The Equivariant B model

In this work, we introduce an equivariant deformation of the $B$ model on the sphere with a $U(1)$-action. We present the deformed supersymmetry transformations and corresponding Lagrangians and study observables in the supercharge cohomology. The inclusion of equivariance allows for the introduction of novel, position-dependent observables on the sphere, which have no counterparts in the conventional $B$ model. Two specific cases we explore in detail are position-dependent superpotentials and complex structure deformations. In both instances, the theory exhibits notable differences from the standard $B$ model, revealing intriguing new features.