Algebraic Geometry and Infinite-dimensional Spaces

点击翻译按钮获取中文摘要

Kapranov proposes to study infinite-dimensional spaces such as spaces of pathsby using algebro-geometric techniques. The traditional analytic approach to suchspaces leads to many difficulties. The algebro-geometric approach has theadvantage of bypassing these difficulties and at the same time preserving andin fact emphasizing the main conceptual results of the study. He proposes todevelop an approach to Floer cohomology based on the algebro-geometric concept of an ind-scheme. In fact, the very definition of an ind-scheme (known forsome time) is not far removed from Floer's ideas about cycles of ``semi-infinite"dimension. Kapranov proposes to apply this approach to various spaces offormal paths and loops in finite-dimensional varieties. Among other things, heproposes to understand the elliptic cohomology theory by using these spaces,in particular to relate the elliptic cohomology and the derived category ofcoherent sheaves, two objects of recent interest.By studying various "anomalies" on such infinite-dimensional spaces, Kapranovproposes to prove a new Riemann-Roch type theorem involving families ofreal, not complex varieties. The motivation for study of infinite-dimensional spaces comes from physics(string theory) where the fundamental object is not a punctual particle but a string propagating in the space-time. The number of degrees of freedom ofsuch a string is clearly infinite. But working with infinite-dimensional spacesis difficult. The usual problems of convergence familiar from multivariable calculus becomein many cases overwhelming when the number of variables becomes infinite. The algebraic approach proposed by Kapranovcan capture the essense of many problems while maintaining the mathematical rigor andthus preventing one from making mistakes. One example of such a problem is thedeterminantal anomaly: determinants of infinite matrices do not behave as expectedfrom the finite-dimensional case but obey different rules. This leads to a wealth of consequences forthe topology and geometry of infinite-dimensional spaces. The algebraic approachallows one to arrive at and generalize these consequences while minimizing theconsiderable effort needed to even define the infinite determinants.