. The resulting blocks are then applied to study the decomposition of one-point functions of the scalar ϕ 2 and the stress tensor T for a three-dimensional free scalar field ϕ . We are able to read off averaged OPE coefficients into exchanged fields of high weight and spin for a complete set of tensor structures. We also extract an asymptotic behaviour of conformal blocks and use it to analyse the density of heavy-heavy-light OPE coefficients for spinning operators, comparing it with semi-classical predictions, such as the dimensions of operators at large charge.
Abstract In d-dimensional CFTs with a large number of degrees of freedom an important set of operators consists of the stress tensor and its products, multi stress tensors. Thermalization of such operators, the equality between their expectation values in heavy states and at finite temperature, is equivalent to a universal behavior of their OPE coefficients with a pair of identical heavy operators. We verify this behavior in a number of examples which include holographic and free CFTs and provide a bootstrap argument for the general case. In a free CFT we check the thermalization of multi stress tensor operators directly and also confirm the equality between the contributions of multi stress tensors to heavy-heavy-light-light correlators and to the corresponding thermal light-light two-point functions by disentangling the contributions of other light operators. Unlike multi stress tensors, these light operators violate the Eigenstate Thermalization Hypothesis and do not thermalize.
We study conformal field theories at finite temperature in the presence of a temporal conformal line defect, wrapping the thermal circle, akin to a Polyakov loop in gauge theories. Although several symmetries of the conformal group are broken, the model can still be highly constrained from its features at zero temperature. In this work we show that the defect and bulk one and two-point correlators can be written as functions of zero-temperature data and thermal one-point functions (defect and bulk). The defect one-point functions are new data and they are induced by thermal effects of the bulk. For this new set of data we derive novel sum rules and establish a bootstrap problem for the thermal defect one-point functions from the KMS condition. We also comment on the behaviour of operators with large scaling dimensions. Additionally, we relate the free energy and entropy density to the OPE data through the one-point function of the stress-energy tensor. Our formalism is validated through analytical computations in generalized free scalar field theory, and we present new predictions for the O(N) O ( N ) model with a magnetic impurity in the \varepsilon ε -expansion and the large N N limit.
We consider defect conformal field theories (DCFTs) at finite temperature, where a
p-dimensional conformal defect wraps the thermal circle. Extending the previous study of line defects to defects of general dimensions, we determine the general forms of the thermal one-point functions of bulk scalars, conserved currents and the stress tensor by imposing the residual symmetry and the conservation laws. The low temperature expansion of the thermal one-point functions is shown to be reproduced by the bulk-defect operator expansion, allowing us to read off the thermal one-point coefficients of defect operators. We examine the general results in four classes of examples with free bulk theories: the trivial defect, free scalar and fermion theories with a boundary, the free scalar theory with a localized
ϕ deformation in
d=2p+2 dimensions and the free
O(N) model with a localized
ϕ2 deformation in
d=p+2−ε dimensions. In the last example with
ε=1, where the defect becomes an interface, we find that the thermal one-point functions coincide with those of the free scalar theory with the Dirichlet boundary condition, in accordance with the conjectured factorization of an interface CFT into two decoupled boundary CFTs.
We explore the idea that the mass parameter
m can be used as a diagnostic tool for identifying the critical points of thermal two-point functions of massive free scalar fields in odd
d dimensions. For general values of
m these correlators admit an expansion in thermal conformal blocks associated with symmetric traceless operators of definite scaling dimension. For each spin the higher-twist contributions can be resummed into a closed-form dressing factor of the leading-twist coefficient. At special values of
m for which the thermal trace of the energy-momentum tensor vanishes they are identified as two-point functions of thermal CFTs. Through a dimensional uplift, valid for all
m, we map these correlators to thermal two-point functions of higher-derivative massive theories in
d+2 dimensions. We show that the thermal trace of the energy-momentum tensor of the
d+2-dimensional theory is proportional to the free energy of the original
d-dimensional theory. When that thermal trace vanishes for
m=0 the uplifted correlators are identified as two-point functions of a higher-derivative nonunitary thermal CFT
d+2, but they differ from the uplifted nontrivial thermal two-point functions of the original unitary CFT
d.