On quantum corrections in higher-spin theory in flat space

We consider an interacting theory of an infinite tower of massless higher-spin fields in flat space with cubic vertices and their coupling constants found previously by Metsaev. We compute the one-loop bubble diagram part of the self-energy of the spin 0 member of the tower by summing up all higher-spin loop contributions. We find that the result contains an exponentially UV divergent part and we discuss how it could be cancelled by a tadpole contribution depending on yet to be determined quartic interaction vertex. We also compute the tree-level four-scalar scattering amplitude due to all higher-spin exchanges and discuss its inconsistency with the BCFW constructibility condition. We comment on possible relation to similar computations in AdS background in connection with AdS/CFT.Comment: 34 pages, minor corrections and references adde

Conformal anomaly c-coefficients of superconformal 6d theories

We propose general relations between the conformal anomaly and the chiral (R-symmetry and gravitational) anomaly coefficients in 6d (1,0) superconformal theories. The suggested expressions for the three type B conformal anomaly c-coefficients complement the expression for the type A anomaly a-coefficient found in arXiv:1506.03807. We check them on several examples -- the standard (1,0) hyper and tensor multiplets as well as some higher derivative short multiplets containing vector fields that generalize the superconformal 6d vector multiplet discussed in arXiv:1506.08727. We also consider a family of higher derivative superconformal (2,0) 6d multiplets associated to 7d multiplets in the KK spectrum of 11d supergravity compactified on S^4. In particular, we prove that (2,0) 6d conformal supergravity coupled to 26 tensor multiplets is free of all chiral and conformal anomalies. We discuss some interacting (1,0) superconformal theories, predicting the c-coefficients for the "E-string" theory on multiple M5-branes at E_8 9-brane and for the theory describing M5-branes at an orbifold singularity. Finally, we elaborate on holographic computation of subleading corrections to conformal anomaly coefficients coming from R^2+R^3 terms in 7d effective action, revisiting, in particular, the (2,0) theory case.Comment: 32 pages, v4: Added footnotes 8, 25, 26 clarifying that the available data leads to a 1-parameter family of the conformal anomaly coefficients c_1, c_2 as functions of chiral anomaly coefficients; the results for c_i in recent arXiv:1702.03518 also belong to this famil

On boundary correlators in Liouville theory on AdS2_{2}

We consider the Liouville theory in fixed Euclidean AdS$_2$ background. Expanded near the minimum of the potential the elementary field has mass squared 2 and (assuming the standard Dirichlet b.c.) corresponds to a dimension 2 operator at the boundary. We provide strong evidence for the conjecture that the boundary correlators of the Liouville field are the same as the correlators of the holomorphic stress tensor (or the Virasoro generator with the same central charge) on a half-plane or a disc restricted to the boundary. This relation was first observed at the leading semiclassical order (tree-level Witten diagrams in AdS$_2$) in arXiv:1902.10536 and here we demonstrate its validity also at the one-loop level. We also discuss arguments that may lead to its general proof.Comment: 23 pages, 1 pdf figure. v2: minor change

CTC_T for conformal higher spin fields from partition function on conically deformed sphere

We consider the one-parameter generalization $S^4_q$ of 4-sphere with a conical singularity due to identification $\tau=\tau + 2 \pi q$ in one isometric angle. We compute the value of the spectral zeta-function at zero $z(q) = \zeta(0, q)$ that controls the coefficient of the logarithmic UV divergence of the one-loop partition function on $S^4_q$. While the value of the conformal anomaly a-coefficient is proportional to $z(1)$, we argue that in general the second $c = C_T$ anomaly coefficient is related to a particular combination of the second and first derivatives of $z(q)$ at $q=1$. The universality of this relation for $C_T$ is supported also by examples in 6 and 2 dimensions. We use it to compute the c-coefficient for conformal higher spins finding that it coincides with the "$r=-1$" value of the one-parameter Ansatz suggested in arXiv:1309.0785. Like the sums of $a_s$ and $c_s$ coefficients, the regularized sum of $z_s(q)$ over the whole tower of conformal higher spins $s=1,2, ...$ is found to vanish, implying UV finiteness on $S^4_q$ and thus also the vanishing of the associated Re'nyi entropy. Similar conclusions are found to apply to the standard 2-derivative massless higher spin tower. We also present an independent computation of the full set of conformal anomaly coefficients of the 6d Weyl graviton theory defined by a particular combination of the three 6d Weyl invariants that has a (2,0) supersymmetric extension.Comment: 29 pages. v2: minor change

