Spectrum of D=6, N=4b Supergravity on AdS_3 x S^3

TL;DR

This work provides a complete spectrum for $D=6$, $N=4b$ supergravity coupled to $n$ tensor multiplets on $AdS_{3}\times S^{3}$, with symmetry $SU(1,1|2)_{L}\times SU(1,1|2)_{R}$ and unbroken $SO(4)_{R}\times SO(n)$. By performing a meticulous harmonic analysis on $S^{3}$ and analytic continuation to $S^{3}_{E}$, the authors diagonalize the linearized bosonic and fermionic equations, organizing states into towers of spin-$2$, spin-$1$ (both $SO(n)$ singlet and vector reps), and a spin-$\tfrac{1}{2}$ multiplet, all transforming under the stated superalgebra. The spectrum exhibits two spin-$2$ towers, two spin-$1$ towers, and a spin-$\tfrac{1}{2}$ matter tower, with Yang–Mills states occupying the $\ell=1$ level in the spin-$2$ and spin-$1$ sectors, respectively. The results provide a detailed bulk realization of the AdS$_{3}$/CFT$_{2}$ correspondence for this $6D$ supergravity, including the identification of boundary operators via conformal weights $E_{0},s_{0}$ and the corresponding $SO(4)_{R}\times SO(n)$ representations.

Abstract

The complete spectrum of D=6, N=4b supergravity with n tensor multiplets compactified on AdS_3 x S^3 is determined. The D=6 theory obtained from the K_3 compactification of Type IIB string requires that n=21, but we let n be arbitrary. The superalgebra that underlies the symmetry of the resulting supergravity theory in AdS_3 coupled to matter is SU(1,1|2)_L x SU(1,1|2)_R. The theory also has an unbroken global SO(4)_R x SO(n) symmetry inherited from D=6. The spectrum of states arranges itself into a tower of spin-2 supermultiplets, a tower of spin-1, SO(n) singlet supermultiplets, a tower of spin-1 supermultiplets in the vector representation of SO(n) and a special spin-1/2 supermultiplet also in the vector representation of SO(n). The SU(2)_L x SU(2)_R Yang-Mills states reside in the second level of the spin-2 tower and the lowest level of the spin-1, SO(n) singlet tower and the associated field theory exhibits interesting properties.

Figures (3)

• Figure 1: The spin $2$ supermultiplet structure for $\ell\geq0$. A given representation is denoted by $D^{(\ell_1,\ell_2)}\ (E_0,s_{0}) (R\times S\,)$ where $(\ell_1,\ell_2)$ label the $S^3$ isometry group $SO(4)$; $(E_0,s_{0})$ label the representation of the AdS group $SO(2,2)$; $R$ denotes the representation of $SO(4)_R$ and $S$ denotes a representation of $SO(n)$. The supercharge $Q_{+}^{(1/2,\,1/2)}(-{{{ 1} \over { 2}}},\,-{{{ 1} \over { 2}}})(2_{+},0)$ acts to the southwest and the supercharge $Q_{-}^{(1/2,\,-1/2)}({{{ 1} \over { 2}}},\,-{{{ 1} \over { 2}}})(2_{-},0)$ acts to the southeast. The full multiplet structure is obtained by adding the conjugate tower of multiplets in which the replacements $\ell_{2}\rightarrow -\ell_{2}$, $s_{0}\rightarrow -s_{0}$, $2_{\pm}\rightarrow 2_{\mp}$ are made. The multiplet at level $\ell$ thus contains $16(\ell+1)(\ell+3)$ Bose and that many Fermi states. At level $\ell=-1$ there are $SU(2)_{L}$ Yang-Mills states with $(E_{0},\,s_{0})=(1,1)$ and $SU(2)_{R}$ Yang-Mills states with $(E_{0},\,s_{0})=(1,-1)$ which are pure gauge. At level $-1$ one also finds the non-propagating graviton and gravitini. At level $\ell=0$, the states on the lower right diagonal are absent and the resulting special spin $2$ multiplet consists of 48 Bose and 48 Fermi degrees of freedom. At level $\ell=1$ there are physical $SU(2)_{L}$ Yang-Mills states with $(E_{0},\,s_{0})=(5,1)$ and physical $SU(2)_{R}$ Yang-Mills states with $(E_{0},\,s_{0})=(5,-1)$. For $\ell\geq 1$ the structure of the multiplets is generic, and no other Yang-Mills states arise.
• Figure 2: The spin $1$, $SO(n)$ singlet supermultiplet structure for $\ell\geq0$. The multiplet is self-conjugate and contains $8(\ell+2)^{2}$ Bose and that many Fermi states at level $\ell$. At level $\ell=-1$ there is an unphysical spin ${{{ 1} \over { 2}}}$ multiplet residing at the left diamond. At level $\ell=0$ there is triplet of $SU(2)_{R}$ Yang-Mills states with $(E_{0},\,s_{0})=(3,1)$ and a triplet of $SU(2)_{L}$ Yang-Mills states with $(E_{0},\,s_{0})=(3,-1)$. For $\ell\geq 0$ the structure of the multiplets is generic and no other Yang-Mills states arise. See caption of Figure \ref{['httf']} for furher notations.
• Figure 3: The structure of the spin $1$ supermultiplet in the vector representation of $SO(n)$ for $\ell\geq-1$. The multiplet is self-conjugate and contains $8\,n\,(\ell+2)^2$ Bose and that many Fermi states at level $\ell$. For the special value $\ell=-1$ one finds a spin ${{{ 1} \over { 2}}}$ multiplet consisting of $8\,n$ Bose and $8\,n$ Fermi states residing at the left diamond. For $\ell\geq 0$ the structure of the multiplets is generic. These are matter spin $1$ multiplets. See the caption of Figure \ref{['httf']} for further notations.