MIT Infinite Dimensional Algebra Seminar (Fall 2024)

In the geometric approach to quantum theory that I suggested several years ago we take as a starting point the set of states. This viewpoint is more general than the standard approach where the states are considered as density matrices and the algebraic approach where the states are identified with positive functionals on an associative algebra with involution. A large class of examples including conventional quantum mechanics can be constructed from classical theory where our devices can measure only a part of observables.

In a geometric approach one can derive the formulas for probabilities analyzing interaction with a random environment. If the theory is translation-invariant we can define a notion of particle and scattering of particles.

The quantum difference equations are K-theoretic analogs of Dubrovin connection in quantum cohomology. In my talk I show that over $p$-adic fields the quantum difference equations of Nakajima quiver varieties are equipped with the Frobenius automorphism $z\to z^p$.

I show that the corresponding Frobenius intertwiner is a partition function of quasimaps with special boundary conditions. I describe an explicit formula for the degree zero term of the intertwiner and explain the connection with works of Dwork, Sperber and Kedlaya.

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Let S be a topological surface with holes. Let M(S,L) be the moduli space parametrising hyperbolic structures on S with geodesic boundary, and a given set L of lengths of the boundary circles. It carries the Weil-Peterson volume form. The volumes of spaces M(S,L) are finite. M.Mirzakhani proved remarkable recursion formulas for them, related to several areas of Mathematics.

However if S is a surface P with polygonal boundary, e.g. just a polygon, similar volumes are infinite. We consider a variant of these moduli spaces, and show that they carry a canonical exponential volume form. We prove that exponential volumes are finite, and satisfies unfolding formulas generalizing Mirzalkhani's recursions.

This part of the talk is based on the joint work with Zhe Sun.

There is a generalization of these moduli spaces for any split simple real Lie group G, with canonical exponential volume forms. When the modular group of the surface P is finite, our exponential volumes are finite for any G. When P are polygons, they provide a commutative algebra of positive Whittaker functions for the group G. The tropical limits of the positive Whittaker are the (zonal) spherical functions for the group G.

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The first part of the talk will be given by Vasily Krylov and is based on the joint work with Ivan Perunov. We will discuss the geometry of Coulomb branches corresponding to finite Dynkin quivers. These Poisson varieties are known to be isomorphic to the so-called generalized slices in affine Grassmannians. Using the multiplication morphisms introduced by Braverman, Finkelberg, and Nakajima, we will construct coverings of resolutions of certain slices by affine spaces and, in particular, obtain explicit Darboux coordinates on them as well as characters of tangent spaces at torus fixed points.

The second part of the talk, given by Hunter Dinkins and based on joint work with Andrey Smirnov and Jiwu Jang, will discuss the 3d mirror dual picture. The mirror dual to the Coulomb branches of the first part are ADE type Nakajima quiver varieties. The quasimap vertex functions of such varieties are expected to encode information about the tangent spaces of the dual varieties. These vertex functions can be computed explicitly as power series using the combinatorics of minuscule posets. We will explain how to sum these series in certain cases and directly match characters of tangent spaces of the first part.

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This is based on arXiv:2408.02485, joint with Bezrukavnikov. We construct certain functors on categories of representations of rational Cherednik algebras associated with symmetric groups in zero and large positive characteristic. Our functors are indexed by pairs of a partition and a rational number, the slope. For a given slope, the functors give an action of the positive half of the Heisenberg Lie algebra, while when the slope varies and the characteristic is positive we get a categorical action of the positive half of the elliptic Hall algebra. In this talk I will explain necessary notions, sketch the construction, and provide the motivation, partly coming from the double affine representation theory.