$DIFF(s^1)$, TeichmÜller Space and Period Matrices: Canonical Mappings via String Theory

Abstract

There is a completely natural and intimate relationship between the diffeomorphism group of the circle and the Teichmüller spaces of Riemann surfaces discovered by us in 1988. Such a relationship had been sought-after by physicists from conjectures connecting the loop-space approach to string theory with the path-integral approach. Precisely, the remarkable homogeneous space Diff$(S^1)$/$SL(2,R)$ (which is one of the two possible quantizable coadjoint orbits of Diff$(S^1)$), embeds as a complex analytic and Kähler submanifold of the universal Teichmüller space. Furthermore, this very homogeneous space, Diff$(S^1)$/$SL(2,R)$, considered by the previous work as a Kähler submanifold of the universal Teichmüller space, allows on it a natural holomorphic period mapping, $Π$, that generalises the classical map associating to a genus $g$ Riemann surface its period matrix. Utilising the fact that the group of quasiconformal homeomorphisms of $S^1$ acts symplectically on the Sobolev space of order $1/2$ on the circle, we (with Dennis Sullivan) have recently extended $Π$ to the entire universal Teichmüller space. All this is related to non-perturbative string theory.