Spin structures in enumerative geometry
Workshop in Leiden, 16-18 June 2025

Overview. The goal of the workshop is to gather different points of view on spin structures, especially from a modular and enumerative geometry perspective. In particular, this includes the following topics:

• FJRW theory;
• Topological recursion of Ω-classes;
• Refinement of invariants according to sign/parity;
• Choice of spin structures in Oh-Thomas formalism (e.g. DT4 invariants);
• Invariants of surfaces of positive genus;
• BKP hierarchy;
• Birationnal geometry of connected components of moduli spaces defined by parity

• 13h30-14h30: Alessandro Giacchetto
• 14h30-15h30: Sergej Monavari
  (Break)
• 16h00-17h45: Short talks session
  - Catherine Cannizzo
  - Georgios Politopoulos
  - Amira Tlemsani
  - David Klompenhouwer
  (Pizzas and drinks)

• 9h00-10h30: Melissa Liu
  (Break)
• 11h00-12h30: Ran Tessler
  (Lunch)
• 14h00-15h00: Andrei Bud
  (Break)
• 15h30-16h30: Paolo Rossi
• 16h30-17h30: Markus Upmeier
  (Social dinner)

• 9h00-10h00: Alessandro Chiodo
• 10h00-11h00: Elba Garcia-Failde
  (Break)
• 11h30-12h30: Riccardo Ontani

• Andrei Bud (Goethe-Universität, Frankfurt). The Kodaira dimension of strata of differentials
  Abstract: The Kodaira dimension is a fundamental invariant that measures the birational complexity of algebraic varieties. In this talk, we will investigate the Kodaira dimension of strata of Abelian differentials, highlighting differences between even and odd spin parities, as well as between low and high genera. This is based on joint work with Dawei Chen and Martin Möller.
  
  
• Catherine Cannizzo (University of California, Berkeley). Mirror Symmetry for Theta Divisors.
  Abstract: Enumerative invariants of symplectic manifolds are crucial in the string theory-inspired duality of mirror symmetry. Building on Floer cohomology, Fukaya et al. constructed an algebraic invariant of symplectic manifolds (the Fukaya category), whose structure maps are built from counting curves. Kontsevich conjectured this invariant is homologically mirror to D^b Coh of the B-model, meaning the two invariants are equivalent. In this talk, we will state our result on core homological mirror symmetry for theta divisors, emphasizing the role of Gromov-Witten invariants in the proof. This is joint forthcoming work with Haniya Azam, Heather Lee, and Chiu-Chu Melissa Liu.
  
  
• Alessandro Chiodo (Sorbonne Université, Paris). The double ramification cycle via Grothendieck-Riemann-Roch
  Abstract: Given a family of smooth curves C -> S with a line bundle L on C, it is natural to study the locus of points x in S where L_x is trivial on C_x. When the family is stable, the definition can be extended, not directly on the base scheme S, but more naturally on a modification S' of S (a logarithmic blow-up). The problem is in many ways analogous to the problem of defining a Néron model on the moduli space of stable curves (instead of a DVR). In 2014, David Holmes put forward a new approach, constructing a Néron model over a birational modification of the entire moduli space of curves. Similar ideas were applied by Holmes, Marcus and Wise to extending the double ramification cycle. When applied to moduli of rth roots, via Grothendieck Riemann–Roch, this point of view yields the Hodge-DR conjecture (work in collaboration with David Holmes).
  
