Wei-Ping Li, *Hong Kong University of Science and Technology*

Junwu Tu, *Shanghai Tech University*

On Calabi–Yau threefolds there are two types of integral invariants, quantum K-invariants and Gopakumar–Vafa invariants. In this talk, I will explain a joint project (with You-Cheng Chou) which aims to show that the quantum K-invariants and Gopakumar invariants are equivalent. At genus zero, this is a conjecture by Jockers–Mayr and Garoufalidis–Scheidegger (for the quintic), and a proof of the JMGS conjecture will be presented.

Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We constructed an algebraic virtual cycle. A key step is a localisation of Edidin-Graham’s square root Euler class for $SO(2n,\mathbb{C})$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We also develop a theory of complex Kuranishi structures on projective schemes which 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 the moduli spaces to prove the two virtual 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.

Givental and Lee introduced quantum K-theory, a K-theoretic generalization of Gromov–Witten theory. It studies holomorphic Euler characteristics of coherent sheaves on moduli spaces of stable maps to given target spaces. In this talk, I will introduce the quantum K-theory for orbifold target spaces which generalizes the work of Tonita-Tseng. In genus zero, I will define a quantum K-ring which specializes to the full orbifold K-ring introduced by Jarvis-Kaufmann-Kimura. As an application, I will give a detailed description of the quantum K-ring of weighted projective spaces, which generalizes a result by Goldin-Harada-Holm-Kimura. This talk is based on joint work with Yang Zhou.

In this talk, I will explain quantum spectrum and asymptotic expansions in FJRW theory of invertible singularities of general type. Inspired by Galkin-Golyshev-Iritani’s Gamma conjectures for quantum cohomology of Fano manifolds, we propose Gamma conjectures for FJRW theory of general type. Here the Gamma structures are essential to understand the connection between algebraic structures of the singularities (such as Orlov’s semiorthogonal decompositions of matrix factorizations) and the analytic structures, such as asymptotic expansions in FJRW theory. The talk is based on the work joint with Ming Zhang.

The DT/PT correspondence is a formula which relates Donaldson-Thomas invariants counting closed subschemes in Calabi-Yau 3-folds, and Pandharipnade-Thomas invariants counting stable pairs on them. It gives an economical way of formulating GW/DT correspondence by Maulik-Nekrasov-Okounkov-Pandharipande. The DT/PT correspondence was proved by wall-crossing in the derived category, where on the wall we have a special type of Ext-quivers called DT/PT quivers. In this talk, I will give a categorical wall-crossing formula for categories of matrix factorizations associated with DT/PT quivers. The resulting formula is a semiorthogonal decomposition which involves quasi-BPS categories, and is a categorical analogue of numerical DT/PT correspondence. This is a joint work with Tudor Padurariu.

Caldararu-Costello-Tu defined Categorical Enumerative Invariants (CEI) as a set of invariants associated to a cyclic A-infinity category (with some extra conditions/data), that resemble the Gromov-Witten invariants in symplectic geometry. In this talk I will explain how one can define these invariants for Calabi-Yau A-infinity categories - a homotopy invariant version of cyclic - and then show the CEI are Morita invariant. This has applications to Mirror Symmetry and Algebraic Geometry.

We define integer valued invariants of an orbifold Calabi-Yau threefold $X$ with transverse ADE orbifold points. These invariants contain equivalent information to the Gromov-Witten invariants of $X$ and are related by a Gopakumar-Vafa like formula which may be regarded as a universal multiple cover / degenerate contribution formula for orbifold Gromov-Witten invariants. We also give sheaf theoretic definitions of our invariants. As examples, we give formulas for our invariants in the case of a (local) orbifold K3 surface. These new formulas generalize the classical Yau-Zaslow and Katz-Klemm-Vafa formulas. This is joint work with S. Pietromonaco.

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 in the sense of Behrend-Fantechi 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. The method presented can also be applied to the moduli space of stable map spaces from semi-log-canonical surfaces to projective varieties.