Mini-courses

Alexey Oblomkov - 3D TQFT, topological invariants and categorical representation theory.

In my lectures I outline recent developments in setting  mathematical theory of 3D TQFT conjectured by Kapustin-Rozansky-Saulina (KRS). I show some applications to knot homology and categorical representation theory. The lectures are based on the joint papers with Lev Rozansky.

In details, the first lecture is dedicated to setting KRS theory, more generally TQFTs with defects, and explaining the details of the theory in the case when the target of the Sigma model is the Hilbert scheme of points on the plane. In the second lecture, I explain why  HOMFLYPT homology of a link is a value of the partition function for the above mentioned KRS theory and derive a construction of a coherent sheaf on the Hilbert scheme for every braid. In the last lecture I explain how the gl(m|n) knot homology can be computed within our setting. The last construction also leads to a model for a categorification of the quantum super-group gl(m|m).

Denis Nesterov - Unramified Gromov-Witten theory

Unramified Gromov-Witten theory is a higher-dimensional generalisation of Hurwitz theory,  constructed by Kim, Kresch and Oh. It is related to the standard Gromov-Witten theory by an interpolating stability condition, which gives rise to wall-crossing formulas. The aim of the minicourse is to explain this wall-crossing phenomenon together with its relation to Gopakumar-Vafa invariants and, if time permits, the Gromov-Witten/Hurwitz correspondence.

Plan:

1. Introduction to unramified GW theory

2. GW/uGW wall-crossing

3. GV invariants and GW/H correspondence

Abstracts

Yannik Schuler - Maps, sheaves, and membranes.

There are essentially two perspectives on enumerating curves in a threefold X: either by viewing them as being parametrised by a map or as being cut out by an ideal sheaf. By the MNOP conjecture these two approaches are linked via an explicit (but conceptually very surprising) change of coordinates in the generating series of curve counting invariants. For X a Calabi-Yau admitting the action of a torus we will discuss a refinement of this conjecture: On the maps side there will be the equivariant Gromov-Witten theory of X \times C^2 and on the sheaf side we have Nekrasov-Okounkov's K-theoretic Pandharipande-Thomas theory. We will discuss consequences and evidence for the correspondence. Moreover, I will explain how it is motivated by a link to the (conjectural) index of M-theory on X \times C^2. This is joint work with Andrea Brini.

Veronica Fantini - Strong-weak symmetry and resurgent fermionic traces of local P2.

Quantizing the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent formal power series in the Planck constant and its inverse.  In this talk, we show that a full-fledged strong-weak resurgent symmetry is at play, exchanging the perturbative/nonperturbative contributions to the holomorphic and anti-holomorphic blocks in the factorization of the spectral trace. This relies on a global net of relations connecting the perturbative series and the discontinuities in the dual regimes, which is built upon the analytic properties of the L-functions with coefficients given by the Stokes constants and the q-series acting as their generating functions. Based on a joint work with C. Rella arXiv:2404.10695.

Ben Davison - Okounkov's conjecture via BPS Lie algebras I.

Given a finite quiver Q, using their theory of stable envelopes and resulting R matrices, Maulik and Okounkov defined a new type of Yangian algebra.  This algebra is defined as a subalgebra of the endomorphism algebra of the (equivariant) cohomology of all the Nakajima quiver varieties associated to Q. If Q is an orientation of a type ADE Dynkin diagram, the Maulik-Okounkov algebra recovers the usual Yangian algebra, a deformation of the universal enveloping algebra of the current algebra of the associated ADE type Lie algebra.  If Q is the one-loop quiver, their theory also recovers a well-known Yangian algebra, and the Grojnowski-Nakajima action of an infinite-dimensional Heisenberg algebra on the cohomology of Hilbert schemes of C^2 is recovered as a part of the theory. 

 For general quivers, the picture is less clear.  Although, as in the cases above, there is a Lie algebra g_{MO} which generates the whole of the Maulik-Okounkov algebra, even determining the dimensions of the graded pieces remained an open question until quite recently. Okounkov's conjecture states that these dimensions are given by coefficients of Kac's polynomials, which count isomorphism classes of Q-representations over finite fields.  

