The Complete KLT-Map Between Gravity and Gauge Theories

TL;DR

The work provides a complete 4D catalog of KLT mappings between super Yang–Mills theories and supergravity theories across $ ext{N}=4,3,2,1,0$ by using the S-kernel and on-shell superamplitudes. A diamond-diagram formalism facilitates organizing states and deriving the resulting gravity multiplets from tensor products of gauge theories, revealing equivalences such as $ ext{N}=3 ext{ YM} o ext{N}=4 ext{ YM}$ and $ ext{N}_G=7 ext{,8}$ gravity equivalence. The authors systematically derive symmetry groups, including $SU( ext{N}_G)_R$ and extra $U(1)$ factors, and show how vanishing identities arise from R-symmetry and charge conservation constraints. The results illuminate how maximal and non-maximal supersymmetric theories relate under KLT, and provide a framework for projecting to minimal gravity multiplets with or without matter content, with potential extensions to matter-rich YM theories and loop-level considerations.

Abstract

We present the complete map of any pair of super Yang-Mills theories to supergravity theories as dictated by the KLT relations in four dimensions. Symmetries and the full set of associated vanishing identities are derived. A graphical method is introduced which simplifies counting of states, and helps in identifying the relevant set of symmetries.

Figures (16)

• Figure 1: Diamond diagrams for superfields of super Yang-Mills theories with increasing amount of supersymmetry. The $SU(\mathcal{N})_R$ indices $a,b,c$ are labeled as superscripts, where $a<b<c$ with $a,b,c=1,2,\ldots,\mathcal{N}$. The hidden indices, which refer to where they originated from in the maximally supersymmetric multiplet are indicated in parentheses. The numbers inside the diamonds show the number of corresponding states on each horizontal line.
• Figure 2: Diamond diagrams that demonstrates the matching of states in (supergravity)$_{\mathcal{N}_G=8}$ = (super Yang-Mills)$_{\widetilde{\mathcal{N}}=4}\otimes$ (super Yang-Mills)$_{\mathcal{N}=4}$. The numbers inside the diamonds indicate the number of states on each line, and the number next to the dots indicate the helicities. Only the highest, lowest and zero helicities have been labeled explicitly.
• Figure 3: Diamond diagram for (supergravity)$_{\mathcal{N}_G=7}$ = (super Yang-Mills)$_{\widetilde{\mathcal{N}}=4}\otimes$(super Yang-Mills)$_{\mathcal{N}=3}$. The two diamonds represent the $\Phi^{\mathcal{N}_G=7}$ and $\Psi^{\mathcal{N}_G=7}$ superfields, respectively. Note that there is a hidden $SU(8)_R$ index in the $\Psi$ field, for instance, the negative helicity graviton state is $-2^{1234567(8)}$.
• Figure 4: Diamond diagram for (supergravity)$_{\mathcal{N}_G=6}$ = (super Yang-Mills)$_{\widetilde{\mathcal{N}}=4}\otimes$ (super Yang-Mills)$_{\mathcal{N}=2}$. The two diamonds represent the $\Phi^{\mathcal{N}_G=6}$ and $\Psi^{\mathcal{N}_G=6}$ superfields, respectively. There are two hidden indices $(78)$ for the $\Psi$ field.
• Figure 5: Diamond diagram for (supergravity)$_{\mathcal{N}_G=6}$ = (super Yang-Mills)$_{\widetilde{\mathcal{N}}=3}\otimes$ (super Yang-Mills)$_{\mathcal{N}=3}$. The four diamonds correspond to the four superfields $\Phi^{\mathcal{N}_G=6}$, $\Theta^{\mathcal{N}_G=6}$, $\Gamma^{\mathcal{N}_G=6}$ and $\Psi^{\mathcal{N}_G=6}$. The hidden indices are $(4)$ for $\Theta$, $(8)$ for $\Gamma$ and $(48)$ for $\Psi$ superfield.