Virtual invariants via shifted microlocalization

I will suggest a general recipe for the construction of virtual invariants of moduli spaces, such as the virtual fundamental class, using derived algebraic geometry. In order to realize in this manner virtual invariants of (-1)-shifted and (-2)-shifted symplectic stacks (such as critical cohomology, vanishing cycle sheaves, Borisov-Joyce virtual classes, etc.), one is led to develop a (-1)-shifted analogue of microlocal sheaf theory. This is the subject of my recent joint work with Kinjo, Park, and Safronov, which I will attempt to give a gentle introduction to.

Parabolic AGT correspondence from critical stable envelopes

Negut formulated and proved an parabolic AGT correspondence, namely, certain q-deformed W-algebra acts on the equivariant K-theory of the moduli space of parabolic sheaves on \(P^1\times P^1\) (a.k.a. affine Laumon space). The cohomological version remains open. In this talk I will outline a proof of the cohomological version of parabolic AGT in the rectangular case. A key ingredient in the proof is the critical stable envelope developed in my recent joint work with Yalong Cao, Andrei Okounkov, and Zijun Zhou.

A derivation of \((q,T)\)-deformed vertex algebras from geometry and wall-crossing

Borcherds' bicharacter construction of vertex algebras was realized by Joyce on the homology of moduli stacks of sheaves. Building upon prototypical examples and counterexamples, I will isolate the appropriate definition of equivariant generalized homology theories for stacks. This construction produces adic-complete modules. In homology, this provides precisely the framework for \(T\)-deformed vertex algebras, which I previously termed additive deformations. These refine Haisheng Li's axioms by controlling the poles. In K-homology, the bicharacter construction yields quantum vertex algebras in the sense of Etingof-Kazhdan. Combining these observations produces a natural definition of \((q,T)\)-deformations closely related to the work of Frenkel-Reshetikhin. Towards the end, I will discuss the deformed Virasoro algebras arising from our construction. This is based on joint work with Emile Bouaziz.

tba

tba

tba

tba

K-theoretic DT/PT counts for local curves in 3-folds are expected to exhibit a very nontrivial and powerful duality as a consequence of the conjectural 3-dimensional mirror symmetry for the Hilbert schemes of points in the plane, as well as from their conjectural relations to counts of membranes in 5-folds. Talks by Rahul Pandharipande, Andrey Smirnov, Yannik Schuler, Zijun Zhou, Tommaso Botta, Andrei Okounkov, and others will present various approaches to these conjectures and related topics. \(\Rightarrow\)website \(\Rightarrow\)poster

Bridging affine vertex algebras and affine Springer fibres

I would like to explain a program which aims to build relationships between affine Springer theory and representations of simple affine vertex algebras. We will discuss some explicit correspondence for admissible levels, and formulate some new conjectures about representations of simple affine vertex algebras at integer nonadmissible levels and their associated varieties. This is based on joint work with Dan Xie, Wenbin Yan and Qixian Zhao.

Donaldson-Thomas invariants of \(\left[\mathbb{C}^4/\mathbb{Z}_r\right]\)

In this talk I will review the DT4 theory and talk about the computations of zero-dimensional DT invariants of the quotient stack \(\left[\mathbb{C}^4/\mathbb{Z}_r\right]\), confirming a conjecture of Cao-Kool-Monavari. Our main theorem is established through an orbifold analogue of Cao-Zhao-Zhou's degeneration formula combined with the zero-dimensional Donaldson-Thomas invariants for \( \mathbf{A}_{r-1}×\mathbb{C}^2\) and an explicit determination of orientations of Hilbert schemes of points on \(\left[\mathbb{C}^4/\mathbb{Z}_r\right]\).

The Hamiltonian reduction of hypertoric mirror symmetry

Consider a circle acting on a symplectic manifold X by hamiltonian transformations. There is an induced topological action on the Fukaya category, and one can take the `quotient' by this topological action at different moment parameters. For nonsingular moment parameters, one expects to obtain the Fukaya category of the symplectic reduction. I will discuss joint work with Peng Zhou and Vivek Shende which describes the quotient at a singular moment parameter, when X is a multiplicative hypertoric variety. We obtain the Fukaya category of a toroidal arrangement, connecting with conjectures of Lekili and Segal.