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It is presented a method of construction of sigma-models with target space geometries different from conformally flat ones. The method is based on a treating of a constancy of a coupling constant as a dynamical constraint following as an equation of motion. In this way we build N=4 and N=8 supersymmetric four-dimensional sigma-models in d=1 with hyper-Kahler target space possessing one isometry, which commutes with supersymmetry.
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New heterotic torsional geometries are constructed as orbifolds of T^2 bundles over K3. The discrete symmetries considered can be freely-acting or have fixed points and/or fixed curves. We give explicit constructions when the base K3 is Kummer or algebraic. The orbifold geometries can preserve N=1,2 supersymmetry in four dimensions or be non-supersymmetric.
Using the harmonic superspace techniques in D=2 N=4, we present an explicit derivation of a new hyper-Kahler metric associated to the Toda like self interaction $H ^{4+}(omega, u)= (frac{xi^{++}}{lambda})^{2}exp(2lambda omega)$. Some important features are also discussed.
We analyse the geometry of four-dimensional bosonic manifolds arising within the context of $N=4, D=1$ supersymmetry. We demonstrate that both cases of general hyper-Kahler manifolds, i.e. those with translation or rotational isometries, may be supersymmetrized in the same way. We start from a generic N=4 supersymmetric three-dimensional action and perform dualization of the coupling constant, initially present in the action. As a result, we end up with explicit component actions for $N=4, D=1$ nonlinear sigma-models with hyper-Kahler geometry (with both types of isometries) in the target space. In the case of hyper-Kahler geometry with translational isometry we find that the action possesses an additional hidden N=4 supersymmetry, and therefore it is N=8 supersymmetric one.
Two results regarding Kahler supermanifolds with potential $K=A+Cthetabartheta$ are shown. First, if the supermanifold is Kahler-Einstein, then its base (the supermanifold of one lower fermionic dimension and with Kahler potential $A$) has constant scalar curvature. As a corollary, every constant scalar curvature Kahler supermanifold has a unique superextension to a Kahler-Einstein supermanifold of one higher fermionic dimension. Second, if the supermanifold is itself scalar flat, then its base satisfies the equation $$ phi^{bar ji}phi_{ibar j}=2Delta_0 S_0 + R_0^{bar ji}R_{0ibar j} - S_0^2, $$ where $Delta_0$ is the Laplace operator, $S_0$ is the scalar curvature, and $R_{0ibar j}$ is the Ricci tensor of the base, and $phi$ is some harmonic section on the base. Remarkably, precisely this equation arises in the construction of certain supergravity compactifications. Examples of bosonic manifolds satisfying the equation above are discussed.
We address the construction of four-dimensional N=2 supersymmetric nonlinear sigma models on tangent bundles of arbitrary Hermitian symmetric spaces starting from projective superspace. Using a systematic way of solving the (infinite number of) auxiliary field equations along with the requirement of supersymmetry, we are able to derive a closed form for the Lagrangian on the tangent bundle and to dualize it to give the hyperkahler potential on the cotangent bundle. As an application, the case of the exceptional symmetric space E_6/SO(10) times U(1) is explicitly worked out for the first time.
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