Workshop on moduli spaces, virtual invariants and shifted symplectic structures


June 20-30, 2023                             Soorim Cultural Foundation 204 /  KIAS 8101


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Monday (June 26)

Marco Robalo (Sorbonne Universite)


Title: Derived Foliations and Local Models


Abstract: In this talk we will recall the formalism of derived foliations introduced by Toen-Vezzosi. We will then explain the construction of a moduli stack parametrizing local data of Lagrangian foliations with Liouville structures. We call it the Darboux stack. In the case of (-1)-shifted derived scheme, the Darboux stack parametrizes local presentations as a derived critical loci. In the case of (-2)-shifted symplectic forms, tit describes the local models of Borisov-Joyce for moduli spaces of coherent sheaves Calabi-Yau 4folds. This talk provides the preliminary materials for B. Hennion’s talk.



Benjamin Hennion (University of Paris-Saclay)


Title: Glueing matrix factorizations


Abstract: We will explain how categorified Donaldson--Thomas invariants of Calabi--Yau 3-folds can be obtained by glueing singularity invariants from local models of a suitable moduli space endowed with a (-1)-shifted symplectic structure.

The process goes through a careful study of the moduli of such local models.

We will show how to recover Brav--Bussi--Dupont--Joyce--Szendroi's perverse sheaf categorifying the DT-invariants, as well as how to glue more evolved singularity invariants, such as matrix factorizations (thus answering a conjecture of Kontsevich and Soibelman).

This is joint work with M. Robalo and J. Holstein.



Younghan Bae (ETH Zurich)


Title: Counting surfaces on Calabi-Yau 4-folds


Abstract: Let X be a smooth projective Calabi-Yau 4-fold. The Hilbert scheme of 2-dimensional subschemes on X has a Boris-Joyce/Oh-Thomas virtual cycle. When the Hodge locus of the given surface class has positive codimension, the virtual cycle vanishes. In this talk I will explain a way to fix this issue using (-1)-shifted 1-forms on the Hilbert scheme. The resulting reduced virtual cycle has both geometric meaning and a Hodge theoretic consequence. This is a joint work with Martijn Kool and Hyeonjun Park.



Hyeonjun Park (Korea Institute for Advanced Study)


Title: A Darboux theorem and virtual Lagrangian cycles for (-2)-shifted symplectic fibrations


Abstract: There are two crucial discoveries for (-2)-shifted symplectic derived schemes: (1) Darboux theorem of Brav-Bussi-Joyce/Bouaziz-Grojnowski (2) virtual cycles of Borisov-Joyce/Oh-Thomas. In this talk, we extend these to families of (-2)-shifted symplectic derived schemes.

Firstly, we show that (-2)-shifted symplectic fibrations are (-1)-shifted Lagrangians on the derived critical loci of functions on the bases when the obstructions of exactness come from the functions. Based on this, we provide a Darboux theorem for (-2)-shifted symplectic fibrations that are not necessarily exact.

Secondly, we construct virtual Lagrangian cycles for (-2)-shifted exact symplectic fibrations. A key technical lemma is a generalized cone reduction for (-p)-shifted exact p-forms. We also present a virtual pullback formula for Lagrangian correspondences.

As an application, we prove that the Donaldson-Thomas invariants of Calabi-Yau 4-folds are invariant along the deformations for which the (0,4)-Hodge pieces of the second Chern characters remain zero. This assures that the reduced virtual cycles for counting surfaces detect the variational Hodge conjecture.

Finally, we propose a refined Donaldson-Thomas theory of Calabi-Yau 4-folds in terms of the vanishing cycle cohomology of the Hodge loci, relying on the Joyce conjecture for (-1)-shifted Lagrangians.






Tuesday (June 27)

Martijn Kool (Utrecht University)


Title: K-theoretic DT/PT correspondence for surfaces on Calabi-Yau fourfolds


Abstract: We introduce K-theoretic virtual invariants for certain moduli spaces of "surfaces on Calabi-Yau fourfolds". The most obvious moduli space to consider is the Hilbert scheme of 2-dimensional subschemes. However, unlike the case of "curves on Calabi-Yau threefolds", there are now _two_ types of stable pairs moduli spaces. We discuss how these three moduli spaces are related by varying GIT stability and how they parametrize objects in the derived category. For toric Calabi-Yau fourfolds, and one of the two stable pairs theories, we introduce a conjectural vertex DT/PT correspondence which we can verify in examples. Joint work with Y. Bae and H. Park.



Yalong Cao (RIKEN iTHEMS)


Title: From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials


Abstract: I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau 4-folds. For non-compact CY4 like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.



Woonam Lim (ETH Zurich)


Title: Virasoro constraints, vertex operator algebras, and wall-crossing.


