Abstract:
The exact renormalization group is used to study the RG flow of
quantities in field theories. The basic idea is to write an evolution
operator for the RG flow and evaluate it in perturbation theory. This
is easier than directly solving the differential equation. This is illustrated by reproducing known results in the four dimensional φ 4 field
theory and the two dimensional Sine-Gordon theory. It is shown that
the calculation of beta function is somewhat simplified. The technique
is also used to calculate the c-function in two dimensional Sine-Gordon
theory. This agrees with other prescriptions for calculating c-functions
in the literature. If one extrapolates the connection between central
charge of a CFT and entanglement entropy in two dimensions, to the
c-function of the perturbed CFT, then one gets a value for the en-
tanglement entropy in Sine-Gordon theory that is in exact agreement
with earlier calculations. Next, the Sine Gordon theory is generalized
to include many scalar fields and several cosine terms. This is similar
to the world sheet description of a string propagating in a tachyon
background. This model is studied as a (boundary) 2d euclidean field
theory and also using an AdS 3 holographic bulk dual. The beta functions for the cosine vertex of this modified theory are first computed
in the boundary using techniques based on the exact RG. The beta
functions are also computed holographically using position space and
momentum space techniques. The results are in agreement with each
other and with earlier computations. The cosine perturbation is of
the form cos bX. Due to wave function renormalisation the parameter
b, and thus the dimension of the cosine, get renormalised. The beta
function for this parameter is thus directly related to the anomalous
dimension of the X field. We compute this beta function in position
space. They match with the earlier results in [22].
