Euclidean supersymmetry, twisting and topological sigma models

Abstract

We discuss two dimensional N-extended supersymmetry in Euclidean signature and its R-symmetry. For N = 2, the R-symmetry is SO(2) × SO(1, 1), so that only an A-twist is possible. To formulate a B-twist, or to construct Euclidean N = 2 models with Hflux so that the target geometry is generalised Kahler, it is necessary to work with a complexification of the sigma models. These issues are related to the obstructions to the existence of non-trivial twisted chiral superfields in Euclidean superspace.

2D and 3D topological field theories for generalized complex geometry

“…One can show that our 2D theory with generalized complex structure corresponding to an ordinary complex structure is, upon gauge fixing, equivalent to the B-model [35], while the 2D theory for a symplectic structure is equivalent to the A-model [35]. The more general 2D models on a generalized Kähler manifold should correspond to a topological twist of the N = (2, 2) nonlinear sigma model [21,18]. We will present the detailed analysis of the gauge fixing for these models in a forthcoming work [9].…”

Generalized complex geometry

“…One can show that our 2D theory with generalized complex structure corresponding to an ordinary complex structure is, upon gauge fixing, equivalent to the B-model [35], while the 2D theory for a symplectic structure is equivalent to the A-model [35]. The more general 2D models on a generalized Kähler manifold should correspond to a topological twist of the N = (2, 2) nonlinear sigma model [21,18]. We will present the detailed analysis of the gauge fixing for these models in a forthcoming work [9].…”

Semichiral fields on S 2 and generalized Kähler geometry

“…The target space geometry of these models is the complexification of the target space geometry of the corresponding models defined in Lorentzian signature. See [48] for a discussion of these issues.…”