On the maximal superalgebras of supersymmetric backgrounds

Abstract

Abstract. In this note we give a precise definition of the notion of a maximal superalgebra of certain types of supersymmetric supergravity backgrounds, including the Freund-Rubin backgrounds, and propose a geometric construction extending the well-known construction of its Killing superalgebra. We determine the structure of maximal Lie superalgebras and show that there is a finite number of isomorphism classes, all related via contractions from an orthosymplectic Lie superalgebra. We use the structure theory to show that maximally supersymmetric waves do not possess such a maximal superalgebra, but that the maximally supersymmetric Freund-Rubin backgrounds do. We perform the explicit geometric construction of the maximal superalgebra of AdS 4 ×S 7 and find that is isomorphic to osp(1|32). We propose an algebraic construction of the maximal superalgebra of any background asymptotic to AdS 4 ×S 7 and we test this proposal by computing the maximal superalgebra of the M2-brane in its two maximally supersymmetric limits, finding agreement.

Supersymmetric Yang-Mills theory on conformal supergravity backgrounds in ten dimensions

“…The proof is virtually identical to the one for the Cahen-Wallach vacua of eleven-dimensional and type IIB Poincaré supergravities in [110]. As explained in ( [110], section 3), for any maximal superalgebra m, k0 acts trivially on the radical k ⊥ 1 of the skewsymmetric bilinear form ω characterising m. Now, inspecting equation (2.34) we see that ζ = ∂ − acts semisimply on k1 with nonzero eigenvalues, so that k k0 1 = 0. Therefore ω, having trivial radical, must be symplectic and hence, from equation (2.49), it follows that m must have trivial centre.…”

“…In [110] the notion of the maximal superalgebra of a supergravity background was introduced, generalising to non-flat backgrounds the M-algebra of [111]. Given a supergravity background with Killing superalgebra k = k0 ⊕ k1, the maximal superalgebra (should it exist) is defined to be Lie superalgebra m = m0 ⊕ m1, satisfying the following properties 1. m1 = k1 and k0 is a Lie subalgebra of m0; 2. the odd-odd bracket is an isomorphism ⊙ 2 m1 ∼ = m0; and 3. the projection ⊙ 2 m1 → k0 coincides with the odd-odd bracket of k and the restriction to k0 of the bracket m0 ⊗ m1 → m1 is the k-bracket.…”

“…We start by showing that the maximal superalgebra of k C is indeed isomorphic to osp(1|16) C , following the construction in [110], mutatis mutandis. First of all, let us de- , this is precisely the odd-odd bracket of k C .…”

“…First of all, let us de- , this is precisely the odd-odd bracket of k C . It follows from the discussion in ( [110], section 4.3) that the action of k0 on k1, when viewed through the lens of the cone construction, is via the Clifford action of the parallel 2-forms on the cones of AdS p and S q to which the special Killing 1-forms in k0 lift. Therefore we extend this Clifford action to all of m0, which also lift as parallel forms to the cones.…”

Conformal symmetry superalgebras

“…A well-established feature of supersymmetric supergravity backgrounds is that they inherit a rigid Lie superalgebra structure, known as the symmetry superalgebra of the background [16][17][18][19][20]. The even part of this superalgebra contains the Killing vectors which generate isometries of the background.…”