Program

New ideas in string theory, in particular D-branes and their relevance to open strings, have in many ways revolutionized modern quantum field theory, but this subject is currently highly heuristic: its formalization and mathematical development has barely begun. The geometric naturality and flexibility of these concepts has fostered rapid development, but their codification is completely open. Orbifolds, gerbes, and stacks are all topics with well-established classical literatures, but the idea that they should be grouped together, and that the various kinds of twistings they manifest are relevant to physics, is a new idea in mathematics. Stacks and orbifolds are related concepts, generalizing the classical notion of a quotient by a compact group with finite isotropy, but which are studied by different mathematical constituencies (algebraic vs. differential geometers); while gerbes, which generalize both, allow quotients by more exotic topological groups (such as higher Eilenberg-MacLane spaces). String theorists [Dixon, Vafa, et al] realized very early that pretty much anything they wanted to do with smooth manifolds would generalize to orbifolds, whose utility increased with the realization that mirrors of smooth Calabi-Yau manifolds are often proper orbifolds; but the roles played by singularities on opposite sides of the mirrors is only now beginning to be systematically studied. The motivic integration program, in which singular varieties are deformed over function fields (corresponding to the topological study of the fixed points of a free loopspace, under its natural circle action) is beginning to shed considerable geometric light on these issues [cf eg Lupercio’s work on the (exotic, rational) grading of orbifold cohomology in terms of normal bundles in the free loopspace]. Work of Sullivan, Chas, Cohen, and Stacey has outlined a potential purely topological approach to the study of generalized Gromov-Witten invariants. In fact loop groups and free loopspaces have from the beginning been foundation stones for the topological understanding of string theory, and elliptic cohomology and (twisted) K-theory now play a central role in the theory of positive-energy representations [Hopkins, Freed, Teleman]. In another direction, the Drinfel’d-Beilinson chiral algebra program [Gorbounov, Malikov, Schecht-man; Kapranov, Vasserot, . . . ] globalizes the theory of vertex algebras to the context of general manifolds, and leads to a better understanding of some mysterious questions about relations between local geometry (eg spin structures) and global analysis (modularity of the elliptic genus). There has been great technical progress recently in the study of free loopspaces of orbifolds, and in equivariant elliptic cohomology [Ando]. On the other hand, vertex operator algebras were to some extent invented for the study of finite groups, and the site of interaction between that subject, the theory of orbifolds, and the study of these mysterious higher kinds of twisting is one of the most promising places for new conceptual breakthroughs. Physicists encounter such twistings in the context of differential geometry, and mathematicians [Hopkins, Freed, Singer] have developed new tools (eg hybrids of Deligne cohomology and K-theory, and new (and fundamentally important) kinds of orientations) which have greatly clarified the geometric and analytic roles of these structures. Related ideas (involving von Neumann algebras) are important in recent work [Stolz-Teichner, Ando-Berenstein] on relations between open string theory and elliptic cohomology. There is reason to think that all these developments may eventually be seen as part of an emerging theory of commutative quotients for noncommutative spaces. Another important recent development, the solution of Mumford’s conjecture by Madsen and Weiss, highlights the relevance of algebraic topology in problems related to moduli spaces of inherent interest in geometry and physics. D-branes themselves have roots in analysis, as well as deep connections to the differential geometry of special structures. Current thinking links them to geometric cycles in K-theory, decorated with extra structures [holonomy, generalized connections, bundles] of various sort, whose general theory is not yet clear. We expect this to be a subject of lively interest by the time of the program. New Topological Structures web site NTS Seminars