Topological B-Model on Weighted Projective Spaces and Self-Dual Models in Four Dimensions

TL;DR

This paper explores topological B-models on weighted projective spaces as CY supermanifolds, deriving four-dimensional self-dual field theories via twistor techniques, extending known equivalences involving supermanifolds and gauge theories.

Contribution

It introduces new target spaces for the B-model, weighted projective spaces, and establishes their connection to self-dual and twisted N=4 SYM theories using twistor methods.

Findings

Weighted projective spaces are CY supermanifolds for p+q=4.

Holomorphic Chern-Simons theory on these spaces relates to self-dual N=4 SYM.

Derived self-dual truncations of N=4 SYM from these geometries.

Abstract

It was recently shown by Witten on the basis of several examples that the topological B-model whose target space is a Calabi-Yau (CY) supermanifold is equivalent to holomorphic Chern-Simons (hCS) theory on the same supermanifold. Moreover, for the supertwistor space CP^{3|4} as target space, it has been demonstrated that hCS theory on CP^{3|4} is equivalent to self-dual N=4 super Yang-Mills (SYM) theory in four dimensions. We consider as target spaces for the B-model the weighted projective spaces WCP^{3|2}(1,1,1,1|p,q) with two fermionic coordinates of weight p and q, respectively - which are CY supermanifolds for p+q=4 - and discuss hCS theory on them. By using twistor techniques, we obtain certain field theories in four dimensions which are equivalent to hCS theory. These theories turn out to be self-dual truncations of N=4 SYM theory or of its twisted (topological) version.