Beijing Institute of Mathematical Sciences and Applications

In this talk, we introduce a novel algorithm of methodically generating 2D lattice models undergoing phase transitions. These 2D models are realised by giving critical boundary conditions to 3D topological orders (symTOs/symTFTs) described by string-net models, often called the strange correlators. We engineer these critical boundary conditions by introducing a commensurate amount of non-commuting anyon condensates. The non-invertible symmetries preserved at the critical point can be controlled by studying a novel ``refined condensation tree''. Our structured method generates an infinite family of critical lattice models, including the A-series minimal models, and uncovers previously unknown critical points. Notably, we find at least three novel critical points (c≈1.3, 1.8, and 2.5 respectively) preserving the Haagerup symmetries, in addition to recovering previously reported ones. The condensation tree, together with a generalised Kramers-Wannier duality, predicts precisely large swathes of phase boundaries, fixes almost completely the global phase diagram, and sieves out second order phase transitions. This is not only illustrated in well-known examples (such as the 8-vertex model related to the A5 category) but also further verified with precision numerics, using our improved (non-invertible) symmetry-preserving tensor-network RG, in novel examples involving the Haagerup symmetries. We show that critical couplings can be precisely encoded in the categorical data (Frobenius algebras and quantum dimensions in unitary fusion categories), providing a powerful, systematic route to discovering and potentially classifying new conformal field theories. If time allows, we will also look into the RG flows of these models and discuss the relation between fixed points and continuous field theories.<br>

I consider a non-common type of 4D/2D correspondence, relating vortices in 4D supersymmetric QCD with superstrings, compactified on non-compact Calabi-Yau spaces. In particular it allows to compute spectrum of 4D theory at strong coupling by means of 2D superconformal world-sheet theory, corresponding to supersymmetric version of 2D black hole. The conjecture survives under few tests on state's counting, reduced on field theory side to certain exercises in representation theory.

It is well known that the beta-function, which determines the energy dependence of the coupling constants, depends on the regularization scheme in which it is defined. For N=2 supersymmetric sigma models, the beta-function in the minimal subtraction scheme is known up to five loops. A remarkable fact is that for certain models it is possible to turn to a renormalization scheme in which the renormalization group flow equation takes the form of the Ricci flow equation. In this talk we focus on the complete T-duals of the eta-deformed SU(n)/U(n-1) sigma models and the lambda-deformed SU(n)/U(n-1) sigma models, for which such a change of scheme can be implemented.<br>We will also discuss the relation between these models and present the derivation of the Kaehler potential for the SU(2)/U(1) lambda-deformed sigma model.

The Zamolodchikov tetrahedron equation plays a central role in the theory of integrable systems in three dimensions. It governs the integrability of quantum field theories in (2+1) dimensions and statistical mechanical models on three-dimensional lattices, much as the Yang–Baxter equation does for integrable (1+1)-dimensional quantum field theories and two-dimensional lattice models. Despite its fundamental importance and long history, our understanding of the tetrahedron equation remains relatively limited compared with that of the Yang–Baxter equation. In this talk, I will discuss two approaches to constructing solutions of the tetrahedron equation. The first is a cluster algebraic approach, proposed in collaboration with Junya Yagi [arXiv:2211.10702] and developed with Rei Inoue, Atsuo Kuniba, Yuji Terashima, and Junya Yagi [arXiv:2403.08814]. This framework produces a new class of solutions that includes most previously known solutions as special limits. Moreover, it reveals a novel connection between our R-matrices and partition functions of 3d N=2 supersymmetric gauge theories. The second approach is based on topological quantum field theory (TQFT). In recent joint work with Myungbo Shim, Hao Wang, and Junya Yagi [arXiv:2602.22060], we construct three-dimensional lattice models using line defects in state-integral models associated with shaped triangulations of 3-manifolds. The resulting Boltzmann weights satisfy a variant of the tetrahedron equation. Together, these results suggest new links between three-dimensional integrable systems, cluster algebras, supersymmetric gauge theories, and TQFT.

