Generating Functional in CFT on Riemann Surfaces II: Homological Aspects |
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Authors: | Ettore Aldrovandi Leon A Takhtajan |
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Institution: | S.I.S.S.A. International School for Advanced Studies, Via Beirut 2/4, 34013 Trieste, Italy.?E-mail: ettore@fm.sissa.it, IT Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794-3651, USA.?E-mail: leontak@math.sunysb.edu, US
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Abstract: | We revisit and generalize our previous algebraic construction of the chiral effective action for Conformal Field Theory on
higher genus Riemann surfaces. We show that the action functional can be obtained by evaluating a certain Deligne cohomology
class over the fundamental class of the underlying topological surface. This Deligne class is constructed by applying a descent
procedure with respect to a Čech resolution of any covering map of a Riemann surface. Detailed calculations are presented
in the two cases of an ordinary Čech cover, and of the universal covering map, which was used in our previous approach. We
also establish a dictionary that allows to use the same formalism for different covering morphisms.
The Deligne cohomology class we obtain depends on a point in the Earle–Eells fibration over the Teichmüller space, and on
a smooth coboundary for the Schwarzian cocycle associated to the base-point Riemann surface. From it, we obtain a variational
characterization of Hubbard's universal family of projective structures, showing that the locus of critical points for the
chiral action under fiberwise variation along the Earle–Eells fibration is naturally identified with the universal projective
structure.
Received: 29 June 2000 / Accepted: 16 January 2002 |
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