April 18, 2014
We study N=(0,2) deformed (2,2) two-dimensional sigma models. Such heterotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional 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 Yang-Mills. Instanton calculus provides a straightforward method for the derivation. The anomalous dimensions are calculated up to two loops implying that one of the beta functions is explicitly known up 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 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.
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November 28, 2011
We consider two-dimensional $\mathcal{N}=(0,2)$ 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 it...
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We discuss various questions which emerge in connection with the Lie-algebraic deformation of $\mathbb{CP}^1$ sigma model in two dimensions. First we supersymmetrize the original model endowing it with the minimal ${\cal N}=(0,2)$ and extended ${\cal N}=(2,2)$ supersymmetries. Then we derive the general hypercurrent anomaly in the both cases. In the latter case this anomaly is one-loop but is somewhat different from the standard expressions one can find in the literature beca...
January 28, 2014
We review diverse two-dimensional models emerging on the world sheet of non-Abelian strings in the low-energy limit. Non-Abelian strings are supported in a class of four-dimensional bulk theories with or without supersymmetry. In supersymmetric bulk theories we are mostly interested in BPS-saturated strings. Some of these two-dimensional models, in particular, heterotic models, were scarcely studied in the past, if at all. Our main emphasis is on the heterotic N=(0,2) models....
September 22, 2010
In this paper we begin the study of renormalizations in the heterotically deformed N=(0,2) CP(N-1) sigma models. In addition to the coupling constant g^2 of the undeformed N=(2,2) model, there is the second coupling constant \gamma describing the strength of the heterotic deformation. We calculate both \beta functions, \beta_g and \beta_\gamma at one loop, determining the flow of g^2 and \gamma. Under a certain choice of the initial conditions, the theory is asymptotically fr...
July 10, 2023
In this paper we consider perturbation theory in generic two-dimensional sigma models in the so-called first-order formalism, using the coordinate regularization approach. Our goal is to analyze the first-order formalism in application to $\beta$ functions and compare its results with the standard geometric calculations. Already in the second loop, we observe deviations from the geometric results that cannot be explained by the regularization/renormalization scheme choices. M...
April 25, 2006
In this paper, we study the perturbative aspects of a twisted version of the two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal E$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted in terms of an obstruction to a global definition of the associated sheaf...
January 31, 2008
We explore the nonperturbative aspects of the chiral algebras of N = (0,2) sigma models, which perturbatively are intimately related to the theory of chiral differential operators (CDOs). The grading by charge and scaling dimension is anomalous if the first Chern class of the target space is nonzero. This has some nontrivial consequences for the chiral algebra. As an example, we study the case where the target space is CP^1, and show that worldsheet instantons trivialize the ...
January 7, 2019
The NSVZ $\beta$ functions in two-dimensional $\mathcal N=(0,2)$ supersymmetric models are revisited. We construct and discuss a broad class of such models using the gauge formulation. All of them represent direct analogs of four-dimensional ${\mathcal N} =1$ Yang-Mills theories and are free of anomalies. Following the same line of reasoning as in four dimensions we distinguish between the holomorphic and canonical coupling constants. This allows us to derive the exact two-di...
April 8, 2005
Certain perturbative aspects of two-dimensional sigma models with (0,2) supersymmetry are investigated. The main goal is to understand in physical terms how the mathematical theory of ``chiral differential operators'' is related to sigma models. In the process, we obtain, for example, an understanding of the one-loop beta function in terms of holomorphic data. A companion paper will study nonperturbative behavior of these theories.
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We study the non-minimal supersymmetric heterotically deformed $\mathcal{N}=(0,2)$ sigma model with the Grassmannian target space $\mathcal{G}_{M,N}$. To develop the appropriate superfield formalism, we begin with a simplified model with flat target space, find its beta function up to two loops, and prove a non-renormalization theorem. Then we generalize the results to the full model with the Grassmannian target space. Using the geometric formulation, we calculate the beta fu...