NCMS · A new approach in curve counting theories and mirror symmetry

A Marie Sklodowska-Curie IF at ETH-Zurich will lead to major developments of the PI's current research.The PI's research focuses on better understanding the mathematical implications of the physical dualities that arise in the study of string theory. Mirror symmetry, which is a kind of duality in string theory, equates two physical theories called the A-model and B-model. Mirror symmetry predicts that the A-model (resp. the B-model) of a space/variety is equivalent to the B-model (resp. the A-model) of itsmirror space/variety. In mathematics, the A-model corresponds to Gromov--Witten (GW) theory, which is one of the first modern curve counting theories in enumerative geometry. To a physicist, a complex curve represents the worldsheet of a string propagating through space-time.In 2018, the PI initiated a new research program which provides a novel approach of using orbifold techniques to count curves in algebraic varieties with tangency conditions along co-dimension one sub-varieties (divisors). This novel approach defines a generalization of relative GW theory which plays a central role in mirror symmetry. Several major advances have been achieved in the past two years and a new research direction has been created.This proposal focuses on this new research program and its applications to curve counting theories and mirror symmetry. We expect to build a firm foundation for our new theory and expand this program along various directions. The proposal is divided into three main projects. The first project focuses on structural properties of the new GW theory and its relation with punctured GW theory. The second project explores applications of the new theory to several aspects of mirror symmetry including Gross--Siebert program, the Strominger--Yau--Zaslow (SYZ) conjecture and the Doran--Harder--Thompson (DHT) conjecture. The third application focuses on its connections with other curve counting theories.