Abstract: In enumerative geometry, Virasoro constraints were first conjectured for the moduli of stable curves (the Witten conjecture) and stable maps. Recently, the analogous constraints were conjectured in several sheaf theoretic contexts; stable pairs on 3-folds and torsion-free sheaves on surfaces. In joint work with A. Bojko and M. Moreira, we generalize and reinterpret Virasoro conjecture in sheaf theory using Joyce’s vertex algebra. A new interpretation makes use of a conformal element and primary states of vertex algebras which are classical subjects in representation theory. As an application, we prove the constraints for any moduli of torsion-free sheaves on curves and surfaces via Joyce's wall-crossing formulas.



Arkadij Bojko (ETH Zurich)


Title: Wall-crossing for Calabi-Yau fourfolds and applications


Abstract: Joyce's vertex algebras are a powerful new ingredient added to the existing theory of wall-crossing for sheaves on surfaces. My work focuses on proving wall-crossing in two dimensions higher - for Calabi-Yau fourfolds. It is desirable that the end result can have many concrete applications to existing conjectures. For this purpose, I introduce yet another new structure into the picture - formal families of vertex algebras. Apart from being a natural extension of the theory, they allow to wall-cross with insertions instead of the plain virtual fundamental classes. To make the whole machinery work with (polynomial) Bridgeland stability conditions and sheaf-counting classes for fourfolds, I require a different approach compared to the surface case. In the talk, I will discuss the main difficulties that I encountered, and I will present examples using the complete package.



Dominic Joyce (University of Oxford)- online


Title: The structure of invariants counting coherent sheaves on complex surfaces.


Abstract: Let X be a complex projective surface with geometric genus p_g > 0. We can form moduli spaces  of Gieseker (semi)stable coherent sheaves on X with Chern character (r,a,k), where we take the rank r to be positive. In the case in which stable = semistable, there is a (reduced) perfect obstruction theory on , giving a virtual class  in homology.

By integrating universal cohomology classes over this virtual class, one can define enumerative invariants counting semistable coherent sheaves on X. These have been studied by many authors, and include Donaldson invariants, K-theoretic Donaldson invariants, Segre and Verlinde invariants, part of Vafa-Witten invariants, and so on.
In my paper, in a more general context, I extended the definition of the virtual class
to allow strictly semistables, proved wall-crossing formulae for these classes and associated “pair invariants”, and gave an algorithm to compute the invariants  by induction on the rank r, starting from data in rank 1, which is the Seiberg-Witten invariants of X and fundamental classes of Hilbert schemes of points on X. This is an algebro-geometric version of the construction of Donaldson invariants from Seiberg-Witten invariants; it builds on work of Mochizuki 2008.

This talk will report on a project to implement this algorithm, and actually compute the invariants  for all ranks r > 0. I prove that the  for fixed r and all a,k with a fixed mod r can be encoded in a generating function involving the Seiberg-Witten invariants and universal functions in infinitely many variables. I will spend most of the talk explaining the structure of this generating function, and what we can say about the universal functions, the Galois theory and algebraic numbers involved, and so on. This proves several conjectures in the literature by Lothar Göttsche, Martijn Kool, and others, and tells us, for example, the structure of U(r) and SU(r) Donaldson invariants of surfaces with ((b^2_+ > 1)) for any rank r ≥ 2.




Wednesday (June 28)

Adeel Khan (Academia Sinica)


Title: Derived specialization and microlocalization


Abstract: I will discuss a derived analogue of specialization and microlocalization of sheaves, which give rise to certain sheaf-theoretic invariants of a derived stack. In the quasi-smooth case, we will see that they give a microlocal perspective on the virtual fundamental class. We thus propose derived microlocalization on non-quasi-smooth derived moduli stacks as a potential mechanism for extracting interesting new numerical invariants.



Tasuki Kinjo (Kyoto University)


Title: Microlocal methods in enumerative geometry


Abstract: In this talk, I will present a new method for constructing virtual fundamental classes of quasi-smooth derived schemes, using the perverse sheaves of vanishing cycles. This approach is motivated by the interplay between derived geometry and microlocal geometry. Additionally, we will demonstrate how this idea leads to the construction of the critical cohomological Hall algebras for the canonical bundle of algebraic surfaces, refining the cohomological Hall algebras of Kapranov-Vasserot. This talk is partially based on joint work with Adeel Khan.




Thursday (June 29)

Jeroen Hekking (University of Regensburg)


Title: Blow-ups and normal bundles in nonconnective derived geometry


Abstract: A current development in derived algebraic geometry is the theory of derived rings (developed by Antiau, Bhatt, Mathew, Raksit, et al), which are nonconnective algebras that recover animated rings after restricting to the connective part. It turns out that this is a remarkably convenient framework for constructing derived Rees algebras.

