# The Sheffield Geometry and Physics Seminar

**The Sheffield Geometry and Physics Seminar (SGaPS) runs two-talks seminars on a fortnightly basis, starting the first week of February.**

## Time and venue

The seminars will take place in Room J-11, Hicks Building.

The first seminar will take place on Monday 7 February, at 2.00pm.

After that, we will have a seminar every other Monday, at 2.00pm, starting on 28 February.

## Confirmed speakers

Murad Alim (Universität Hamburg)

Christian Böhning (University of Warwick)

Francesca Carocci (EPFL)

Nadir Fasola (University of Sheffield)

Lotte Hollands (Heriot-Watt University)

Qingyuan Jiang (University of Edinburgh)

Martijn Kool (Utrecht University)

Naoki Koseki (University of Edinburgh)

William Elbæk Mistegård (Centre for Quantum Mathematics)

Navid Nabijou (University of Cambridge)

Jeongseok Oh (Imperial College London)

Kento Osuga (University of Warsaw)

Qaasim Shafi (Imperial College London)

Angelica Simonetti (University of Cambridge)

Balázs Szendrői (University of Oxford)

Dimitri Zvonkine (Laboratoire Mathématiques de Versailles)

## Organisers

Andrea Brini, Nadir Fasola and Nikita Nikolaev.

## Talks

### Skew matrices of linear forms, matrix factorisations and intermediate Jacobians of cubic threefolds

**Speaker: **Christian Böhning (Warwick)**Date: **6 June 2022**Time: **2.00pm

**Abstract:** I will report on some ongoing joint work with Hans-Christian von Bothmer (Hamburg) and Lukas Buhr (Mainz). Results due to Druel and Beauville show that the blowup of the intermediate Jacobian of a smooth cubic threefold X in the Fano surface of lines can be identified with a moduli space of semistable sheaves of Chern classes c_1=0, c_2=2, c_3=0 on X. We identify this space with a space of matrix factorisations. This has the advantage that this description naturally generalises to singular and even reducible cubic threefolds. In this way, given a degeneration of X to a reducible cubic threefold X_0, we obtain an associated degeneration of the above moduli spaces of semistable sheaves.

### Roots and Logs in the Enumerative Forest

**Speaker: **Navid Nabijou (Cambridge)**Date: **6 June 2022**Time: **3.30pm

**Abstract:** Logarithmic and orbifold structures provide two independent ways to model curves in a variety tangent to a divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify "birational invariance" as the key property distinguishing the two theories. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is much better understood. Connections to local curve counting will also be discussed. No prior knowledge of Gromov-Witten theory will be assumed.

### Virtual cycles on projective completions and quantum Lefschetz formula

**Speaker: **Jeongseok Oh (Imperial College London)**Date: **7 February 2022**Time: **2.00pm

**Abstract: **For a compact quasi-smooth derived scheme M with (-1)-shifted cotangent bundle N, there are at least two ways to localise the virtual cycle of N to M via torus and cosection localisations, introduced by Jiang-Thomas. We produce virtual cycles on both the projective completion and projectivisation of N and show the ones on the former push down to Jiang-Thomas cycles and the one on the latter computes the difference. Using the idea we study the difference between quintic and formal quintic Gromov-Witten invariants.

### Gromov-Witten invariants of Blow-Ups

**Speaker: **Qaasim Shafi (Imperial College London)**Date: **7 February 2022**Time: **3.30pm

**Abstract: **Gromov-Witten invariants play an essential role in mirror symmetry and enumerative geometry. Despite this, there are few effective tools for computing Gromov-Witten invariants of blow-ups. Blow-ups of X can be rewritten as subvarieties of Grassmann bundles over X. In joint work with Tom Coates and Wendelin Lutz, we exploit this fact and extend the abelian/non-abelian correspondence, a modern tool in Gromov-Witten theory. Combining these two steps allows us to get at the genus 0 invariants of a large class of blow-ups.

### The Automorphism Equivariant Hitchin Index

**Speaker: **William Elbæk Mistegård (Centre for Quantum Mathematics)**Date: **3 March 2022**Time: **10.00am

**Abstract: **The moduli space of Higgs bundles on a compact Riemann surface was introduced by Hitchin in his study of the self-duality equations on a Riemann surface. This is a quasi-projective hyper-Kähler variety, which supports an algebraic torus-action and a torus-equivariant line bundle generating the Picard group. This line bundle is called the determinant line bundle of cohomology, or determinant line bundle for short. Given an automorphism of the Riemann surface, there is an induced lift to the determinant line bundle. We define and study the automorphism equivariant Hitchin index (AEHI). This the trace of the derived action on the torus-weight spaces of the cohomology of the determinant line bundle. Our study is motivated by topological quantum field theory and complex Chern-Simons theory via non-abelian Hodge theory, which identifies the moduli space of Higgs bundles with the moduli space of flat complex connections on the Riemann surface. We prove that the AEHI is a topological invariant of the three-manifold obtained as the mapping torus of the automorphism of the Riemann surface and we provide an explicit formula for the AEHI in terms of: cohomological pairings of the Atiyah-Bott generators on the moduli space of parabolic bundles on the quotient Riemann surface (i.e. the orbit space of the automorphism), cohomological pairings on symmetric powers of the quotient Riemann surface and Seifert invariants of the mapping torus. These results provides new links between algebraic geometry and quantum topology, and our topological invariant can be seen as a generalization of the Witten-Reshetikhin-Turaev quantum invariant of the mapping torus. This is joint work with J.E. Andersen and T. Hausel.

