Deformations of complex supermanifolds

Deformations of Complex Structures

In this section, we discuss briefly how to compare different almost complex structures on a manifold. Let M be a fixed compact manifold, TM its tangent bundle. Let J ∈ End(TM ) be an almost complex structure on M , with associated decomposition TM ⊗ C ' TM 0,1 ⊕ TM 1,0, and projections π0,1 and π1,0 to the two summands. Now suppose J ′ is a second almost complex structure. The reader should thi...

Supermanifolds Corresponding to the Trivial Vector Bundle over Complex Torus

Metastable Vacua and Complex Deformations

We use the non-normalizable complex deformations to describe the stringy realizations of the metastable vacua in N = 1, SU(Nc) SUSY theories with Nf > Nc massive fundamental flavors. The consideration of the non-normalizable deformations requires a modified Toric duality. The new approach considers the tachyon condensation between pairs of wrapped D5 branes and anti D5 branes and the resulting ...

On Calabi-Yau supermanifolds

We prove that a Kähler supermetric on a supermanifold with one complex fermionic dimension admits a super Ricci-flat supermetric if and only if the bosonic metric has vanishing scalar curvature. As a corollary, it follows that Yau’s theorem does not hold for supermanifolds. Calabi[1] proposed that if a Kähler manifold has vanishing first Chern class, that is, the Ricci-form obeys Rij̄(g) = ∂iv̄j ...

Supermanifolds from Feynman graphs

We generalize the computation of Feynman integrals of log divergent graphs in terms of the Kirchhoff polynomial to the case of graphs with both fermionic and bosonic edges, to which we assign a set of ordinary and Grassmann variables. This procedure gives a computation of the Feynman integrals in terms of a period on a supermanifold, for graphs admitting a basis of the first homology satisfying...