Abstract
The string vertices of closed string field theory are subsets of the moduli spaces of punctured Riemann surfaces that satisfy a geometric version of the Batalin-Vilkovisky master equation. We present a homological proof of existence of string vertices and their uniqueness up to canonical transformations. Using hyperbolic metrics on surfaces with geodesic boundaries we give an exact construction of string vertices as sets of surfaces with systole greater than or equal to L with L ≤ 2 arcsinh 1. Intrinsic hyperbolic collars prevent the appearance of short geodesics upon sewing. The surfaces generated by Feynman diagrams are naturally endowed with Thurston metrics: hyperbolic on the vertices and flat on the propagators. For the classical theory the length L is arbitrary and, as L → ∞ hyperbolic vertices become the minimal-area vertices of closed string theory.
Highlights
The list of vertices above is precisely that for which the surfaces have negative Euler number and admit hyperbolic metrics of constant negative curvature
The string vertices of closed string field theory are subsets of the moduli spaces of punctured Riemann surfaces that satisfy a geometric version of the Batalin-Vilkovisky master equation
We have shown that Pg,n is isomorphic to Pg,n, which is homotopy equivalent to the moduli space of Riemann surfaces with boundary
Summary
We will review the homological argument of [7] that proves the existence of the string vertices, and their uniqueness up to canonical transformation. (The super-string version of this argument has recently been provided by Moosavian and Zhou [24].) We show how the construction guarantees that the chain constructed as Feynman diagrams represents the fundamental homology class of Mg,n. We will review the homological argument of [7] that proves the existence of the string vertices, and their uniqueness up to canonical transformation. (The super-string version of this argument has recently been provided by Moosavian and Zhou [24].) We show how the construction guarantees that the chain constructed as Feynman diagrams represents the fundamental homology class of Mg,n. The material provides perspective on the problem of finding explicit string vertices, but is not required for the construction of hyperbolic vertices in the subsequent sections
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