### Refinement of Quantum Curves

**Speaker: **Kento Osuga (University of Warsaw)**Date: **3 March 2022**Time: **11.30am

**Abstract: **Topological recursion is a recursive formalism that takes an algebraic curve as the initial data, and computes a variety of invariants such as Kontsevich-Witten intersection numbers or knot invariants. Another interesting application of topological recursion is a quantisation of algebraic curves which are often called quantum curves. For degree two curves, topological recursion admits a suitable 1-parameter refinement so-called refined topological recursion. In this talk I will address properties of refined topological recursion and construct explicit form of refined quantum curves for genus zero curves. If time permits, I will also discuss about a somewhat unexpected relation between topological recursion and BPS structures in the refined setting.

### Non-perturbative quantum geometry, resurgence and BPS structures

**Speaker: **Murad Alim (Universität Hamburg)**Date: **14 March 2022**Time: **2.00pm

**Abstract: **BPS invariants of certain physical theories correspond to Donaldson-Thomas (DT) invariants of an associated Calabi-Yau geometry. BPS structures refer to the data of the DT invariants together with their wall-crossing structure. On the same Calabi-Yau geometry another set of invariants are the Gromov-Witten (GW) invariants. These are organized in the GW potential, which is an asymptotic series in a formal parameter and can be obtained from topological string theory. A further asymptotic series in two parameters is obtained from refined topological string theory which contains the Nekrasov-Shatashvili (NS) limit when one of the two parameters is sent to zero. I will discuss in the case of the resolved conifold how all these asymptotic series lead to difference equations which admit analytic solutions in the expansion parameters. A detailed study of Borel resummation allows one to identify these solutions as Borel sums in a distinguished region in parameter space. The Stokes jumps between different Borel sums encode the BPS invariants of the underlying geometry and are captured in turn by another set of difference equations. I will further show how the Borel analysis of the NS limit connects to the exact WKB study of quantum curves. This is based on joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner.

### Gromov-Witten invariants of complete intersections

**Speaker: **Dimitri Zvonkine (Laboratoire Mathématiques de Versailles)**Date: **14 March 2022**Time: **2.00pm

**Abstract: **We show that there is an effective way to compute all Gromov-Witten (GW) invariants of all complete intersections. The main tool is Jun Li's degeneration formula: it allows one to express GW invariants of a complete intersection from GW invariants of simpler complete intersections. The main difficulty is that, in general, the degeneration formula does not apply to primitive cohomology insertions. To circumvent this difficulty we introduce simple nodal GW invariants. These invariants do not involve primitive cohomology classes, but instead make use of imposed nodal degenerations of the source curve. The algorithm for computing GW invariants relies on two main statements: (i) simple nodal GW invariants can be computed by the degeneration formula, (ii) simple nodal GW invariants determine all GW invariants of a complete intersection. The first statement is geometric; the second uses the invariance of GW invariants under monodromy and some representation theory. This is joint work with Hulya Arguz, Pierrick Bousseau and Rahul Pandharipande.

### Z/2Z-smoothings of cusp singularities

**Speaker: **Angelica Simonetti (University of Cambridge)**Date: **28 March 2022**Time: **2.00pm

**Abstract: **Cusp singularities and their quotients by a suitable action of Z/2Z are among the surface singularities which appear at the boundary of the compactification of the moduli space of surfaces of general type due to Kollar, Shepherd-Barron and Alexeev.

Since only those singularities that admit a smoothing family occur at the boundary of this moduli space, it is useful to find nice conditions under which they happen to be smoothable.

We will describe a sufficient condition for a cusp singularity admitting a Z/2Z action to be equivariantly smoothable. In particular we will see it involves the existence of certain Looijenga (or anticanonical) pairs (Y,D) that admit an involution fixed point free away from D and that reverses the orientation of D.

### Derived projectivizations of complexes

**Speaker: **Qingyuan Jiang (University of Edinburgh)**Date: **28 March 2022**Time: **2.00pm

**Abstract: **In this talk, we will discuss the counterpart of Grothendieck's projectivization construction in the realm of derived algebraic geometry.

We will first discuss the motivations and definitions of derived projectivizations and study their fundamental properties.

