gravity - Vasiliev Higher Spin Theory and Supersymmetry

Recently there is renewed interest in the ideas of Vasiliev, Fradkin and others on generalizing gravity theories on deSitter or Anti-deSitter spaces to include higher spin fields (utilizing known loopholes in the Weinberg-Witten theorem by including infinitely many higher spin fields and by working with asymptotic conditions that do not permit S-matrix to exist). There is also a conjecture for a duality for the theory as formulated in asymptotically AdS space with the O(N) vector model in the large N limit.

So in this context I am curious if there are supersymmetric generalizations of the theory, and how much supersymmetry can be shown to be consistent with this set of ideas (given that the usual restriction to 32 supercharges comes from forbidding higher spin fields).

Answer

A supersymmetric extension for ${\mathrm{AdS}}_4$ background was found by Konstein and Vasiliev in Nucl.Phys.B331:475-499,1990, and later generalised by Vasiliev in hep-th/0404124 to higher dimensions. In 4d, there are three classes of infinite-dimensional extended higher spin superalgebras which generate symmetries of the higher spin equations of motion on ${\mathrm{AdS}}_4$. In each case, the bosonic part contains a subalgebra of the form ${\mathfrak{so}}(3,2) \oplus {\mathfrak{g}}(m) \oplus {\mathfrak{g}}(n)$, comprising the ${\mathrm{AdS}}_4$ isometries and ${\mathfrak{g}}$ being either ${\mathfrak{u}}$, ${\mathfrak{o}}$ or ${\mathfrak{usp}}$. The corresponding higher spin superalgebras are denoted ${\mathfrak{hg}}(m,n|4)$. They contain the usual $N$-extended lie superalgebra ${\mathfrak{osp}}(N|4)$ as a subalgebra only when $m=n$. Indeed, for $m\neq n$, massless unitary irreps of ${\mathfrak{hg}}(m,n|4)$ contain a different number of bosons and fermions. In the simplest class with ${\mathfrak{g}}={\mathfrak{u}}$, bosons have all integer spins $\gt$ 1 and are in the adjoint of ${\mathfrak{u}}(m) \oplus {\mathfrak{u}}(n)$ while fermions have all half-integer spins $\gt$ 3/2 and are in the bifundamental of ${\mathfrak{u}}(m) \oplus {\mathfrak{u}}(n)$. (The standard spin 2 graviton is contained in a diagonal ${\mathfrak{u}}(1)$ factor.) The amount of extended higher spin supersymmetry in this sense is therefore unconstrained.