N = (0,2) deformation of the CP(1) model: Two-dimensional analog of N=1 Yang-Mills theory in four dimensions

We consider two-dimensional N = (0, 2) sigma models with the CP(1) target space. A minimal model of this type has one left-handed fermion. Nonminimal extensions contain, in addition, N-f right-handed fermions. Our task is to derive expressions for the beta functions valid to all orders. To this end we use a variety of methods: (i) perturbative analysis; (ii) instanton calculus; (iii) analysis of the supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, and some other arguments. All these arguments, combined, indicate a direct parallel between the heterotic N = (0, 2) CP(1) models and four-dimensional super-Yang-Mills theories. In particular, the minimal N = (0, 2) CP(1) model is similar to N = 1 supersymmetric gluodynamics. Its exact beta function can be found; it has the structure of the Novikov-Shifman-Vainshtein-Zakharov (NSVZ) beta function of supersymmetric gluodynamics. The passage to nonminimal N = (0, 2) sigma models is equivalent to adding matter. In this case an NSVZ-type exact relation between the beta function and the anomalous dimensions gamma of the "matter" fields is established. We derive an analog of the Konishi anomaly. At large N-f our beta function develops an infrared fixed point at small values of the coupling constant (analogous to the Banks-Zaks fixed point). Thus, we reliably predict the existence of a conformal window. At N-f = 1 the model under consideration reduces to the well-known N = (2, 2) CP(1) model.