The Complex Structure on Teichmüller Space

Abstract We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so-called higher complex structure, we use the punctual Hilbert scheme of the plane. The moduli space of higher complex structures is defined and is shown to be a generalization of the classical Teichmüller space. We give arguments for the conjectural isomorphism between the moduli space of higher complex structures and Hitchin’s component.

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Introduction of a complex structure on the p-integrable Teichmüller space

Annales Academiae Scientiarum Fennicae Mathematica ◽

10.5186/aasfm.2014.3952 ◽

2014 ◽

Vol 39 ◽

pp. 947-971 ◽

Cited By ~ 7

Author(s):

Masahiro Yanagishita

Keyword(s):

Complex Structure ◽

Teichmüller Space ◽

Teichmuller Space ◽

Integrable Teichmüller Space

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QUASISYMMETRIC SEWING IN RIGGED TEICHMÜLLER SPACE

Communications in Contemporary Mathematics ◽

10.1142/s0219199706002210 ◽

2006 ◽

Vol 08 (04) ◽

pp. 481-534 ◽

Cited By ~ 9

Author(s):

DAVID RADNELL ◽

ERIC SCHIPPERS

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Riemann Surfaces ◽

Operator Algebras ◽

Complex Structure ◽

Teichmüller Space ◽

Teichmuller Space ◽

Closed Curves ◽

Teichmüller Spaces ◽

Teichmuller Spaces

One of the basic geometric objects in conformal field theory (CFT) is the moduli space of Riemann surfaces whose n boundaries are "rigged" with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alternative model is the moduli space of n-punctured Riemann surfaces together with local biholomorphic coordinates at the punctures. We refer to both of these moduli spaces as the "rigged Riemann moduli space".By generalizing to quasisymmetric boundary parametrizations, and defining rigged Teichmüller spaces in both the border and puncture pictures, we prove the following results: (1) The Teichmüller space of a genus-g surface bordered by n closed curves covers the rigged Riemann and rigged Teichmüller moduli spaces of surfaces of the same type, and induces complex manifold structures on them; (2) With this complex structure, the sewing operation is holomorphic; (3) The border and puncture pictures of the rigged moduli and rigged Teichmüller spaces are biholomorphically equivalent.These results are necessary in rigorously defining CFT (in the sense of G. Segal), as well as for the construction of CFT from vertex operator algebras.

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