Mathematical Physics

It is easy to prove that d-dimensional complex Hilbert space can contain at most d^2 equiangular lines.  But despite considerable evidence and effort, sets of this size have only been proved to exist for finitely many d.  Such sets are relevant in quantum information theory, where they define optimal quantum measurements known as SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures).  They also correspond to complex projective 2-designs of the minimum possible cardinality.   Numerical evidence points to their existence for all d as orbits of finite Heisenberg groups, the current record being d=844 [Scott-Grassl '17].  However, to date, they are only proven to exist for finitely many d (the current record being d=323 [SG17]) via computer-assisted calculations in number fields of degree increasing with d.  In this talk, I will discuss the structure of these number fields, which turn out to be specific abelian extensions of specific real quadratic number fields [Appleby, Flammia, McConnell, Y. 1604.06098].  Such fields are known to exist by general theorems of class field theory, but until now, had never been found 'explicitly' in Nature.  This contrasts the classical situation for abelian extensions of CM fields, which are generated by the torsion points of abelian varieties with complex multiplication.   All known Heisenberg-covariant SIC-POVMs have unitary symmetries under the associated Weil representation that are intimately related to the structure of the underlying number fields.  A proper understanding of this relationship may ultimately lead to a general proof of their existence in all dimensions, rather than the finite number of examples currently proved to exist. 

I will talk about a recent proof, joint with M. Gröchenig and D. Wyss, of a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The proof, inspired by an argument of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.

Geometric invariant theory (GIT) is an essential tool for constructing moduli spaces in algebraic geometry. Its advantage, that the construction is very concrete and direct, is also in some sense a draw-back, because semistability in the sense of GIT is often more complicated to describe than related intrinsic notions of semistability in moduli problems. Recently a theory has emerged which treats the results and structures of geometric invariant theory in a broader context. The theory of Theta-stability applies directly to moduli problems without the need to approximate a moduli problem as an orbit space for a reductive group on a quasi-projective scheme. I will discuss some new progress in this program: joint with Jarod Alper and Jochen Heinloth, we give a simple necessary and sufficient criterion for an algebraic stack to have a good moduli space. This leads to the construction of good moduli spaces in many new examples, such as the moduli of Bridgeland semistable objects in derived categories. Time permitting, I will also discuss applications to enumerative geometry and wall crossing formulas.

Moore and Tachikawa conjecture that there exists a functor from the category of 2-bordisms to a certain category whose objects are algebraic groups and morphisms between $G$ and $H$ are given by affine symplectic varieties with an action of $G\times H$.  I will explain a proof of this conjecture due to Ginsburg and Kazhdan, and its relation to Coulomb branches of certain quiver gauge theories which allows to make interesting calculations.

I'll do my best to explain my approach to the BFN construction of (quantum) Coulomb branches.  This approach is based on viewing the BFN algebra as an endomorphism algebra in a larger category that's easier to present (and which we can draw some pretty pictures for).  In particular, this approach is helpful in understanding the representation theory of this algebra, and in constructing and analyzing tilting generators on Coulomb branches.

The talk is based on my recent work with Ryan Aziz. We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra in the category of corepresentations of a coquasitriangular Hopf algebra gives a new larger coquasitriangular Hopf algebra, for example taking c_q[SL_2] to c_q[SL_3] for these quantum groups reduced at certain odd roots of unity. As an application we find new generators for c_q[SL2] with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced quantum enveloping algebra u_q(sl2). This allows one to calculate  Fourier transform and other results for such quantum groups. Our method also works for even roots of unity where we obtain new finite-dimensional quantum groups, including an 8-dimensional one at q=-1. Our method
can be used to construct  many other new finite-dimensional  quasitriangular Hopf algebras and their duals that could be fed into applications in quantum gravity and quantum computing.

We use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. From a given plabic graph G we have two coordinate systems: we have a network chart for the A-model Grassmannian, and a cluster chart for the B-model (Landau-Ginzburg model) Grassmannian. On the A-model side, we use the network chart from G and an ample divisor D to define an associated Newton-Okounkov polytope NO_G(D). We give explicit formulas for the lattice points in NO_G(D) in terms of the combinatorics of Young diagrams.   We then reinterpret NO_G(D) in terms of the superpotential and the cluster chart for the B- model Grassmannian. *This is joint work with Konstanze Rietsch. 

Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in single formal automorphism of the cluster variety, called the DT-transformation. An oriented surface S with punctures, and a finite number of special points on the boundary give rise to a moduli space, closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. We determine the DT-transformation of this space. This is a joint work with Alexander Goncharov.

I will present some results on three-dimensional gauge theory from the point of view of extended topological field theory. In this setting a theory is specified by describing its collection of boundary conditions - in our case, a collection of categories (standing in for 2d TFTs) with a prescribed symmetry group G. We will apply ideas from Seiberg-Witten geometry to construct a new commutative algebra of symmetries for categorical representations (or line operators in the gauge theory) -  a categorification of Kostant's description of the center of the enveloping algebra. (Joint with Sam Gunningham and David Nadler)

The notion of singular support for coherent sheaves was introduced by Arinkin and Gaitsgory in order to carefully state the geometric Langlands conjecture.  This is a conjectural equivalence of categories of sheaves on certain moduli spaces: in order to make the conjecture reasonable one needs to restrict to sheaves which satisfy a certain "singular support condition".  In this talk I'll explain how to think about this singular support condition from the point of view of boundary conditions in twisted N=4 gauge theory.  Specifically, Arinkin and Gaitsgory's singular support condition arises by considering only those boundary conditions which are compatible with a natural choice of vacuum state.  By allowing this vacuum state to move away from this natural choice we see aspects of a rich additional structure for the geometric Langlands correspondence.  This work is joint with Philsang Yoo.