The Algebra & Geometry of Modern Physics

The Langlands program predicts a correspondence between finite dimensional representations of Galois groups of extensions of number fields and automorphic representations of groups over fields of adeles. The geometric analogy of this program constructs so-called Hecke eigensheaves associated with flat bundles over Riemann surfaces. Kapustin and Witten have described the Geometric Langlands program by compactifying on a Riemann surface certain version of supersymmetric Yang-Mills theore. The supersymmetries of their model involve spinors from a space of some representation of a Clifford algebra associated with a twisted embedding of the groups Spin(4) into Spin(4)xSpin(6). The analogues of flat bundles are played by so-called electric zerobranes and the analogues of the Hecke operators are so-called 't Hooft operators acting on magnetic branes. In the first talk I will present mathematical introduction to the geometric Langlands program.