The super period matrix with Ramond punctures

We generalize the super period matrix of a super Riemann surface to the case that Ramond punctures are present. For a super Riemann surface of genus gg with 2r2r Ramond punctures, we define, modulo certain choices that generalize those in the classical theory (and assuming a certain generic condition is satisfied), a g|r×g|rg|r×g|r period matrix that is symmetric in the Z2Z2-graded sense. As an application, we analyze the genus 2 vacuum amplitude in string theory compactifications to four dimensions that are supersymmetric at tree level. We find an explanation for a result that has been found in orbifold examples in explicit computations by D’Hoker and Phong: with their integration procedure, the genus 2 vacuum amplitude always vanishes “pointwise” after summing over spin structures, and hence is given entirely by a boundary contribution.