Space Of One-forms Research Articles

Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface Sigma , and the union mathcal {O} of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on mathcal {O} to the Bergman space of holomorphic forms on Sigma is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on Sigma by elements of Bergman space and Dirichlet space on fixed regions in R containing Sigma .

Reviving 3D mathcal{N} = 8 superconformal field theories

We present a Lagrangian formulation for mathcal{N} = 8 superconformal field theories in three spacetime dimensions that is general enough to encompass infinite-dimensional gauge algebras that generally go beyond Lie algebras. To this end we employ Chern-Simons theories based on Leibniz algebras, which give rise to L∞ algebras and are defined on the dual space mathfrak{g} * of a Lie algebra mathfrak{g} by means of an embedding tensor map ϑ: mathfrak{g} * → mathfrak{g} . We show that for the Lie algebra mathfrak{sdif}{mathfrak{f}}_3 of volume-preserving diffeomorphisms on a 3-manifold there is a natural embedding tensor defining a Leibniz algebra on the space of one-forms. Specifically, we show that the cotangent bundle to any 3-manifold with a volume-form carries the structure of a (generalized) Courant algebroid. The resulting mathcal{N} = 8 superconformal field theories are shown to be equivalent to Bandos-Townsend theories. We show that the theory based on S3 is an infinite-dimensional generalization of the Bagger-Lambert-Gustavsson model that in turn is a consistent truncation of the full theory. We also review a Scherk-Schwarz reduction on S2 × S1, which gives the super-Yang-Mills theory with gauge algebra mathfrak{sdif}{mathfrak{f}}_2 , and we construct massive deformations.

An averaging trick for smooth actions of compact quantum groups on manifolds

We prove that, given any smooth action of a compact quantum group (in the sense of [9]) on a compact smooth manifold satisfying some more natural conditions, one can get a Riemannian structure on the manifold for which the corresponding C∞(M)-valued inner product on the space of one-forms is preserved by the action.

DIFFERENTIAL STRUCTURE ON κ-MINKOWSKI SPACE, AND κ-POINCARÉ ALGEBRA

We construct realizations of the generators of the κ-Minkowski space and κ-Poincaré algebra as formal power series in the h-adic extension of the Weyl algebra. The Hopf algebra structure of the κ-Poincaré algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on κ-Minkowski space which is compatible with the action of the Lorentz algebra. In contrast to the conventional bicovariant calculus, the space of one-forms has the same dimension as the κ-Minkowski space.

Noncommutative differential forms on the kappa-deformed space

We construct a differential algebra of forms on the kappa-deformed space. For a given realization of noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of one-forms and nilpotent exterior derivatives. We derive explicit expressions for the exterior derivative and one-forms in covariant and noncovariant realizations. We also introduce higher order forms and show that the exterior derivative satisfies the graded Leibniz rule. The differential forms are generally not graded commutative, but they satisfy the graded Jacobi identity. We also consider the star-product of classical differential forms. The star-product is well defined if the commutator between the noncommutative coordinates and one-forms is closed in the space of one-forms alone. In addition, we show that in certain realizations the exterior derivative acting on the star-product satisfies the undeformed Leibniz rule.