Schedule for: 19w5171 - Moduli and Invariants

In some geometric situations moduli spaces of sheaves can be realized as a degenerate intersection of two (shifted) Lagrangian subspaces in a (shifted) symplectic space. Computations with obstruction theory is one way of dealing with degenerate intersection. A potential alternative is to deform the geometric picture in a non-commutative direction. We present some examples on deformation quantization of sheaves and morphisms between them.

We study non-commutative projective varieties in the sense of Artin-Zhang, which are given by non-commutative homogeneous coordinate rings, which are finite over their centre. We construct moduli spaces of stable modules for these, and construct a symmetric obstruction theory in the CY3-case. This gives deformation invariants of Donaldson-Thomas type. The simplest example is the Fermat quintic in quantum projective space, where the coordinates commute only up to carefully chosen 5th roots of unity. We explore the moduli theory of finite length modules, which mixes features both of the Hilbert scheme of commutative 3-folds, and the representation theory of quivers with potential. This is mostly a report on the work of Yu-Hsiang Liu, with contributions by myself and Atsushi Kanazawa.

Given a virtually smooth quasi-projective scheme $M$, and a morphism from $M$ to a nonsingular quasi-projective variety $B$, we show it is possible to find an affine bundle $M'/ M$ that admits a perfect obstruction theory relative to $B$. We study the resulting virtual cycles on the fibers of $M'/B$ and relate them to the image of the virtual cycle $[M]^{vir}$ under refined Gysin homomorphisms. Our main application is when $M$ is a moduli space of stable codimension 1 sheaves on a nonsingular projective surface or Fano threefold.

In 2014 Gaiotto conjectured a Lagrangian correspondence between holomorphic Lagrangian of opers in the Dolbeault moduli space of Higgs bundles ​and the de Rham moduli space of holomorphic connections. ​The conjecture was solved in 2016 for holomorphic opers in paper with Fredrickson, Kydonakis, Mazzeo, Mulase and Neitzke. ​ By a similar analysis method, Collier and Wentworth, extended the correspondence for more general Lagrangians consisting of stable points. ​ In my talk, I will present an algebraic geometry description of the Lagrangian correspondence of Gaiotto, based on the work of Simpson.​

The moduli stacks of sheaves on Calabi-Yau four-folds carry -2-shifted symplectic structures (Pantev, Toen, Vezzosi, Vaquie). Viewing these stacks as objects in differential geometry, one can construct Lagrangian foliations relative to these symplectic structures, such that quotients by the foliations are perfectly obstructed derived stacks, equipped with globally defined -1-shifted potentials, whose critical loci are the original moduli stacks. This is a joint work with A.Sheshmani and S-T.Yau.

The derived symmetries associated to a 3-fold admitting an Atiyah flop may be organised into an action of the fundamental group of a sphere with three punctures, thought of as a stringy Kaehler moduli space. I extend this to general flops of irreducible curves on 3-folds in joint work with M. Wemyss. This uses certain deformation algebras associated to the curve and its multiples, with applications to Gopakumar-Vafa invariants.

The Geometric Langlands Conjecture (GLC) for a curve $C$ and a group $G$ is a non-abelian generalization of the relation between a curve and its Jacobian. It claims the existence of Hecke eigensheaves on the moduli of $G$-bundles on $C$. The parabolic GLC is a further extension to curves with punctures. After explaining and illustrating the conjectures, I will outline an approach to proving them using non-abelian Hodge theory. A key geometric ingredient is the locus of wobbly bundles: bundles that are stable but not very stable. If time allows, I will discuss two instances where this program has been implemented recently: GLC for $G=GL(2)$ and genus 2 curves (with T. Pantev and C. Simson), and parabolic GLC for $\mathbb{P}^1$ with marked points (with T. Pantev).

In this talk we reframe a collection of well-known comparison results in genus-zero Gromov-Witten theory in order to relate these to integral transforms between derived categories. This implies that various comparisons between Gromov-Witten and FJRW theory are compatible with the integral structure introduced by Iritani. We conclude with a proof that a version of the LG/CY correspondence relating quantum D-modules with Orlov's equivalence is implied by a version of the crepant transformation conjecture.

For the moduli of derived category objects or the partial desingularizations of the moduli stack of semistable sheaves on Calabi-Yau 3-folds, there are no perfect obstruction theories but only semi-perfect obstruction theories. While a semi-perfect obstruction theory is sufficient for the construction of virtual cycles in Chow groups, it seems insufficient for virtual structure sheaves. In this talk, I will introduce the notion of an almost perfect obstruction theory, which lies in between a semi-perfect obstruction theory and an honest perfect obstruction theory. I will show that an almost perfect obstruction theory enables us to construct the virtual structure sheaf and hence K-theoretic virtual invariants. Examples of DM stacks with almost perfect obstruction theories include the Inaba-Lieblich moduli spaces of simple gluable perfect complexes and the partial desingularizations of moduli stacks of semistable sheaves on Calabi-Yau 3-folds. We thus obtain K-theoretic Donaldson-Thomas invariants of derived category objects and K-theoretic generalized Donaldson-Thomas invariants. Based on a joint work with Michail Savvas.

