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We analyze the properly normalized three-point correlator of two protected scalar operators and one higher spin twist-two operator in N=4 super Yang-Mills, in the limit of large spin j. The relevant structure constant can be extracted from the OPE of the four-point correlator of protected scalar operators. We show that crossing symmetry of the four point correlator plus a judicious guess for the perturbative structure of the three-point correlator, allow to make a prediction for the structure constant at all loops in perturbation theory, up to terms that remain finite as the spin becomes large. Furthermore, the expression for the structure constant allows to propose an expression for the all loops four-point correlator G(u,v), in the limit u,v -> 0. Our predictions are in perfect agreement with the large j expansion of results available in the literature.
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We describe an iterative scheme which allows us to calculate any multi-loop correlator for the complex matrix model to any genus using only the first in the chain of loop equations. The method works for a completely general potential and the results contain no explicit reference to the couplings. The genus $g$ contribution to the $m$--loop correlator depends on a finite number of parameters, namely at most $4g-2+m$. We find the generating functional explicitly up to genus three. We show as well that the model is equivalent to an external field problem for the complex matrix model with a logarithmic potential.
Vasilievs higher-spin theories in various dimensions are uniformly represented as a simple system of equations. These equations and their gauge invariances are based on two superalgebras and have a transparent algebraic meaning. For a given higher-spin theory these algebras can be inferred from the vacuum higher-spin symmetries. The proposed system of equations admits a concise AKSZ formulation. We also discuss novel higher-spin systems including partially-massless and massive fields in AdS, as well as conformal and massless off-shell fields.
We give an explicit superspace construction of higher spin conserved supercurrents built out of $4D,mathcal{N}=1$ massless supermultiplets of arbitrary spin. These supercurrents are gauge invariant and generate a large class of cubic interactions between a massless supermultiplet with superspin $Y_1=s_1+1/2$ and two massless supermultiplets of arbitrary superspin $Y_2$. These interactions are possible only for $s_1geq 2Y_2$. At the equality, the supercurrent acquires its simplest form and defines the supersymmetric, higher spin extension of the linearized Bel-Robinson tensor.
We observe that the partition function of the set of all free massless higher spins s=0,1,2,3,... in flat space is equal to one: the ghost determinants cancel against the physical ones or, equivalently, the (regularized) total number of degrees of freedom vanishes. This reflects large underlying gauge symmetry and suggests analogy with supersymmetric or topological theory. The Z=1 property extends also to the AdS background, i.e. the 1-loop vacuum partition function of Vasiliev theory is equal to 1 (assuming a particular regularization of the sum over spins); this was noticed earlier as a consistency requirement for the vectorial AdS/CFT duality. We find that Z=1 is also true in the conformal higher spin theory (with higher-derivative d^{2s} kinetic terms) expanded near flat or conformally flat S^4 background. We also consider the partition function of free conformal theory of symmetric traceless rank s tensor field which has 2-derivative kinetic term but only scalar gauge invariance in flat 4d space. This non-unitary theory has a Weyl-invariant action in curved background and corresponds to partially massless field in AdS_5. We discuss in detail the special case of s=2 (or conformal graviton), compute the corresponding conformal anomaly coefficients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions.
We propose generalised $mathcal{N}=1$ superconformal higher-spin (SCHS) gauge multiplets of depth $t$, $Upsilon_{alpha(n)dot{alpha}(m)}^{(t)}$, with $ngeq m geq 1$. At the component level, for $t>2$ they contain generalised conformal higher-spin (CHS) gauge fields with depths $t-1$, $t$ and $t+1$. The supermultiplets with $t=1$ and $t=2$ include both ordinary and generalised CHS gauge fields. Super-Weyl and gauge invariant actions describing the dynamics of $Upsilon_{alpha(n)dot{alpha}(m)}^{(t)}$ on conformally-flat superspace backgrounds are then derived. For the case $n=m=t=1$, corresponding to the maximal-depth conformal graviton supermultiplet, we extend this action to Bach-flat backgrounds. Models for superconformal non-gauge multiplets, which are expected to play an important role in the Bach-flat completions of the models for $Upsilon^{(t)}_{alpha(n)dot{alpha}(m)}$, are also provided. Finally we show that, on Bach-flat backgrounds, requiring gauge and Weyl invariance does not always determine a model for a CHS field uniquely.
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