Integrable systems and Gromov-Witten theory of non-orientable surfaces

可积系统和不可定向表面的 Gromov-Witten 理论

• 批准号: 0406077
• 负责人: Motohico Mulase
• 金额: $ 10万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406077&HistoricalAwards=false
• 关键词: Integrable; systems; Gromov; Witten; theory

AbstractAward: DMS-0406077Principal Investigator: Motohico MulaseThe proposed project is aimed at generalizing one of the recentdevelopments in the area of Gromov-Witten theory of compactRiemann surfaces to the case of non-orientable surfaces. In thearea of complex and symplectic geometry, integrable systems, andmathematical physics, amazing new developments have been achievedin recent years. At the time of their discovery in 1985,Donaldson invariants of differentiable four manifolds, Jonespolynomials of knots and links, and Gromov's idea of pseudo-holomorphic curves were considered as independent entities.Since the time of Atiyah's provocative lecture in 1988, theirhidden inter-relations have been emerged. Through the work ofmany mathematicians and physicists, a clearer picture of theconnection of these theories is revealed. Most recently, moredirect relations have been discovered, through calculation of thegenerating function of these invariants and their associationwith integrable systems and representation theory. Among them isthe work of Okounkov and Pandharipande, who determined theGromov-Witten invariants of an arbitrary compact Riemann surfaceas the target space. Their fundamental results include a proof ofthe conjectured Virasoro constraints, identification of thegenerating function of the invariants of the Riemann sphere as asolution to the two-dimensional Toda lattice equations throughFermionic Fock representation, and a new proof of theWitten-Kontsevich theory of intersection of cohomology classes onthe moduli space of Riemann surfaces. We propose to establishGromov-Witten theory of real algebraic curves without boundary,and obtain the counterpart of the above theorems for the case ofnon-orientable surfaces.The topological and geometric structures of various kinds ofspaces have attracted intensive research in mathematics andmathematical physics for many decades. Poincare's idea onhomology and homotopy theories have proven to be usefulthroughout the 20th century. The applications of these theoriesare seen in physics, chemistry, and understanding the dynamicalproperties of DNA. Only toward the end of the previous centurymathematics has encountered a true generalization of the ideas ofPoincare, applicable to the particularly interesting spaces(called symplectic manifolds) that appear naturally in any kindsof mechanics and dynamics. The new invariants, known as theGromov-Witten invariants of symplectic spaces, have shownextremely mysterious connections to almost all areas ofmathematics. Mathematicians feel quite happily that theyunderstand the homology and homotopy theories very well. Comparedto that level, our current understanding of the Gromov-Wittentheory is at the best very limited, and the whole subject isstill filled with mysteries. We believe that to have a deeperunderstanding of the theory, we should generalize it further andinclude the consideration of new cases never done before. Theproposed project is aimed at discovering a generalization of thetheory in a particular context, using non-orientable surfaces.

AbstractAward：DMS-0406077首席研究员：Motohico Mulase拟议的项目旨在将Gromov-Witten紧致Riemann曲面理论领域的最新发展之一推广到不可定向曲面的情况。近年来，在复几何、辛几何、可积系统和数学物理等领域都取得了惊人的新进展。在1985年发现它们的时候，可微四流形的唐纳森不变量、纽结和链环的琼斯多项式以及格罗莫夫的伪全纯曲线的思想被认为是独立的实体。通过许多数学家和物理学家的工作，这些理论之间的联系得到了更清晰的揭示。最近，通过计算这些不变量的生成函数，以及它们与可积系统和表示论的联系，发现了更多的关系。其中包括Okounkov和Pandharipande的工作，他们确定了目标空间中任意紧致Riemann曲面的Gromov-Witten不变量。他们的基本结果包括证明了Virasoro约束的存在性，通过Fermionic Fock表示确定了作为二维户田格点方程解的Riemann球面不变量的生成函数，以及证明了Riemann曲面模空间上的上同调类相交的Witten-Kontsevich理论。本文提出了建立无边界真实的代数曲线的Gromov-Witten理论，并在不可定向曲面的情况下得到了上述定理的对应定理。庞加莱的同调思想和同伦理论在整个世纪都被证明是有用的。这些理论在物理学、化学和理解DNA的动力学性质方面都有应用。 直到上个世纪末，数学才遇到了庞加莱思想的真正推广，适用于特别有趣的空间（称为辛流形），这些空间自然地出现在任何力学和动力学中。新的不变量，被称为辛空间的Gromov-Witten不变量，几乎与所有数学领域都有着极其神秘的联系。数学家们感到非常高兴，因为他们很好地理解了同调理论和同伦理论。与这个水平相比，我们目前对Gromov-Witten理论的理解充其量是非常有限的，整个主题仍然充满了谜团。我们认为，为了更深入地理解这一理论，我们应该进一步概括它，并考虑以前从未做过的新案例。 该项目的目的是发现一个推广的理论在一个特定的背景下，使用非定向表面。