Quasimap Theory and Gromov-Witten Invariants of Complete Intersections

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This research is in the field of algebraic geometry, an old and highly developed branch of mathematics, which at its core is the study of geometric shapes defined by polynomial equations. Moduli theory is concerned with how these shapes deform when parameters are varied in a continuous fashion, and compactified moduli spaces describe in particular the kind of degenerate limiting shapes that may appear when deformations are performed. Various geometric properties of interest do not change under deformations and are often easier to analyze if limits adequate for the problem at hand are allowed. The "wall-crossing phenomenon" refers loosely to changing the compactified moduli spaces by disallowing certain limiting shapes and replacing them with different ones. The compactified moduli spaces studied in this project have deep connections with the mirror symmetry phenomenon discovered in string theory, a very active area of theoretical physics. In the last two decades, the results and techniques from algebraic geometry, especially the theory of moduli spaces, have been successfully employed in string theory. On the other hand, ideas from string theory have opened up new directions of research in mathematics by suggesting striking conjectures and at the same time putting old unsolved problems into a new light. This project will continue this fruitful interaction by offering new insights on mirror symmetry at higher genus, via the study of wall-crossing between moduli spaces.This project aims to continue the investigator's study of compactifications of moduli spaces of maps from curves to a large class of GIT quotient targets. These compactifications, called moduli spaces of stable quasimaps, produce new curve-counting invariants, which should be related to Gromov-Witten invariants by wall-crossing formulas. Indeed, such wall-crossing formulas in genus zero were established by the PI with Kim in recent years, and they turn out to provide significant generalizations of Givental's toric mirror theorems. One of the main goals of this project is to vastly extend the wall-crossing formulas by establishing them in higher genus and at the level of virtual classes for many compact targets. These formulas will then have many consequences which will be investigated. An important application is to the Mirror Conjecture at higher genus for complete intersection Calabi-Yau varieties, such as the quintic threefold. In this case, the wall-crossing formula may be viewed as giving a mathematically rigorous interpretation of the physicist's "`holomorphic limit of the B-model partition function" as the generating function for quasimap invariants. Further applications and generalizations that emerge from the study of wall-crossing relate to the so-called Landau-Ginzburg/Calabi-Yau correspondence, and more generally to the new theory of the gauged linear sigma model of Fan-Jarvis-Ruan in higher genus.