Color/kinematics duality for general abelian orbifolds of N=4 super Yang-Mills theory

TL;DR

This paper develops color/kinematics (BCJ) duality for general abelian orbifolds of ${\cal N}=4$ sYM, focusing on theories with adjoint and bi-fundamental fields and providing explicit one-loop four-point amplitudes for both gluons and scalars across ${\cal N}=2$, ${\cal N}=1$, and ${\cal N}=0$ cases. It introduces a graph-based organization where loop integrands are controlled by orbifold representations, derives one-loop kinematic Jacobi relations in orbifold theories, and demonstrates that regular orbifolds yield tadpole-free amplitudes with clean reductions to parent theories. The work also discusses a (non-factorized) double-copy construction for certain orbifold supergravities and shows that existence of duality is closely tied to supersymmetry, with a dimensionality condition $D=3,4,6,10$ emerging for theories with fermions. Overall, the results illuminate how color/kinematics duality extends to quiver/orbifold settings and outline a path toward broader applications in gravity via double-copy techniques.

Abstract

To explore color/kinematics duality for general representations of the gauge group we formulate the duality for general abelian orbifolds of the SU(N), N=4 super Yang-Mills theory in four dimensions, which have fields in the bi-fundamental representation, and use it to construct explicitly complete four-vector and four-scalar amplitudes at one loop. For fixed number of supercharges, graph-organized L-loop n-point integrands of all orbifold theories are given in terms of a fixed set of polynomials labeled by L representations of the orbifold group. In contrast to the standard duality-satisfying presentation of amplitudes of the N=4 super Yang-Mills theory, each graph may appear several times with different internal states. The color and R-charge flow provide a way to deform the amplitudes of orbifold theories to those of more general quiver gauge theories which do not necessarily exhibit color/kinematics duality on their own. Based on the organization of amplitudes required by the duality between color and kinematics in orbifold theories we show how the amplitudes of certain non-factorized matter-coupled supergravity theories can be found through a double-copy construction. We also carry out a comprehensive search for theories with fields solely in the adjoint representation of the gauge group and amplitudes exhibiting color/kinematics duality for all external states and find an interesting relation between supersymmetry and existence of the duality.

Figures (7)

• Figure 1: Quivers for the ${\cal N}=2, 1, 0$ examples. Each node is a gauge group factor and lines joining them are fields/multiplets in bi-fundamental representation. The arrow points from $\bar{\bf N}$ to ${\bf N}$. Lines starting and ending at the same node represent adjoint matter fields.
• Figure 2: The Feynman graphs contributing to the four-fermion amplitude.
• Figure 3: Basis of box integrals at one loop.
• Figure 4: Basis of triangle integrals at one loop.
• Figure 5: Basis of bubble integrals at one loop.