Chandrashekar Devchand

Special complex manifolds

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection ∇ such that ∇J is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a ∇-parallel symplectic form !. This generalises Freeds definition of (affine) special Kahler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and Kahler manifolds. We prove that the above three types of special geometry are completely solvable, in the sense that they are locally defined by free holomorphic data. In fact, any special complex manifold is locally realised as the image of a holomorphic 1-form � : C n → T ∗ C n . Such a realisation induces a canonical ∇-parallel symplectic structure on M and any special sym- plectic manifold is locally obtained this way. Special Kahler manifolds are realised as complex Lagrangian submanifolds and correspond to closed forms �. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which generalise the hyper-Kahler structure on the cotangent bundle of a special Kahler manifold.

Polyvector Super-Poincare Algebras

A class of ℤ2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form where the algebra of generalized translations W=W0+W1 is the maximal solvable ideal of W0 is generated by W1 and commutes with W. Choosing W1 to be a spinorial module (a sum of an arbitrary number of spinors and semispinors), we prove that W0 consists of polyvectors, i.e.all the irreducible submodules of W0 are submodules of We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of invariant valued bilinear forms on the spinor module S.

Killing spinors are Killing vector fields in Riemannian Supergeometry

A supermanifold M is canonically associated to any pseudo-Riemannian spin manifold (M0, g0). Extending the metric g0 to a field g of bilinear forms g(p) on TpM, p ϵ M0, the pseudo-Riemannian supergeometry of (M, g) is formulated as G-structure on M, where G is a supergroup with even part G0 ≊ Spin(k, l); (k, l) the signature of (M0, go). Killing vector fields on (M, g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field Xs on M. Our main result is that Xs is a Killing vector field on (M, g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field Xs.

The supersymmetric Camassa-Holm equation and geodesic flow on the superconformal group

We study a family of fermionic extensions of the Camassa–Holm equation. Within this family we identify three interesting classes: (a) equations, which are inherently Hamiltonian, describing geodesic flow with respect to an H1 metric on the group of superconformal transformations in two dimensions, (b) equations which are Hamiltonian with respect to a different Hamiltonian structure and (c) supersymmetric equations. Classes (a) and (b) have no intersection, but the intersection of classes (a) and (c) gives a system with interesting integrability properties. We demonstrate the Painleve property for some simple but nontrivial reductions of this system, and also discuss peakon-type solutions.

Extended self-dual Yang-Mills from the N = 2 string

We show that the physical degrees of freedom of the critical open string with N = 2 superconformal symmetry on the worldsheet are described by a self-dual Yang-Mills field on a hyperspace parametrised by the coordinates of the target space R2,2 together with a commuting chiral spinor. A prepotential for the self-dual connection in the hyperspace generates the infinite tower of physical fields corresponding to the inequivalent pictures or spinor ghost vacua of this string. An action is presented for this tower, which describes consistent interactions amongst fields of arbitrarily high spin. An interesting truncation to a theory of five fields is seen to have no graphs of two or more loops.

Yang-Mills connections over manifolds with Grassmann structure

Let M be a manifold with Grassmann structure, i.e., with an isomorphism of the cotangent bundle T*M≅E⊗H with the tensor product of two vector bundles E and H. We define the notion of a half-flat connection ∇W in a vector bundle W→M as a connection whose curvature F∈S2E⊗Λ2H⊗End W ⊂ Λ2T*M⊗End W. Under appropriate assumptions, for example, when the Grassmann structure is associated with a quaternionic Kahler structure on M, half-flatness implies the Yang–Mills equations. Inspired by the harmonic space approach, we develop a local construction of (holomorphic) half-flat connections ∇W over a complex manifold with (holomorphic) Grassmann structure equipped with a suitable linear connection. Any such connection ∇W can be obtained from a prepotential by solving a system of linear first order ODEs. The construction can be applied, for instance, to the complexification of hyper-Kahler manifolds or more generally to hyper-Kahler manifolds with admissible torsion and to their higher-spin analogs. It yields solutions...

Supersymmetric Lorentz-covariant hyperspaces and self-duality equations in dimensions greater than (4 vertical bar 4)

We generalise the notions of supersymmetry and superspace by allowing generators and coordinates transforming according to more general Lorentz representations than the spinorial and vectorial ones of standard lore. This yields novel S0(3, 1 )-covariant superspaces, which we call hyperspaces, having dimensionality greater than (414) of traditional super-Minkowski space. As an application, we consider gauge fields on complexifications these superspaces; and extending the concept of self-duality, we obtain classes of completely solvable equations analogous to the four-dimensional self-duality equations. (~) 1997 Elsevier Science B.V.

Oxidation of Self-Duality to 12 Dimensions and Beyond

Using (partial) curvature flows and the transitive action of subgroups of

Matryoshka of Special Democratic Forms

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