The N=1\mathcal{N}=1 algebra W∞[μ]\mathcal{W}_\infty[\mu] and its truncations

The main objective of this work is to construct and classify the most general classical and quantum $\mathcal{N}=1$ $\mathcal{W}_\infty$-algebras generated by the same spins as the singlet algebra of $N$ fermions and $N$ bosons in the vector representation of $O(N)$ in the $N\to\infty$ limit. This type of algebras appears in a recent $\mathcal{N}=1$ version of the minimal model holography. Our analysis strongly suggests that there is a one parameter family $\mathcal{W}_\infty[\mu]$ of such algebras at every given central charge. Relying on this assumption, we identify various truncations of $\mathcal{W}_\infty[\mu]$ with, on the one hand, (orbifolds of) the Drinfel'd-Sokolov reductions of the Lie superalgebras $B(n,n)$, $B(n-1,n)$, $D(n,n)$ and $D(n+1,n)$, and, on the other hand, (orbifolds of) three $\mathcal{N}=1$ cosets. After a closer inspection we show that these cosets can be realized as a Drinfel'd-Sokolov reduction of $B(n,n)$, $D(n,n)$ and $D(n+1,n)$. We then discuss the implications of our findings for the quantum version of the $\mathcal{N}=1$ minimal model holography.Comment: old section 5.4 removed, 46 page

Extended supersymmetry in AdS_3 higher spin theories

We determine the asymptotic symmetry algebra (for fields of low spin) of the $M\times M$ matrix extended Vasiliev theories on AdS$_3$ and find that it agrees with the $\mathcal{W}$-algebra of their proposed coset duals. Previously it was noticed that for $M=2$ the supersymmetry increases from $\mathcal{N}=2$ to $\mathcal{N}=4$. We study more systematically this type of supersymmetry enhancements and find that, although the higher spin algebra has extended supersymmetry for all $M\geq 2$, the corresponding asymptotic symmetry algebra fails to be superconformal except for $M=2$, when it has large $\mathcal{N}=4$ superconformal symmetry. Moreover, we find that the Vasiliev theories based on $shs^E\! \left( \mathcal{N} \vert 2, \mathbb{R} \right)$ are special cases of the matrix extended higher spin theories, and hence have the same supersymmetry properties.Comment: 23 page

Even spin minimal model holography

The even spin W^e_\infty algebra that is generated by the stress energy tensor together with one Virasoro primary field for every even spin s \geq 4 is analysed systematically by studying the constraints coming from the Jacobi identities. It is found that the algebra is characterised, in addition to the central charge, by one free parameter that can be identified with the self-coupling constant of the spin 4 field. We show that W^e_\infty can be thought of as the quantisation of the asymptotic symmetry algebra of the even higher spin theory on AdS_3. On the other hand, W^e_\infty is also quantum equivalent to the so(N) coset algebras, and thus our result establishes an important aspect of the even spin minimal model holography conjecture. The quantum equivalence holds actually at finite central charge, and hence opens the way towards understanding the duality beyond the leading 't Hooft limit.Comment: 32 pages, v2: reference added, minor changes in tex

A lattice approach to the conformal \OSp(2S+2|2S) supercoset sigma model. Part II: The boundary spectrum

We consider the partition function of the boundary $OSp(2S+2|2S)$ coset sigma model on an annulus, based on the lattice regularization introduced in the companion paper. Using results for the action of $OSp(2S+2|2S)$ and $B_L(2)$ on the corresponding spin chain, as well as mini-superspace and small $g_\sigma^2$ calculations, we conjecture the full spectrum and set of degeneracies on the entire critical line. Potential relationship with the $OSp(2S+2|2S)$ Gross-Neveu model is also discussed.Comment: 32 pages, 7 figure

On the coset duals of extended higher spin theories

We study the holographic duality between the M x M matrix extension of Vasiliev higher spin theories on AdS3 and the large N limit of SU(N+M)/SU(N) x U(1) type cosets. We present a simplified proof for the agreement of the spectra and clarify the relation between this duality and the version in which the cosets are replaced by Kazama-Suzuki models of Grassmannian type.Comment: 27 pages, 1 tabl

N=1 extension of minimal model holography

The CFT dual of the higher spin theory with minimal N = 1 spectrum is determined. Unlike previous examples of minimal model holography, there is no free parameter beyond the central charge, and the CFT can be described in terms of a non-diagonal modular invariant of the bosonic theory at the special value of the 't Hooft parameter lambda=1/2. As evidence in favour of the duality we show that the symmetry algebras as well as the partition functions agree between the two descriptions.Comment: 28 page

A lattice approach to the conformal \OSp(2S+2|2S) supercoset sigma model. Part I: Algebraic structures in the spin chain. The Brauer algebra

We define and study a lattice model which we argue is in the universality class of the $OSp(2S+2|2S)$ supercoset sigma model for a large range of values of the coupling constant $g_\sigma^2$. In this first paper, we analyze in details the symmetries of this lattice model, in particular the decomposition of the space of the quantum spin chain $V^{\otimes L}$ as a bimodule over $OSp(2S+2|2S)$ and its commutant, the Brauer algebra $B_L(2)$. It turns out that $V^{\otimes L}$ is a nonsemisimple module for both $OSp(2S+2|2S)$ and $B_L(2)$. The results are used in the companion paper to elucidate the structure of the (boundary) conformal field theory.Comment: 36 pages, 20 figure