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We develop a general formalism of duality rotations for bosonic conformal spin-$s$ gauge fields, with $sgeq 2$, in a conformally flat four-dimensional spacetime. In the $s=1$ case this formalism is equivalent to the theory of $mathsf{U}(1)$ duality-invariant nonlinear electrodynamics developed by Gaillard and Zumino, Gibbons and Rasheed, and generalised by Ivanov and Zupnik. For each integer spin $sgeq 2$ we demonstrate the existence of families of conformal $mathsf{U}(1)$ duality-invariant models, including a generalisation of the so called ModMax Electrodynamics ($s=1$). Our bosonic results are then extended to the $mathcal{N}=1$ and $mathcal{N}=2$ supersymmetric cases. We also sketch a formalism of duality rotations for conformal gauge fields of Lorentz type $(m/2, n/2)$, for positive integers $m $ and $n$.
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Massless conformal scalar field in d=4 corresponds to the minimal unitary representation (minrep) of the conformal group SU(2,2) which admits a one-parameter family of deformations that describe massless fields of arbitrary helicity. The minrep and its deformations were obtained by quantization of the nonlinear realization of SU(2,2) as a quasiconformal group in arXiv:0908.3624. We show that the generators of SU(2,2) for these unitary irreducible representations can be written as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group and apply them to define and study higher spin algebras and superalgebras in AdS_5. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS_5 is simply the enveloping algebra of SU(2,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS_5. Furthermore, the enveloping algebras of the deformations of the minrep define a one parameter family of HS algebras in AdS_5 for which certain 4d covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras SU(2,2|N) and we find a one parameter family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a family of (supersymmetric) HS theories in AdS_5 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 4d. We also discuss the corresponding picture in AdS_4 where the 3d conformal group Sp(4,R) admits only two massless representations (minreps), namely the scalar and spinor singletons.
Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of deformations labelled by the spin t of an SU(2)_T group, which is the 6d analog of helicity in four dimensions. These deformations of the minrep of SO(6,2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6,2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5,1) and apply them to define higher spin algebras and superalgebras in AdS_7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS_7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS_7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS_7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8*|2N) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS_7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.
We study the Thermo-field realization of the duality between the Rindler-AdS higher spin theory and $O(N)$ vector theory. The CFT represents a decoupled pair of free $O(N)$ vector field theories. It is shown how this decoupled domain CFT is capable of generating the connected Rindler-AdS background with the full set of Higher Spin fields.
It is shown that similarly to massless superparticle, classical global symmetry of the Berkovits twistor string action is infinite-dimensional. We identify its superalgebra, whose finite-dimensional subalgebra contains $psl(4|4,mathbb R)$ superalgebra. In quantum theory this infinite-dimensional symmetry breaks down to $SL(4|4,mathbb R)$ one.
The linearized spectrum and the algebra of global symmetries of conformal higher-spin gravity decompose into infinitely many representations of the conformal algebra. Their characters involve divergent sums over spins. We propose a suitable regularization adapted to their evaluation and observe that their characters are actually equal. This result holds in the case of type-A and type-B (and their higher-depth generalizations) theories and confirms previous observations on a remarkable rearrangement of dynamical degrees of freedom in conformal higher-spin gravity after regularization.
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