Holographic Renormalization of Fake Supergravities

The string/gauge theory correspondence allows to calculate correlators in certain strongly coupled gauge theories via solving the equations of motion of supergravity (SUGRA). In this work we propose a method to calculate correlators for theories with logarithmically running gauge couplings, which corresponds to a logarithmically wrapped bulk-metric on the gravity side. One of the most prominent examples is the Klebanov-Strassler background, for which calculations were carried out. However, the proposed method is more general in nature and is applicable for all theories that are known as ”fake” SUGRA theories. Such ”fake” SUGRA theories allow for BPS domain-wall solutions, which are the holographic duals of renormalization group flows. It would be a daunting task to find the full counterterms for such general theories. Thus, at present we content ourselves with calculating the 2-point function and to some extent the 1-point function. Furthermore, we only consider gauge theories living on the flat space-time, which allows us to neglect all the counterterms involving the space-time curvature. We start with the string/gauge correspondence formula e−Son−shell[s] = ∫ DΦ e−SQFT[Φ]+ ∫ Oisi ddx , (1) where Son−shell[s] is the renormalized on-shell bulk action evaluated as a functional of suitably defined boundary values si of the various bulk fields, which are identified with the sources coupling to certain QFT operators Oi. Son−shell[s] is then identified with the generating functional of the connected correlation functions of various QFT operators. In particular, the exact 1-point functions of the QFT operators are given by 〈Oi(x)〉 = − δSon−shell δsi(x) . (2) In order to evaluate the 2-point functions, one has to know the dependence of the right hand side of (1) on the sources up to linear order, which requires to solve the linearized equations of motion. Hence, in order to calculate 1and 2-point functions, it suffices to know the action up to quadratic order in fluctuations. In this work we give a recipe to determine the quadratic terms. In doing so we use the so-called gauge invariant mechanism. In order to calculate the vacuum expectation values we would need to know the boundary terms linear in fluctuations. We do not know yet how to generalize our prescription for those terms. Thus, we calculate only the linear term of the 1-point function, i.e excluding the VEV. However, we make an observation that the response function of the zero mode seems to encode some information about the VEV. One advantage of our approach is that it allows to discuss the scheme dependence of 1and 2-point functions in a rather general way.