On local Calabi–Yau supermanifolds and their mirrors

Abstract

We use local mirror symmetry to study a class of local Calabi-Yau supermanifolds with bosonic sub-variety V b having a vanishing first Chern class. Solving the usual super-CY condition, requiring the equality of the total U (1) gauge charges of bosons Φ b and the ghost like fields Ψ f one b q b = f Q f , as b q b = 0 and f Q f = 0, several examples are studied and explicit results are given for local A r super-geometries. A comment on purely fermionic super-CY manifolds corresponding to the special case where q b = 0, ∀b and f Q f = 0 is also made.

Irreducible holonomy algebras of Riemannian supermanifolds

“…As in Section 2 we may show that V is of the form V = L ⊕ L * , where g ⊂ gl(L) is irreducible. If g ⊂ sp(V ) is a skew-Berger subalgebra, then (g ⊂ gl(L)) [1] = {0}, where g [1] = {ϕ ∈ L * ⊗ g|ϕ(x)y = −ϕ(y)x for all x, y ∈ L} is the skew-symmetric prolongation of the subalgebra g ⊂ gl(L). Irreducible subalgebras g ⊂ gl(L) with g [1] = 0 are classified in [17].…”

“…If the representation g ⊂ sp(2m, R) is not absolutely irreducible, then P ω (g) is isomorphic to (g C ⊂ gl(m, C)) [1] and the proof follows from [16,17].…”

“…For example, for so(n, R) ⊕ sl(2, R) ⊂ sp(2n, R) it holds P ω (sl(2, R) ⊂ sp(V )) = 0, P ω (so(n, R) ⊂ so(V )) = (so(n, C) ⊂ sl(n, C)) [1] .…”

“…Thus if g ⊂ sp(L ⊕ L * ) is a skew-symmetric weak-Berger subalgebra, then (g ⊂ gl(L)) [1] = {0} and g ⊂ gl(L) is given in [17].…”

On Local F-Theory Geometries and Intersecting D7-Branes

“…This has been obtained by blowing up the ordinary ADE singularties of the K3 surface which have a nice physical representation in terms of sigma model with four supercharges [29,30,31].…”