Programme

The talks were in room K2.31 (the Nash Lecture theatre) and S-3.20 of the Strand Campus. The coffee breaks and poster session were held in room K2.40 in the King's Building

Thursday

12.00 - 13.00 Registration

13.00 - 13.05 Welcome

13.05 - 13.55.          L Bottini

13.45 - 14.25 Christian Copetti

14.25 - 14.45 break

14.45 - 15.35 Stefano Negro

15.35 - 16.05 Costanza Benassi

16.05 - 16.15 Poster gong show

Mahesh Balasubramanian

Laura Felder

Faisal Karimi

Hector Puerta Ramisa

Jake Stedman

16.15 - 17.00 poster session

17.00 - 17.30 Simon Ekhammar

17.30 - 18.00 Nat Levine

19.00 - Dinner

Friday

09.00 - 09.50 Jules Lamers

09.50 - 10.20 Max Downing

10.20 - 10.50 break

10.50 - 11.40 Roberto Tateo

11.40 - 12.10 Arpit Das

12.10 - 12.40 Alex Radcliffe

12.40 - 14.10 break

14.10 - 15.00 Paola Ruggiero

15.00 - 15.30 Anup Anand Singh

15.30 - 16.00 Augustus Brown

16.00 end

Titles, Abstracts and Slides of the talks

Benassi  

Random matrix ensembles and integrable differential identities

Random matrix models naturally arise in relation to a great variety of problems in mathematics and physics. In particular, we explore the unitary and orthogonal ensemble and their rich integrable structure. In both cases we focus on a specific reduction characterised by a Hamiltonian with even interactions only. It turns out that the relevant order parameter of the unitary ensemble constitutes a solution to the modified KP equation, whilst the orthogonal ensemble is underpinned by a novel integrable chain. Based on C.B., M. Dell'Atti, A. Moro (arXiv:2504.11296).

Bottini

Categorical Landau paradigm and Haagerup symmetry

In this talk I will present a general framework to study gapped phases and phase transitions of (1+1)d theories in the presence of non-invertible symmetries. This outlines a categorical version of the well known Landau paradigm for second order phase transitions. I will also show how we can use this approach to make progress on the longstanding question of finding a conformal field theory with the exotic Haagerup symmetry.

Brown
Large charge meets semiclassics in N=4 super Yang-Mills

We consider correlation functions in N=4\dagger SU(N) supersymmetric Yang-Mills theory involving superconformal primary operators. In particular, we consider the case where two of the operators are heavy, where the dimension of the heavy operators Delta to \infty with \lambda = \Delta\, g_{\text{YM}}^2 held finite. We find that these correlators can be viewed as lower-point correlators of light operators on the Coulomb Branch. We demonstrate this for three-point (Heavy-Heavy-Light) and four-point (Heavy-Heavy-Light-Light) correlators.

Copetti  

(Phantom) symmetry and defect conformal manifolds

Symmetry is a powerful organizing principle in quantum field theory. In this talk, we explore how symmetries can be realized on dynamical defects, constraining their properties. As an application, we demonstrate that seemingly mysterious interface conformal manifolds can be understood purely through symmetry considerations. In particular, the folded theory exhibits an enhanced continuous symmetry - potentially non-invertible - which may be broken at the interface. We refer to this as a phantom symmetry. We then show how this structure can be used to compute the interface's reflection coefficient RR along the conformal manifold.

Das

Bootstrapping the Frobenius approach to rational CFTs via modular differential equations

Two-dimensional rational CFTs with exactly two independent characters satisfy a second-order modular linear differential equation akin to the Mathur-Mukhi-Sen equation. We introduce a bootstrap method for constructing Frobenius-type power-series solutions to this MLDE. Starting from the MLDE, we derive explicit recursion relations for the series coefficients and, by invoking modular transformation and growth constraints, uncover key structural properties of the resulting quasi-characters. These quasi-characters are known to form a basis of character-like solutions for two-character RCFTs arising from MLDEs. Our bootstrapping technique is expected to facilitate the identification of analogous bases in theories with more than two characters. This framework unifies and extends the earlier analyses of quasi characters by Chandra and Mukhi.

Downing

Boundary Critical Loop Models

We study the continuum limit of critical loop models placed on a disc. In the continuum limit these models are 2d CFTs and hence can be studied using conformal bootstrap. We use the bootstrap approach to systematically study the allowed boundary conditions of such models.

Ekhammar  

Investigating Integrable Regge Trajectories

The non-perturbative spectrum of local operators in N=4 SYM is encoded in the integrability-based Quantum Spectral Curve. However, the usefulness of the Quantum Spectral Curve extends far beyond local operators. I will show how it can be used to understand what happens when we analytically continue local operators to negative spin, into the regime of horizontal Regge trajectories. In particular, upon taking a weak coupling limit, it turns out that the spectrum is controlled by a novel set of Baxter-Bethe equations, strongly hinting at the emergence of a novel spin chain.

