Boundaries and defects in conformal field theories at finite temperature

We consider conformal boundaries and defects in Euclidean conformal field theory (CFT) at finite temperature and infinite volume. We distinguish between defects that wrap the thermal circle and those localized at a point on it. In both set-ups, for any dimension and co-dimension, we classify bulk one-point functions of scalar, vector and spin-2 primaries. We relate the low-temperature expansion of bulk scalar one-point functions to zero-temperature defect CFT data and new finite-temperature data of defect primaries. For localized defects, we derive one-point bootstrap sum rules using periodicity and the defect operator expansion. We consider several examples in free bulk CFTs: unitary and non-unitary boundary CFTs, a wrapping monodromy defect, and a localized magnetic line. We determine thermal defect CFT data in all examples. For defects wrapping the thermal circle, we compute thermodynamic observables. For localized defects, we observe a Hurwitz zeta function decomposition of one-point functions suggestive of an underlying dispersion relation.

The paper is organized as follows:

"In section 2 we fix our conventions and briefly review zero-temperature DCFT and thermal CFT. Section 3 develops the kinematics of both defect configurations. We classify the one-point functions of defect and bulk primaries, identify the thermal DCFT data and derive Euclidean inversion formulas that extract them. In section 4 we turn to the constraints imposed by the KMS condition. We derive novel bootstrap sum rules for bulk one-point functions in the presence of defects localized on the thermal circle. The remaining sections are devoted to examples."

The paper discusses:

"Higher-dimensional defects wrapping the thermal circle. We consider conformal defects of arbitrary dimension and co-dimension. Using the preserved symmetries, we classify the constant thermal one-point functions of defect-local operators."

"Localised defects and a one-point function bootstrap. We extend the analysis to defects localised on a fixed Euclidean time slice. We classify the thermal one-point functions of defect primaries and identify the corresponding thermal DCFT data."

"Explicit examples and thermodynamics. We illustrate these results in free-field examples of increasing complexity: scalar, fermion, hypermultiplet and higher-derivative scalar BCFTs, a monodromy defect wrapping the thermal circle, and a magnetic line defect localized on the thermal circle. We extract thermal DCFT data from these correlators and comment on their analytic structure."

The paper also details:

A Endpoint derivation of the KMS sum rules

Monodromy defect thermal one-point functions of all Tµν components

Line defects in scalar CFTs at finite temperature

Relevant excerpts include:

"In both set-ups, and for any dimension and co-dimension, we classify bulk one-point functions of scalar, vector and spin-2 primaries. We relate the low-temperature expansion of bulk scalar one-point functions to zero-temperature defect CFT data and new finite-temperature data of defect primaries."

For localized defects, we observe a Hurwitz zeta function decomposition of one-point functions suggestive of an underlying arXiv:2609.38130v1 [hep-th] 29 Sep 2026 dispersion relation.

"The KMS condition plays the role of a bootstrap equation for the thermal CFT two-point function. It is similar to the crossing equation for the bulk two-point function in zero-temperature DCFT in that it does not involve positive-definite coefficients."

We derive a set of sum rules for the coefficients entering the thermal block expansion of the one-point function.

The paper concludes with discussions on thermodynamic quantities derived from the stress tensor one-point function, such as energy density, entropy density, and free energy density. For example:

Defining dimensionless energy, entropy and free energy densities as in eqs. (5.19), (5.25) and (5.24), we find for any d E = 2 1 − 2 1−d d κf ζ(d)

The paper also discusses boundary effects, such as:

"We will compute thermal one-point functions of simple bulk operators from which we extract some thermal DCFT data. One of the operators that we consider is the stress tensor, through which we can compute certain thermodynamic quantities."

The boundary contribution to the energy as E∂ = A β d−1 Z ∞ 0 dz E(z) + d − 1 d bT, which is manifestly finite.

The paper's structure covers:

Section 5.1 Free scalar field, Section 5.2 Free Dirac fermion, Section 5.3 Free hypermultiplet, and Section 5.4 Free higher-derivative scalar.

The paper's focus is on how conformal symmetry and the KMS condition constrain thermal observables in the presence of defects (boundaries or impurities). The key results involve deriving bootstrap sum rules for thermal DCFT data, extracting thermal one-point functions of bulk operators using Euclidean inversion formulas, and computing thermodynamic quantities like energy density and free energy density.

