TY - JOUR
T1 - The super period matrix with Ramond punctures
AU - Witten, Edward
N1 - Funding Information:
Research was partly supported by NSF Grant PHY-1314311 . I would like to thank E. D’Hoker, R. Donagi, and D. Phong for discussions, and D’Hoker and Phong for help in reconciling some formulas here with their results. I also thank P. Deligne for detailed comments on an earlier version and for several helpful suggestions.
Publisher Copyright:
© 2015 Elsevier B.V.
PY - 2015/6/1
Y1 - 2015/6/1
N2 - 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.
AB - 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.
KW - Algebraic geometry
KW - Strings and superstrings
KW - Supermanifolds and supergroups
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U2 - 10.1016/j.geomphys.2015.02.017
DO - 10.1016/j.geomphys.2015.02.017
M3 - Review article
AN - SCOPUS:84924726149
SN - 0393-0440
VL - 92
SP - 210
EP - 239
JO - Journal of Geometry and Physics
JF - Journal of Geometry and Physics
ER -