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Gromov Witten Theory and Integrable Systems

Final Report Summary - GROWINT (Gromov Witten Theory and Integrable Systems)

In the last forty years, starting with the discovery of the relevance of Yang-Mills fields in the description of sub-nuclear phenomena, the solution of frontier problems in theoretical high energy physics has required more and more refined mathematical tools. This need has only intensified since the appearance of String Theory as the leading candidate for a unified theory of nature. Besides its phenomenological relevance, and whether or not it holds the key for a complete understanding of quantum gravity, the theory of superstrings has had a truly remarkable impact on sometimes distant areas of Mathematical Physics and Mathematics in general, both as a heuristic guiding principle and as a major unifying framework.
A paradigmatic example of this interaction is given by topological string theory. On the physics side, it arises as a “topological subsector” of type IIA string theory, whose observables are highly constrained by supersymmetry and are thus under good theoretical control. As such it represents a privileged laboratory test general ideas about dualities in string theory and gauge theory, providing at the same time a great deal of information about the effective dynamics of supersymmetric quantum field theories. Its mathematical formalization is given in terms of moduli spaces of maps from a source projective curve (the Euclideanized string world-sheet) to a compact Kaehler manifold. The topological string correlators then correspond to certain intersection numbers on called Gromov–Witten invariants.

The theme of the project lies at the interface between geometry, theoretical physics, and the theory of integrable systems. The main aim of the project was to uncover the existence of infinite–dimensional symmetries in Gromov–Witten theory, and to systematically study their representations in the framework of classical, infinite-dimensional integrable systems. The main body of the project was structured into two main lines of research: a first strand (Task 1) centered on the study and characterization of integrable strucutres in quantum cohomology and its higher genus deformations, and a second strand (Task 2) devoted to the study of applications in Algebraic Geometry (Task 2a) and High Energy Physics, in particular string theories on local Calabi-Yau geometries (Task 2b). Additionally, a wide spectrum of Training Activities for the Fellow were set in place, ranging from focused research-level training on a variety of themes relevant for the project (advanced methods in Gromov-Witten Theory, Donaldson-Thomas and Pandharipande-Thomas theory) to additional professional training in the fields of Team Management, Scientific Leadership and Grant-Writing.

The Research and Training activities carried out by the Fellow under the supervision of the Project Coordinator have been strikingly successful. Most of the Research Objectives detailed in the Research Proposal have been fully accomplished and have had a deep impact on the relevant mathematical community: these include a complete characterization of the integrable hierarchies governing an important class of toric Calabi-Yau threefolds, novel formulations of mirror symmetry for a large class of local Calabi-Yau geometries, new tools to test important conjectures in the field such as the Crepant Transformation Conjecture, and the study of the wall crossing behavior of open Gromov-Witten invariants of Aganagic-Vafa orbibranes. Moreover, new avenues of research have been opened, including a study of the low energy dynamics of N=1 gauge theories in five dimensions with simply laced gauge group from the point of view of integrable systems.