Lamers

A solvable lattice model of non-unitary fermions with long-range interactions and extended symmetry

I will present an integrable long-range version of the \(XX\) model, arising as the free-fermion point of the Haldane-Shastry spin chain. It has a description via non-unitary fermions, based on the free-fermion Temperley-Lieb algebra. Systems of even and odd length behave very differently; I will mostly focus on odd length. I will explicitly give two commuting hamiltonians. While non-unitary, their spectrum is real by PT-symmetry. One hamiltonian is chiral and quadratic in fermions, similar to Fendley's 'free fermions in disguise', while the other is parity-invariant and quartic. Their one-particle spectra have two linear branches, realising a massless relativistic dispersion on the lattice. The appropriate fermionic modes arise from 'quasi-translation' symmetry, which replaces ordinary translation symmetry. The model exhibits exclusion statistics, like the isotropic Haldane-Shastry chain, with even more 'extended symmetry' and larger degeneracies.

Levine

Convexity of scaling dimensions over conformal manifolds

We study conformal field theories (CFTs) with exactly marginal couplings, focusing at first on odd spacetime dimensions. Under general assumptions, we prove that the scaling dimension of the lightest unprotected operator with any given quantum numbers is a concave function over the conformal manifold with respect to the Zamolodchikov metric. More generally, the sum of the lowest \(n\) scaling dimensions is concave for any \(n\). As an example, we consider a class of 3d N=2 SCFTs studied by Strassler in 1998. We discuss the eventual generalisation of these results to even spacetime dimensions (assuming sufficient supersymmetry that the finite part of the sphere partition function is unambiguous).

Negro

RG flows between minimal models: non-invertible symmetries & exact solvability

In recent years there have been radical developments in our understanding of symmetries, with the discovery of generalization to the usual notion of global symmetries, such as higher-form, non-invertible, or, more generally, of higher-categorical symmetries. These extensions replace the familiar group structures with more general algebraic or categorical structures and, just like the standard symmetries we all know, they can be used to constrain the dynamics of physical systems. Here I will focus on a particular class of examples: 2-dimensional Conformal Field Theories (CFTs). I will show how the categorical structures formed by the many non-invertible symmetries in 2D CFT imposes severe constraints on the allowed Renormalisation Group flows between them. Finally I will also show how, quite shockingly, even though most of these flows are quite evidently not integrable, they still admit an integrability-like exact description of their ground-state spectrum, obtained as a deformation of the familiar integrable ones for the {3,1}, {5,1} and {1,2} flows.

Radcliffe

Monodromy Pinning Defects in the Critical O(2N) Model

I will talk about a family of defects in the O(2N) model that preserve conformal symmetry along the defect, whilst breaking symmetry under rotations transverse to the defect. We construct these defects by starting with a monodromy defect in the O(2N) model and looking at the IR fixed points of RG flows triggered by relevant defect operators with non-zero transverse spin. We will explore the consequences of breaking transverse rotational symmetry on these defects, and how various observables can be computed in the large N and 4-epsilon expansions. This is based on ongoing work with Petr Kravchuk.

Ruggiero

Full counting statistics after quantum quenches as hydrodynamic fluctuations

The statistics of fluctuations on large regions of space encodes universal properties of many-body systems. At equilibrium, it is described by thermodynamics. However, away from equilibrium such as after quantum quenches, the fundamental principles are more nebulous. In particular, although exact results have been conjectured in integrable models, a correct understanding of the physics is largely missing. In this talk, I will discuss these principles, taking the example of the number of particles within a large interval in one-dimensional interacting systems. These are based on simple hydrodynamic arguments from the theory of ballistically transported fluctuations, and in particular the Euler-scale transport of long-range correlations. This allows to obtain a formula for the full counting statistics in terms of thermodynamic and hydrodynamic quantities, whose validity though depends on the structure of hydrodynamic modes. In fermionic-statistics interacting integrable models with a continuum of hydrodynamic modes, such as the Lieb-Liniger model for cold atomic gases, the formula reproduces previous conjectures, but is in fact not exact: more specifically, it gives the correct cumulants up to, including, order 5, while long-range correlations modify higher cumulants.

Singh

Lagrangian multiforms, 3d mixed BF theory, and Hitchin integrable systems

First introduced in 2009, Lagrangian multiforms provide a variational framework for describing integrable hierarchies using a generalised variational principle applied to an appropriate generalisation of a classical action. Much progress has been achieved in recent years to extend this framework to various possible contexts. This talk deals with a novel geometric approach to Lagrangian one-forms for an important class of finite-dimensional integrable hierarchies: those of Hitchin systems associated with a compact Riemann surface of arbitrary genus. After discussing the general framework, I will illustrate how our construction also yields explicit Lagrangian one-forms for the rational and elliptic Gaudin hierarchies and the elliptic spin Calogero-Moser hierarchy in certain special cases. In our approach, these hierarchies arise from a certain gauge-fixing of three-dimensional mixed topological-holomorphic BF theory, establishing their gauge-theoretic origin within the entirely variational framework of Lagrangian multiforms. These results are based on joint work with V. Caudrelier, D. Harland, and B. Vicedo.

