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Integrability in (Quantum) Field Theory
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Description
We are excited to announce that the Durham meeting of the Classical and Quantum Integrability Network will take place on 21 and 22 May 2026.
Speakers will be:
- Ioana Coman-Lohi (University of Edinburgh)
- Clare Dunning (University of Leeds)
- Simon Ekhammar (King's College London)
- Stefano Negro (University of York)
- Dan Thompson (Swansea University)
- Alessandro Torrielli (University of Surrey)
- Benoît Vicedo (University of York)
Research talks will take place on Thursday afternoon and Friday morning. There will also be two introductory talks on Thursday morning on Integrability from 4d Chern-Simons and higher Chern-Simons and Integrability for AdS3 strings.
Please register below to participate either in-person or online. The meeting will be held in the Scott Logic Lecture Theatre (MCS 0001) of the Mathematical and Computer Sciences building and online on Teams:
https://teams.microsoft.com/meet/35252278272934?p=cGRHbk7ZW7PhjJPPCb
Titles and Abstracts
Ioana Coman-Lohi - Chiralization of quiver varieties
Considering a type of Nakajima quiver varieties, which we will take to define the associated varieties of some vertex algebras, I will explain how to construct these vertex algebras from the quiver varieties using two chiral quantization procedures and BRST reduction. The starting point will be a larger vertex algebra, which is canonically associated to the representation space of the quiver, and where the two variants of chiral quantization refer to the global vs. local BRST reduction. In this setting, I will clarify how the resulting vertex algebras are related, and in particular which conditions imply that the natural vertex algebra map between them is injective.
The broader context of this discussion is the construction and analysis of vertex algebras supported on the boundary of topologically twisted 3D N=4 supersymmetric quantum field theories, which provides a new, interesting analogue of established 3D/2D correspondences and which relates boundary vertex algebras to protected sectors of the SQFTs. Here, the chiral quantization procedure provides a powerful tool for the study of dualities and relations, as well as 3D mirror symmetry from the perspective of the boundary theory. Recent results have furthermore identified categories of modules of the boundary vertex algebras and corresponding quantum groups, with categories of line operators in the 3D theory.
Clare Dunning - Partitions and Wronskian polynomials
Partitions of integers play a role in a variety of fields including number theory, representation theory and random matrix theory. They are also of independent interest in enumerative combinatorics.
I will use partitions to label certain Wronskian polynomials that arise in various applications including
- the excited-state sine-Gordon model in its massless limit
- various families of rational solutions associated with Painleve IV and Painleve V.
Curiously, it turns out that various properties of the partitions play a much more important role in these applications than simple labels.
Simon Ekhammar - Functional Equations and 2D Fishnets
Integrability is a powerful tool for understanding the AdS/CFT correspondence from weak to strong coupling. In particular, the Quantum Spectral Curve (QSC) allows one to compute the non-perturbative spectrum of planar N=4 supersymmetric Yang-Mills theory. It has long been hoped that the remaining CFT data can also be computed from the building blocks of the QSC, but an efficient all-loop formalism is still lacking.
This motivates the search for simpler models that share many features of the N=4 QSC. One such example is provided by bi-scalar fishnet models. Originally defined in four dimensions, these models have since been generalised to any dimension. I will discuss recent work on the 2D bi-scalar conformal fishnet theory using both operator-based and functional approaches. The spectrum of this model is captured by two coupled sl(2) Baxter equations, providing a remarkably simple setting that may lead to new insights into N=4 and the ever-mysterious integrability structure of AdS3/CFT2.
Stefano Negro - Advances in the ODE/IM Correspondence: the Q-functions without Stokes
The ODE/IM correspondence identifies spectral data of ordinary differential equations with eigenvalues of commuting operators in quantum integrable models. In the best-known examples, such as quantum KdV, the standard construction of Q- and T-functions relies crucially on the presence in the ODE of an irregular singularity at infinity: the existence of Stokes sectors leads to connection coefficients satisfying the characteristic TQ- and QQ-relations. This picture has recently been placed in the broader framework of affine Gaudin models and affine opers.
In this talk I will discuss a different mechanism for producing Q-type functions that does not rely on the presence of irregular singularities. Using the characteristic gauge freedom of opers, one can identify distinguished forms of the oper that produce distinguished parallel sections -- i.e. solutions to specific forms of the ODE. I will explain how the initial conditions relating these different sections define specific functions on the space of opers, and how these resulting functions satisfy the expected QQ-relations. I will present this picture in two examples. In the case of perturbed generalised hypergeometric opers -- relevant to the Fateev model -- functional relations arise from regular-singularity monodromy rather than Stokes phenomena. The other case is that of λ-opers, associated with quantum KdV; here the construction connects naturally with the more familiar Q-functions arising from the Stokes phenomenon.
This suggests a broader understanding of the origin of Q-functions, in which Stokes phenomena provide one important realisation rather than the defining mechanism.
Dan Thompson - Symmetries and Phase Space in the Integrable Diamond
Six-dimensional holomorphic Chern–Simons theory on twistor space provides a unifying origin for a broad class of lower-dimensional integrable field theories. In this talk, I will review this “integrable diamond” perspective, explaining how choices of meromorphic differential, boundary conditions, and reductions lead to familiar structures such as WZW4, integrable deformations, gauged models, and dualised theories.
I will then describe recent progress in exploring the symmetry structures embedded in this picture using the covariant phase-space method. Via a short detour through minitwistor space, this provides a natural origin for rich quasi-local symmetries, analogous to the affine current algebras of two-dimensional WZW models. I will conclude with an outlook toward a four-dimensional Maillet algebra for general self-dual/Lax connections.
Alessandro Torrielli - Boundary Scattering in AdS3
We will review some recent work done in constructing a boundary scattering theory for AdS_3 integrable strings. We will mainly focus on the massless sector as a paradigm to display the representation theory associated with the reflection algebra. We will introduce all the ingredients needed to perform the algebraic Bethe ansatz equations and obtain the spectrum. Based on work with D. Bielli, V. Gautam, V. Moustakis and A. Prinsloo.
Network and Funding
This is part of a series of workshops on Classical and Quantum Integrability, involving Durham, Edinburgh, Glasgow, Heriot-Watt, Leeds, Loughborough, Northumbria and York Universities.
We gratefully acknowledge funding from the London Mathematical Society, the Isaac Newton Institute and Heilbronn Institute (Additional Funding Programme for Mathematical Sciences, delivered by EPSRC, grant number EP/V521917/1) and UKRI, grant number UKRI2067.
The LMS also have funds to help parents and carers.
The Durham meeting is organised by Patrick Dorey, Ben Hoare and Parita Shah.
Ben Hoare
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