Award: DMS-0705233, 0705381 Principal Investigator: Matthew Ando, Eric R. Sharpe
Ando and Sharpe will investigate several questions in elliptic cohomology suggested by the physics of open string theory. In physics, elliptic genera arise in closed string theory, but the recent work of Stolz and Teichner and of Borisov and Libgover and Ganter strongly suggests that open string theory plays an important role in elliptic cohomology. Ando and Sharpe propose to introduce recent developments in open string theory, particularly ideas about D-branes and K-theory, to the study of elliptic cohomology. For example they hope to use open string theory to investigate the relationship between elliptic cohomology and the K-theory of K-theory, as studied by Baas, Dundas, Richter, and Rognes. They also hope to give a unified account of the McKay correspondence for elliptic genera, due to Borisov and Libgober, and for sheaves, due to Bridgeland, King, and Reid.
Elliptic cohomology arises from topology, and it signals a profound relationship between physics, particularly the physics of string theory, and at least two branches of mathematics, topology and number theory. Precisely how these things are related through elliptic cohomology remains mysterious: the physicist Edward Witten called elliptic cohomology "a piece of 21st century mathematics that happened to fall into the 20th century." Until very recently, the study of the physics of elliptic cohomology has focused mainly on "closed" string theory, in which particles are replaced by loops of string. There is now a rich theory of "open" strings, in which particles may also be unlooped strands of string. Ando and Sharpe suggest that the physics of open string theory has deep implications in elliptic cohomology. They propose to investigate several of these implications, which they hope will shed light on some of the important mysteries in the subject.