Wall-crossings in quasimap theory and applications

该项目旨在将研究者从曲线映射的模空间的紧化研究扩展到一类大的GIT商目标。这些紧化，称为稳定拟映射的模空间，产生了新的曲线计数不变量，这些不变量有望通过与Gromov-Witten不变量的壁交公式联系起来。在最近与Kim的合作中，PI发现了这种零属的过墙公式，并证明了它们对Givental的环面镜像定理的重要推广。PI将致力于在几个方向上扩展这一理论。首先，将wall-crossing公式推广到更高格的曲线上，由此得到的Calabi-Yau三叠面的理论在理论上等于镜面Calabi-Yau的BCOV b模型理论，但不需要通过镜面映射改变坐标。PI计划至少在当地的卡拉比-丘目标上证明这一猜想。其他项目将提供对轨道目标的扩展（应用于Crepant Resolution Conjecture），以及对Landau-Ginzburg模型的扩展（应用于Calabi-Yau/LG通信）。进一步的应用，如在Gromov-Witten理论的一般阿贝尔/非阿贝尔对应的证明也被期待。这项研究属于代数几何领域，代数几何是数学的一个古老而高度发展的分支，其核心是研究由多项式方程定义的几何形状。模量理论关注的是这些形状如何变形。研究者研究的模空间与弦理论中发现的镜像对称现象有着深刻的联系，弦理论是理论物理中一个非常活跃的领域。近二十年来，代数几何的成果和技术，特别是模空间理论，已经成功地应用于弦理论。另一方面，弦理论的思想通过提出惊人的猜想，开辟了数学研究的新方向，同时使旧的未解决的问题有了新的认识。该项目将继续这种富有成效的互动，为高等属的镜像对称提供新的见解。该奖项由代数和数论以及拓扑学项目共同资助。

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.