Analytic torsion and Faddeev-Popov ghosts
Date
2002
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Abstract
The regularized determinant of the Laplacian on n-differentials on
a hyperbolic Riemann surface is studied. The main result is an intrinsic
characterization of the connection form for the determinant
line bundle, endowed with the Quillen metric, over the Teichmüller
space, in terms of the Green’s function of the Cauchy-Riemann operator.
Further, an explicit series representation of that Green’s
function, on a Schottky uniformization of the surface, is established.
This is a rigorous version of physical heuristics due to
Martinec and Verlinde & Verlinde, relating the determinant to the
stress-energy tensor of Faddeev-Popov ghost fields on the Riemann
surface. One corollary is a simpler proof of the rigorous hyperbolic
Belavin-Knizhnik formula, due to Zograf and Takhtajan, which is
an intrinsic characterization of the curvature form of the determinant
line bundle with Quillen metric. Another corollary is a proof
of an explicit holomorphic factorization formula for n = 1 and
genus greater than 1, due to Zograf, which generalizes the well known
formula for n = 1 and genus 1 relating the determinant of
the Laplacian to the Dedekind eta function.
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Keywords
Teichmüller space, Laplacian, Riemann surface, Cauchy-Riemann, Quillen metric, Green’s function, Schottky space