In this talk, we will review the theory of derived rings, and explain how it relates to animated rings and commutative ring spectra. We will then look at a way to geometrize the theory to nonconnective derived stacks. This will be the framework in which we define derived blow-ups in terms of derived Rees algebras, and where we exhibit a derived deformation to the normal bundle.

The theory of derived rings is actually an axiomatic system that covers more than nonconnective derived algebraic geometry. A salient example is derived analytic geometry (in terms of Banach algebras), which we can review if time permits. A main application of derived blow-ups is a derived reduction of stabilizers algorithm, leading to sheaf-theoretic counting invariants.

This is based on joint work with Khan–Rydh, with Ben-Bassat, and with Rydh–Savvas.



Michail Savvas (University of Texas at Austin)


Title: Stabilizer reduction and sheaf-theoretic invariants


Abstract: I will discuss a canonical blowup procedure for Artin stacks in derived algebraic geometry, which eliminates positive-dimensional stabilizer groups and generalizes the partial desingularization algorithm developed by Kirwan for smooth varieties in the context of Geometric Invariant Theory. This procedure applies in particular to moduli stacks of sheaves, allowing us to construct intersection-theoretic generalized sheaf-counting invariants on surfaces and Calabi-Yau threefolds. Based on joint work with Jeroen Hekking and David Rydh.



Jeongseok Oh (Imperial College)


Title: Complex Kuranishi Structures and counting sheaves on Calabi-Yau 4-folds


Abstract: We develop a theory of complex Kuranishi structures on projective schemes. These are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts.

We apply the theory to projective moduli spaces M of stable sheaves on Calabi-Yau 4-folds. Using real derived differential geometry, Borisov- Joyce produced a virtual homology cycle on M. In the prequel work we constructed an algebraic virtual cycle on M. We prove the cycles coincide in homology after inverting 2 in the coefficients.

In particular, when Borisov-Joyce’s real virtual dimension is odd, their virtual cycle is torsion.

This is a joint work with Richard Thomas.



Yunfeng Jiang (University of Kansas) - online


Title: The virtual fundamental class for the moduli space of general type surfaces


Abstract: Sir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class on the moduli stack of lci (locally complete intersection) covers over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the virtual fundamental class. This can be taken as a generalization of the tautological invariants on the moduli space of stable curves to the moduli space of stable surfaces. If time permits, we also talk about the possible methods to construct a virtual fundamental class on the moduli space of stable maps from semi-log-canonical surfaces to projective varieties, and especially Calabi-Yau 4-folds.




Friday (June 30)

Tony Pantev (University of Pennsylvania) 


Title: Derived moduli of D-branes and superpotentials


Abstract: Moduli of D-branes on to Calabi-Yau manifolds are naturally equipped with enhanced geometric structures which play important role in classical field theory and are an essential input for the quantization problem. I will explain how one can recognize when such enhanced structures arise from a local or global superpotential and how Gromov-Witten invariants introduce derived and non-commutative corrections to the geometry of moduli spaces of branes. I will discuss applications to higher dimensional Chern-Simons functionals and to non-abelian Hodge theory. The talk is based on joint works with Calaque, Katzarkov, Kontsevich, Toen, Vaquie, and Vezzosi.



Christopher Brav (Moscow Institute of Physics and Technology)


Title: The cyclic Deligne conjecture and Calabi-Yau structures


Abstract:The Deligne conjecture, many times a theorem, states that for a dg category C, the dg endomorphisms End(Id_C) of the identity functor-- that is, the Hochschild cochains-- carries a natural structure of 2-algebra. When C is endowed with a Calabi-Yau structure, then Hochschild cochains and Hochschild chains are identified up to a shift, and we may transport the circle action from Hochschild chains onto Hochschild cochains. The cyclic Deligne conjecture states that the 2-algebra structure and the circle action together give a framed 2-algebra structure on Hochschild cochains. We establish the cyclic Deligne conjecture, as well as a variation that works for relative Calabi-Yau structures on dg functors D --> C, more generally for functors between stable infinity categories. We discuss examples coming from oriented manifolds with boundary, Fano varieties with anticanonical divisor, and doubled quivers with preprojective relation. This is joint work with Nick Rozenblyum.



Pavel Safronov (University of Edinburgh)


Title: Critical cohomology and deformation quantization


Abstract: For any stack equipped with a d-critical structure the DT sheaf is a mixed Hodge module which allows one to define the critical cohomology of the stack. In this talk I will explain a relationship between critical cohomology and deformation quantization, namely, the theory of DQ modules and BV quantization. As an example of these ideas, I will explain a relationship between the critical cohomology of the character stack of a closed oriented 3-manifold, complexified Floer homology of Abouzaid--Manolescu and skein modules of Przytycki--Turaev. This is a report on work in progress, joint with Sam Gunningham.