Global symmetries of Yang-Mills squared in various dimensions
A. Anastasiou, L. Borsten, M. J. Hughes, S. Nagy
TL;DR
The paper develops a unified, division-algebra–driven framework to describe the global symmetries of supergravity theories obtained by tensoring two on-shell super Yang–Mills multiplets in dimensions $3 \le D \le 10$. It introduces three central algebras—R-symmetry $\mathfrak{ra}(\mathcal{N},D)$, the maximal compact subgroup $\mathfrak{h}$, and the U-duality algebra $\mathfrak{g}$—each expressed in terms of the left and right Yang–Mills internal symmetries and a division-algebra label $\mathds{D}$ associated with $D$. The authors provide explicit formulae: $\mathfrak{ra}(\mathcal{N}_L+\mathcal{N}_R,D)=\mathfrak{a}(\mathcal{N}_L,\mathds{D})\oplus\mathfrak{a}(\mathcal{N}_R,\mathds{D})\oplus\mathds{D}[\mathcal{N}_L,\mathcal{N}_R]$, $\mathfrak{h}(\mathcal{N}_L+\mathcal{N}_R,D)=\mathfrak{int}(\mathcal{N}_L,D)\oplus\mathfrak{int}(\mathcal{N}_R,D)\oplus\delta_{D,4}\mathfrak{u}(1)\oplus\mathds{D}[\mathcal{N}_L,\mathcal{N}_R]$, and $\mathfrak{g}(\mathcal{N}_L+\mathcal{N}_R,D)=\mathfrak{h}(\mathcal{N}_L+\mathcal{N}_R,D)\oplus\mathds{D}_*[\mathcal{N}_L]\otimes\mathds{D}_*[\mathcal{N}_R]\oplus\mathds{D}[\mathcal{N}_L,\mathcal{N}_R]\oplus\mathds{R}_L\otimes\mathds{R}_R+i\delta_{D,4}\mathds{R}_L\otimes\mathds{R}_R$. These constructions systematically reproduce the U-duality pyramids (and the $D=3$ Freudenthal magic square) by encoding left/right internal symmetries and the associated division-algebra content. The framework accommodates matter couplings via non-supersymmetric tensor factors and clarifies how scalar moduli populate symmetric cosets $G/H$, with explicit examples (e.g., in $D=5$). The work thus provides a compact, symmetry-based atlas for the global symmetries of gravity theories arising from Yang–Mills squaring, with implications for understanding M-theory structures and UV properties of half-maximal and lower-supersymmetry theories.
Abstract
Tensoring two on-shell super Yang-Mills multiplets in dimensions $D\leq 10$ yields an on-shell supergravity multiplet, possibly with additional matter multiplets. Associating a (direct sum of) division algebra(s) $\mathbb{D}$ with each dimension $3\leq D\leq 10$ we obtain formulae for the algebras $\mathfrak{g}$ and $\mathfrak{h}$ of the U-duality group $G$ and its maximal compact subgroup $H$, respectively, in terms of the internal global symmetry algebras of each super Yang-Mills theory. We extend our analysis to include supergravities coupled to an arbitrary number of matter multiplets by allowing for non-supersymmetric multiplets in the tensor product.