The Sheffield Geometry and Physics Seminar

Given a Riemann surface C and a central charge c, one can define the notion of "Virasoro block" on C, introduced in the context of conformal field theory. The space of Virasoro blocks carries various interesting algebraic and geometric structures. I will recall this story and then describe a new scheme for constructing Virasoro blocks at central charge c=1, by relating them to simpler "abelianized" blocks on a branched double cover of C. This is joint work in progress with Qianyu Hao, inspired by work of Coman-Longhi-Pomoni-Teschner, Iwaki, Marino, Bridgeland and others. 

Bridgeland stability conditions have been constructed on curves, surfaces, and in some higher dimensional examples. In several cases, there are only so-called “geometric” stability conditions which are constructed using slope stability for sheaves, whereas in other cases there are more (e.g. coming from an equivalence with representations of a quiver). Lie Fu -- Chunyi Li -- Xiaolei Zhao were the first to provide a general result explaining this phenomena. In particular, they showed that if a variety has a finite map to an abelian variety, then all stability conditions are geometric. In this talk, we test the converse on surfaces that arise as free quotients by finite groups. To do this, we will develop a method to study stability conditions on any triangulated category with a group action. This is joint work with Edmund Heng and Tony Licata. 

Every Enriques surface Y has an elliptic pencil, and every elliptic pencil on Y has two multiple fibers, whose reduced support is called a half-fiber. The non-degeneracy invariant of an Enriques surface is defined to be the maximum number of half-fibers meeting each other at exactly one point. 

This invariants influences the projective geometry of Y, as well as the structure of its derived category. In collaboration with R. Moschetti and L. Schaffler, we study techniques to compute non-degeneracy. These mix computer algebra and classical geometric methods. I'll illustrate our results in a few examples and I'll outline future directions. 

In this talk, I will introduce the concept of Symmetry Topological Field Theory (SymTFT); this is a (d+1)-dimensional topological theory associated to a d-dimensional theory T, which neatly encodes its symmetry properties and has emerged as a key tool to study generalized symmetries. In particular, I will show how the SymTFT can be used to determine gapped infra-red phases of 2d theories that have a symmetry S described in general by a fusion category. This approach gives concrete computational tools to extract information on the gapped phase, such as the symmetry breaking pattern, the number of vacua, and the action of the symmetry on such vacua.  Moreover, I will introduce a generalization of the usual SymTFT framework that allows us to characterize phase transitions between such gapped phases. The SymTFT also manifestly encodes the order parameters for the phases, thus providing a generalized version of the Landau paradigm for symmetries that go beyond the standard group-like case.

An old theorem of Mikhalkin says that the number of rational plane curves of degree d through 3d-1 points is equal to a count of tropical curves, combinatorial objects which are more amenable to computations. One can try to generalise this result in two directions, either by allowing for higher genus curves or allowing for different conditions than solely passing through points. I’ll discuss a generalisation which does both, using intersection theory on the moduli space of curves and integrable hierarchies, as well as ongoing work connecting this thread to quantum scattering diagrams coming from log Calabi-Yau surfaces. This is joint work with Patrick Kennedy-Hunt and Ajith Urundolil Kumaran.

In this talk, I will give a brief introduction to the theory of spectral networks on a marked Riemann surface. I will describe how the GMN non-abelianization map can be understood in terms of Lagrangian Floer theory of the spectral curve in the cotangent bundle of  the Riemann surface, for the quadratic differentials case. If time permits, I will describe one way to generalize this to the higher rank situation.

Previous work has given proof and evidence that BPS states in local Calabi-Yau 3-folds can be described and counted by exponential networks on the punctured plane, with the help of a suitable non-abelianization map to the mirror curve. This provides an appealing elementary depiction of moduli of special Lagrangian submanifolds, but so far only a handful of examples have been successfully worked out in detail. In this talk, I will present an explicit correspondence between torus fixed points of the Hilbert scheme of points on C^2\subset C^3 and anomaly free exponential networks attached to the quadratically framed pair of pants. This description realizes an interesting, and seemingly novel, "age decomposition'' of linear partitions. We also provide further details about the networks' perspective on the full D-brane moduli space.(Joint work with Sibasish Banerjee, Mauricion Romo, Raphael Senghaas)

For a compact Lie group G, its massless Coulomb branch algebra is the G-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this talk, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian G-manifolds when the symplectic manifolds are open and convex. This is a generalization of the closed case where symplectic cohomology is replaced with quantum cohomology. Following Teleman, we also explain how it relates to the Coulomb branch algebra of cotangent-type representations. This is joint work with Eduardo González and Dan Pomerleano.

We consider the A-model obtained by a partial topological twist of a 3d N=2 supersymmetric gauge theory. The twisted indices are the Witten indices over the Hilbert space of the 3d theory compactified on a closed Riemann surface, which generalise the Verlinde formulae. (In the special case of pure Chern-Simons theories – that is, 3d gauge theory without matter--, they give us the number of conformal blocks of the corresponding WZW model on the Riemann surface.) Previous physical methods only computed the twisted indices for a gauge group G that is simply connected and/or unitary. We generalise the supersymmetric computation to G any real compact Lie group, using the notion of higher-form symmetries. I will give a pedagogical presentation of our results. [Work to appear with Elias Furrer and Osama Khlaif.]

Hodge number duality is one of the most fundamental phenomena in mirror symmetry. In the 1990s, Batyrev and Borisov introduced a combinatorial mirror construction for nef toric complete intersections of Calabi-Yau varieties, verifying Hodge number duality for these cases. Clarke has recently expanded this construction, broadening its scope to include a wide range of examples of mirror pairs. 

In this talk, I will discuss ongoing work with Andrew Harder, where we establish Hodge number duality for a large class of orbifold Clarke mirror pairs. We achieve this by developing a new tropical geometric tool to compute Hodge numbers. Our results not only confirm the result of Batyrev and Borisov but also lead to a proof of a conjecture by Katzarkov, Kontsevich, and Pantev for orbifold toric complete intersections. If time permits, I will also describe several applications, including the functoriality in Fano mirror symmetry and mirror symmetry for singular varieties.

I'll recall some basics about Slodowy slices, generalized slices in the affine Grassmannian, and quantizations thereof called W-algebras and Yangians, respectively, as well as their analogues for affine Lie algebras which are naturally described using the theory of vertex algebras. Then I'll explain a construction of vertex algebras associated to divisors in toric Calabi-Yau threefolds, which include affine W-algebras in type A for arbitrary nilpotents, and outline a dictionary between the geometry of the threefolds and the representation theory of these algebras. I'll also explain the physical interpretation of these results, as an example of twisted holography for M5 branes in the omega background.