Perturbative aspects of heterotically deformedCP(N−1)sigma model

Abstract

In this paper we begin the study of renormalizations in the heterotically deformed $\mathcal{N}=(0,2)$ $CP(N\ensuremath{-}1)$ sigma models. In addition to the coupling constant ${g}^{2}$ of the undeformed $\mathcal{N}=(2,2)$ model, there is the second coupling constant $\ensuremath{\gamma}$ describing the strength of the heterotic deformation. We calculate both $\ensuremath{\beta}$ functions, ${\ensuremath{\beta}}_{g}$ and ${\ensuremath{\beta}}_{\ensuremath{\gamma}}$ at one loop, determining the flow of ${g}^{2}$ and $\ensuremath{\gamma}$. Under a certain choice of the initial conditions, the theory is asymptotically free. The $\ensuremath{\beta}$ function for the ratio $\ensuremath{\rho}={\ensuremath{\gamma}}^{2}/{g}^{2}$ exhibits an infrared fixed point at $\ensuremath{\rho}=1/2$. Formally this fixed point lies outside the validity of the one-loop approximation. We argue, however, that the fixed point at $\ensuremath{\rho}=1/2$ may survive to all orders. The reason is the enhancement of symmetry---emergence of a chiral fermion flavor symmetry in the heterotically deformed Lagrangian---at $\ensuremath{\rho}=1/2$. Next we argue that ${\ensuremath{\beta}}_{\ensuremath{\rho}}$ formally obtained at one loop, is exact to all orders in the large-$N$ (planar) approximation. Thus, the fixed point at $\ensuremath{\rho}=1/2$ is definitely the feature of the model in the large-$N$ limit.