1993/10/08 by Subhashis Nag, Nag, Subhashis
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9310051
42 pages, TEX
arxiv created 1993/10/08 · arxiv updated 2009/11/30
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(S1)/SL(2,R) (which is one of the two possible quantizable coadjoint orbits of Diff(S1)), embeds as a complex analytic and Kähler submanifold of the universal Teichmüller space. Furthermore, this very homogeneous space, Diff(S1)/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 S1 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.