Permutation-Equivariant Quantum K-Theory in Higher Genus

弦理论在探索自然界的终极定律时，将被称为代数曲线的数学对象置于基础物理学现代框架的中心，以惊人的速度和持久性产生新的数学问题，并指出看似合理的答案。在这个项目中要调查的问题位于过去两个世纪的数学两个主要途径的十字路口。其中之一是深入追求代数曲线的复杂性质，其形式继承自高斯、阿贝尔、雅可比、黎曼、克莱因和庞加莱的著作。另一个是数学物理的广阔概念图景，由经典力学、统计力学和量子力学的进步所支配，并且经常与汉密尔顿、麦克斯韦、吉布斯、庞加莱、希尔伯特、爱因斯坦和外尔的名字联系在一起。一些正在研究的问题是由弦理论产生的数学问题所激发的;本项目的具体目标是发展Gromov-Witten理论的新篇章，即哈密顿系统相空间的拓扑不变量理论，其中置换群的特征，研究并计算了标记点的重新编号对全纯曲线到目标Kahler相空间的稳定映射的模空间上相干层的Cech上同调的作用。对这类置换等变K-理论Gromov-Witten不变量的研究包括：（a）构造这类不变量并探索其一般性质：（B）发展用生成函数表示的辛环空间量子化形式;（c）建立相应的量子Riemann-Roch定理，给出置换等变K理论Gromov的完全的代数刻画。维滕不变量的上同调Gromov-维滕不变量;（d）发展用于计算置换等变Gromov-维滕不变量的不动点局部化技术;（e）应用该技术以获得镜像公式的K-理论类似物（即，以识别具有环面流形的某些亏格0置换等变Gromov-Witten不变量的环面q-超几何函数）;（f）引入并研究K-理论镜像（即，这类q-超几何函数的复振荡表示和相应的D_q-模）：（g）阐明了在置换等变量子K-理论中隐对称作用的q-差分算子群的作用，并利用这些对称从相应的q-超几何函数重构环面流形的所有亏格0不变量;（h）将玻色子-费米子对应与高等亏格中的阿德尔特征相结合，用无扭不变量表示扭置换-等变K理论Gromov-Witten不变量;（i）探讨点目标空间的量子K-理论与可积系统KdV-族的q-类似物之间的关系，通过与Witten-Kontsevich定理的类比预测点的上同调Gromov-Witten不变量。

String theory, in the search for the ultimate laws of nature, places mathematical objects known as algebraic curves at the center of the modern framework of fundamental physics, generating new mathematical questions and pointing to plausible answers with an amazing pace and persistence. Questions to be investigated in this project lie at the crossroads of two major pathways in mathematics of the past two centuries. One of them is the in-depth pursuit of the intricate properties of algebraic curves, in the form inherited from works of Gauss, Abel, Jacobi, Riemann, Klein, and Poincare. The other is the broad conceptual landscape of mathematical physics, dictated by the progress of classical, statistical, and quantum mechanics, and often associated with the names of Hamilton, Maxwell, Gibbs, Poincare, Hilbert, Einstein, and Weyl. Some of the questions under study are motivated by mathematical questions arising out of string theory; in turn, the research is expected to provide feedback to string theorists inspiring previously unanticipated directions of research.The specific aim of this project is to develop a new chapter of the Gromov-Witten theory, that is, the theory of topological invariants of phase spaces of Hamiltonian systems, where the characters of permutation groups, acting by renumbering of marked points on the Cech cohomology of coherent sheaves over moduli spaces of stable maps of holomorphic curves to a target Kahler phase space, are studied and computed. The ongoing and forthcoming research of such permutation-equivariant K-theoretic Gromov-Witten invariants is to include: (a) constructing these invariants and exploring their general properties; (b) developing the symplectic loop-space quantization formalism for representing the invariants by generating functions; (c) establishing the appropriate Quantum Riemann-Roch Theorems to provide the complete adelic characterization of permutation-equivariant K-theoretic Gromov-Witten invariants in terms of cohomological Gromov-Witten invariants; (d) developing the fixed-point-localization techniques for computing permutation-equivariant Gromov-Witten invariants; (e) applying the techniques in order to obtain K-theoretic analogues of the mirror formulas (i.e., to identify toric q-hypergeometric functions with certain genus-0 permutation equivariant Gromov-Witten invariants of toric manifolds); (f) introducing and studying the K-theoretic mirrors (i.e., complex oscillatory representations of such q-hypergeometric functions, and the corresponding D_q-modules); (g) elucidating the role of the groups of q-difference operators acting by hidden symmetries in the permutation-equivariant quantum K-theory, and exploiting these symmetries to reconstruct all the genus-0 invariants of toric manifolds from the respective q-hypergeometric functions; (h) expressing twisted permutation-equivariant K-theoretic Gromov-Witten invariants in terms of the untwisted ones by combining the boson-fermion correspondence with the adelic characterization in higher genus; and (i) exploring the relationships between quantum K-theory of the point target space and the q-analogues of the KdV-hierarchy of integrable systems, anticipated by analogy with the Witten-Kontsevich theorem for cohomological Gromov-Witten invariants of the point.