Bosonization of Matter βγ CFT, Non-compact Coordinate and Noncritical String Theory

We consider a (supersymmetric) bosonization of general commuting matter $\beta\gamma$ CFT. Unlike the conventional ghost $\beta\gamma$ CFT bosonization, which is typically described in terms of one scalar and one set of $\xi\eta$ CFT, the matter $\beta\gamma$ CFT is naturally bosonized to two scalar CFTs. Surprisingly, there exists an independent subsector, which satisfies a constraint necessary for consistent bosonization, while the usual sector has a description in terms of two compact coordinates. The subsector itself is a complete system which is equivalent to the original $\beta\gamma$ CFT. Furthermore, a non-compact coordinate naturally emerges in this "non-compact subsector". As an immediate application, we consider the bosonization of matter $\beta\gamma$ CFT in the context of full string theory and find the connection to noncritical string theories in the light-cone gauge.