Workshop

I will discuss a class of Riemann-Hilbert problems which arise naturally from the wall-crossing formula in Donaldson-Thomas theory. As well as giving an explicit description of the simplest example, I will try to explain the motivating analogy with the deformed flat connection in the theory of Frobenius manifolds.

I'll describe a conjectural way to understand the COHA of certain toric Calabi-Yau three-folds in terms of a 5-dimensional gauge theory given by compactifying M-theory on the Calabi-Yau. An analysis of the 5-dimensional gauge theory leads to a conjectural expression for the COHA as the positive part of the double current algebra built from a supergroup. This expression reproduces known formulae for the refined DT invariants of the toric Calabi-Yau

The theory of quasi-maps, developed in recent work of Ciocan-Fontanine and Kim, is a generalization of Gromov-Witten theory that depends on an additional stability parameter varying over positive rational numbers. When that parameter tends to infinity, Gromov-Witten theory is recovered, while when it tends to zero, the resulting theory encodes information related to the "B-model." Ciocan-Fontanine and Kim proved a wall-crossing formula exhibiting how the theory changes with the stability parameter, and in this talk, we discuss an alternative proof of their result as well as a generalization to other gauged linear sigma models. This is joint work with Felix Janda and Yongbin Ruan.

: Let X be a Calabi-Yau 3-dimensional orbifold, and let Y be a crepant resolution of the coarse moduli space of X. When X satisfies the "hard Lefschetz condition" (that is, when the fibres of the resolution are at most 1-dimensional), the DT crepant resolution conjecture of Bryan-Cadman-Young gives a precise relation between the DT curve counts of X and Y. I will explain a proof of this conjecture via wall crossing and the motivic Hall algebra. This is joint work with Sjoerd Beentjes and John Calabrese.

I will explain how, using microlocal sheaf theory, one can associate categories to exact symplectic manifolds, objects to exact lagrangians, and functors to exact lagrangian correspondences.  Time permitting I will also discuss comparison with the Fukaya category, and applications to homological mirror symmetry. 

Let K be a field, and M be the “projective linear" moduli stack of objects in a suitable K-linear abelian category A (such as the coherent sheaves coh(X) on a smooth projective K-scheme X) or triangulated category T (such as the derived category D^bcoh(X)). I will explain how to define a Lie bracket [ , ] on the homology H_*(M) (with a nonstandard grading), making H_*(M) into a graded Lie algebra. This is a new variation on the idea of Ringel-Hall algebra. There is also a differential-geometric version of this: if X is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, G2 manifold, Spin(7) manifold) then we can define Lie brackets both on the homology of the moduli spaces of all U(n) or SU(n) connections on X for all n, and on the homology of the moduli spaces of instanton U(n) or SU(n) connections on X for all n. All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wall-crossing formulae under change of stability condition. Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with b^2_+=1), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for CY 3-folds, Fano 3-folds, and CY 4-folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear" moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs. I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra (H_*(M), [ , ]).

The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti (BKMP) relates all genus open and closed Gromov-Witten invariants of a symplectic toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. It is a version of all genus open-closed mirror symmetry. The goal of this talk is to describe implications of the Remodeling Conjecture on structures of higher genus open-closed Gromov-Witten invariants.

The example of K3xE suggests that computing the partition function of a Calabi-Yau threefold requires a good understanding of both its Gromov-Witten and Donaldson-Thomas theory. I will explain two results, one for each theory, related to this discussion.

By the work of H.-L. Chang and J. Li, the (arbitrary genus) Gromov-Witten invariants of a quintic threefold can be computed using the relatively simple moduli space of stable maps to P^4 with a p-field.

This is an example of a GLSM moduli space. Another example is the moduli of r-spin curves with a field, which by the work of Chang-Li-Li computes Witten's r-spin class. One subtlety in using these moduli spaces for computations lies in the cosection localized virtual class that is necessary to produce invariants from these non-compact moduli spaces.

In my talk, I will discuss joint work in progress with Q. Chen, Y.

Ruan on how to derive the same invariants from an ordinary virtual class on a compactified moduli space constructed using logarithmic geometry. This virtual class is amenable to torus localization methods.

In the case of a quintic threefold, in joint work in progress with S.

Guo and Y. Ruan, we apply this method to prove the holomorphic anomaly equations. There are further applications in joint work with A.

Sauvaget and D. Zvonkine, and with X. Wang.

Recently there has been progress in evaluating the series of Segre integrals associated with tautological vector bundles on Hilbert schemes of points on surfaces. The talk will explain some of the new ideas in the subject, including the connection between Segre theory and Verlinde theory in the Hilbert scheme context. It will be mostly based on joint work with Dragos Oprea and Rahul Pandharipande.

This will be a report on a joint work in progress with Mina Aganagic.

Its goal is to prove the equality of curve counts in dual geometries, whenever both counts can be defined using present-day technology.