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Superstring amplitudes meet surfaceology

Qu Cao, Jin Dong, Song He, Fan Zhu

TL;DR

The paper develops a novel framework that recasts tree-level open superstring amplitudes in terms of curve-integral stringy Tr$(\phi^3)$ data, via a Pfaffian expansion that yields a sum over shifted amplitudes. Upon scaffolding residues, the n-gluon superstring amplitude becomes a manifestly gauge-invariant linear combination of mixed bosonic string amplitudes with gluons and scalars, with nested commutator prefactors that reduce to traces of field-strengths. These constructions lead to cancellations of tachyon poles and F^3 vertices to all orders in the alpha' expansion and expose new relations between superstring and bosonic string building blocks, with extensions to heterotic and Type II via double copy. The results are derived in both worldsheet and curve-integral formalisms and hold promise for further applications to loops, external fermions, and BRST-invariant formulations. Altogether, the work provides a unifying, gauge-invariant bridge between stringy amplitudes across theories and opens new avenues for exact alpha'-level relations and closed-string generalizations.

Abstract

We reformulate tree-level amplitudes in open superstring theory (type-I) in terms of stringy Tr$(φ^3)$ amplitudes with various kinematical shifts in the "curve-integral" formulation: while the bosonic-string amplitude with $n$ pairs of "scaffolding" scalars comes from a particularly simple shift of the Tr$(φ^3)$ one (corresponding to $n$ length-$2$ cycles), the analogous superstring amplitude requires "correction" terms given by bosonic-string amplitudes with longer, even-length "cycles", which are also Tr$(φ^3)$ ones at shifted kinematics dictated by the cycles; in total it is expressed as a sum of $(2n{-}3)!!$ shifted amplitudes originated from the expansion of a reduced Pfaffian. Upon taking $n$ scaffolding residues, this leads to a new formula of the $n$-gluon superstring amplitude, which is manifestly symmetric in $n{-}1$ legs, as a gauge-invariant combination of mixed bosonic string amplitudes with gluons and scalars, which come from length-$2$ cycles and longer ones respectively (the total sum is associated with the expansion a $n\times n$ symmetrical determinant); the corresponding prefactors are nested commutators of $2n$-gon kinematical variables, which nicely become traces of field-strengths for those legs corresponding to scalars in the mixed amplitudes. These interesting linear combinations of bosonic string amplitudes must guarantee the cancellation of tachyon poles and $F^3$ vertices ${\it etc.}$, and they give new relations between the superstring amplitude and its bosonic-string building blocks to all orders in the $α'$ expansion (the first order gives a new formula for gluon amplitudes with a single $F^3$ insertion in terms of Yang-Mills-scalar amplitudes). We provide both the worldsheet and "curve-integral" derivations, and discuss applications to heterotic and type II cases.

Superstring amplitudes meet surfaceology

TL;DR

The paper develops a novel framework that recasts tree-level open superstring amplitudes in terms of curve-integral stringy Tr data, via a Pfaffian expansion that yields a sum over shifted amplitudes. Upon scaffolding residues, the n-gluon superstring amplitude becomes a manifestly gauge-invariant linear combination of mixed bosonic string amplitudes with gluons and scalars, with nested commutator prefactors that reduce to traces of field-strengths. These constructions lead to cancellations of tachyon poles and F^3 vertices to all orders in the alpha' expansion and expose new relations between superstring and bosonic string building blocks, with extensions to heterotic and Type II via double copy. The results are derived in both worldsheet and curve-integral formalisms and hold promise for further applications to loops, external fermions, and BRST-invariant formulations. Altogether, the work provides a unifying, gauge-invariant bridge between stringy amplitudes across theories and opens new avenues for exact alpha'-level relations and closed-string generalizations.

Abstract

We reformulate tree-level amplitudes in open superstring theory (type-I) in terms of stringy Tr amplitudes with various kinematical shifts in the "curve-integral" formulation: while the bosonic-string amplitude with pairs of "scaffolding" scalars comes from a particularly simple shift of the Tr one (corresponding to length- cycles), the analogous superstring amplitude requires "correction" terms given by bosonic-string amplitudes with longer, even-length "cycles", which are also Tr ones at shifted kinematics dictated by the cycles; in total it is expressed as a sum of shifted amplitudes originated from the expansion of a reduced Pfaffian. Upon taking scaffolding residues, this leads to a new formula of the -gluon superstring amplitude, which is manifestly symmetric in legs, as a gauge-invariant combination of mixed bosonic string amplitudes with gluons and scalars, which come from length- cycles and longer ones respectively (the total sum is associated with the expansion a symmetrical determinant); the corresponding prefactors are nested commutators of -gon kinematical variables, which nicely become traces of field-strengths for those legs corresponding to scalars in the mixed amplitudes. These interesting linear combinations of bosonic string amplitudes must guarantee the cancellation of tachyon poles and vertices , and they give new relations between the superstring amplitude and its bosonic-string building blocks to all orders in the expansion (the first order gives a new formula for gluon amplitudes with a single insertion in terms of Yang-Mills-scalar amplitudes). We provide both the worldsheet and "curve-integral" derivations, and discuss applications to heterotic and type II cases.
Paper Structure (16 sections, 103 equations, 3 figures, 3 tables)

This paper contains 16 sections, 103 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A unity of amplitudes for stringy Tr$(\phi^3)$, bosonic string and superstring
  3. $2n$-gon superstring as a sum of shifted stringy Tr$(\phi^3)$
  4. A new formula for the $n$-gluon superstring amplitude from scaffolding residues
  5. Properties and new structures of superstring amplitudes
  6. Cancellations of $F^3$ vertices, tachyon poles, etc.
  7. New relations for the $\alpha'$ expansions
  8. Double copy: heterotic and type II closed superstring
  9. Conclusions and outlook
  10. Worldsheet derivations
  11. Review of tree-level open string amplitudes
  12. Derivation of the new formula for superstring
  13. The closure of the new formula for superstring
  14. Curve integrals: review and derivations
  15. A lightening review of curve integrals for the disk
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Example of $6$-gon superstring amplitude. Red line denotes the perfect matching $\frac{1}{z_{i,j}}$. Black line denotes overall factor $\frac{1}{z_{2i-1,2i}}$. Here we have omitted $s_{1,2}{-}1$ and $s_{3,4}{-}1$.
  • Figure 2: Examples for the perfect macthings of $n=4$.
  • Figure 3: Scaffolding triangulation of a disk with 6 marked points (left), the corresponding fat graph (middle), and the word associated with the chord $(1,3)$ (right).