We will then focus on complexes of perfect-amplitude contained in [0,1]. In this case, the derived projectivizations enjoy special pleasant properties. For example, they satisfy the generalized Serre's theorem and the derived version of Beilinson's relations, and there are structural decompositions for their derived categories.

Finally, we will discuss some applications of this framework, including

applications to classical situations, such as derived categories of certain reducible schemes and irreducible singular schemes

applications to Hecke correspondence moduli, focusing on the cases of surfaces

applications to moduli of pairs and moduli of extensions, focusing on the cases of curves, surfaces, and threefolds

If time allows, we might also discuss the generalizations of these results to the cases of derived Grassmannians and some other types of derived Quot schemes.

### Cohomological chi-independence for Gopakumar-Vafa invariants

**Speaker: **Naoki Koseki (University of Edinburgh)**Date: **25 April 2022**Time: **2.00pm

**Abstract: **Recently, Maulik and Toda gave a mathematical definition of the Gopakumar-Vafa(GV) invariants, which are the virtual counts of curves inside a Calabi-Yau 3-fold. GV invariants are conjecturally equivalent to other curve counting theories such as Gromov-Witten invariants (GV=GW conjecture). One of the mysterious features of GV invariants is a chi-independence conjecture, which is expected from GV=GW conjecture.

In my recent work with Tasuki Kinjo (Tokyo), we proved the chi-independence of GV invariants for a certain class of non-compact Calabi-Yau 3-dolds, called local curves. I will explain our result and other recent developments of the theory.

### Surface defects in Vafa-Witten theory and flags of sheaves on the projective plane

**Speaker: **Nadir Fasola (The University of Sheffield)**Date: **25 April 2022**Time: **2.00pm

**Abstract: **Nested Hilbert schemes of points and curves on smooth projective surfaces encode interesting information about enumerative problems and physical theories. Their virtual fundamental classes have been shown to recover both the virtual classes of Seiberg-Witten and reduced stable pair theories, while their obstruction theories can be used to obtain information about Vafa-Witten and reduced Donaldson-Thomas invariants. Motivated by a D-brane construction arising in supersymmetric String Theory, I’ll study the representation theory of an enhancement of the ADHM quiver. This, in a particular case, models nested Hilbert schemes of points on the affine plane or, more generally, flags of framed torsion-free sheaves on the projective plane. The partition function of the theory determined by such a quiver naturally computes generating functions of virtual invariants of moduli spaces of stable representations, which, in turn, are conjectured to carry some information about the cohomology of character varieties via a work of Hausel, Letellier and Rodriguez-Villegas.

### Counting surfaces on Calabi-Yau fourfolds

**Speaker: **Martijn Kool (Utrecht University)**Date: **9 May 2022**Time: **2.00pm

**Abstract: **I will introduce invariants for counting surfaces on Calabi-Yau fourfolds. In a family, they are deformation invariant along Hodge loci. If non-zero, the variational Hodge conjecture for the family under consideration holds. Time permitting, I will discuss DT/PT wall-crossing and relations to Nekrasov's Magnificent Four. Joint work with Y. Bae and H. Park.

### ADE singularities, Quot schemes and generating functions

**Speaker: **Balázs Szendrői (University of Oxford)**Date: **9 May 2022**Time: **2.00pm

**Abstract: **Starting with an ADE singularity C^2/Gamma for Gamma a finite subgroup of SL(2,C), one can build various moduli spaces of geometric and representation-theoretic interest as Nakajima quiver varieties. These spaces depend in particular on a stability parameter; quiver varieties at both generic and non-generic stability are of geometric interest. We will explain some of these connections, focusing in particular on generating functions of Euler characteristics at different points in stability space. Based on joint papers and projects with Craw, Gammelgaard, Gyenge, and Nemethi.

### BPS invariant from non Archimedean integrals

**Speaker: **Francesca Carocci (EPFL)**Date: **23 May 2022**Time: **2.00pm

**Abstract: **We consider moduli spaces M(ß,χ) of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes. Working over a non-archimedean local field F, we define a natural measure on the F-points of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on M(ß,χ) with respect to this measure is independent of the Euler characteristic. Analogous statements hold for (meromorphic or not) Higgs bundles. Recent results of Maulik-Shen and Kinjo-Coseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles. This is a joint work with Giulio Orecchia and Dimitri Wyss.

### Resurgence, partition functions and BPS states for N=2 theories

**Speaker: **Lotte Hollands (Heriot-Watt University)**Date: **23 May 2022**Time: **2.00pm

**Abstract: **Recently, there have been various exciting developments in the interplay between BPS structures, topological string partition functions and exact WKB analysis. In this talk I will report on this from the perspective of four-dimensional N=2 field theory and its lift to five dimensions. I will try to explain how the non-perturbative open and closed partition functions for these theories may be obtained from the exact WKB analysis applied to the associated differential/ difference equations, and how these partition functions encode their BPS states. This talk is based on 2109.14699, 2203.08249 and work in progress.