We prove the existence of topological rings in $(0,2)$ theories containing non-anomalous left-moving $U(1)$ currents by which they may be twisted. While the twisted models are not topological, their ground operators form a ring under non-singular OPE which reduces to the $(a,c)$ or $(c,c)$ ring at $(2,2)$ points and to a classical sheaf cohomology ring at large radius, defining a quantum sheaf cohomology away from these special loci. In the special case of Calabi–Yau compactifications, these rings are shown to exist globally on the moduli space if the rank of the holomorphic bundle is less than eight.

In this note, we point out the striking relation between the conditions arising within geometric quantization and the non-perturbative Poisson sigma model. Starting from the Poisson sigma model, we analyze necessary requirements on the path integral measure which imply acertain integrality condition for the Poisson cohomology class $[\alpha]$. The same condition was considered before by Crainic and Zhu but in a different context. In the case when $[\alpha]$ is in the image of the sharp map, we reproduce the Vaisman’s condition for prequantizable Poisson manifolds. For integrable Poisson manifolds, we show, with a different procedure than in Crainic and Zhu, that our integrality condition implies the prequantizability of the symplectic groupoid. Using the relation between prequantization and symplectic reduction, we construct the explicit prequantum line bundle for a symplectic groupoid. This picture supports the program of quantization of Poisson manifold via symplectic groupoid. At the end, we discuss the case of a generic coisotropic $D$-brane.

It is suggested that topological membranes play a fundamental role in the recently proposed topological $M$-theory. We formulate a topological theory of membranes wrapping associative three-cycles in a sevendimensional target space with $G_2$ holonomy. The topological BRST rules and BRST invariant action are constructed via the Mathai–Quillen formalism. In a certain gauge, we show this theory to be equivalent to a membrane theory with two BRST charges found by Beasley and Witten. We argue that at the quantum level, an additional topological term should be included in the action, which measures the contributions of membrane instantons. We construct a set of local and non-local observables for the topological membrane theory. As the BRST cohomology of local operators turns out to be isomorphic to the de Rham cohomology of the $G_2$ manifold, our observables agree with the spectrum of $d = 4$, $N = 1 G_2$ compactifications of $M$-theory.