Tateo

Exploring Graded Integrable Models: S-Matrices, TBA Equations, and the ODE/IM Correspondence

We introduce a \(\mathbb{Z}_n\) grading structure into a broad class of integrable quantum field theories. By decomposing the two-body scattering amplitudes into cyclically shifted components, we define graded scattering theories in which asymptotic states are organised into internal symmetry sectors labelled by \(\mathbb{Z}_n\) flavours. This framework preserves a generalised notion of braiding unitarity and crossing symmetry, and possesses an infinite tower of conserved charges with fractional spin. We develop the associated Thermodynamic Bethe Ansatz and analyse the finite-volume spectrum across twisted sectors. In the ultraviolet limit, preliminary analytical and numerical results are consistent with the spectrum of a cyclic orbifold, \(\mathrm{CFT}^{\otimes n} /\mathbb{Z}_n\), of the original conformal field theory. We also discuss a potential general connection between the graded TBA and the ODE/IM correspondence. This is joint work with Nicolò Brizio, Alessandro Cortassa, Tommaso Morone, and Nicolò Primi.

Titles and Abstracts of the posters, and the posters themselves.

Balasubramanian  

Anomalous symmetries and gapped RG flows in 3d

We study properties of symmetry preserving RG flows to gapped phases in 3d, using structure theorems of tensor categories. We present rigorous examples using 3d N=4 SCFTs.

Felder

Higher Chiral Algebras via a Polysimplicial Model

Vertex algebras are a great tool to make the concept of operator product expansions precise. Yet their definition through Borchard's identity is conceptually not straightforward. Although this can be partially fixed via, for example, Frenkel and Ben-Zwi's approach, one is still left with a coordinate dependent description. An elegant reformulation is given through Beilinson and Drinfeld's notion of chiral algebras that lift the construction of vertex algebras into the realm of Lie operads using the theory of D-modules. So far, all of this only works in one complex dimension, what could be seen as a major obstacle. However, when one is willing to go to the higher/derived setting, we can define higher chiral algebras for complex dimensions greater than one. This is comparatively unexplored so far. We do this by replacing the Lie operad with the Lie infinity operad (i.e. the Jacobi identity is relaxed to only hold up to higher homotopy expressed through higher operations). In my presentation, I will present joint work with Zhengping Gui and Charles Young where we introduce an example of a higher chiral algebra. We generalise the unit chiral algebra on the affine line by looking at a polysimplicial model of the space of derived sections of the structure sheaf on configuration space. In there, we find an operadic structure and proof that it is isomorphic to the Lie infinity operad. These findings will set the ground for future research on the particular model and its concrete implications as well as its potential generalisations and lead to a better understanding of operator product expansions in particular and, more generally, quantum field theory in higher dimensions.

Karimi

Modular Properties of Generalised Gibbs Ensembles

When presented with an integrable hierarchy in a QFT, it is natural to consider extended partition functions (GGEs) of that theory. In the case of 2D CFT, it is then natural to ask "What are the modular properties of this object?". We have investigated this question in various cases, such as the KdV hierarchy when one has access to the Thermodynamic Bethe Ansatz; and the \(c=-2\) logarithmic theories, where more than one hierarchy can be realised. One finds that: the GGE is not a modular invariant but has complicated modular properties that are not apparent perturbatively; in the asymptotic modular transformation, one finds additional non-commuting charges from outside of the hierarchy; and in the special case of \(c=1/2 \, ,-2\), we stay within the hierarchy after the transformation.

Puerta Ramisa

Superconformal Weight Shifting Operators

We develop a framework for constructing superconformal blocks for correlators of general supermultiplets in theories with SU(m,m|2n) symmetry, such as four-dimensional N=2 and N=4 conformal theories. We use analytic superspace, viewed as the super-Grassmannian Gr(m|n,2m|2n), which includes 4D Minkowski space (m=2, n=0). In this formalism, superblocks for non-half-BPS correlators are analogous to non-supersymmetric conformal blocks for correlators of fields with spin. We construct SU(m,m|2n)-covariant differential operators which generalise the existing conformal weight-shifting operators, and thus allow us to derive all superconformal blocks from the known half-BPS blocks. Our results provide a framework from which to advance the conformal bootstrap in 4D supersymmetric settings, with potential extensions to lower and higher-dimensional SCFTs. The Grassmannian formalism is also seen to offer a natural and often simpler alternative to the embedding space formalism of non-supersymmetric CFTs.

Stedman

Integrable Field Theories with Several Coupled Lax Connections

During the last 5 years it has been found integrable field theories, and their Lax connections, can be naturally constructed from a gauge theory called 4d Chern-Simons (4dCS). In previous work of our's it was shown a large class of gauged sigma models (GSMs), such as the gauged WZW model, arise when two copies of 4dCS are coupled together. In this poster we present the theory's Hamiltonian analysis, proving the aforementioned GSMs have an infinite set of Poisson commuting charges, guaranteeing integrability.