Monday 20 January 2025, 14:00-15:30
The low-energy theory of 4d N = 2 supersymmetric gauge theories is encoded in the Seiberg-Witten (SW) curve which is naturally related to an integrable system. For SU(2) theories, in presence of Ω-background this integrable system is given by Painlevé equations. In this talk we we will show that the Nekrasov partition functions of 4d SU(2) susy gauge theories on the blowup of spacetime are tau functions which solve the Painlevé equations in Hirota bilinear form. These solutions can be expressed as expansions in terms of the moduli of the quantum SW curve and the construction can be applied also to non-Lagrangian theories. Furthermore, these solutions have manifest modular properties which provide a natural non-perturbative completion of the corresponding topological string partition function and directly lead to the BCOV holomorphic anomaly equations of the topological string.
Monday 27 January 2025, 14:00-15:30
Strongly coupled quantum field theories (QFTs) are notoriously hard to study. Supersymmetry (S), if present, provides powerful tools to access non-perturbative physics exactly. Stringy constructions of SQFTs - such as Hanany-Witten style brane systems - help us uncover their physics even in regimes completely inaccessible via current field theory methods. After some introduction I will focus on 5d N = 1 theories realised on fivebrane webs ending on sevenbranes in the presence of an O7+ orientifold plane in Type IIB String Theory. In particular SO(K) theories with matter in the vector representation and SU(K) theories with matter in the fundamental representation as well as one second rank symmetric, and their infinite coupling UV completions. We will see how to use so-called magnetic quivers to study the moduli spaces and partial Higgsings of these theories.
Monday 3 February 2025, 14:00-15:30
Goncharov-Kenyon (GK) integrable systems are deeply connected with the Seiberg-Witten theory of 5D supersymmetric theories compactified on a circle, and (conjecturally) with Coulomb branches of theories compactified further to 3D. In the original construction, the GK systems are labelled by integral convex polygons. We propose an extension of this class through the reductions of GK systems that are labelled by decorated polygons. Isomorphisms of reduced GK integrable systems are given by mutations in a dual cluster structure. These involve the polynomial mutations of the spectral curve equations and polygon mutations of the corresponding decorated Newton polygons.
Based on joint work with P. Gavrylenko, A. Marshakov, and M. Semenyakin.
Monday 10 February 2025, 14:00-15:30
For a 3d N=4 supersymmetric gauge theory, the Coulomb branch and Higgs branch are symplectic dual to each other. They admit a stratification into symplectic leaves. In this work, a subtraction algorithm to work out the stratification for the Higgs branch of an arbitrary unitary quiver is introduced. All minimal Higgs patterns and corresponding local singularities are classified. Related concepts like Namikawa Weyl groups and isometries of slices induced by monodromy on the base leaves are determined in the subtraction. Interesting properties in analytic structure in the Hilbert series are also discussed when Higgsing a Theory. The presentation is based on the previous work arxiv:2409.16356 and an ongoing work with Marcus Sperling.
Monday 24 February 2025, 14:00-15:30
In 11d supergravity, there are known solutions with superisometry algebra d(2,1;γ)+d(2,1;γ) which are holographically dual to the 6d maximally superconformal field theory with 2d superconformal defects. In this talk I will present our work from 2402.11745 in which we show that a limit of these solutions, for which γ →−∞, reproduces another known class of solutions, holographically dual to small N=(4,4) superconformal surface defects. Notably, our construction produces a finite Ricci scalar, in contrast to the known small N=(4,4) solutions. Using the Ryu-Takayanagi prescription, one can calculate the Entanglement Entropy of a spherical region centered around the defect. I will show how this was done for these new small N=(4,4) solutions, allowing us to extract a linear combination of defect Weyl anomaly coefficients. Finally, I will argue why the standard stress-energy tensor one-point function computation can’t be done here, explaining why we can't separate out the A-type and B-type coefficients using these computations alone.
Monday 3 March 2025, 14:00-15:30
I will report on a recent development in the program to describe partition functions of supersymmetric gauge theories in various dimensions using intertwiners of quantum toroidal algebras. The combination of intertwiners reproducing a gauge theory can be read off from the corresponding Type IIB brane picture (e.g. of Hanany-Witten type). This formalism naturally implies that partition functions of this kind are eigenfunctions of difference equations which turn out to be Hamiltonians of quantum integrable systems. In particular this gives new results about elliptic deformations of the Ruijsenaars-Schneider integrable models.
