  The Sine Gordon theory is generalized to include 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 beta
functions of the field strength renormalization are computed in position space.
They match with the earlier results in \cite{Oak:2017trw}.
