63 research outputs found
Equivalence of Geometric h<1/2 and Standard c>25 Approaches to Two-Dimensional Quantum Gravity
We show equivalence between the standard weak coupling regime c>25 of the
two-dimensional quantum gravity and regime h<1/2 of the original geometric
approach of Polyakov [1,2], developed in [3,4,5].Comment: 10 pages, late
On real projective connections, V.I. Smirnov's approach, and black hole type solutions of the Liouville equation
We consider real projective connections on Riemann surfaces and corresponding
solutions of the Liouville equation. It is shown that these solutions have
singularities of special type (of a black hole type) on a finite number of
simple analytical contours. The case of the Riemann sphere with four real
punctures, considered in V.I. Smirnov's thesis (Petrograd, 1918), is analyzed
in detail.Comment: 13 pages, final versio
Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces
We formulate and solve the analog of the universal Conformal Ward Identity
for the stress-energy tensor on a compact Riemann surface of genus , and
present a rigorous invariant formulation of the chiral sector in the induced
two-dimensional gravity on higher genus Riemann surfaces. Our construction of
the action functional uses various double complexes naturally associated with a
Riemann surface, with computations that are quite similar to descent
calculations in BRST cohomology theory. We also provide an interpretation for
the action functional in terms of the geometry of different fiber spaces over
the Teichm\"{u}ller space of compact Riemann surfaces of genus .Comment: 38 pages. Latex2e + AmsLatex2.1. One embedded figure. One section on
the relation with the geometry of fiber spaces on the Teichmueller space and
several important references adde
Generating Functional in CFT on Riemann Surfaces II: Homological Aspects
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 \v{C}ech resolution of any covering map of a
Riemann surface. Detailed calculations are presented in the two cases of an
ordinary \v{C}ech 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
Teichm\"uller 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.Comment: Latex, xypic, and AMS packages. 53 pages, 1 figur
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