Titles and abstracts

Daniele Artico: Boundaries and defects in thermal conformal field theories

In this talk, I develop a framework for studying conformal field theories with boundaries and defects at finite temperature, with particular emphasis on the interplay between thermal and defect geometries.
The resulting non-trivial geometry strongly constrains correlation functions. I derive the allowed one-point functions of scalar, vector, and spin-two operators for defects both wrapping and localized on the thermal circle, and obtain KMS sum rules constraining defect CFT data. I then present explicit one- and two-point functions in free theories, illustrating the analytic structures that emerge at finite temperature.
I conclude by outlining how these results can be used to formulate a defect-bootstrap programme for thermal CFTs, highlighting open problems and exciting directions. 

Nathan Benjamin: From Z to a

The free energy of any CFT, log Z(\beta; \omega_i), admits two expansions: high temperature (\beta to 0) and fast rotation ("semi-universal", i.e. \omega_i to 1). In 4d, locality of the thermal effective action implies the coefficient of \beta^{-3+2q} in the high temperature expansion is a symmetric polynomial in \omega_1^2 and \omega_2^2 of degree q, which naively gives ~q^2/4 free coefficients. We show that locality additionally implies ~q^2/12 further relations on these coefficients (leaving only ~q^2/6 degrees of freedom at each order).

All relations above are homogeneous, except at order \beta^1 due to the Weyl anomaly. From this, the a-anomaly can be extracted directly from the partition function in a striking formula: a = 3/2 log Z(\omega_1^2 = e^{2\pi i /3}, \omega_2^2 = e^{4\pi i /3})_{\beta^1} where _{\beta^1} means we extract the \beta^1 coefficient in the small \beta expansion.

The counting is governed by a collapse of the effective action on the rotating background to three generators. The relations resum in the fast-spinning expansion into differential equations in \beta obeyed by the semi-universal (\omega to 1) limit and its corrections. We verify the relations in a variety of CFTs. We generalize to any even d, but show no similar relations exist at odd d.

Based on arXiv:2609.03013.

Ivan Gusev: Stringy resolution of bouncing singularities

Black-hole singularities leave characteristic signatures in the analytic structure of correlation functions and scattering amplitudes. In the classical limit, null geodesics that bounce off the black-hole singularity give rise to “bouncing singularities” at complex time. In this talk, I will investigate their fate at finite string length in the near-horizon geometry of near-extremal NS5-branes, whose cigar factor is described by the exact SL(2,\mathbb{R})_k/U(1) worldsheet conformal field theory. After relating the classical reflection amplitude to bouncing null geodesics, I will show that the exact string-corrected amplitude admits a time-domain continuation that is an entire function of complex time, completely resolving the bouncing singularity. This resolution is nonperturbative in the inverse coset level (1/k): the singularity persists at every finite order of the expansion.

Based on joint work with Ilija Burić and Andrei Parnachev.

Johan Henriksson: Anomalous dimensions of operators with any number of fields - in one go 

In perturbative conformal field theory, anomalous dimensions of composite operators can be computed using Feynman diagram techniques. For towers of operators, for instance primaries of \phi^k type in Wilson--Fisher theories, a strategic way to organise the computation is to separate the k-dependent combinatorics from the evaluation of a set of diagrams which can be made independent of k. This allows to compute the anomalous dimension of entire towers of operators in one go, giving results on closed form in k. I will explain how this computation can be practically achieved at multi-loop level using the background field method. Our result is a set of master anomalous dimensions which carry the information about multi-loop results for operators with arbitrary number of fields transforming in arbitrary representations of global symmetry. 

Based on 2511.16740 and work in progress.

Jakob Hollweck: Constructing Conformal Blocks from Oscillator Representations

I will discuss a constructive approach to four-dimensional conformal blocks based on oscillator representations of SU(2,2). In this framework, conformal blocks are obtained by gluing a set of oscillator wavefunctions, and their evaluation reduces to contractions involving SU(2) Clebsch–Gordan coefficients. After illustrating the method for scalar vacuum and thermal blocks, I will discuss the additional representation-theoretic difficulties introduced by spin, in particular multiplicities, and recent progress towards resolving them. I will finish with an outlook on extending the construction to more general backgrounds and other dimensions.

Stefanos Robert Kousvos: A Bootstrap Study of Confinement in AdS

Yang-Mills theory in AdS${}_4$ with Dirichlet boundary conditions is expected to undergo a transition as the AdS radius varies, since the boundary data is incompatible with confinement in flat space. Various mechanisms have been proposed for the disappearance of the Dirichlet boundary condition, which we discuss. From the boundary viewpoint, the associated CFT is a deformation of a generalised free theory of non-Abelian conserved currents. We test the various proposed scenarios by deriving non-perturbative constraints from the numerical conformal bootstrap of the four-point function of non-Abelian conserved currents. We rule out the scenario in which the boundary current decouples, and find favourable evidence that the Dirichlet boundary condition disappears via fixed point annihilation.

