Making the hyper-Kähler structure ofN=2quantum string manifest

Abstract

We show that the Lorentz covariant formulation of Nϭ2 string in a curved space reveals an explicit hyper-Kähler structure. Apart from the metric, the superconformal currents couple to a background two-form. By superconformal symmetry the latter is constrained to be holomorphic and covariantly constant and allows one to construct three complex structures obeying a ͑pseudo͒quaternion algebra.

Supersymmetry of the chiral de Rham complex

“…The superconformal structures exhibited in this paper are inspired by the wellknown supersymmetries of σ-models. Zumino [Zum79] and Alvarez-Gaumé and Freedman [AGF81] showed that nonlinear σ-models on general Riemannian, complex and hyperkähler targets carry N = 1, 2, 4 super-Poincaré symmetries, respectively (see [HKLR87,Fre99,IAS99] for excellent discussions, and [BGL04] for a related recent development). In the N = 2 case, this structure is superconformal when the target is Calabi-Yau.…”

Conformally Flat Pseudo-riemannian Homogeneous Ricci Solitons 4-spaces

“…Theorem 5.1. Among conformally flat homogeneous non-locally symmetric fourdimensional pseudo-Riemannian non-trivial Ricci soliton with the Ricci operator of Segre types [22], [1,12] or [11,2], the Ricci Solitons examples are listed in the following Tables 3,4 and 5, where the checkmark means that X is invariant for all Lie algebras of that form. Proof.…”

Four-dimensional Osserman–Ivanov–Petrova metrics of neutral signature

“…From now on, we fix an orthonormal basis {e 1 = x, e 2 = J x, e 3 , e 4 = J e 3 } for V such that y = (sinh ϕ 0 )J x + (cosh ϕ 0 )e 3 for some ϕ 0 . Taking the family of oriented nondegenerate mixed 2-planes π ϕ = {x, (sinh ϕ)J x + (cosh ϕ)e 3 } , a straightforward calculation from (4) shows that A(π ϕ ), when expressed with respect to the fixed orthornormal basis, takes the form…”

“…Pseudo-Riemannian metrics of signature other than Lorentzian have received considerable attention in mathematical physics since the work of Ooguri and Vafa [22] on N = 2 strings [4,8,17,19] to list but a few of the many possible references. Applications of pseudo-Riemannian metrics in braneworld cosmology have been discussed by Sahni and Shtanov [24].…”