Twisted multiplets and new supersymmetric non-linear σ-models

D=2,N=2, Supersymmetric theories on Non(anti)commutative Superspace

“…Further, in two dimensions one can have new σ-models due to the possibility of defining twisted multiplets [59], in addition to the chiral and antichiral multiplets. These σ-models have established various connections between D = 4, N = 1 and D = 2, N = 2 models.…”

Bihermitian metrics on Hopf surfaces

“…However, their construction starts by producing a family of positive smooth functions f β,m,λ satisfying the first condition and lim λ→0 f β,m,λ = f β m ,β , where f β m ,β is the function f α,β with α = β m , defined above. Since f β m ,β is H-invariant (it is, in fact, U (1) × U (1)-invariant as one can see from the equation (10) in [14]) and since H commutes with γ 0 , by replacing f β,m,λ with its average over H, we can assume without loss that f β,m,λ are H-invariant. Thus, still following [14], the (1, 1)-form F β,m,λ := dd c f β,m,λ f β,m,λ descends to the respective complex manifold (M λ , J λ ).…”

“…It turns out [16] that a generalized Kähler structure is equivalent to the data of a Riemannian metric g and two g-orthogonal complex structures (J + , J − ), satisfying the relations d c + F g + + d c − F g − = 0, dd c ± F g ± = 0, where F g ± (·, ·) = g(J ± ·, ·) are the fundamental 2-forms of the hermitian structures (g, J ± ), and d c ± are the associated i(∂ ± − ∂ ± ) operators. (These conditions on a pair of hermitian structures were, in fact, first described in the physics paper [10] as the general target space geometry for a (2, 2) supersymmetric sigma model.) In four dimensions we obtain a bihermitian structure, provided that the generalized Kähler structure is of even type (which corresponds to our assumption that J ± induce the same orientation on M ) and F g + = ±F g − (i.e.…”

Symplectic sigma models in superspace

“…If such a term is allowed, one can construct a twisted supersymmetric sigma model, which enjoys N = 4 supersymmetry 1 even though the target space is not hyper-Kähler (neither it is Kähler) [2]. 2 If the second condition is not satisfied and we are dealing not with field theory, but with quantum mechanics, even a larger variety of supersymmetric sigma models can be constructed. A special interesting subclass of such (0 + 1) sigma models is formed by those which cannot be directly reproduced by dimensional reduction from the higher d ones.…”

“…The simplest model of this type is defined on a conformally flat 3-dimensional manifold with fermions and bosons belonging, respectively, to the fundamental and adjoint representations of SO(3) ∼ USp (2). The Lagrangian in components was constructed in [4] and the superfield description was given in [5] (see also [6]).…”