Periodic Reporting for period 2 - ROGW (Real and open Gromov-Witten theory)
Période du rapport: 2022-03-01 au 2023-08-31
The ROGW project aims to establish new relations between natural objects in symplectic geometry with other fields of mathematics including knot and representation theory and the theory of integrable systems. All of these relations are motivated by theoretical physics. The main objects of study are moduli spaces of pseudo-holomorphic maps giving rise to real and open Gromov-Witten invariants. The classical Gromov-Witten invariants were introduced by Gromov at the birth of symplectic topology giving rise to obstructions to symplectic embeddings. Their interpretation by Witten as the coefficients of a partition function of a field theory placed them in a new light : striking dualities understood in physics relate them to mathematical objects of completely different nature and on completely different manifolds. This has and continues to generate enormous amount of high-level research aimed at understanding these relations better.
The project is concerned with establishing such relations in the context of the recently introduced, by the PI and A. Zinger, real Gromov-Witten theory. In particular, it aims to determine the systems of differential equations governing the real Gromov-Witten theory, that parallel the KdV and Toda hierarchies in the classical case. We will further study the real Gromov-Witten theory of toric Calabi-Yau threefolds with the aim of establishing a connection with SO/Sp Chern-Simons theory and a real version of the remodeled mirror symmetry. Finally, we will seek to establish foundational results of open Gromov-Witten theory in a general context.
The project is concerned with establishing such relations in the context of the recently introduced, by the PI and A. Zinger, real Gromov-Witten theory. In particular, it aims to determine the systems of differential equations governing the real Gromov-Witten theory, that parallel the KdV and Toda hierarchies in the classical case. We will further study the real Gromov-Witten theory of toric Calabi-Yau threefolds with the aim of establishing a connection with SO/Sp Chern-Simons theory and a real version of the remodeled mirror symmetry. Finally, we will seek to establish foundational results of open Gromov-Witten theory in a general context.
The project has advanced in several of the proposed directions. Perhaps the most surprising one is the fact that the (signed) real Hurwitz numbers satisfy the CKP hierarchy as established by Thomas Guidoni, doctoral student within the framework of the project. This in turn would imply the corresponding statement for real GW invariants as outlined in the proposal and invites questions regarding the underlying dualities. Significant advancements are also made towards establishing the connection between the real GW invariants and the Kauffman polynomial in a joint work in progress with Tobias Ekholm and Vivek Shende as well as establishing the existence of open GW invariants in certain situations.
In the second half of the project, the PI will continue the study of the proposed directions, and in particular those related to topological recursion and mirror symmetry.