## Abstract

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 g with 2. r 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. r period matrix that is symmetric in the Z2-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.

Original language | English (US) |
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Pages (from-to) | 210-239 |

Number of pages | 30 |

Journal | Journal of Geometry and Physics |

Volume | 92 |

DOIs | |

State | Published - Jun 1 2015 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematical Physics
- General Physics and Astronomy
- Geometry and Topology

## Keywords

- Algebraic geometry
- Strings and superstrings
- Supermanifolds and supergroups