193 research outputs found
Genus Two Meromorphic Conformal Field Theory
We construct the genus two (or two loop) partition function for meromorphic
bosonic conformal field theories. We use a sewing procedure involving two genus
one tori by exploiting an explicit relationship between the genus two period
matrix and pinching modular parameters. We obtain expressions for the partition
function for the chiral bosonic string, even rank lattice theories and
self-dual meromorphic conformal field theories including the Moonshine Module.
In particular, we find that for self-dual theories with central charge 24, the
genus two partition function multiplied by a universal holomorphic function of
the moduli is given by a meromorphic Siegel modular form of weight 2 where this
universal function includes ghost contributions. We also discuss a novel
expansion for certain Siegel modular forms.Comment: 25 pages, AMS Latex2e, 2 figures, Talk presented at Workshop on
Moonshine, CRM, Montreal, May 29 to June 4, 199
The geometry of maximal representations of surface groups into SO(2,n)
In this paper, we study the geometric and dynamical properties of maximal
representations of surface groups into Hermitian Lie groups of rank 2.
Combining tools from Higgs bundle theory, the theory of Anosov representations,
and pseudo-Riemannian geometry, we obtain various results of interest.
We prove that these representations are holonomies of certain geometric
structures, recovering results of Guichard and Wienhard. We also prove that
their length spectrum is uniformly bigger than that of a suitably chosen
Fuchsian representation, extending a previous work of the second author.
Finally, we show that these representations preserve a unique minimal surface
in the symmetric space, extending a theorem of Labourie for Hitchin
representations in rank 2.Comment: 56 pgs, section 3 has been reorganized , former sections 4.2 and 4.3
have been merged into section 4.2 and rewritten to avoid reference to maximal
surfaces and Higgs bundles, appendix added on strong version of
Ahlfors-Schwarz-Pick lemma. To appear in Duke Math Journa
Higher Teichm\"uller Spaces: from SL(2,R) to other Lie groups
The first part of this paper surveys several characterizations of
Teichm\"uller space as a subset of the space of representation of the
fundamental group of a surface into PSL(2,R). Special emphasis is put on
(bounded) cohomological invariants which generalize when PSL(2,R) is replaced
by a Lie group of Hermitian type. The second part discusses underlying
structures of the two families of higher Teichm\"uller spaces, namely the space
of maximal representations for Lie groups of Hermitian type and the space of
Hitchin representations or positive representations for split real simple Lie
groups.Comment: The file uploaded on May 12th was the wrong one and did not contain
the Section 4.6 that was added. This is the version to appear in the Handbook
of Teichm\"uller theor
Genus 0, 1, 2 actions of some almost simple groups of lie rank 2
Please see the paper.
Thanks
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