$DIFF(s^1)$, Teichm\"Uller Space and Period Matrices: Canonical Mappings via String Theory

TL;DR

This paper explores the deep connections between the diffeomorphism group of the circle, Teichmüller spaces, and period mappings, revealing new insights relevant to non-perturbative string theory and extending classical concepts to the universal Teichmüller space.

Contribution

It introduces a natural holomorphic period mapping on the universal Teichmüller space, generalizing classical period matrices, and links it to string theory via the diffeomorphism group of the circle.

Findings

The homogeneous space Diff(S^1)/SL(2,R) embeds as a Kähler submanifold of the universal Teichmüller space.

A natural holomorphic period mapping extends classical period matrices to the universal Teichmüller space.

Connections between string theory, Teichmüller theory, and the diffeomorphism group of the circle are established.

Abstract

There is a completely natural and intimate relationship between the diffeomorphism group of the circle and the Teichm\"uller 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 K\"ahler submanifold of the universal Teichm\"uller space. Furthermore, this very homogeneous space, Diff$(S^1)$/$SL(2,R)$, considered by the previous work as a K\"ahler submanifold of the universal Teichm\"uller space, allows on it a natural holomorphic period mapping, $\Pi$, 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 $\Pi$ to the entire universal Teichm\"uller space. All this is related to non-perturbative string theory.