$N=1$ Supersymmetry, Weil-Petersson Volume Recursion, and a Spectral Curve

Abstract

The Stanford-Witten-Norbury generalization of Mirzakhani's volume recursion computes $V^{(2m)}_{g,n}(\{b_i\})$, the Weil-Petersson volumes of the moduli space of $N=1$ supersymmetric Riemann surfaces of genus $g$ with $n$ Neveu-Schwarz boundaries of geodesic lengths $b_i$ ($i{=}1,\ldots,n$), and $2m$ Ramond punctures. Recently, a spectral curve has been derived that allows their Laplace transforms $W^{(2m)}_{g,n}(\{{\hat z}_i\})$ to be computed using topological recursion. We prove that the Stanford-Witten-Norbury volume recursion is directly derivable from the spectral curve. An alternative volume recursion can also be derived from it. The difference comes from whether the Ramond information is in the initial data, or in the volume recursion's kernels. The latter invites a geometrical understanding.