New Perspectives on Hyperkähler Manifolds - A Celebration of Dimitri Markushevich's (60+2)nd Birthday

Wirthmüller's theorem and modular differential equations for basic Jacobi forms

Jun 15, 2022, 11:30 AM

1h

BellaVista Relax Hotel

In-person Talk

Speaker

Valery Gritsenko (Université de Lille, France, and HSE University, Moscow, Russia)

Description

The graded rings of Weyl-invariant weak Jacobi forms play an important role in the theory of Frobenius varieties.
Basis generators of these rings are also generating functions of some quantities (indices of vector bundles, Gromov-Witten invariants, multiplicities of positive roots of Lorentzian Kac-Moody algebras). The case of the root system $D_n$ is very interesting from this point of view.
With my formal Ph.D. student Dimitri Adler, we generalise the idea of the $D_8$-tower of Jacobi forms coming for the theory of reflective modular form, and give a simple constructive proof of the Wirthmüller theorem for the root system $D_n$. Then we study the modular differential equations for the most important generators. A surprise is that there are unusual anomalies for some n and reasonable differences for varieties  such as $K3$ surfaces and $CY_3$.