The Sheffield Geometry and Physics Seminar

The variety of commuting matrices is an important space in algebraic geometry, and has been studied from various perspectives. The stack of commuting matrices M is the same as the stack of zero dimensional sheaves in the plane, and can be used to define various Hall algebras that act on cohomologies of the Hilbert scheme of points in the plane. In this talk, I will talk about cohomologies of the stack M. First, I will recall results of Davison and Davison-Meinhardt about the Borel-Moore homology of M. These results are proved via an analysis of the BPS sheaves of points in the three dimensional affine space. Next, I will discuss (partial) analogues of these results for the K-theory and the category of coherent sheaves on M. The central objects are a categorical replacement of the BPS sheaves. The talk is based on joint results with Yukinobu Toda. 

Recently, a relationship between BPS invariants of four-dimensional supersymmetric QFTs (equivalently, Donaldson-Thomas invariants of certain 3CY triangulated categories) and the Eynard-Orantin topological recursion (which computes invariants of "spectral curves'' originally appearing in the theory of matrix models), was observed for a class of fundamental examples.  We review both formalisms and explain how to modify the topological recursion to obtain the "$\beta$-deformed'' or "refined" topological recursion when the initial data is sufficiently nice. For the simplest such examples, we show how the corresponding free energies can be expressed in terms of a new collection of refined BPS invariants which, unlike the unrefined case, do not seem to have appeared in the Donaldson-Thomas theory to date.  Based on joint works with K. Osuga.

The Alday-Gaiotto-Tachikawa (AGT) conjecture in physics predicts a relationship between 2d conformal field theories and certain 4d gauge theories. A precise mathematical version (proved by Maulik-Okounkov, Schiffmann-Vasserot and others) states that the equivariant cohomology of the moduli space of instantons (4d side) is a module of a certain W-algebra (2d side), and that the fundamental class of the moduli space is a Whittaker vector in the W-algebra module, known as the Gaiotto vector. I will show how one can realize this Gaiotto vector as the partition function of an Airy structure, and thereby relate it to the topological recursion formalism of Eynard and Orantin. This means that one can compute the Nekrasov instanton partition function (which is the norm squared of the Gaiotto vector) using topological recursion techniques. Time permitting, I will discuss some possible applications (all work-in-progress) of this relationship including extensions to Argyres-Douglas theories, relations to Hurwitz theory and matrix models, and connections to integrability.

The talk is based on joint work with Vincent Bouchard, Gaetan Borot and Thomas Creutzig.

I will report on joint work in progress with Luca Giovenzana. I will describe some developments in the abstract theory of quasi del Pezzo pseudolattices, before showing how this theory arises naturally in the contexts of type II degenerations of K3 surfaces and elliptically fibred K3 surfaces. This can be thought of as a manifestation of mirror symmetry; I will discuss what it could tell us about mirror symmetry for K3 surfaces and the 2-dimensional Fano/LG correspondence.

In recent years various unifying frameworks for understanding 2d integrable field theories have emerged. In this talk I will review the approach based on 4d Chern-Simons theory, due to Costello and Yamazaki, and describe recent progress towards extracting general 2d integrable fields theories from 4d Chern-Simons theory.

Placing D3-branes at conical Calabi-Yau threefold singularities produces many AdS5/CFT4 duals. Recent progress in differential geometry has produced a technique, called K-stability, to recognize which singularities admit conical Calabi-Yau metrics. On the other hand, the algebraic technique of noncommutative crepant resolutions (NCCRs) has been developed to associate a quiver to a singularity. In favorable situations (such as the hypersurface case), producing an NCCR is equivalent to finding suitable matrix factorizations of the hypersurface. I will put together K-stability and NCCRs to produce new AdS5/CFT4 pairs, beyond the well-known toric setup.

In this talk I shall discuss recent work with Tom Coates and Sara Veneziale in which we successfully recover the dimension of a Fano variety X directly from the regularized quantum period of X via machine learning. We apply machine learning to the question: does the quantum period of a Fano variety X know the dimension of X? Note that there is as yet no theoretical understanding of this. We show that machine learning techniques can recover the dimension with > 80% accuracy, demonstrating that machine learning can pick out structure from complex mathematical data in situations where we lack a theoretical understanding. It also gives positive evidence for the assertion (which is proven for smooth Fanos in low dimensions, but unknown in general) that the quantum period of a Fano variety determines that variety. 

 Fix a Calabi-Yau 3-fold X of Picard rank one satisfying the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as the quintic 3-fold. I will first describe explicit formulae relating rank zero Donaldson-Thomas (DT) invariants to Pandharipande-Thomas (PT) invariants using wall-crossing with respect to weak Bridgeland stability conditions on X. As applications, I will find sharp Castelnuovo-type bounds for PT invariants, and explain how combining these explicit formulae with S-duality in physics enlarges the known table of Gopakumar-Vafa (GV) invariants. The second part is joint work with string theorists Sergei Alexandrov, Albrecht Klemm, Boris Pioline and Thorsten Schimannek. 

I will discuss how to use Laurent inversion, a technique coming from mirror symmetry which constructs toric embeddings, to study the local structure of the K-moduli space of a K-polystable toric Fano variety. More specifically, starting from a given toric Fano 3-fold X of anticanonical volume 28 and Picard rank 4, and combining a local study of its singularities with the global deformation provided by Laurent inversion, we are able to conclude that the K-moduli space is rational around X. This is joint work with Andrea Petracci.