We consider general-symmetry higher spin fields in AdS 5 and derive the expressions for their one-loop corrections to vacuum energy E c and the associated 4d boundary conformal anomaly a-coefficient. We propose a similar expression for the second conformal anomaly c-coefficient. We show that all the three quantities (E c , a, c) computed for N = 8 gauged 5d supergravity are equal to − 1 2 of their values for N = 4 conformal 4d supergravity and also to twice the values for N = 4 Maxwell multiplet. This gives a 5d derivation of the fact that the system of N = 4 conformal supergravity and four N = 4 Maxwell multiplets is anomaly free. The values of (E c , a, c) for the states at level p of Kaluza-Klein tower of 10d type IIB supergravity compactified on S 5 turn out to be equal to those for p copies of N = 4 Maxwell multiplets. This may be related to the fact that these states appear in the tensor product of p superdoubletons. Under a natural regularization of the sum over p, the full 10d supergravity contribution is then minus that of one Maxwell multiplet, in agreement with the standard adjoint AdS/CFT duality (SU (N ) SYM contribution is N 2 −1 times that of one Maxwell multiplet). We also verify the matching of (E c , a, c) for spin 0 and 1 2 boundary theory cases of vectorial AdS/CFT duality. The consistency conditions for vectorial AdS/CFT turn out to be equivalent to the cancellation of anomalies in the closely related 4d conformal higher spin theories. In addition, we study novel example of the vectorial AdS/CFT duality when the boundary theory is described by free spin 1 fields and is dual to a particular higher spin theory in AdS 5 containing fields in mixed-symmetry representations. We also discuss its supersymmetric generalizations. 1 Also at Lebedev Institute, Moscow arXiv:1410.3273v3 [hep-th] 18 Nov 2014 8 The boundary operator becomes local only for special values of ∆ (see, e.g., a discussion of the scalar case in [10]). In general, we shall assume analytic continuation in ∆. 9 In the AdS/CFT context this should be equal to the generating functional for correlators of bilinear currents J ∼ Φ * ∂ s Φ in the boundary CFT, Z(ϕ) = dΦ exp[−S4(Φ) + J · ϕ]. Integrating over N fields Φ gives induced action for ϕ starting with N ϕKϕ ∼ N log ε ϕÕϕ + ... where ε is playing the role of a UV 4d cutoff.

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 × 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 on the sphere 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.

We study the leading quantum string correction to the dressing phase in the asymptotic Bethe Ansatz system for superstring in AdS 3 × S 3 × T 4 supported by RR flux. We find that the phase should be different from the phase appearing in the AdS 5 ×S 5 case. We use the simplest example of a rigid circular string with two equal spins in S 3 and also consider the general approach based on the algebraic curve description. We also discuss the case of the AdS 3 × S 3 × S 3 × S 1 theory and find the dependence of the 1-loop correction to the effective string tension function h(λ) (expected to enter the magnon dispersion relation) on the parameters α related to the ratio of the two 3-sphere radii. This correction vanishes in the AdS 3 × S 3 × T 4 case.

Following Polchinski and Sully (arXiv:1104.5077), we consider a generalized Wilson loop operator containing a constant parameter ζ in front of the scalar coupling term, so that ζ = 0 corresponds to the standard Wilson loop, while ζ = 1 to the locally supersymmetric one. We compute the expectation value of this operator for circular loop as a function of ζ to second order in the planar weak coupling expansion in N = 4 SYM theory. We then explain the relation of the expansion near the two conformal points ζ = 0 and ζ = 1 to the correlators of scalar operators inserted on the loop. We also discuss the AdS 5 × S 5 string 1-loop correction to the strong-coupling expansion of the standard circular Wilson loop, as well as its generalization to the case of mixed boundary conditions on the five-sphere coordinates, corresponding to general ζ. From the point of view of the defect CFT 1 defined on the Wilson line, the ζ-dependent term can be seen as a perturbation driving a RG flow from the standard Wilson loop in the UV to the supersymmetric Wilson loop in the IR. Both at weak and strong coupling we find that the logarithm of the expectation value of the standard Wilson loop for the circular contour is larger than that of the supersymmetric one, which appears to be in agreement with the 1d analog of the F-theorem.

We consider the 1-loop correction to the energy of folded spinning string solution in the AdS 3 part of AdS 5 ×S 5. The classical string solution is expressed in terms of elliptic functions so an explicit computation of the corresponding fluctuation determinants for generic values of the spin appears to be a non-trivial problem. We show how it can be solved exactly by using the static gauge expression for the string partition function (which we demonstrate to be equivalent to the conformal gauge one) and observing that all the corresponding second order fluctuation operators can be put into the standard (single-gap) Lamé form. We systematically derive the small spin and large spin expansions of the resulting expression for the string energy and comment on some of their applications.

The most general large N = 4 superconformal W ∞ algebra, containing in addition to the superconformal algebra one supermultiplet for each integer spin, is analysed in detail. It is found that the W ∞ algebra is uniquely determined by the levels of the two su(2) algebras, a conclusion that holds both for the linear and the non-linear case. We also perform various cross-checks of our analysis, and exhibit two different types of truncations in some detail.

If the top is very heavy, m t ≫ M Z , the dominant radiative correction effects in all electroweak precision tests can be exactly characterized in terms of two quantities, the ρ-parameter and the GIM violating Z → bb coupling. These quantities can be computed using the Standard Model Lagrangian with vanishing gauge couplings. This is done here up to two loops for arbitrary values of the Higgs mass.

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