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Chandrashekar Devchand

Publications and source records attributed to Chandrashekar Devchand.

At least 19 recordsLinked to original sources
arXiv2020

Hyperkähler cones and instantons on quaternionic Kähler manifolds

We present a novel approach to the study of Yang-Mills instantons on quaternionic Kähler manifolds, based on an extension of the harmonic space method of constructing instantons on hyperkähler manifolds. Our results establish a bijection between local equivalence classes of instantons on quaternionic Kähler manifolds M and equivalence classes of certain holomorphic maps on an appropriate SL_2(C)-bundle over the Swann bundle of M.

math.DG↗
arXiv2018

Instantons on hyperkähler manifolds

An instanton $(E, D)$ on a (pseudo-)hyperkähler manifold $M$ is a vector bundle $E$ associated to a principal $G$-bundle with a connection $D$ whose curvature is pointwise invariant under the quaternionic structures of $T_x M, \ x\in M$, and thus satisfies the Yang-Mills equations. Revisiting a construction of solutions, we prove a local bijection between gauge equivalence classes of instantons on $M$ and equivalence classes of certain holomorphic functions taking values in the Lie algebra of $G^\mathbb{C}$ defined on an appropriate $SL_2(\mathbb{C})$-bundle over $M$. Our reformulation affords a streamlined proof of Uhlenbeck's Compactness Theorem for instantons on (pseudo-)hyperkähler manifolds.

math.DG↗
arXiv2016

On pseudo-hyperkähler prepotentials

An explicit surjection from a set of (locally defined) unconstrained holomorphic functions on a certain submanifold of (Sp_1(C) \times C^{4n}) onto the set HK_{p,q} of local isometry classes of real analytic pseudo-hyperkähler metrics of signature (4p,4q) in dimension 4n is constructed. The holomorphic functions, called prepotentials, are analogues of Kähler potentials for Kähler metrics and provide a complete parameterisation of HK_{p,q}. In particular, there exists a bijection between HK_{p,q} and the set of equivalence classes of prepotentials. This affords the explicit construction of pseudo-hyperkähler metrics from specified prepotentials. The construction generalises one due to Galperin, Ivanov, Ogievetsky and Sokatchev. Their work is given a coordinate-free formulation and complete, self-contained proofs are provided. An appendix provides a vital tool for this construction: a reformulation of real analytic G-structures in terms of holomorphic frame fields on complex manifolds.

math.DG↗
arXiv2013

Oxidation of self-duality to 12 dimensions and beyond

Using (partial) curvature flows and the transitive action of subgroups of O(d,Z) on the indices {1,...,d} of the components of the Yang-Mills curvature in an orthonormal basis, we obtain a nested system of equations in successively higher dimensions d, each implying the Yang-Mills equations on d-dimensional Riemannian manifolds possessing special geometric structures. This `matryoshka' of self-duality equations contains the familiar self-duality equations on Riemannian 4-folds as well as their generalisations on complex Kähler 3-folds and on 7- and 8-dimensional manifolds with G_2 and Spin(7) holonomy. The matryoshka allows enlargement (`oxidation') to a remarkable system in 12 dimensions invariant under Sp(3). There are hints that the underlying geometry is related to the sextonions, a six-dimensional algebra between the quaternions and octonions.

hep-th↗
arXiv2009

Ternutator Identities

The ternary commutator or ternutator, defined as the alternating sum of the product of three operators, has recently drawn much attention as an interesting structure generalising the commutator. The ternutator satisfies cubic identities analogous to the quadratic Jacobi identity for the commutator. We present various forms of these identities and discuss the possibility of using them to define ternary algebras.

hep-th↗
arXiv2008

Matryoshka of Special Democratic Forms

Special p-forms are forms which have components ϕ_{μ_1...μ_p} equal to +1,-1 or 0 in some orthonormal basis. A p-form ϕ\in Λ^p R^d is called democratic if the set of nonzero components {ϕ_{μ_1...μ_p}} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1,...,d}. Knowledge of these symmetry groups allows us to define mappings of special democratic p-forms in d dimensions to special democratic P-forms in D dimensions for successively higher P \geq p and D \geq d. In particular, we display a remarkable nested stucture of special forms including a U(3)-invariant 2-form in six dimensions, a G_2-invariant 3-form in seven dimensions, a Spin(7)-invariant 4-form in eight dimensions and a special democratic 6-form Ωin ten dimensions. The latter has the remarkable property that its contraction with one of five distinct bivectors, yields, in the orthogonal eight dimensions, the Spin(7)-invariant 4-form. We discuss various properties of this ten dimensional form.

math-ph↗
arXiv2007

Special Graphs

A special p-form is a p-form which, in some orthonormal basis {e_μ}, has components ϕ_{μ_1...μ_p} = ϕ(e_{μ_1},..., e_{μ_p}) taking values in {-1,0,1}. We discuss graphs which characterise such forms.

