The geometric deformation of curved L∞L_\infty algebras and Lie algebroids

We investigate the deformation of an $L_\infty$ structure that encodes the geometric information about a smooth vector bundle. We establish that this specific type of deformation endows the bundle with a Lie algebroid structure. Additionally, we compute the Atiyah class and Atiyah-Chern classes in the geometrically deformed case. The precise relationship between the first Atiyah-Chern class and the modular class is shown. In the case of action Lie algebroids, the Atiyah-Chern classes at the leading order are given by the equivariant Chern characters.Comment: Comments are welcome

N=(0, 2) Deformation of (2, 2) Sigma Models: Geometric Structure, Holomorphic Anomaly and Exact Beta Functions

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.Comment: 38 pages, 6 figures, minor changes in the text and references, the journal versio

N=(0,2) Supersymmetry and a Nonrenormalization Theorem

In this paper we continue the study of perturbative renormalizations in an $\mathcal{N}=(0,2)$ supersymmetric model. Previously we analyzed one-loop graphs in the heterotically deformed CP$(N-1)$ models. Now we extend the analysis of the $\beta$ function and appropriate $Z$ factors to two, and, in some instances, all loops in the limiting case $g^2\to 0$. The field contents of the model, as well as the heterotic coupling, remain the same, but the target space becomes flat. In this toy $\mathcal{N}=(0,2)$ model we construct supergraph formalism. We show, by explicit calculations up to two-loop order, that the $\beta$ function is one-loop-exact. We derive a nonrenormalization theorem valid to all orders. This nonrenormalization theorem is rather unusual since it refers to (formally) $D$ terms. It is based on the fact that supersymmetry combined with target space symmetries and "flavor"? symmetries is sufficient to guarantee the absence of loop corrections. We analyze the supercurrent supermultiplet (i.e., the hypercurrent) providing further evidence in favor of the absence of higher loops in the $\beta$ function.Comment: 21 pages, 5 figures, a new section on supercurrent analysis added, published versio