| Date/Time | Speaker | Title | Abstract |
|---|---|---|---|
| Monday 9:30, 11:00 | Cristina Manolache | Introduction to virtual invariants | In the first lecture I will give examples of moduli spaces of curves on a given variety and I will define virtual invariants (such as GW and DT invariants). In the second lecture I will discuss virtual classes and their properties in a number of simple examples. I will give an application to genus one GW invariants. |
| Monday 14:00 | Pavel Putrov | 4-manifolds and (topological) modular forms | It is known that Vafa-Witten partition function on a four-manifold, that is the generating function of (virtual) Euler characteristics of instanton moduli spaces, exhibits nice modular transformation properties. In my talk will discuss a conjectural generalization of this statement motivated by physics of 6d superconformal field theories and present a couple of simple concrete examples. My talk is based on a joint work with Gukov, Pei and Vafa. |
| Monday 15:30 | Dominic Joyce | Vertex algebra and Lie algebra structures on the homology of moduli spaces | I give an overview of a programme which defines the structure of a graded vertex algebra (a complicated algebraic structure coming from Conformal Field Theory), or a graded Lie algebra, on large classes of moduli spaces (stacks) in both Algebraic Geometry and Differential Geometry. On the differential-geometric side these include moduli spaces of connections satisfying instanton-type curvature conditions, possibly with extra data such as Higgs fields. I expect that these algebraic structures control phenomena to do with enumerative invariants, such as wall-crossing formulae or (Seiberg-Witten invariants) => (Donaldson invariants), and may enable the computation of homology of moduli spaces using representation theory. |
| Tuesday 9:30 | Richard Thomas | Refined Vafa-Witten invariants for projective surfaces | I’ll sketch a definition of refined VW invariants for projective surfaces. (Joint work with Yuuji Tanaka and Davesh Maulik.) |
| Tuesday 11:00 | Richard Thomas | Computing Vafa-Witten invariants via degeneracy loci | VW invariants of projective surfaces are made up of “instanton contributions” and “monopole contributions”. For the former see many other talks in this conference. I’ll explain how to compute the latter in terms of degeneracy loci. (Joint work with Amin Gholampour.) |
| Tuesday 14:00 | Nicolo Piazzalunga | The 4-vertex | I plan to talk about the 4-vertex: it computes K-theoretic Donaldson-Thomas invariants for toric Calabi-Yau 4-folds by counting bound states of D0-D2-D4-D6-D8 branes. |
| Tuesday 15:30 | Pietro Longhi | Exponential networks and 5d BPS spectra | We develop a systematic framework to study BPS states of 5d N=1 gauge theories. The key idea, inspired by 2d-4d wall-crossing, is to probe the BPS spectrum using co-dimension two defects. 3d-5d systems feature a new BPS sector of mixed “3d-5d BPS states’’, whose (framed) wall-crossing phenomena encode information about the 5d BPS spectrum. We will explain how a non-abelianization map for exponential networks, motivated by physics, computes 3d-5d states and their wall-crossing. We will discuss applications of this framework to 3d-5d systems arising from geometric engineering, and briefly comment on a relation to enumerative geometry. This talk is based on joint work with S. Banerjee and M. Romo |
| Wednesday 9:30, 11:00 | Maxim Zabzine | Generalising the Donaldson-Witten theory and transversely elliptic complex | I will review different versions of N=2 supersymmetric Yang-Mills theories on 4D manifolds, e.g. the (equivariant) Donaldson-Witten theory, Pestun’s calculation on 4-sphere, etc. I will explain that all these theories can be understood in unified framework. I will explain that from QFT point of view the underlying moduli problem should be transversely elliptic (not necessary elliptic). I will explain physical and mathematical features of the construction (based on works 1812.06473 and 1904.12782). |
| Wednesday 14:00 | Yukinobu Toda | On categorical Donaldson-Thomas theory for local surfaces | I will introduce the notion of categorical Donaldson-Thomas theories for moduli spaces of stable sheaves on the total space of a canonical line bundle on a smooth projective surface. They are constructed as certain gluings of locally defined triangulated categories of matrix factorizations, via the linear Koszul duality together with the theory of singular supports for coherent sheaves on quasi-smooth derived stacks. |
