Permutation-Equivariant Quantum K-Theory in Higher Genus

在寻找自然界终极法则的过程中，弦理论将被称为代数曲线的数学对象置于现代基础物理框架的中心，以惊人的速度和坚持不懈的速度产生新的数学问题，并指出合理的答案。在这个项目中要研究的问题位于过去两个世纪数学的两条主要道路的十字路口。其中之一是对代数曲线的复杂性质的深入追求，其形式继承了高斯、阿贝尔、雅各比、黎曼、克莱因和庞加莱的作品。另一个是数学物理的广泛概念图景，由经典、统计和量子力学的进步决定，经常与汉密尔顿、麦克斯韦、吉布斯、庞加莱、希尔伯特、爱因斯坦和韦尔的名字联系在一起。这个项目的具体目的是发展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，其中通过对全纯曲线到目标Kahler相空间的稳定映射的凝聚层上的切赫上同调的标记点的重新编号来研究和计算置换群的特征。目前和即将进行的这类置换等变K-理论Gromov-Witten不变量的研究包括：(A)构造这些不变量并探索它们的一般性质；(B)发展用母函数表示不变量的辛循环空间量子化形式；(C)建立适当的量子Riemann-Roch定理，以上同调Gromov-Witten不变量的形式提供置换等变K-理论Gromov-Witten不变量的完整刻画；(D)发展计算置换等变Gromov-Witten不变量的不动点局部化技术；(E)应用这些技巧获得镜像公式的K-理论类比(即，证明具有环流形的某些亏格-0置换等变Gromov-Witten不变量的环Q-超几何函数)：(F)引入和研究K-理论镜像(即此类Q-超几何函数的复振荡表示，以及相应的D_q-模)；(G)阐明由隐对称作用的q-差算子群在置换等变量子K-理论中的作用，并利用这些对称性从相应的q-超几何函数重建环流形的所有亏格-0不变量；(H)通过结合玻色子-费米子对应和更高亏格中的adelic刻画，用非扭曲置换等变K-理论Gromov-Witten不变量表示；以及(I)通过类似于点的上同调Gromov-Witten不变量的Witten-Kontsevich定理，探索点目标空间的量子K-理论与可积系统的KdV-族的Q-类似物之间的关系。

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.