Chiral algebras in Landau-Ginzburg models

Chiral algebras in the cohomology of the ${\bar{Q}}_{+}$ supercharge of two-dimensional $\mathcal{N}=(0,2)$ theories on flat spacetime are discussed. Using the supercurrent multiplet, we show that the answer is renormalization group invariant for theories with an R-symmetry. For $\mathcal{N}=(0,2)$ Landau-Ginzburg models, the chiral algebra is determined by the operator equations of motion, which preserve their classical form, and quantum renormalization of composite operators. We study these theories and then specialize to the $\mathcal{N}=(2,2)$ models and consider some examples.