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Title: The positive orthogonal Grassmannian and loop amplitudes of ABJM

Journal Article · · Journal of Physics. A, Mathematical and Theoretical
 [1];  [2];  [3]
  1. Institute for Advanced Study, Princeton, NJ (United States). School of Natural Sciences; National Taiwan Univ., Taipei (Taiwan)
  2. Queen Mary Univ. of London (United Kingdom)
  3. Institute for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
+ Show Author Affiliations

Herein we study the combinatorics associated with the positive orthogonal Grassmannian OGk and its connection to ABJM scattering amplitudes. We present a canonical embedding of OGk into the Grassmannian Gr(k, 2k), from which we deduce the canonical volume form that is invariant under equivalence moves. Remarkably the canonical forms of all reducible graphs can be converted into reduced ones with products of dlog forms. Unlike super Yang–Mills, here the Jacobian plays a crucial role to ensure the dlog form of the reduced representation. Furthermore in the positive region, we identify the functional map that arises from the triangle equivalence move as a 3-string scattering S-matrix which satisfies the tetrahedron equations by Zamolodchikov, implying (2 + 1)-dimensional integrability. We discuss the solution to the BCFW recursion relation for loop amplitudes, and demonstrate the presence of all physical singularities as well as the absence of all spurious ones. The on-shell diagram solution to the loop recursion relation exhibits manifest two-site cyclic symmetry and reveals that, to all loop, four and six-point amplitudes only have logarithmic singularities.

Research Organization:
Institute for Advanced Study, Princeton, NJ (United States)
Sponsoring Organization:
USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF); Science and Technology Facilities Council
Grant/Contract Number:
SC0009988; PHY-1314311; ST/J000469/1
OSTI ID:
1599344
Journal Information:
Journal of Physics. A, Mathematical and Theoretical, Vol. 47, Issue 47; ISSN 1751-8113
Publisher:
IOP PublishingCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 24 works
Citation information provided by
Web of Science

References (26)

A twistor string for the ABJ(M) theory journal June 2014
Conformal anomaly for amplitudes in $\mathcal {N}=6$ superconformal Chern–Simons theory journal November 2012
The two-loop six-point amplitude in ABJM theory journal March 2013
Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond journal October 2005
Reseaux électriques planaires II journal December 1996
The geometry of on-shell diagrams journal August 2014
Scattering amplitudes/Wilson loop duality in ABJM theory journal January 2012
Yangian-Invariant Scattering Amplitudes in Supersymmetric Chern-Simons Theory journal October 2010
Positroid stratification of orthogonal Grassmannian and ABJM amplitudes journal September 2014
All one-loop amplitudes in $ \mathcal{N}=6 $ superconformal Chern-Simons theory journal October 2012
Tetrahedron equations and the relativisticS-matrix of straight-strings in 2+1-Dimensions journal December 1981
The Amplituhedron journal October 2014
An all order identity between ABJM and $ \mathcal{N} $ = 4 SYM four-point amplitudes journal April 2012
One loop amplitudes in ABJM journal July 2012
Dualities for loop amplitudes of N = 6 Chern-Simons matter theory journal November 2011
Tree-level recursion relation and dual superconformal symmetry of the ABJM theory journal March 2011
A note on amplitudes in $$ \mathcal{N} = {6} $$ superconformal Chern-Simons theory journal July 2012
Scattering amplitudes and toric geometry journal September 2013
Circular planar graphs and resistor networks journal November 1998
Functional tetrahedron equation journal December 1998
A duality for the S matrix journal March 2010
Scattering Amplitudes/Wilson Loop Duality In ABJM Theory text January 2011
One Loop Amplitudes In ABJM text January 2012
Scattering Amplitudes and Toric Geometry text January 2013
The Geometry of On-Shell Diagrams text January 2013
Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond text January 2005

Cited By (3)

Unified formalism for 6D superamplitudes based on a symplectic Grassmannian journal August 2019
Integrable amplitude deformations for N = 4 super Yang-Mills and ABJM theory journal January 2015
Unified Formalism for 6D Superamplitudes Based on a Symplectic Grassmannian text January 2019

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Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
supersymmetric gauge theory
scattering amplitudes
Grassmannian manifolds
combinatorics