Logarithmic Gromov-Witten theory virtually counts the number of holomorphic curves with prescribed tangency condition along boundary divisors. In this talk I will introduce a variant of logarithmic maps called the punctured logarithmic maps. They naturally appear in a generalization of the gluing formulas of Li-Ruan and Jun Li. The punctured invariants play the role of relative invariants in these classical gluing formulas. They extend logarithmic Gromov-Witten theory by allowing negative tangency conditions with boundary divisors. This talk is based on a joint work with Dan Abramovich, Mark Gross and Bernd Siebert.

The Hilbert scheme parameterizing $n$ points on a K3 surface $X$ is a holomorphic symplectic manifold with rich properties. In the 90s it was discovered that the generating function for the Euler characteristics of the Hilbert schemes is related to both modular forms and the enumerative geometry of rational curves on $X$. We show how this beautiful story generalizes to K3 surfaces with a symplectic action of a group $G$. Namely, the Euler characteristics of the "$G$-fixed Hilbert schemes” parametrizing $G$-invariant collections of points on $X$ are related to modular forms of level $|G|$ and the enumerative geometry of rational curves on the stack quotient $[X/G]$ . These ideas lead to some beautiful new product formulas for theta functions associated to root lattices. The picture also generalizes to refinements of the Euler characteristic such as $\chi_y$ genus and elliptic genus leading to connections with Jacobi forms and Siegel modular forms.

Virtual correspondences are constructed between smooth moduli spaces of certain stable 1-dimensional sheaves on surfaces, leading to a representation of a Lie algebra on the direct sum of the cohomologies of these moduli spaces. This representation commutes with the $SL_2 \times SL_2$ representation described by Gopakumar and Vafa using $M$-theory, constructed mathematically via perverse sheaves and hard Lefschetz. In the case of a rational elliptic surface with a type II^* fiber (the $E$-string of physics), a representation of the affine $E_8$ Lie algebra is obtained. Since the cohomologies determine BPS numbers of the associated local surface, the generating function of these BPS numbers or their $SL_2 \times SL_2$ refinements is the character of a representation of affine $E_8$, as predicted by Huang, Klemm, and Poretschkin via physics. This talk is based on joint work with Davesh Maulik.

We study moduli space of holomorphic triples $f: E_{1} \rightarrow E_{2}$, composed of (possibly rank $>1$) torsion-free sheaves $(E_{1}, E_{2})$ and a holomorphic map between them, over a smooth complex projective surface $S$. The triples are equipped with a Schmitt stability condition. We prove that when the Schmitt stability parameter becomes sufficiently large, the moduli space of triples benefits from having a perfect relative and absolute obstruction theory in some cases (depending on Chern character of $E_{1}$). We further generalize our construction to higher-length flags of higher rank sheaves by gluing triple moduli spaces, and extend earlier work, with Gholampur and Yau, where the obstruction theory of nested Hilbert schemes over the surface was studied. Here we extend the earlier results to the moduli space of flags $E_{1}\rightarrow E_{2}\rightarrow \cdots \rightarrow E_{n}$, where the maps are injective (by stability). There is a connection, by wall-crossing in the master space, developed by Mochizuki, between the theory of such higher rank flags, and the theory of Higgs pairs on the surface, which provides the means to relate the flag invariants to the local DT invariants of any threefold given by a line bundle over the surface, $X :={\rm Tot}(L \rightarrow S)$. The latter DT invariants, when L is the canonical bundle of S, contribute to Vafa-Witten invariants. Joint work with Shing-Tung Yau, arXiv:1911.00124.

Gromov-Witten invariants of Calabi-Yau one-folds (including elliptic curves and elliptic orbifold curves) are quasimodular forms. This can be proved using tautological relations and some ordinary differential equations in the theory of quasimodular forms, with minimal calculations. Such a method is also applicable to the Fan-Jarvis-Ruan-Witten theory of simple elliptic singularities. This allow us to prove the LG/CY correspondence for all CY one-folds using Cayley transformation of quasimodular forms, where GW/FJRW invariants are coefficients of Fourier/Taylor expansions of the same quasimodular forms. This talk is based on joint work with Jie Zhou, and Jun Li, Jie Zhou.

