MIT Infinite Dimensional Algebra Seminar (Fall 2025)

Some additive monoidal categories admit an abelian envelope, which is a tensor category containing them in a universal way. Recently, a lot of theory has been developed to identify and construct abelian envelopes. This has been utilised to define the (higher) Verlinde categories ${\rm Ver}_{p^n}$, abelian envelopes of quotients of the category of tilting modules of ${\rm SL}_2$ in positive characteristic, and their union ${\rm Ver}_{p^\infty}$. The Verlinde categories are incompressible, meaning they cannot be expressed as the representations of an affine group scheme in any smaller category. In fact, the subcategories of ${\rm Ver}_{p^\infty}$ comprise all currently-known incompressible categories, making them a conjectural foundation for all symmetric tensor categories. I will give an overview of the construction of ${\rm Ver}_{p^\infty}$ by Benson-Etingof-Ostrik, and then discuss some adaptations to the construction which re-prove and elucidate some properties of these categories. This will give an alternate description of ${\rm Ver}_{p^\infty}$ in terms of the perfection of ${\rm SL}_2$, and an interpretation of ${\rm Ver}_{p^n}$ as a Serre quotient.

Semisimplification functors on tensor categories underlie many useful constructions in representation theory, including the Frobenius functor in modular representation theory and the Duflo-Serganova functor from Lie superalgebras. I will discuss how semisimplification functors and generalizations thereof can be applied to the study of modular representations and degenerate categorical Heisenberg actions. In the latter case, the functors will always define (non-exact!) morphisms of categorical actions, and in particular cases, categorify a certain element of the mod-p centre of affine sl_p. These functors admit a convenient diagrammatic description which clarifies their properties. Finally, I will explain how these functors naturally globalize known equivalences of subcategories of representations of S_n, originally due to Henke-Koenig and also studied by Harman.

iQuantum groups are a remarkable family of coideal subalgebras of Drinfeld-Jimbo quantum groups introduced originally by Letzter, generalized by Kolb, and studied further by Bao and Wang (who introduced this terminology). I will focus only on the family of quasi-split iquantum groups, which have a particularly elegant presentation involving just one type of relation. This relation is similar to the usual quantum Serre relation but with an additional lower order term. I will explain how to categorify quasi-split iquantum groups, giving a representation theoretic interpretation for this lower order term. The categorification of Lusztig's modified form for quantum groups discovered in 2008 by Khovanov, Lauda and Rouquier is a special case of our construction. The talk is based on joint work with Weiqiang Wang and Ben Webster.

I will discuss a generalization of the two-dimensional Poisson sigma model to a holomorphic-topological field theory in three dimensions. It is defined from the data of a Poisson vertex algebra and when specialized to well-known PVAs, the field theory can be identified with field theories such as Chern-Simons theory and (higher spin) gravity. I'll explain how PVA modules correspond to line defects in this theory, and finally, I'll aim to explain how this field theory sheds light on the deformation quantization problem of Poisson vertex algebras.

Microformal or thick morphisms, introduced by Th. Voronov, are a generalisation of smooth maps between manifolds that still give rise to pullbacks on functions. These pullbacks are in general nonlinear and formal, and in special cases they define L-infinity morphisms between the algebras of functions on homotopy Poisson or homotopy Schouten manifolds. In this talk, I will give a brief introduction to thick morphisms (of which there are so called classical and quantum versions), and describe a graphical calculus which calculates all terms in the formal power series that result from their pullbacks. The method is heavily inspired by the work of Cattaneo-Dherin-Felder on formal symplectic groupoids, which itself is based on techniques from the numerical analysis of ODEs, with the resulting expansions resembling the expansions over Feynman diagrams of perturbative QFT.

Higgs and Coulomb branches of quiver gauge theories form two rich families of Poisson varieties that are expected to be exchanged by 3D mirror symmetry. The Hikita conjectures relate the algebra and geometry of these branches in a nontrivial way, allowing one to extract new structures on one side by studying its mirror. In this talk I will attempt to give an overview of the current state of the Hikita conjecture, with an emphasis on its interplay with other important structures such as representations of loop groups, the geometric Satake equivalence, the Riemann–Roch isomorphism, and twisted traces (time permitting). I will illustrate how the picture works in the ADE case (where a lot can be proved) and formulate conjectures that generalize these observations to arbitrary quivers. Time permitting, I will explain a construction that extends some results beyond ADE. Talk will be based on joint works with Dinkins, Dumanski, Karpov, Lance, and Perunov as well as the results of many great mathematicians working in this field.

The category of Kac-Moody modules is equivalent (modulo issues with homological algebra) to factorization modules over a factorization (aka vertex) algebra. We will explain a higher categorical counterpart of this, where we embed fully faithfully categorical representations of the loop group into factorization modules over the affine Grassmannian. This realizes an old vision due to Beilinson. This is a joint work with Lin Chen, Charles Fu and David Yang.

The property (T) conjecture for mapping class groups predicts that finite-dimensional unitary representations of mapping class groups of surfaces of genus at least 3 have no infinitesimal deformations (are rigid). This rigidity question has been extensively studied for finite image representations, where it is known as the Ivanov conjecture. A natural direction is to investigate infinite image unitary representations. Natural examples arise from unitary modular fusion categories via the Reshetikhin-Turaev construction, yielding what are called quantum representations of mapping class groups.

For closed surfaces of genus at least 7, I will discuss rigidity of quantum representations arising from SU(2) and SO(3) modular categories at prime roots of unity. The strategy is to relate the problem to Ocneanu rigidity—a result asserting that modular fusion categories admit no non-trivial deformations. Ocneanu rigidity reduces our problem to constructing an injective map from deformations of the mapping class group representation in question to deformations of the modular fusion category. The construction uses modular functor theory and harmonic representatives from Hodge theory. Injectivity follows from a partial homological stability result for first cohomology groups of mapping class groups with coefficients in adjoints of SU(2) and SO(3) quantum representations.

We discuss the physical meaning of modular invariance for two-dimensional unitary conformal quantum field theories. For QFT models, while T invariance is necessary for locality, S invariance is not mandatory. The S invariance is a form of completeness of the theory that has a precise meaning as Haag duality for arbitrary multi-interval regions. We present a mathematical proof and its physical interpretation. For rational CFT's, the failure of modular invariance or Haag duality can be measured by an index, related to the quantum dimensions of the model. We show how to compute this index from the modular transformation matrices. This is a joint work with V. Benedetti, H. Casini, R. Longo, and J. M. Magan.