Boundaries and defects in conformal field theories at finite temperature

summary

The gist

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

In short

The paper studies conformal boundaries and defects in Euclidean CFT at finite temperature and infinite volume. It classifies bulk one-point functions for various primaries, relating low-temperature expansions to zero-temperature data. It derives bootstrap sum rules for localized defects and computes thermodynamic observables like energy density from thermal correlators.

Key concepts

Conformal Field Theory (CFT): A type of quantum field theory where the physics remains unchanged under conformal transformations. This symmetry is crucial in describing the behavior of particles and fields at specific energy scales, allowing for precise calculations at finite temperatures.
Defects: These are localized disturbances or boundaries within the CFT setup, such as defects wrapping a thermal circle or being fixed at a point on it. The paper analyzes how these structural features affect the field theory's properties and correlations.
KMS Condition: This condition acts as a bootstrap equation for the two-point function in thermal CFT. It imposes constraints on the correlators, similar to crossing symmetry in zero-temperature theories, helping to determine the structure of thermal correlation functions.
Bootstrap Sum Rules: These are mathematical relationships derived from physical principles (like periodicity and operator expansions) that constrain the coefficients of correlation functions. The paper uses these rules to derive new constraints on thermal data for bulk one-point functions.

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This episode discusses

The paper

Boundaries and defects in conformal field theories at finite temperature · Read on arXiv

Dipartimento SMFI, Università di Parma · INFN Gruppo Collegato di Parma · Dipartimento di Fisica, Università di Torino · INFN Sezione di Torino

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.

Transcript

Introduction to the show: ident: High Energy Physics: Theory. The day in high-energy theory: gravity, strings, fields and the questions left open.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Boundaries and defects in conformal field theories at finite temperature".

Mira: Here's the gist of the paper. We consider conformal boundaries and defects in Euclidean conformal field theory (CFT) at finite temperature and infinite volume.

Kai: We distinguish between defects that wrap the thermal circle and those localized at a point on it.

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

Kai: 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.

Paper summary: Kai: So, this paper we're looking at is called "Boundaries and defects in conformal field theories at finite temperature". It tackles how boundaries and defects show up in Euclidean conformal field theory, but specifically when the temperature is finite and the volume is infinite.

Mira: Exactly. The core idea here is to look at two types of defects: those that wrap around the thermal circle, and those that are just localized right on that circle itself.

Kai: In both cases, they classify all the one-point functions for scalar, vector, and spin-two primaries in both bulk and defect setups <ref:2609.38130#pg1,scalar, vector, and spin-2 primaries>. And they connect the low-temperature behavior of bulk scalar one-point functions to zero-temperature data and new finite temperature data for these defect primaries.

Lev: From an engineering standpoint, relating low temperature expansions to zero temperature data is useful because it gives us a way to use existing knowledge about vacuum physics to predict what happens when you add thermal effects or defects.

Kai: For the localized defects, they derive one-point bootstrap sum rules using periodicity and the defect operator expansion. That's a new set of rules for how those localized things behave thermally.

Mira: It’s interesting because in free bulk CFTs—like unitary or non-unitary ones—they manage to determine all this thermal data for various examples, including a wrapping monodromy defect and a magnetic line defect localized on the circle.

Lev: Getting thermal data for those specific examples is important because it tells us what kind of constraints we have when we try to build real hardware, where you need these finite temperature properties to match up with your cooling cycles.

Kai: For the wrapping defects, they compute thermodynamic observables directly from that data. For the localized ones, they find a Hurwitz zeta function decomposition for the one-point functions which suggests some kind of underlying dispersion relation.

Mira: So, basically, this paper is using conformal symmetry and thermal constraints to extract concrete numerical data for these defects across different field theories and boundary conditions at finite temperature.

Lev: It’s an application of QFT machinery to constrain physical behavior under real-world conditions, which is always a good use of the tools we have.

Conclusion: Kai: So, looking at the whole paper "Boundaries and defects in conformal field theories at finite temperature", it seems the main point is using these conformal constraints to map out how thermal effects modify systems with boundaries or defects.

Mira: Right, they aren't just studying abstract math; they are connecting zero-temperature data with finite-temperature properties for these defect systems, which is a crucial link.

Kai: The implications seem to be that we can use these bootstrap sum rules and the extracted thermal one-point functions to find concrete thermodynamic quantities like energy density and free energy density.

Lev: From my side, it means if you were designing some kind of quantum system with topological defects, you have a formal way to calculate how the thermal environment affects its properties.

Mira: It tells us that all these thermodynamic quantities coincide with the no-boundary results in this context, which is a pretty strong statement about the structure of these theories at these limits.

Kai: It’s about using symmetry to pin down the thermal behavior of things that are physically real, like boundaries or impurities in a field theory.

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