Recently, in joint work with Tommaso Botta, we proved this conjecture as a consequence of an isomorphism between the Maulik-Okounkov Lie algebra and a quite different Lie algebra (a special case of a "BPS Lie algebra") built via the Kontsevich-Soibelman theory of cohomological Hall algebras for quivers with potential.  This isomorphism in turn follows from a careful analysis of nonabelian stable envelopes.  In this talk I will motivate the constructions of these two Lie algebras, as well as introducing Okounkov's conjecture.

Oscar Kivinen - A Lie-theoretic generalization of some Hilbert schemes.

I will introduce several varieties attached to a complex reductive group, generalizing for example Hilb^n(C^2) and Haiman’s isospectral Hilbert scheme, which pertain to the GL_n-case. I will then explain what is currently known about these varieties (including some low rank examples) and what else one would like to know about them.

Robert Hanson - Twistors and the geometric Langlands program 

Geometric Langlands correspondences come in three main flavours: de Rham, Dolbeault, and Betti, formulated respectively on moduli stacks of Higgs bundles, flat connections, and monodromy representations. The notion of a lambda-connection interpolates between the de Rham and Dolbeault theories. In this talk, I will use lambda-connections to construct a twistor stack for Higgs bundles and propose a twistoral flavour of geometric Langlands correspondence. I will try to convince you that twistors provide a natural home for the action of S-duality on hyperkähler boundary conditions, introduced by Kapustin and Witten in the context of 4D supersymmetric Yang-Mills. This is based on joint work with Emilio Franco, alongside ongoing work. 

Anna Barbieri (cancelled) - Moduli spaces of Bridgeland stability conditions and of quadratic differentials, and compactification

I will discuss a correspondence between two different moduli problems: one arising in the theory of Bridgeland stability condition, the other from meromorphic multi-differentials, and applications, in particular the construction of a smoth compactification of the stability manifold for A_n-type categories. Based on joint works with Martin Moeller, Jeonghoon So, Yu Qiu.

Tommaso Botta - Okounkov's conjecture via BPS Lie algebras II. 

Building on Ben Davison's talk, I will explain how to identify the Maulik-Okounkov Lie algebra of a quiver Q with the BPS Lie algebra of the tripled quiver \tilde{Q} with its canonical cubic potential. The bridge to compare these seemingly diverse words is the theory of non-abelian stable envelopes,  which we use to interpolate between representations of the MO Lie algebra and representations of the BPS Lie algebra. The general philosophy underlying the non-abelian stable envelopes, and in particular their relation with stability conditions and wall-crossing will be discussed.

In conclusion, I will explain how to use these results to deduce Okounkov's conjecture, equating the graded dimensions of the MO Lie algebra with the coefficients of Kac polynomials. The talk is based on joint work with Ben Davison.

Ana Peón-Nieto - Classification of very stable Higgs bundles. 

The seminal work of Hausel and Hitchin provides examples of dual BBB and BAA branes on the moduli space of Higgs bundles, the latter being given by upward flows from very stable regular nilpotent Higgs bundles. I will speak about ongoing work on the classification of components of the global nilpotent cone into very stable and wobbly, which are respectively those components containing or not a very stable Higgs bundle. I will show that, essentially, the only very stable Hodge bundles correspond to the regular (by Hausel—Hitchin)  and subregular nilpotent orbits. For the latter, the computation of the virtual equivariant multiplicities defined by Hausel and Hitchin yields the multiplicity of the corresponding component of the nilpotent cone, expected to be the rank of the dual BBB brane. Finally, I will analyse  the obstruction that these invariants provide to the existence of very stable points in a given component. 

Nadir Fasola - Tetrahedron instantons via Donaldson-Thomas theory.

Tetrahedron instantons have recently been introduced in String Theory by Pomoni-Yan-Zhang to describe the dynamics of systems of intersecting D-branes in $\mathbb C^4$. According to a general philosophy, the associated partition function can be interpreted in terms of a sheaf counting problem. I will explain how tetrahedron instantons moduli spaces can be studied in terms of Quot schemes, in the framework of a Donaldson-Thomas type theory. This is based on joint work with S. Monavari.