Superstring amplitudes meet surfaceology

We reformulate tree-level amplitudes in open superstring theory (type-I) in terms of stringy Tr (\phi^3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> amplitudes with various kinematical shifts in the “curve-integral” formulation: while the bosonic-string amplitude with n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> pairs of “scaffolding” scalars comes from a particularly simple shift of the Tr (\phi^3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> one (corresponding to n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> length- 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>2</mml:mn> </mml:math> cycles), the analogous superstring amplitude requires “correction” terms given by bosonic-string amplitudes with longer, even-length “cycles”, which are also Tr (\phi^3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> ones at shifted kinematics dictated by the cycles; in total it is expressed as a sum of (2n-3)!! <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>!</mml:mi> <mml:mi>!</mml:mi> </mml:mrow> </mml:math> shifted amplitudes originated from the expansion of a reduced Pfaffian. Upon taking n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> scaffolding residues, this leads to a new formula of the n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> -gluon superstring amplitude, which is manifestly symmetric in n-1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> legs, as a gauge-invariant combination of mixed bosonic string amplitudes with gluons and scalars, which come from length- 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>2</mml:mn> </mml:math> cycles and longer ones respectively (the total sum is associated with the expansion a n× n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> symmetrical determinant); the corresponding prefactors are nested commutators of 2n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>F</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> vertices etc., and they give new relations between the superstring amplitude and its bosonic-string building blocks to all orders in the \alpha' <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>α</mml:mi>