In this talk, I will talk about the morse theory on the path space of a complexified manifold. In the beginning, I will review the finite dimensional theory of exponential integrals. Then I will go to the infinite dimensional case，the path integral. We start with a Riemannian manifold $(X,g)$, and consider its complexification $(X_{\mathbb{C}}$,$g_{\mathbb{C}})$ , then study Morse flow on the path space of $X_{\mathbb{C}}$. Finally, we will talk about a little about the relation to heat kernel and resurgence theory. This talk is based on the joint work with Si Li and Yong Li.

In this talk we will discuss the renormalization group equation for N=2 supersymmetric sigma models in two-dimensional spacetime. The presence of this supersymmetry results in the Kaehler property of the manifold describing the space of fields, which allows us to reformulate the renormalization group equation in terms of the Kaehler potential. It will be shown that the complete T-dual eta-deformed CP^(n-1) sigma models and the lambda-deformed SU(2)/U(1) and SU(3)/U(2) sigma models provide solutions of this equation up to and including five-loop order in a special renormalization scheme. The complex structure of the lambda-deformed SU(2)/U(1) sigma model will be shown to lead to the Kaehler potential becoming the sum of the Kaehler potentials of the eta-deformed SU(2)/U(1) model and its T-dual for appropriate choice of target space complex coordinates. In conclusion we will discuss the implications of this observation in the context of Poisson-Lee symmetries of the eta- and lambda-deformed models.

Any homogeneous convex cone can be represented as the cone of positive definite Hermitian matrices in some matrix algebra with involution. We say that a cone has rank n if it lies in an n×n matrix algebra. Among cones of rank 3,there is a particularly interesting class, namely the special Vinberg cones. They are constructed using Clifford modules and are cones over projective special real manifolds. Projective special real manifolds describe supergravity theories in 5 dimensions. The definition of special Vinberg cones generalizes to arbitrary rank. I will discuss the classification of special Vinberg cones of rank 4. This result was published in the paper: Alekseevsky, D. V., and Osipov, P. "Special Vinberg cones of rank 4." Journal of High Energy Physics 2026.1 (2026): 1.

A classic result of Hartogs says that, in complex dimension $n>1$, holomorphic function on a punctured $n$-disc can be analytically continued to a function on the unpunctured $n$-disc. This motivated the use of derived differential geometry in, for instance, Faonte-Hennion-Kapranov, as a way to recover the "missing" negative modes in higher-dimensional current algebras. Following this line of thinking, together with the recent surge of interest in 3d topological-holomorphic (TH) field theories, the TH foliated version of the "formal punctured disc" was studied by Garner-Williams in 2023; this is known as the formal raviolo. In this talk, I will introduce a 3d TH derived version of the Wess-Zumino-Witten model, which realizes an affine derived current higher-algebra, described as a centrally-extended infinite-dimensional $L_{\infty}$-algebra. This theory is quantized into an affine raviolo vertex algebra, which may exhibit interesting higher-dimensional integrable properties.

I will review how affine Yangian symmetry describes integrable systems that appear in two-dimensional conformal field theory. One of the advantages of this approach is that it automatically provides equations for the spectrum (Bethe anzatz equations). Then I will present new results on affine Yangian description of certain coset CFTs.

We investigate the holonomies of Knizhnik-Zamolodchikov connections with n regular and 1 or 2 irregular singular points. We show how Associators can be computed perturbatively and nonperturbatively, and introduce novel types of invariants when braiding in the presence of irregular defects is considered.

The Calogero-Moser model is a celebrated example of a completely integrable system, with numerous connections to several areas of mathematics and physics. It describes a system of $n$ of identical particles scattering on the line with inverse-square potential. There are also trigonometric, hyperbolic and elliptic version of this model. The integrability of the system can be shown in different ways, for example, constructing the higher Hamiltonans via Dunkl operators. We propose an R-matrix generalization of the quantum elliptic Calogero-Moser system, based on the Baxter--Belavin elliptic R-matrix. This is achieved by introducing R-matrix-valued Dunkl operators so that commuting quantum spin Hamiltonians can be obtained from symmetric combinations of those. Using the freezing procedure, we construct integrable long-range spin chains. The talk is based on the joint work with Oleg Chalykh arXiv:2509.18989.