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 $$S^1\times \Sigma _g$$ S 1 × Σ 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 paper (Gukov in J Geom Phys 168, 104311, 2021), where the equivariant index formula for the dimension of the Hilbert space of the Rozansky–Witten theory is interpreted as a kind of Verlinde formula. In this interpretation, the fixed points of the target hyperkähler geometry correspond to certain ‘states’. We extend the formalism of part one to incorporate equivariance on the target geometry. For certain non-compact hyperkähler geometry, we can apply the tilting theory to the derived category of coherent sheaves, whose objects label the Wilson loops, allowing us to pick a basis for the latter. We can then compute the fusion products in this basis and we show that the objects that have diagonal fusion rules are intimately related to the fixed points of the geometry. Using these objects as basis to compute the dimension of the Hilbert space leads back to the Verlinde formula, thus answering the question that motivated the paper.

Transverse hilbert schemes, bi-hamiltonian systems and hyperkähler geometry

Dedicated to the memory of Sir Michael Francis Atiyah (1929-2019) We give a characterization of Atiyah’s and Hitchin’s transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces.

AN EQUIVARIANT DESCRIPTION OF CERTAIN HOLOMORPHIC SYMPLECTIC VARIETIES

Varieties of the form$G\times S_{\!\text{reg}}$, where$G$is a complex semisimple group and$S_{\!\text{reg}}$is a regular Slodowy slice in the Lie algebra of$G$, arise naturally in hyperkähler geometry, theoretical physics and the theory of abstract integrable systems. Crooks and Rayan [‘Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices’,Math. Res. Let., to appear] use a Hamiltonian$G$-action to endow$G\times S_{\!\text{reg}}$with a canonical abstract integrable system. To understand examples of abstract integrable systems arising from Hamiltonian$G$-actions, we consider a holomorphic symplectic variety$X$carrying an abstract integrable system induced by a Hamiltonian$G$-action. Under certain hypotheses, we show that there must exist a$G$-equivariant variety isomorphism$X\cong G\times S_{\!\text{reg}}$.

Collections of Orbits of Hyperplane Type in Homogeneous Spaces, Homogeneous Dynamics, and Hyperkähler Geometry

Consider the space M = O(p, q)/O(p) × O(q) of positive p-dimensional subspaces in a pseudo-Euclidean space V of signature (p, q), where p > 0, q > 1 and $(p,q)\neq (1,2)$, with integral structure: $V = V_{\mathbb{Z}} \otimes \mathbb{Z}$. Let Γ be an arithmetic subgroup in $G = O(V_{\mathbb{Z}})$, and $R \subset V_{\mathbb{Z}}$ a Γ-invariant set of vectors with negative square. Denote by R⊥ the set of all positive p-planes W ⊂ V such that the orthogonal complement W⊥ contains some r ∈ R. We prove that either R⊥ is dense in M or Γ acts on R with finitely many orbits. This is used to prove that the squares of primitive classes giving the rational boundary of the Kähler cone (i.e., the classes of “negative” minimal rational curves) on a hyperkähler manifold X are bounded by a number which depends only on the deformation class of X. We also state and prove the density of orbits in a more general situation when M is the space of maximal compact subgroups in a simple real Lie group.

Implosion for hyperkähler manifolds

We introduce an analogue in hyperkähler geometry of the symplectic implosion, in the case of$\mathrm{SU} (n)$actions. Our space is a stratified hyperkähler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

Real Moment Map and Hyperkähler Geometry Techniques: The Cotangent Bundle to the 2-Sphere and SL(2, ℂ)-Adjoint Orbits

Going back to Kirwan and others, there is an established theory that uses moment map techniques to study actions by complex reductive groups on Kähler manifolds. Work of P. Heinzner, G. Schwarz, and H. Stötzel has extended this theory to actions by real reductive groups. In this paper, we apply these techniques to actions of the real group SU(1,1) ⊂ SL(2, ℂ) on a certain complex manifold of dimension two. More precisely, because of the SU(2)-invariant hyperkähler structure on this manifold, we are able to study a family of actions which includes and “interpolates” two well-known actions of SL(2, ℂ): the adjoint action on the orbit of a semisimple element of 𝔰𝔩(2, ℂ), and the action of SL(2, ℂ) on the cotangent bundle of the flag variety of SL(2, ℂ).