Chiral de Rham complex over locally complete intersections

Given a locally complete intersection $X\hookrightarrow Y$ we define a version of a derived chiral De Rham complex, thereby “chiralizing” a result by Illusie and Bhatt. A similar construction attaches to a graded ring a dg vertex algebra, which we prove to be Morita equivalent to a dg algebra of differential operators. For example, the dg vertex algebra associated to a fat point, which also arises in the Landau–Ginzburg model, is shown to be derived rational.