We study $\mathcal{N}=(0,2)$ deformed (2, 2) two-dimensional sigma models. Such hete-rotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional $\mathcal{N}=1$ theories. We study geometric aspects and holomorphic properties of these models and derive a number of exact expressions for the $\beta$ functions in terms of the anomalous dimensions analogous to the NSVZ $\beta$ function in four-dimensional gauge theories. Instanton calculus provides a straightforward method for the derivation verified at two loops by explicit calculations. Obtained two-loop anomalous dimensions give an explicit expression for one of the $\beta$ functions valid to three loops. The fixed point in the ratio of the couplings found previously at one loop is not shifted at two loops. We also consider the $\mathcal{N}=(0,2)$ supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, as well as the “Konishi anomaly.” This gives us another method for finding exact $\beta$ functions. We prove that despite the chiral nature of the models under consideration, quantum loops preserve isometries of the target space.

• Received 29 April 2014

DOI:https://doi.org/10.1103/PhysRevD.90.045014