Calculation of Rozansky-Witten invariants on the Hilbert schemes of points on a K3 surface and the Generalised Kummer varieties

Rozansky-Witten invariants via Atiyah classes

Recently, L.Rozansky and E.Witten (hep-th/9612216) associated to any hyperKaehler manifold X a system of "weights" (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a very simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyperKaehler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work …

Rozansky-Witten theory, Localised then Tilted

The paper has two parts, in the first part, we apply the localisation technique to the Rozansky-Witten theory on compact HyperKähler targets. We do so via first reformulating the theory as some supersymmetric sigma-model. We obtain the exact formula for the partition function with Wilson loops on $S^1\timesΣ_g$ and the lens spaces, the results match with earlier computations using Feynman diagrams on K3. The second part is motivated by a very curious preprint \cite{Gukov:2020lqm}, where the equivariant index formula for the dimension of the Hilbert space the theory is interpreted as a kind of …

Kummer surfaces associated with Seiberg-Witten curves

By carrying out a rational transformation on the base curve $\mathbb{CP}^1$ of the Seiberg-Witten curve for $\mathcal{N}=2$ supersymmetric pure $\mathrm{SU}(2)$-gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17. The isogeny relating the Seiberg-Witten curve for pure $\mathrm{SU}(2)$-gauge theory to the one for $\mathrm{SU}(2)$-gauge theory with $N_f=2$ massless hypermultiplets extends to define a Nikulin involution on each K3 surface in the family. We show that the desingularization of the quotient of the K3 surface by the involution is isomorphic to a Kummer …

Extended TQFTs and Algebraic Geometry

We study a potential method for constructing the Rozansky--Witten TQFT as an extended $(1+1+1)$-TQFT. We construct a $2$-category consisting of schemes, complexes of sheaves and sheaf morphisms and show that there are $(1+1)$-TQFTs valued in the truncation of this category which have state spaces that agree with the Rozansky--Witten TQFT. However, we also show that if such a TQFT is based on a reduced Noetherian scheme, it cannot be extended upwards to a $(1+1+1)$-TQFT.

Lie algebras in symmetric monoidal categories

We study algebras defined by identities in symmetric monoidal categories. Our focus is on Lie algebras. Besides usual Lie algebras, there are examples appearing in the study of knot invariants and Rozansky-Witten invariants. Our main result is a proof of Westbury's conjecture for K3-surface: there exists a Lie algebra homomorphism from Vogel's universal simple Lie algebra to the Lie algebra describing the Rozansky-Witten invariants of a K3-surface. Most of the paper involves setting up a proper language to discuss the problem and we formulate nine open questions as we proceed.

全纯辛4-流形的Gopakumar-Vafa型不变量

Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

本文利用约化Gromov-Witten理论为全纯辛4-流形定义了新的Gopakumar-Vafa型不变量，猜想其整数性并给出层论解释，在多个例子中验证猜想，并得到K3^{[2]}型4-流形上亏格2曲线计数的猜想公式及Fujiki常数的新闭公式。