Abstract
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.
Highlights
Riemann surfaces using the associated Teichmüller spaces [12,13,14,15,16,17,18]
It is not known that such corrected string vertices always exist. They are intriguing in their own rights, we see that two proposals for string vertices above suffer from either missing the proof of existence or failing to satisfy the geometric master equation exactly, falling short of providing a consistent string field theory
In order to have a consistent string field theory we must guarantee that the string vertices exist on the moduli spaces of Riemann surfaces while exactly satisfying the geometric master equation
Summary
We describe the problem of finding an explicit description of the hyperbolic metric on the three-holed sphere with geodesic boundaries of lengths Li on the Riemann sphere, which will help us obtain the shapes and locations of the geodesic boundaries and the local coordinates later on. In order to do that, let the factor φ denote a solution of Liouville’s equation (2.1) and define the (holomorphic) stress-energy tensor associated with φ as follows [37]: Tφ(z). Φ(z,z) 2 is a solution of (2.5), even though the scaled ones don’t define a hyperbolic metric with K = −1 because the Liouville’s equation (2.1) is non-linear We can fix such C once and for all as follows. We have already seen that the conformal factor (2.13) always defines a (possibly singular) hyperbolic metric for any given f (z), but the boundaries of X are going to be geodesics only when we relate it to a particular set of solutions for the Fuchsian equation through the relation (2.14), as we shall see. We will cut open appropriate holes around the punctures in X to return back to X ⊂ C and graft flat semi-infinite cylinders to these holes to construct the local coordinates for the hyperbolic three-string vertex
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