In 2011, Moore and Tachikawa conjectured the existence of certain complex Hamiltonian varieties which generate two-dimensional TQFTs where the target category has complex reductive groups as objects and holomorphic symplectic varieties as arrows. It was solved by Ginzburg and Kazhdan using an ad hoc technique which can be thought of as a kind of "symplectic reduction by a group scheme." We clarify their construction by introducing a general notion of "symplectic reduction by a groupoid along a submanifold," which generalizes many constructions at once, such as standard symplectic reduction, preimages of Slodowy slices, the Mikami-Weinstein reduction, and the Ginzburg-Kazhdan examples. This is joint work with Peter Crooks.

In this talk, I will show a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized K3 surfaces of degree two. For this, we use the theory of good moduli spaces, whose study was initiated by Alper. As a corollary, we obtain the generic Strange Duality for K3 surfaces of degree two, extending the results of Marian and Oprea on the generic Strange Duality for K3 surfaces.

This talk is joint work with Nan-Kuo Ho, Paul Selick and Eugene Xia. We describe the space of conjugacy classes of representations of the fundamental group of a genus 2 oriented 2-manifold into $G:= SU(2)$.

• We identify the cohomology ring and a cell decomposition of a space homotopy equivalent to the space of commuting pairs in $SU(2)$.
• We compute the cohomology of the space $M:=\mu^{-1}(-I)$, where $\mu:G^4 \to G$ is the product of commutators.
• We give a new proof of the cohomology of $A:= M/G$, both as a group and as a ring. The group structure is due to Atiyah and Bott in their landmark 1983 paper. The ring structure is due to Michael Thaddeus 1992.
• We compute the cohomology of the total space of the prequantum line bundle over $A$.
• We identify the transition functions of the induced $SO(3)$ bundle $M\to A$.

To appear in QJM (Atiyah memorial special issue). arXiv:2005.07390

References:

[1] M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A308 (1983) 523-615.

[2] T. Baird, L. Jeffrey, P. Selick, The space of commuting n-tuples in $SU(2)$, Illinois J. Math. 55 (2011), no. 3, 805–813.

[3] M. Crabb, Spaces of commuting elements in $SU(2)$, Proc. Edin. Math. Soc. 54 (2011), no. 1, 67–75.

[4] N. Ho, L. Jeffrey, K. Nguyen, E. Xia, The $SU(2)$-character variety of the closed surface of genus 2. Geom. Dedicata 192 (2018), 171–187.

[5] N. Ho, L. Jeffrey, P. Selick, E. Xia, Flat connections and the commutator map for $SU(2)$, Oxford Quart. J. Math., to appear (in the Atiyah memorial special issue).

[6] L. Jeffrey, A. Lindberg, S. Rayan, Explicit Poincar´e duality in the cohomology ring of the $SU(2)$ character variety of a surface. Expos. Math., to appear.

[7] M.S. Narasimhan and C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface. Ann. of Math. 82 (1965) 540–567.

[8] P. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241–262.

[9] C.T.C Wall, Classification problems in differential topology. V. On certain 6-manifolds. Invent. Math. 1 (1966), 355–374; corrigendum, ibid., 2 (1966) 306.

[10] M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles. J. Differential Geom. 35 (1992) 131–149.

[11] E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303-368.

Kashiwara crystals are combinatorial gadgets associated to a representation of a reductive algebraic group that enable us to understand the structure of the representation in purely combinatorial terms. We will describe a type-independent construction of crystals of certain representations, using the heap associated to a fully commutative element in the Weyl group. Then we will discuss how these heaps also lead us to the construction of modules for the preprojective algebra of the Dynkin quiver. Using the rank-nullity theorem, we will see how the Kashiwara operators have a surprisingly nice description in terms of these preprojective algebra modules. This is work in progress joint with Anne Dranowski, Joel Kamnitzer, Tanny Libman and Calder Morton-Ferguson.

An important enumerative invariant of a symplectic variety $X$ is its vertex function. The vertex function is the analog of J-function in Gromov-Witten theory: it is the generating function for the numbers of rational curves in $X$.

In representation theory these functions feature as solutions of various $q$-difference and differential equations associated with $X$, with examples including qKZ and quantum dynamical equations for quantum loop groups, Casimir connections for Yangians and other objects.

In this talk I explain how these equations can be extracted from algebraic topology of symplectic dual variety $X^!$, also known as $3D$-mirror of $X$. This can be summarized as "identity" $$ \text{Enumerative geometry of }X = \text{algebraic topology of }X^! $$ The talk is based on work in progress with Y.Kononov arXiv:2004.07862; arXiv:2008.06309.