Open Hurwitz Flat F manifolds
Guilherme Feitosa de Almeida
TL;DR
The paper develops a general method to derive Open WDVV equations from any Hurwitz Dubrovin Frobenius manifold using the associated Landau-Ginzburg superpotential. It proves a key identity that the primitive of the LG superpotential satisfies the Open WDVV equations alongside the closed WDVV potential, thereby enabling systematic computation of Open WDVV data for Hurwitz spaces. The authors provide explicit low-genus and higher-genus examples, including Open H_{0,1}, Open H_{0,2}, Open Quantum Cohomology, and Open genus-1 Hurwitz spaces, illustrating the practical procedure via residues and Theta-function data. They also discuss integration-constant issues in appendices and point to potential connections with Topological Recursion and monodromy phenomena, suggesting avenues for further research. Overall, the work broadens the scope of Open WDVV to a wide class of Hurwitz-based Frobenius manifolds and offers a concrete computational framework for open-descendant potentials.
Abstract
In this paper, we derive Open WDVV equations starting from any Hurwitz Dubrovin Frobenius manifold. The WDVV equations play a crucial role in the structure of Frobenius manifolds, quantum cohomology, and integrable systems. Extending these ideas, Open WDVV equations provide a framework to incorporate boundary conditions, making them fundamental in Open Gromov-Witten theory. Using Dubrovin's construction of Landau Ginzburg superpotentials associated with Hurwitz spaces, we demonstrate that their primitives satisfy Open WDVV equations. Our approach provides an efficient method for computing Open WDVV equations associated with any Hurwitz Dubrovin Frobenius manifold.