Hilbert schemes and stable pair moduli spaces on compact Calabi-Yau 4-folds have a virtual class, which was constructed in special cases by Cao-Leung and in general by Borisov-Joyce. We conjecture a DT/PT correspondence for the resulting invariants. On toric Calabi-Yau fourfolds, we conjecture a K-theoretic enhancement of this correspondence using invariants which were recently discovered in the case of Hilbert schemes of points by Nekrasov. Using dimensional reduction, we recover Nekrasov-Okounkov's K-theoretic DT/PT correspondence for toric threefolds. Joint work with Y. Cao and S. Monavari.

I will talk about a class of algebras arising from a quiver with potential, called Jacobi algebras. To such algebras we can associate a collection of invariants, called BPS invariants, by lifting constructions from 3-dimensional complex geometry. A conjecture of Wemyss and Brown states, approximately, that in fact the finite-dimensional Jacobi algebras are in bijective correspondence with local isomorphism classes of flopping curves in 3-dimensional complex varieties. A consequence of the conjecture would be that for such algebras, all of the refined invariants are positive (in marked contrast with general Jacobi algebras). I will discuss a recent proof of this positivity result, along with a monodromy/purity conjecture for the cohomologically refined Gopakumar-Vafa invariants of flopping curves.

For a large class of GIT quotients X=W//G, Ciocan-Fontanine-Kim-Maulik and many others have developed the theory of epsilon-stable quasimaps. The conjectured wall-crossing formula of cohomological epsilon-stable quasimap invariants for all targets in all genera has been recently proved by Yang Zhou. In this talk, we will introduce permutation-equivariant K-theoretic epsilon-stable quasimap invariants with level structure and prove their wall-crossing formulae for all targets in all genera. In physics literature, these invariants are related to the $3d N = 2$ supersymmetric gauge theories studied by Jockers-Mayr, and the wall-crossing formulae can be interpreted as relations between invariants in the UV and the IR phases of the $3d$ gauge theory. It is based on joint work in progress with Yang Zhou.

A driving question in Gromov-Witten theory is to relate the invariants of a complete intersection to the invariants of the ambient variety. In genus-zero this can often be done with a ``twisted theory,'' but this fails in higher genus. Several years ago, Chang-Li presented the moduli space of p-fields as a piece of the solution to the higher-genus problem, constructing the virtual cycle on the space of maps to the quintic 3-fold as a cosection localized virtual cycle on a larger moduli space (the space of p-fields). Their result is analogous to the classical statement that the Euler class of a vector bundle is the class of the zero locus of a generic section. I will discuss work joint with Qile Chen and Felix Janda where we extend Chang-Li's result to a more general setting, a setting that includes standard Gromov-Witten theory of smooth orbifold targets and quasimap theory of GIT targets. This work is joint with Qile Chen and Felix Janda.

I will present a moduli space of "logarithmic R-maps" (joint work with Q. Chen and Y. Ruan), which together with its virtual cycles are the key ingredient toward an approach to proving many "mirror" conjectures about higher genus Gromov-Witten invariants of quintic threefolds (joint work with S. Guo and Y. Ruan).

The moduli of stable log varieties or stable pairs $(X,D)$ are the higher dimensional analogue of the compactified moduli of stable pointed curves. The existence of a proper moduli space has been established thanks to the last several decades of advancements in the minimal model program. However, the notion of a family of stable pairs remains quite subtle, and in particular a deformation-obstruction theory for these moduli is not known. When the boundary divisor $D$ is empty, Abramovich and Hassett gave an approach to stable varieties that replaces $X$ with an associated orbifold. They show in this setting that the quite subtle notion of family of stable varieties becomes simply a flat family of the associated orbifolds. We extend this approach to the case where there is a nonempty but reduced boundary divisor $D$ with the hopes of producing a deformation-obstruction theory for these moduli spaces. As an application we show that this approach leads to functorial gluing morphisms on the moduli spaces, generalizing the clutching and gluing morphisms that describe the boundary strata of the moduli of curves. This is joint work with G. Inchiostro.

We define a localised Euler class for isotropic sections, and isotropic cones, in $SO(N)$ bundles. We use this to give an algebraic definition of Borisov-Joyce's sheaf counting invariants on Calabi-Yau 4-folds. When a torus acts, we prove a localisation result. This talk is based on the joint work with R. P. Thomas.

We will show by "generic smoothness" that the big quantum cohomology ring of isotropic Grassmannians $IG(2,2n)$ is generically semisimple but the small quantum cohomology ring is not. That non-semisimplicity leads to a decomposition of the small quantum cohomology ring that relates to a certain decomposition of the derived category of $IG(2,2n)$ in a so-called Lefschetz exceptional collection. This is based on joint work with A. Mellit, A. Kuznetsov, N. Perrin and M. Smirnov.