The momentum amplituhedron of SYM and ABJM from twistor-string maps
Song He, Chia-Kai Kuo, Yao-Qi Zhang
Abstract
A bstract We study remarkable connections between twistor-string formulas for tree amplitudes in $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 SYM and $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 6 ABJM, and the corresponding momentum amplituhedron in the kinematic space of D = 4 and D = 3, respectively. Based on the Veronese map to positive Grassmannians, we define a twistor-string map from G + (2, n ) to a (2 n− 4)-dimensional subspace of the 4d kinematic space where the momentum amplituhedron of SYM lives. We provide strong evidence that the twistor-string map is a diffeomorphism from G + (2 , n ) to the interior of momentum amplituhedron; the canonical form of the latter, which is known to give tree amplitudes of SYM, can be obtained as pushforward of that of former. We then move to three dimensions: based on Veronese map to orthogonal positive Grassmannian, we propose a similar twistor-string map from the moduli space $$ {\mathrm{\mathcal{M}}}_{0,n}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℳ</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> to a ( n− 3)-dimensional subspace of 3d kinematic space. The image gives a new positive geometry which conjecturally serves as the momentum amplituhedron for ABJM; its canonical form gives the tree amplitude with reduced supersymmetries in the theory. We also show how boundaries of compactified $$ {\mathrm{\mathcal{M}}}_{0,n}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℳ</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> map to boundaries of momentum amplituhedra for SYM and ABJM corresponding to factorization channels of amplitudes, and in particular for ABJM case the map beautifully excludes all unwanted channels.