math.DG↗
arXiv2004

Polyvector Super-Poincare Algebras

A class of Z_2-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 g = g_0 + g_1, with g_0 = so(V) + W_0 and g_1 = W_1, where the algebra of generalized translations W = W_0 + W_1 is the maximal solvable ideal of g, W_0 is generated by W_1 and commutes with W. Choosing W_1 to be a spinorial so(V)-module (a sum of an arbitrary number of spinors and semispinors), we prove that W_0 consists of polyvectors, i.e. all the irreducible so(V)-submodules of W_0 are submodules of ΛV. We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of so(V)-invariant Λ^k V-valued bilinear forms on the spinor module S.

hep-th↗
arXiv2003

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\cong E\otimes H with the tensor product of two vector bundles E and H. We define the notion of a half-flat connection \nabla^W in a vector bundle W\to M as a connection whose curvature F\in S^2E\otimes\wedge^2 H\otimes W \subset\wedge^2 T^*M\otimes W. Under appropriate assumptions, for example, when the Grassmann structure is associated with a quaternionic Kaehler 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 \nabla^W over a complex manifold with (holomorphic) Grassmann structure equipped with a suitable linear connection. Any such connection \nabla^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-Kaehler manifolds or more generally to hyper-Kaehler manifolds with admissible torsion and to their higher-spin analogues. It yields solutions of the Yang-Mills equations.

math.DG↗
arXiv2002

Partially-flat gauge fields on manifolds of dimension greater than four

We describe two extensions of the notion of a self-dual connection in a vector bundle over a manifold M from dim M=4 to higher dimensions. The first extension, Omega-self-duality, is based on the existence of an appropriate 4-form Omega on the Riemannian manifold M and yields solutions of the Yang-Mills equations. The second is the notion of half-flatness, which is defined for manifolds with certain Grassmann structure T^C M \cong E \otimes H. In some cases, for example for hyper-Kaehler manifolds M, half-flatness implies Omega-self-duality. A construction of half-flat connections inspired by the harmonic space approach is described. Locally, any such connection can be obtained from a free prepotential by solving a system of linear first order ODEs.

hep-th↗
arXiv2001

Flows on quaternionic-Kaehler and very special real manifolds

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M x N of a quaternionic-Kaehler manifold M of negative scalar curvature and a very special real manifold N of dimension n >=0. Such gradient flows are generated by the `energy function' f = P^2, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kaehler manifolds. For the homogeneous quaternionic-Kaehler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point p in M such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kaehler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kaehler manifolds we find degenerate local minima.

hep-th↗
arXiv2001

Super self-duality for Yang-Mills fields in dimensions greater than four

Self-duality equations for Yang-Mills fields in d-dimensional Euclidean spaces consist of linear algebraic relations amongst the components of the curvature tensor which imply the Yang-Mills equations. For the extension to superspace gauge fields, the super self-duality equations are investigated, namely, systems of linear algebraic relations on the components of the supercurvature, which imply the self-duality equations on the even part of superspace. A group theory based algorithm for finding such systems is developed. Representative examples in various dimensions are provided, including the Spin(7) and G(2) invariant systems in d=8 and 7, respectively.

hep-th↗
arXiv2000

Democratic Supersymmetry

We present generalisations of N-extended supersymmetry algebras in four dimensions, using Lorentz covariance and invariance under permutation of the N supercharges as selection criteria.

hep-th↗
arXiv1998

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 H^1 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 flow equations. Classes (a) and (b) have no intersection, but the intersection of classes (a) and (c) gives a candidate for a new supersymmetric integrable system. We demonstrate the Painlevé property for some simple but nontrivial reductions of this system.

solv-int↗
arXiv1998

String-induced Yang-Mills coupling to self-dual gravity

By considering N=2 string amplitudes we determine the (2+2)-dimensional target space action for the physical degrees of freedom: self-dual gravity and self-dual Yang-Mills, together with their respective infinite towers of higher-spin inequivalent picture states. Novel `stringy' couplings amongst these fields are essential ingredients of an action principle for the effective target space field theory. We discuss the covariant description of this theory in terms of self-dual fields on a hyperspace parametrised by the target space coordinate and a commuting chiral spinor.

hep-th↗
arXiv1998

Lorentz covariant spin two superspaces

Superalgebras including generators having spins up to two and realisable as tangent vector fields on Lorentz covariant generalised superspaces are considered. The latter have a representation content reminiscent of configuration spaces of (super)gravity theories. The most general canonical supercommutation relations for the corresponding phase space coordinates allowed by Lorentz covariance are discussed. By including generators transforming according to every Lorentz representation having spin up to two, we obtain, from the super Jacobi identities, the complete set of quadratic equations for the Lorentz covariant structure constants. These defining equations for spin two Heisenberg superalgebras are highly overdetermined. Nevertheless, non-trivial solutions can indeed be found. By making some simplifying assumptions, we explicitly construct several classes of these superalgebras.

hep-th↗