Wall-crossings in quasimap theory and applications

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

The project aims to expand on 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 are expected to be related by wall-crossing formulas with Gromov-Witten invariants. In recent work with Kim, the PI has found such wall-crossing formulas in genus zero and they turn out to give a significant generalization of Givental's toric mirror theorems. The PI will work to extend the theory in several directions. First, the wall-crossing formulas should extend to curves of higher genus, and the resulting theory for Calabi-Yau three-folds is conjecturally equal to the BCOV B-model theory of the mirror Calabi-Yau, but WITHOUT changing coordinates by the mirror map. The PI plans to prove this conjecture at least for local Calabi-Yau targets. Other projects will provide extensions to orbifold targets (with applications to the Crepant Resolution Conjecture), and to Landau-Ginzburg models (with applications to the Calabi-Yau/LG correspondence). Further applications, such as a proof of the general Abelian/Non-abelian Correspondence in Gromov-Witten theory are also expected. 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. The moduli spaces studied by the investigator 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. The project will continue this fruitful interaction by offering new insights on mirror symmetry at higher genus. This award is co-funded by the Algebra and Number Theory and the Topology programs.