Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two-Loop Superstrings V: Gauge Slice Independence of the N-Point Function

Eric D'Hoker, D. H. Phong

TL;DR

This work establishes gauge-slice independence for two-loop N-point superstring amplitudes by projecting supergeometries to the super period matrix ${\hat{\Omega}}_{IJ}$ and incorporating corrected vertex operators. The authors show that chiral amplitudes are closed forms whose changes under slice variations are Dolbeault- and de Rham-exact, enabling holomorphic, slice-independent left-moving and right-moving pairings after GSO projection. A key technical advance is reformulating vertex operators to include a deformed volume form, yielding a consistent ${\cal V}$ with both $(1,0)$ and $(0,1)$ components and a non-renormalized super-period matrix. They further prove that all slice-dependence can be captured by exact differential terms, which vanish upon integration, thereby delivering gauge-slice–independent, physically meaningful amplitudes with only physical kinematical singularities and setting the stage for explicit calculations of higher-point functions."

Abstract

A systematic construction of superstring scattering amplitudes for $N$ massless NS bosons to two loop order is given, based on the projection of supermoduli space onto super period matrices used earlier for the superstring measure in the first four papers of this series. The one important new difficulty arising for the $N$-point amplitudes is the fact that the projection onto super period matrices introduces corrections to the chiral vertex operators for massless NS bosons which are not pure (1,0) differential forms. However, it is proved that the chiral amplitudes are closed differential forms, and transform by exact differentials on the worldsheet under changes of gauge slices. Holomorphic amplitudes and independence of left from right movers are recaptured after the extraction of terms which are Dolbeault exact in one insertion point, and de Rham closed in the remaining points. This allows a construction of GSO projected, integrated superstring scattering amplitudes which are independent of the choice of gauge slices and have only physical kinematical singularities.

Two-Loop Superstrings V: Gauge Slice Independence of the N-Point Function

TL;DR

This work establishes gauge-slice independence for two-loop N-point superstring amplitudes by projecting supergeometries to the super period matrix and incorporating corrected vertex operators. The authors show that chiral amplitudes are closed forms whose changes under slice variations are Dolbeault- and de Rham-exact, enabling holomorphic, slice-independent left-moving and right-moving pairings after GSO projection. A key technical advance is reformulating vertex operators to include a deformed volume form, yielding a consistent with both and components and a non-renormalized super-period matrix. They further prove that all slice-dependence can be captured by exact differential terms, which vanish upon integration, thereby delivering gauge-slice–independent, physically meaningful amplitudes with only physical kinematical singularities and setting the stage for explicit calculations of higher-point functions."

Abstract

A systematic construction of superstring scattering amplitudes for massless NS bosons to two loop order is given, based on the projection of supermoduli space onto super period matrices used earlier for the superstring measure in the first four papers of this series. The one important new difficulty arising for the -point amplitudes is the fact that the projection onto super period matrices introduces corrections to the chiral vertex operators for massless NS bosons which are not pure (1,0) differential forms. However, it is proved that the chiral amplitudes are closed differential forms, and transform by exact differentials on the worldsheet under changes of gauge slices. Holomorphic amplitudes and independence of left from right movers are recaptured after the extraction of terms which are Dolbeault exact in one insertion point, and de Rham closed in the remaining points. This allows a construction of GSO projected, integrated superstring scattering amplitudes which are independent of the choice of gauge slices and have only physical kinematical singularities.

Paper Structure

This paper contains 23 sections, 110 equations.

Table of Contents

  1. Introduction
  2. Corrections to the Vertex Operators
  3. The Chiral Amplitudes ${\cal B}[\delta]$
  4. The Structure of Chiral and Holomorphic Blocks
  5. Summary of the Algorithm for the $N$-point Function
  6. Chiral Splitting and Chiral Blocks
  7. Chiral Splitting
  8. Deformation to $\hat{\Omega}_{IJ}$: the Chiral Measure
  9. Deformation to $\hat{\Omega}_{IJ}$: the Vertex Operators
  10. Non-Renormalization of the Super-Period Matrix
  11. The $dx_+^\mu$ Formulation of Vertex Operators
  12. Preliminaries
  13. Reformulation
  14. The Amplitudes ${\cal B}[\delta]$ as Closed Differential Forms
  15. $\hat{\mu}$-Dependence
  16. ...and 8 more sections