Quantum spinning strings in AdS_4 x CP^3: testing the Bethe Ansatz proposal

Recently, an asymptotic Bethe Ansatz that is claimed to describe anomalous dimensions of "long" operators in the planar N=6 supersymmetric three-dimensional Chern-Simons-matter theory dual to quantum superstrings in AdS_4 x CP^3 was proposed. It initially passed a few consistency checks but subsequent direct comparison to one-loop string-theory computations created some controversy. Here we suggest a resolution by pointing out that, contrary to the initial assumption based on the algebraic curve considerations, the central interpolating function h(\lambda) entering the BMN or magnon dispersion relation receives a non-zero one-loop correction in the natural string-theory computational scheme. We consider a basic example which has already played a key role in the AdS_5 x S^5 case: a rigid circular string stretched in both AdS_4 and along an S^1 of CP^3 and carrying two spins. Computing the leading one-loop quantum correction to its energy allows us to fix the constant one-loop term in h(\lambda) and also to suggest how one may establish a correspondence with the Bethe Ansatz proposal, including the non-trivial one-loop phase factor. We discuss some problems which remain in trying to match a part of world-sheet contributions (sensitive to compactness of the string direction) and their Bethe Ansatz counterparts.Comment: 37 pages; v2: references added and typos fixe

Supergravity one-loop corrections on AdS_7 and AdS_3, higher spins and AdS/CFT

As was shown earlier, one-loop correction in 10d supergravity on AdS_5 x S^5 corresponds to the contributions to the vacuum energy and boundary 4d conformal anomaly which are minus the values for one n=4 Maxwell supermultiplet, thus reproducing the subleading term in their N^2-1 coefficient in the dual SU(N) SYM theory. We perform similar one-loop computations in 11d supergravity on AdS_7 x S^4 and 10d supergravity on AdS_3 x S^3 x T^4. In the AdS_7 case we find that the corrections to the 6d conformal anomaly a-coefficient and the vacuum energy are again minus the ones for one (2,0) tensor multiplet, suggesting that the total a-anomaly coefficient for the dual (2,0) theory is 4 N^3 - 9/4 N - 7/4 and thus vanishes for N=1. In the AdS_3 case the one-loop correction to the vacuum energy or 2d central charge turns out to be equal to that of one free (4,4) scalar multiplet, i.e. is c=+6. This reproduces the subleading term in the central charge c= 6(Q_1 Q_5 +1) of the dual 2d CFT describing decoupling limit of D5-D1 system. We also present the expressions for the 6d anomaly a-coefficient and vacuum energy for a general-symmetry higher spin field in AdS_7 and consider their application to tests of vectorial AdS/CFT with the boundary conformal 6d theory represented by free scalars, spinors or rank 2 antisymmetric tensors.Comment: 28 pages, v2 minor addition

Partition function of free conformal higher spin theory

We compute the canonical partition function Z of non-interacting conformal higher spin (CHS) theory viewed as a collection of free spin s CFT's in R^d. We discuss in detail the 4-dimensional case (where s=1 is the standard Maxwell vector, s=2 is the Weyl graviton, etc.), but also present a generalization for all even dimensions d. Z may be found by counting the numbers of conformal operators and their descendants (modulo gauge identities and equations of motion) weighted by scaling dimensions. This conformal operator counting method requires a careful analysis of the structure of characters of relevant (conserved current, shadow field and conformal Killing tensor) representations of the conformal algebra so(d,2). There is also a close relation to massless higher spin partition functions with alternative boundary conditions in AdS_{d+1}. The same partition function Z may also be computed from the CHS path integral on a curved S^1 x S^{d-1} background. This allows us to determine a simple factorized form of the CHS kinetic operator on this conformally flat background. Summing the individual conformal spin contributions Z_s over all spins we obtain the total partition function of the CHS theory. We also find the corresponding Casimir energy and show that it vanishes if one uses the same regularization prescription that implies the cancellation of the total conformal anomaly a-coefficient. This happens to be true in all even dimensions d >= 2.Comment: 39 pages, v2 typos corrected and comments adde