| Wednesday 15:00 | Andrea Ricolfi | Virtual classes and virtual motives of Quot schemes on 3-folds | We show that the Quot scheme of finite length quotients of a locally free sheaf on a 3-fold carries, under suitable conditions, a 0-dimensional virtual fundamental class. Using results with S. Beentjes, we solve the associated enumerative theory for Calabi-Yau 3-folds. We present a conjectural formula in the general case, and we develop the parallel motivic theory for these Quot schemes, providing a higher rank version of the work of Behrend-Bryan-Szendroi on the Hilbert scheme of points of a 3-fold. |
| Thursday 9:30, 11:00 | Sergei Gukov | Vertex algebras from moduli problems | |
| Thursday 14:00 | Yukinobu Toda | On categorical wall-crossing of Pandharipande-Thomas theory for local surfaces | I will show that the moduli stack of D0-D2-D6 bound states on the total space of a canonical line bundle is isomorphic to the dual obstruction cone over the moduli stack of pairs on the surface. This result is used to define the categorical Pandharipande-Thomas theory on the local surface. I will propose several conjectural wall-crossing formulas of categorical PT theory, motivated by d-critical analogue of D/K conjecture in birational geometry. Among them, I will show the wall-crossing formula of categorical PT theory with irreducible curve classes. |
| Thursday 15:30 | Michele Del Zotto | Geometric Engineering, 6d SCFTs, and Topological Strings | I will discuss some intriguing consequences of the relation in between topological strings for local elliptic CY three-folds and the omega-deformation of six-dimensional superconformal field theories. Based on joint works with G. Lockhart, as well as A.-K. Kashani-Poor, A. Klemm, J. Gu, and M.-X. Huang. |
| Friday 9:30 | Lothar Göttsche | Verlinde and Lehn formulas for moduli of sheaves on algebraic surfaces | This talk is about joint work with Martijn Kool. Lehn's conjecture was a conjectural formula for the top Segre classes of a tautological bundle associated to line bundles on Hilbert schemes of points on a surface. It was shown by Marian, Oprea and Pandharipande, and generalized to other tautological vector bundles on Hilbert schemes of points. They also proved Verlinde formulas computing generating functions for holomorphic Euler characteristics of line bundles on Hilbert schemes of points, and showed that the generating functions for Lehn and Verlinde formulas are related by an explicit change of variables. In this work we conjecturally extend all these results to moduli spaces of rank 2 stable coherent sheaves on algebraic surfaces with $p_g>0$. |
| Friday 11:00 | Noah Arbesfeld | K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface | I'll explain how to use K-theoretic Donaldson-Thomas theory to deduce certain symmetries of (K-theoretic) Nekrasov functions, and how to use such symmetries to study tautological bundles on Hilbert schemes of points on surfaces. |
| Friday 14:00 | Alessandro Tanzini | Surface defects and nested instantons | We introduce and study a surface defect in four dimensional gauge theories supporting nested instantons with respect to the parabolic reduction of the structure group at the defect. This is engineered from a D7/D3 brane system on a local surface S. For a single D7 brane and S the direct product of an elliptic curve times a Riemann surface with marked points, we get conjectural explicit formulae for the virtual equivariant elliptic genus of a certain bundle over the moduli space of the nested Hilbert scheme of points on the affine plane. A connection with elliptic cohomology of character varieties and an elliptic version of modified Macdonald polynomials will be mentioned. |
| Friday 15:30 | Nina Morishige | Genus Zero Gopakumar-Vafa invariants of the Banana manifold | The Banana manifold is a compact Calabi-Yau threefold constructed as the conifold resolution of the fiber product of a generic rational elliptic surface with itself. We compute Katz’s genus 0 Gopakumar-Vafa invariants of fiber curve classes on the Banana manifold. The weak Jacobi form of weight -2 and index 1 is the associated generating function. The invariants are shown to be an actual count of structure sheaves of certain possibly nonreduced genus 0 curves on the universal cover of the singular locus of the fibers of the manifold. |