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Hyperbolic three-string vertex
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We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville's equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.
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Mixed Solutions to the Liouville Equation
Explicit mixed-monodromy Liouville solutions are constructed and their large-hyperbolic limits are used to describe time-symmetric slices of 2+1 gravity with particles and black holes.
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