Global symmetries of Yang-Mills squared in various dimensions

A Anastasiou,M J Hughes,L Borsten,S Nagy

doi:10.1007/jhep01(2016)148

A Anastasiou, M J Hughes + Show 2 more

Open Access

https://doi.org/10.1007/jhep01(2016)148

Copy DOI

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.

Highlights

Which takes as its argument a pair of division algebras ANL, ANR = R, C, H, O, where we have adopted the convention that dim AN = N
The non-compact generators p, in a manifest int(NL, D) ⊕ int(NR, D) basis, can be read off from those tensor products which yield scalars, which are schematically given by: Aμ ⊗ Aν, λa ⊗ λa, φi ⊗ φi. To recast this observation into the language used for h(NL, NR, D), we summarise here the corresponding division algebraic characterisation of the (D, N ) super Yang-Mills multiplet (Aμ, λa, φi): (i) The gauge potential Aμ is a sa(N, D) singlet valued in R. (ii) The N gaugini λa transform in the defining representation of sa(N, D) and are valued in DN . (iii) The scalars φi span a subspace D∗[N ] ⊆ D[N ] ∼= DN ⊗ DN since they are quadratic in the supersymmetry charges valued in DN
We have shown that the U-duality algebras g for all supergravity multiplets obtained by tensoring two super Yang-Mills multiplets in D ≥ 3 can be written in a single formula with three arguments, g(NL + NR, D)

Summary

Global symmetries of super Yang-Mills squared

Tensoring NL-extended and NR-extended super Yang-Mills multiplets, [NL]V and [NR]V , yields an (NL + NR)-extended supergravity multiplet, [NL + NR]grav,. We consider on-shell space-time little group super Yang-Mills multiplets with global symmetry algebra so(D − 2)ST ⊕ int(N , D),. Where a, i and a , i are indices of the appropriate int(NL, D) and int(NR, D) representations, respectively. The detailed form of these tensor products for D > 3 are summarised in table 2 and table 3, where for a given little group representation we have collected the int(NL, D) ⊕ int(NR, D) representations into the appropriate representations of h(NL + NR, D). Consider the square of the D = 5, N = 2 super Yang-Mills multiplet, which has global symmetry algebra so(3)ST ⊕ sp(2), Aμ (3; 1).

R-symmetry algebras

H algebras

G algebras

Discussion