The paper is organized as follows:

"In section 2 we fix our conventions and briefly review zero-temperature DCFT and thermal CFT. Section 3 develops the kinematics of both defect configurations. We classify the one-point functions of defect and bulk primaries, identify the thermal DCFT data and derive Euclidean inversion formulas that extract them. In section 4 we turn to the constraints imposed by the KMS condition. We derive novel bootstrap sum rules for bulk one-point functions in the presence of defects localized on the thermal circle. The remaining sections are devoted to examples."

"Section 5 studies free BCFTs at finite temperature: free scalars, free fermions, a 4d hypermultiplet and a non-unitary higher-derivative scalar. For each of them, we extract thermal BCFT data and compute thermodynamic quantities."

The paper is organized as follows:

In section 1 Introduction, Section 2 Review, Section 3 Kinematics of conformal defects at finite temperature, Section 4 Bootstrap sum rules, Section 5 Boundaries in CFTs at finite temperature.

We consider conformal boundaries and defects in Euclidean conformal field theory (CFT) at finite temperature and infinite volume.

"In both set-ups, and for any dimension and co-dimension, we classify bulk one-point functions of scalar, vector and spin-2 primaries. We relate the low-temperature expansion of bulk scalar one-point functions to zero-temperature defect CFT data and new finite-temperature data of defect primaries."

For localized defects, we derive one-point bootstrap sum rules using periodicity and the defect operator expansion.

"We determine thermal defect CFT data in all examples. For defects wrapping the thermal circle, we compute thermodynamic observables. For localized defects, we observe a Hurwitz zeta function decomposition of one-point functions suggestive of an underlying arXiv:2609.38130v1 [hep-th] 29 Sep 2026 dispersion relation."

The KMS condition plays the role of a bootstrap equation for the thermal CFT two-point function.

We find that all thermodynamic quantities coincide with the no-boundary results.

We consider conformal boundaries and defects in Euclidean conformal field theory (CFT) at finite temperature and infinite volume.

"In both set-ups, and for any dimension and co-dimension, we classify bulk one-point functions of scalar, vector and spin-2 primaries.

1. Improved system can perform thermal one-point function calculations for bulk scalar operators in free field examples like scalar, fermion, hypermultiplet, and higher-derivative scalar theories using point splitting regularisation (eq. 5.4) and extracting boundary thermal one-point functions from the resulting series expansions (eq. 5.14).

2. Improved system can derive bootstrap sum rules for bulk one-point functions in the presence of defects localized on the thermal circle by treating the KMS condition as a crossing equation, which allows for constraints on coefficients entering the thermal block expansion (eq. 4.6).

3. Improved system can compute thermodynamic observables like dimensionless energy density E, entropy density s, and free energy density f for various CFTs at finite temperature using expressions derived from the stress tensor one-point function (eq. 5.20) and related relations (eq. 5.21), including results for d=6 higher-derivative scalar theories.

4. Improved system can extract thermal boundary one-point functions for bulk operators by dividing out the zero-temperature bulk boundary two-point functions, providing an infinite amount of thermal BCFT data (eq. 5.15).

5. Improved system can analyze the behavior of correlation functions under KMS constraints, specifically showing that for a scalar with bosonic statistics localized on the thermal circle, the KMS condition implies FO(w, z) = FO(1 - w, z) (eq. 3.22).

6. Improved system can determine the tensor structures appearing in one-point functions of spin-2 primary operators in the presence of a defect wrapping the thermal circle by analyzing Table 4 and its associated constraints imposed by KMS and parity symmetries (Page 17).

7. Improved system can compute the thermal one-point function of a vector current for a conserved symmetry, showing that c q=1 J must equal the defect thermal one-point function of the tilt operator (eq. 3.9) when q = 1.

8. Improved system can determine the functional form of bulk correlation functions for specific defects, such as a magnetic line defect, by observing Hurwitz zeta function decompositions analogous to those of thermal two-point functions (Page 16).

We consider conformal boundaries and defects in Euclidean conformal field theory (CFT) at finite temperature and infinite volume. We distinguish defects that wrap the thermal circle and defects that are localised at a point on it. In both set-ups, and for any dimension and co-dimension, we classify bulk one-point functions of scalar, vector and spin-2 primaries. We relate the low-temperature expansion of bulk scalar one-point functions to zero-temperature defect CFT data and new finite-temperature data of defect primaries. For localised defects, we derive one-point bootstrap sum rules using periodicity and the defect operator expansion. We consider several examples in free bulk CFTs: unitary and non-unitary boundary CFTs, a wrapping monodromy defect, and a localised magnetic line. We determine thermal defect CFT data in all examples. For defects wrapping the thermal circle, we compute thermodynamic observables. For localised defects, we observe a Hurwitz zeta function decomposition of one-point functions suggestive of an underlying dispersion relation.