Modification of Matrix Models by Square Terms of Scaling Operators

Abstract

We study one (or two) matrix models modified by terms of the form $g(ρ(P))^2 + g'(ρ'({\cal{O}}))^2$, where the matrix representation of the puncture operator $P$ and the one of a scaling operator ${\cal{O}}$ are denoted by $ρ(P)$ and $ρ'({\cal{O}})$ respectively. We rewrite the modified models as effective theories of baby universes. We find an upper bound for the gravitational dimension of ${\cal{O}}$ under which we can fine tune the coupling constants to obtain new critical behaviors in the continuum limit. The simultaneous tuning of $g$ and $g'$ is possible if the representations $ρ(P)$ and $ρ'({\cal{O}})$ are chosen so that the non-diagonal elements of the mass matrix of the effective theory vanish.