  
• Elba Garcia-Failde (Sorbonne Université, Paris). Volumes of moduli spaces of bordered Klein surfaces
  Abstract: In 2006, Mirzakhani produced a recursion to effectively compute the Weil–Petersson volumes of moduli spaces of genus g hyperbolic surfaces with n marked geodesic boundaries. I will present a generalisation of her recursion that allows to compute volumes of moduli spaces of bordered Klein surfaces. Combining Mirakhani’s recursion with ours, produces a recursion for total volumes of orientable and non-orientable hyperbolic surfaces. When the surface is non-orientable, the volume is considered with respect to a top-form introduced by Norbury in 2008. However, these volumes diverge when the lengths of 1-sided geodesics approach 0; in 2017, Gendulphe proposed to consider regularised volumes whose systole of 1-sided geodesics is greater than epsilon, for epsilon small enough. Making use of a generalisation of McShane identity due to Norbury, we are able to obtain a simple, exact expression for the volume of the moduli space of Klein bottles and a recursion for generic topologies that fully capture the dependence on the geometric regularisation parameter epsilon. I will finish by briefly presenting the relation to a refinement of the universal procedure of topological recursion.
  This talk will be based on work in progress with P. Gregori and K. Osuga.
  
  
• Alessandro Giacchetto (ETH, Zürich). A new spin on Gromov–Witten and Hurwitz
  Abstract: Spin Gromov–Witten invariants were introduced by Kiem and Li to determine the ordinary Gromov–Witten invariants of surfaces with smooth canonical divisors. Conjecturally, these invariants can be expressed as linear combinations of spin Hurwitz numbers, which are themselves computable via representation theory—a relationship known as the spin GW/Hurwitz correspondence. In this talk, I will present a proof of the correspondence in the case of target P^1, and explain how the general case follows from a conjectural degeneration formula for spin GW invariants. Time permitting, I will also discuss new directions related to the Virasoro conjecture for such targets, as well as connections to the GW/PT correspondence. Based on joint works with R. Kramer, D. Lewański, and A. Sauvaget.
  
  
• David Klompenhouwer (Università di Padova, Padova). Meromorphic differentials, spin structures, and the BKP hierarchy
  Abstract: A meromorphic differential on a curve with even orders at zeros and poles induces a spin structure on the curve. This gives a spin refinement of cohomology classes of strata of meromorphic differentials with vanishing residues, which form a partial Cohomological Field Theory (CohFT). We apply the DR hierarchy construction of Buryak to this partial CohFT and show that the resulting system of evolutionary PDEs coincides with the BKP hierarchy, after a reduction. This result relies on some intersection-theoretic computations that are made possible by recent work by Costantini-Sauvaget-Schmitt and the recently proved DR/DZ correspondence. This is based on joint work with Stijn Velstra.
  
  
• Melissa Liu (Columbia University, New-York). Extended Landau-Ginzburg/Calabi-Yau correspondence and open/closed correspondence for the quintic threefold
  Abstract: Chiodo-Ruan proved genus-zero Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for the quintic Calabi-Yau threefold, which relates genus-zero Fan-Jarvis-Ruan-Witten (FJRW) invariants of the Fermat quintic polynomial in five variables and genus-zero Gromov-Witten (GW) invariants of the quintic Calabi-Yau threefold. This correspondence can be viewed as an example of GIT wall-crossing in a gauged linear sigma model (GLSM). I will explain results and conjectures on (1) extending this correspondence to genus-zero open FJRW invariants recently constructed by Tessler-Zhao and genus-zero open GW invariants of the quintic Calabi-Yau threefold, and (2) open/closed correspondence relating the above genus-zero open GW (resp. FJRW) invariants and closed GLSM invariants of a specific GLSM, the extended quintic GLSM, in the positive (resp. negative) phase. This is based on joint work with Konstantin Aleshkin.
  
  
• Sergej Monavari (EPFL, Lausanne). The refined local Donaldson-Thomas theory of curves
  Abstract: The Maulik-Nekrasov-Okounkov-Pandharipande correspondence predicts an equivalence between the partition functions of (numerical) Gromov-Witten and Donaldson-Thomas invariants of smooth projective threefolds. It was recently proposed by Pardon a solution of this conjectural correspondence by reducing to the simpler case of local curves, which are more amenable for computations by means of TQFT methods. Even more recently, inspired by the seminal work of Nekrasov-Okounkov on the index in M-theory, Brini-Schuler proposed a refined GW/DT correspondence. In this talk, I will present a full solution for the Donaldson-Thomas side of the refined GW/DT correspondence in the case of local curves. In particular, I will explain how to derive the refined DT partition function without relying on degeneration techniques and TQFT methods, and how our formulas recover string-theoretic prediction of Aganagic-Schaeffer. Based on work in progress.
  