Monday 10 March 2025, 14:00-15:30
I will describe work on a universal relevant deformation that takes local unitary 3d N=4 SCFTs to TQFTs. In particular, I will describe how Abelian mirror symmetry is related to generalisations of level/rank duality and also touch on some of the more general statements that can be made via 't Hooft anomaly matching.
Monday 17 March 2025, 14:00-15:30
Conformal Field Theories (CFTs) occupy a crucial position between Topological Quantum Field Theories (TQFTs), which are mathematically relatively well understood, and general Quantum Field Theories (QFTs), for which a rigorous non-perturbative formulation remains elusive.
Two-dimensional chiral CFTs are closely related to three-dimensional TQFTs by a bulk-boundary correspondence. Unitary chiral CFTs admit three distinct mathematical formulations: in terms of unitary Vertex Operator Algebras (VOAs), Conformal Nets, and the Segal (functorial) approach where they appear as projective representations of the conformal cobordism category. We discuss progress towards constructing a fully extended functorial field theory description of chiral CFTs, focusing on assignment to points and one-dimensional cobordisms. We will review some basics of von Neumann algebras and Connes Fusion of von Neumann algebra modules, categorifications of which appear in this setting.
Monday 21 October 2024, 14:00-15:30
Diophantine problems concern integer and rational solutions to polynomial equations with integer coefficients. These problems have been studied for thousands of years and remain a very active field of research. I will introduce local-global principles underpinning modern approaches to these Diophantine problems, and the Brauer–Manin obstruction which can explain failures of these local-global principles. Finally, I will describe some joint work with Martin Bright concerning the wild part of the Brauer–Manin obstruction.
Monday 28 October 2024, 14:00-15:30
I will discuss work in preparation which relates two quantities in N = 2 supersymmetric theories: the superconformal index, which contains information about protected local operators, and the twisted index. I will show that under appropriate restrictions on the theory and the space of supersymmetric deformations, these quantities coincide. I will then discuss some implications for the physics of the theory and its holographic dual.
Monday 4 November 2024, 14:00-15:30
I will present a notion of spin structure on a perfect complex in characteristic zero, generalizing the classical notion for an (algebraic) vector bundle. For a complex E on X with an oriented quadratic structure one obtains an associated ℤ/2ℤ-gerbe over X which obstructs the existence of a spin structure on E. This situation arises naturally on moduli spaces of Calabi-Yau fourfolds. Using spin structures as orientation data, we construct a categorical refinement of a K-theory class constructed by Oh-Thomas on such moduli spaces.
Monday 18 November 2024, 14:00-15:30
Conformal blocks are the fundamental building blocks of Conformal Field Theories and play an important role in several areas of mathematical physics from random geometry to black hole physics. Starting from their probabilistic formulation in terms of the Gaussian Multiplicative Chaos (GMC) measure by Promit Ghosal, Guillaume Remy, Xin Sun, Yi Sun, I will prove certain conjectures posed by Zamolodchikov regarding the semiclassical behaviour of conformal blocks and show their relation to the Lamé equation, and other associated integrable structures. This talk is based on a joint work with Promit Ghosal and Andrei Prokhorov (arXiv: 2407.05839).
Monday 25 November 2024, 14:00-15:30
We present a new algorithm to extract the quiver and superpotential of a broad class of threefolds that fall under simple threefold flops. These geometries are generally non-toric and can be viewed as monodromic fibrations over a complex plane of deformed ADE singularities. We illustrate how the quantum field theory of a D2-brane probing these spaces captures their non-commutative crepant resolution (NCCR).
Monday 2 December 2024, 14:00-15:30
It is well known that AdS/CFT provides a microscopic account of the macroscopic black hole entropy. The entropy can be extracted holographically from the study of the supersymmetric index of the dual field theory. In this talk I will discuss some recent results on the Cardy-like limit of the index for 4d N = 4 SYM with real gauge groups SO(2n+1) and USp(2n). I will present the results for the saddles contributing to the index both for the orthogonal and symplectic case and discuss how S-duality manifests in the Cardy-like limit. Subleading contributions reveal crucial to preserve S-duality at the level of the index in this limit. Such contributions are associated to the emergence of pure Chern-Simons theories around each saddle in an effective field theory (EFT) picture. I will show how S-duality is recovered non-trivially through a matching between saddles and corresponding EFTs. Lastly, I will show that the saddles organise themselves according to the inequivalent charge lattices of lines that condense under the unbroken global 1-form subgroups of the centre of the gauge groups.
Monday 9 December 2024, 14:00-15:30
In this talk, we explore integrated correlators in 4-dimensional