# Bulk/boundary duality for matter fields?

Usually, in gauge/gravity duality we have some CFT on a boundary that is dual to a gravity in the bulk. Although CFT is never written in Lagrangian form, it seems for me, that AdS/CFT correspondence relates matter(spin 0, 1/2) and gauge (spin 1) field theory in $$N$$ dimensions to pure gravity (spin 2) in $$N+1$$ dimensions.

Do you know of some examples, if it is possible at all, to have a correspondence between matter-gauge theory (only spin 0, 1/2 or 1) on the boundary and another matter-gauge theory (only spin 0, 1/2, 1) in the bulk?

1. AdS/CFT correspondence relates a string theory in the AdS bulk to a CFT living on the boundary. Quantization of a closed superstring with appropriate GSO projection leads to a tower of half integer and integer spin states, from $$0$$ to $$\infty$$ (which includes the graviton with spin-2), and other massive higher spin states. The boundary values of these fields in the bulk acts as sources for dual CFT operators, which have the same spin as the bulk fields.
You might be confused as you are thinking about $$N=4$$ Super Yang Mills, whose supermultiplet basically consists of particles of spin-0,1/2 and 1. It was shown that the beta function of N=4 SYM vanishes, and consequently it is a CFT, You can construct the higher spin operators of the CFT, using various combinations of these fields, like the current operator $$J_{\mu}$$ and the stress energy $$T_{\mu\nu}$$. You would further need to match the symmetries as well as the spectrum on both the bulk and the boundary side for the duality to work.
2. Consider the case of how the duality works for the gauge field $$A_M$$ in the bulk. One can set the gauge $$A_z =0$$, and solve for the remaining components $$A_{\mu}$$, which take a value $$A_{\mu}^{(0)}$$ at the boundary. This $$A_{\mu}^{(0)}$$ now acts as a source for the conserved current $$J^{\mu}$$ in the bulk, which is a CFT operator. One can similarly show that boundary value of the bulk metric perturbation $$h_{\mu \nu}$$ acts as a source for the stress energy $$T_{\mu\nu}$$ of the boundary.