Sylvain Lacroix: The ODE/IM correspondence as a quantum/classical duality between integrable CFTs on various geometries

In this talk, we will discuss an interpretation of the ODE/IM correspondence as a quantum/classical duality between two-dimensional Integrable Conformal Field Theories (ICFT) on different geometries. One side of the duality concerns a quantum ICFT on the cylinder and more precisely the finite-size spectrum of the infinitely many commuting operators characterising its integrable structure. On the other side, the ODE/IM correspondence then allows to encode this quantum spectrum in the conserved charges of a classical ICFT integrated along cycles of a non-trivial auxiliary Riemann surface. The geometry of this surface is determined by the quantum ICFT and the parameters of its integrable structure. We will illustrate this relation with the example of su(2) coset CFTs on the quantum side and KdV charges on hyperelliptic curves on the classical side.

Francesco Mangialardi: Integrability of d-dimensional conformal blocks on non trivial geometries

I will discuss conformal block expansions of CFT correlation functions defined on a vast class of d-dimensional manifolds which can be built by gluing together higher dimensional cylinders and spheres. These manifolds carry a notion of genus, the case of genus zero corresponding to ordinary correlators on the sphere, and that of genus one to thermal correlation functions, which recently have also attracted great interest.

I will then explain how conformal blocks on these backgrounds admit a characterisation in terms of quantum Hitchin integrable systems. After introducing the relevant integrable models through the theory of 2d Wess-Zumino-Novikov-Witten models, from which they naturally arise, I will describe the limiting process that reduces the integrable Hamiltonians to the differential equations defining the d-dimensional conformal blocks. Simple examples will be used throughout to help build intuition.

This provides an extension of the already established relation between multipoint conformal blocks on the sphere and Gaudin integrable models, which is recovered as a particular case, and embeds the theory of higher dimensional conformal blocks in the landscape of 2d CFTs and supersymmetric gauge theories.

Based on joint work with Ilija Burić, Francesco Russo and Volker Schomerus (to appear)

Enrico Marchetto: Bootstrapping thermal one-point functions: numerical and analytic approaches

Conformal field theories are powerful laboratories for studying QFT: conformal invariance and radial quantization encode the theory-dependent information in a set of CFT data that can be extracted using numerical and analytic bootstrap methods. At finite temperature, additional state-dependent data arise as new thermal one-point functions. These quantities encode the conformal invariance breaking and determine thermal correlation functions. Recent progress in the thermal bootstrap program has shown that thermal one-point functions can be accessed using both numerical and analytic techniques. In this talk, I will explain how the KMS condition can be used to set up a thermal bootstrap problem as a set of sum rules. Their numerical treatment requires control over the contributions of heavy operators, and I will show how Tauberian methods and dispersion relations grant such control. I will illustrate this framework in critical O(N) vector models at finite temperature and highlight the difference with holographic CFTs.

Alessio Miscioscia: A Path to the Analytic Thermal Bootstrap

I will discuss how thermal data—especially thermal two-point functions—can be constrained by a combination of the KMS condition, analyticity, clustering, and the OPE. In particular, I will propose KMS-symmetric dispersion relations for two-point functions. Crucially, only operators contributing to the discontinuity appear in these formulas, and the blocks are explicitly KMS-symmetric, in analogy with Polyakov blocks at zero temperature. The price to pay is an unknown arc function, which I will show can be completely determined using the remaining properties of correlation functions.

Owing to the manifest periodicity of the expansion, I will also discuss the momentum-space version of this construction, making its connection with quasinormal modes (QNMs) explicit. These equations provide a complete bootstrap problem for the QNMs. I will illustrate the framework with free, perturbative, and large-\(N\) examples, including holographic correlators, and mention applications to the 3D Ising model.

Sridip Pal: From Z to all

It has been shown that the free energy of any CFT \log Z(\beta, \omega_i) in even dimension satisfies an infinite set of constraints. We show that they arise due to the presence of a large conformal diffeomorphism isomorphic to \mathbb{Z}_{d/2} and all of them can be encapsulated by a strikingly simple equation \sum_{j} T^j [log Z(\beta,\omega_i)/\beta] = (-1)^{d/2} (d/2)! a, where T is the generator of \mathbb{Z}_{d/2} and a is the a-anomaly term. This recovers the a-anomaly equation advertised in the first talk when we set \omega_j^2= e^{2\pi i j/(d/2+1)} along with other constraints. 

Based ongoing work to appear soon.

Balt van Rees: QFT in AdS and Haag-Ruelle theory

Francesco Russo: Bootstrapping non-rational 2D CFTs

We search for non-rational CFTs, namely unitary 2D Virasoro CFTs characterised by a discrete spectrum and a twist gap in the spectrum of primaries. We use numerical modular bootstrap techniques to explore the space of theories with both invertible and non-invertible symmetries. We find sharp features in our bounds, and test their stability with convergence analysis and by trying different numerical setups.