Group manifold approach to higher spin theory

We consider the group manifold approach to higher spin theory. The deformed local higher spin transformation is realized as the diffeomorphism transformation in the group manifold $\textbf{M}$. With the suitable rheonomy condition and the torsion constraint imposed, the unfolded equation can be obtained from the Bianchi identity, by solving which, fields in $\textbf{M}$ are determined by the multiplet at one point, or equivalently, by $(W^{[a(s-1),b(0)]}_{\mu},H)$ in $AdS_{4}\subset \textbf{M}$. Although the space is extended to $\textbf{M}$ to get the geometrical formulation, the dynamical degrees of freedom are still in $AdS_{4}$. The $4d$ equations of motion for $(W^{[a(s-1),b(0)]}_{\mu},H)$ are obtained by plugging the rheonomy condition into the Bianchi identity. The proper rheonomy condition allowing for the maximum on-shell degrees of freedom is given by Vasiliev equation. We also discuss the theory with the global higher spin symmetry, which is in parallel with the WZ model in supersymmetry.Comment: 35 pages,v2: revised version, v3: 38 pages, improved discussion on global HS symmetry, clarifications added in appendix B, journal versio

U-duality transformation of membrane on TnT^{n} revisited

The problem with the U-duality transformation of membrane on $T^{n}$ is recently addressed in [arXiv:1509.02915 [hep-th]]. We will consider the U-duality transformation rule of membrane on $T^{n}\times R$. It turns out that winding modes on $T^{n}$ should be taken into account, since the duality transformation may bring the membrane configuration without winding modes into the one with winding modes. With the winding modes added, the membrane worldvolume theory in lightcone gauge is equivalent to the $n+1$ dimensional super-Yang-Mills (SYM) theory in $\tilde{T}^{n}$, which has $SL(2,Z)\times SL(3,Z)$ and $SL(5,Z)$ symmetries for $n=3$ and $n=4$, respectively. The $SL(2,Z)\times SL(3,Z)$ transformation can be realized classically, making the on-shell field configurations transformed into each other. However, the $SL(5,Z)$ symmetry may only be realized at the quantum level, since the classical $5d$ SYM field configurations cannot form the representation of $SL(5,Z)$.Comment: 19 pages; v2: 20 pages, reference corrected, extended discussion in section 5, journal versio

Quantum Spectrum of BPS Instanton States in 5d 5d Guage Theories

We construct $1/2$ BPS 1-instanton states in the Hilbert space of $5d$ gauge theories. The states are local with the scale size $\rho=0$. With the degeneracy from the gauge orientation taken into account, the BPS spectrum consists of an infinite number of states in the definite representations of the gauge group and the $(J,0)$ representation of the $SO(4)\cong SU(2)_{L} \times SU(2)_{R}$ rotation group. In the brane webs realization of the $5d$ $\mathcal{N}=1$ gauge theory, BPS states are string webs inside the 5-brane webs. For a theory with the gauge group $SU(N)$ and the Chern-Simons level $\kappa$, we give a classification of string webs in the Coulomb branch and show that for each web, one can always find an instanton state carrying the same spin and the same electric charge. The action of supercharges on instanton states gives the instanton supermultiplets. For the $\mathcal{N} =1$ $SU(2)$ gauge theory with $N_{f} \leq 4$, we construct the instanton vector and hyper multiplets, which, together with the original multiplets of the theory, furnish the complete representation of the $E_{N_{f}+1}$ group at UV.Comment: 58 pages, 5 figure