  
• Riccardo Ontani (Imperial College, London). Jeffrey-Kirwan localisation in the virtual setting
  Abstract: The localisation formula of Atiyah-Bott-Berline-Vergne expresses integrals over a space X acted on by a group G in terms of integrals over the fixed locus X^G. Jeffrey-Kirwan localisation expresses integrals over a quotient X/G in terms of integrals over (certain components of) the fixed locus X^G. The resulting Jeffrey-Kirwan residue formula has become a widely used computational tool in physics. I will discuss a virtual Jeffrey-Kirwan localisation formula for the virtual classes of both Behrend-Fantechi and Oh-Thomas. This is based on an ongoing project with Richard Thomas.
  
  
• Georgios Politopoulos (Universiteit Leiden, Leiden). Title TBA
  Abstract: The strata of k-differentials are spaces parametrizing stable curves admitting a k-log differential with prescribed zeros and poles. When k and all the zeros and poles are odd, each point of such a stratum naturally carries a spin structure; consequently the stratum decomposes into two components according to the parity of the associated spin structures. A natural question to ask is whether the Chow classes of these components are tautological and computable. In this mini-talk, we will investigate how the theory of spin double ramification cycles reduces the computation to the case k=1 and we will sketch the computation for this remaining case, obtaining an affirmative answer to the aforementioned question.
  This is joint work with Adrien Sauvaget and David Holmes.
  
  
• Paolo Rossi (Università di Padova, Padova). Moduli spaces of curves and the classification of integrable systems.
  Abstract: : I will present several results and conjectures on the classification of different classes of integrable systems of evolutionary PDEs, up to the appropriate transformation groups. These include Hamiltonian systems, tau symmetric systems and systems of conservation laws. I will then explain in what sense we expect that integrable systems arising from intersection theory on the moduli space of stable curves are universal objects with respect to these classifications. In the rank one case I will present strong evidence in support of these claims. This is a joint work with A. Buryak.
  
  
• Ran Tessler (Weizmann Institute, Tel-Aviv). Open FJRW theory-geometry, integrable hierarchies and mirror symmetry
  Abstract: We will start by reviewing FJRW theory, and then define its open-string analogue. We will show that the resulting intersection theories satisfy interesting mirror symmetry statements and are closely related to integrable hierarchies.
  Based on joint works with A. Buryak, E. Clader, M. Gross, T. Kelly and Y. Zhao.
  
  
• Amira Tlemsani (Universiteit Leiden, Leiden). DR cycles in higher relative dimensions.
  Abstract: The double ramification cycle of a line bundle on a smooth curve over a scheme S is a cycle in S roughly corresponding to the set of points where the restriction of the line bundle to the fiber is trivial. The problem of computing these cycles (in the more general setting of stable curves) was first posed by Eliashberg in 2001, and has gained a lot of popularity since then. In this talk we will define and compute DR cycles for smooth projective schemes of any relative dimension, by taking inspiration from a suitable proof of the DR cycle formula for smooth curves.
  
  
• Markus Upmeier (University of Aberdeen). Stable homotopy methods for studying spin structures on moduli spaces
  Abstract: I will introduce a new framework for analyzing and constructing spin structures on moduli spaces using tools from stable homotopy theory. Unlike orientations, which could be treated using ordinary category theory (specifically, Picard groupoids, which model stable homotopy 1-types), the case of spin structures requires working with stable homotopy 2-types.
  After reviewing the algebraic topology background that underlies this approach, I will discuss my ongoing research project and some preliminary results.