Bosonic Fradkin-Tseytlin equations unfolded

O V Shaynkman

doi:10.1007/jhep12(2016)118

Abstract

We test infinite-dimensional extension of algebra $$ \mathfrak{s}\mathfrak{u}\left(k,\;k\right) $$ proposed by Fradkin and Linetsky as the candidate for conformal higher spin algebra. Adjoint and twistedadjoint representations of $$ \mathfrak{s}\mathfrak{u}\left(k,\;k\right) $$ on the space of this algebra are carefully explored. For k = 2 corresponding unfolded system is analyzed and it is shown to encode Fradkin-Tseytlin equations for the set of all integer spins 1, 2, . . . with infinite multiplicity.

Highlights

[2] Fradkin and Linetsky proposed a number of candidates for the role of infinite-dimensional 4d conformal higher spin gauge symmetry algebra, which extends ordinary 4d conformal algebra1 so(4, 2) ∼ su(2, 2)
In the present paper we analyze unfolded system corresponding to algebra iu(2, 2) and show that it describes a collection of Fradkin-Tseytlin equations that corresponds to all bosonic spins with infinite degeneracy
We have proposed unfolded system (5.16) that describes linear conformal dynamics of spin s gauge field

Summary

Unfolded formulation: preliminary remarks

Let Md be some d-dimensional manifold with coordinates x1, . . . , xd. Any dynamical system on Md can be reformulated in unfolded form of the first order differential equations [17]. System (2.8), (2.10) and (2.11) is locally invariant with respect to gauge transformation (2.9) of connection 1-form W A(x) and the following gauge transformations of fields ωa(x) and Ci(x) δωa = − A(TA)abωb + AW BHABaiCi , δCi = − A(TA)ijCj. of zero curvature condition (2.8) is fixed, the gauge symmetry above breaks down to the global symmetry that keeps W0A stable. To analyze dynamical content of system (2.22), (2.23) let us first suppose that the righthand side of (2.22) is zero In this case equations (2.22), (2.23) are independent and both have form of covariant constancy conditions. Summarizing, the dynamical content of equation (2.22) with the zero right-hand side is described by Hσ0−, Hσ1−, Hσ2− which correspond to differential gauge parameters, dynamical fields and differential equations on the dynamical fields respectively. Before it we explore structure of underlying su(k, k)-modules

Structure of adjoint module

Structure of twist-adjoint module

Unfolded formulation of Fradkin-Tseytlin equations

Conclusion

Gauge sector

Weyl sector