Non-extremal gravity duals for fractional D3-branes on the conifold

The world volume theory on N regular and M fractional D3-branes at the conifold singularity is a non-conformal N = 1 supersymmetric SU(N + M) x SU(N) gauge theory. In previous work the extremal Type IIB supergravity dual of this theory at zero temperature was constructed. Regularity of the solution requires a deformation of the conifold: this is a reflection of the chiral symmetry breaking. To study the non-zero temperature generalizations non-extremal solutions have to be considered, and in the high temperature phase the chiral symmetry is expected to be restored. Such a solution is expected to have a regular Schwarzschild horizon. We construct an ansatz necessary to study such non-extremal solutions and show that the simplest possible solution has a singular horizon. We derive the system of second order equations in the radial variable whose solutions may have regular horizons.