Applications of Renormalisation Group in Holography [HBNI Th247]

Abstract

The renormalisation group is a powerful set of ideas that relate physical theories at different length scales. The Polchinski’s exact renormalisation group (ERG) equation is a differential equation that enables us to perform an RG transformation on the action of a QFT exactly, i.e., keeping track of all the couplings in the action. There are two parts in this thesis. In the first part, we describe the programme initiated in [1] by Sathiapalan and Sonoda. This programme gives a systematic procedure to obtain the bulk action that is dual to a CFT from the ERG equation corresponding to the CFT. The CFT action is perturbed by source terms, which causes the theory to flow, and the bulk radial direction emerges out of this flow in energy. As the ERG equation depends only on the form of the fixed point action, this equation encodes the bulk theory. The ERG equation has a corresponding evolution operator, which is a D + 1 dimensional functional integral, with the additional dimension being supplied by the energy scale. An action can be read off from this functional integral, which is holographic. But this action is nonlocal, and we describe a systematic field redefinition that makes the kinetic term of this action local. This field redefinition encodes the regularisation scheme of the ERG equation of the boundary theory. We demonstrate this procedure for the case of the boundary scalar, vector, and spin 2 currents in large N vector model in 3D. We obtain the bulk kinetic terms for each of them by mapping described above, and also the cubic vertices for scalar, vector, and graviton-scalar-scalar. We show that the former and the latter can be made local by suitable choice of boundary regulator. Our procedure regularises the boundary propagator differently for different currents, but we describe a way to get around it. In the second part, we construct composite operators for the scalar theory in six dimensions using renormalisation group methods with dimensional regularisation. We express bare scalar operators in terms of renormalised composite operators of lower dimension, then do this with traceless tensor operators. We then express the bare energy momentum tensor in terms of the renormalised composite operators, with some terms having divergent coefficients. We subtract these away and obtain a manifestly finite energy momentum tensor. The subtracted terms are transverse, so this does not affect the conservation of the energy momentum tensor. The trace of this finite improved energy momentum tensor vanishes at the fixed point indicating conformal invariance. Interestingly it is not RG invariant except at the fixed point, but can be made RG invariant everywhere by further addition of finite transverse terms, whose coefficients vanish at the fixed point.