DSpace Collection:http://hdl.handle.net/2440/10872016-07-25T15:58:48Z2016-07-25T15:58:48ZLocalized index and L2-lefschetz fixed-point formula for orbifoldsWang, B.L.Wang, H.http://hdl.handle.net/2440/998272016-06-22T03:20:03Z2016-01-01T00:00:00ZTitle: Localized index and L2-lefschetz fixed-point formula for orbifolds
Author: Wang, B.L.; Wang, H.
Abstract: We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly, and isometrically. These localized indices, generalizing the L²-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operators along conjugacy classes of the discrete group, subject to some trace assumption. Applying the local index technique, we also obtain an L²-version of the Lefschetz fixed-point formulas for orbifolds. These cohomological formulas for the localized indices give rise to a class of refined topological invariants for the quotient orbifold.2016-01-01T00:00:00ZAn Oka principle for equivariant isomorphismsKutzschebauch, F.Larusson, F.Schwarz, G.W.http://hdl.handle.net/2440/997372016-06-16T23:20:25Z2015-01-01T00:00:00ZTitle: An Oka principle for equivariant isomorphisms
Author: Kutzschebauch, F.; Larusson, F.; Schwarz, G.W.
Abstract: Let G be a reductive complex Lie group acting holomorphically on normal Stein spaces X and Y, which are locally G-biholomorphic over a common categorical quotient Q. When is there a global G-biholomorphism X → Y? If the actions of G on X and Y are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that X and Y are G-biholomorphic if X is K-contractible, where K is a maximal compact subgroup of G, or if X and Y are smooth and there is a G-diffeomorphism Ψ : X → Y over Q, which is holomorphic when restricted to each fibre of the quotient map X → Q. We prove a similar theorem when psi is only a G-homeomorphism, but with an assumption about its action on G-finite functions. When G is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of G-biholomorphisms from X to Y over Q. This sheaf can be badly singular, even for a low-dimensional representation of SL₂(ℂ). Our work is in part motivated by the linearisation problem for actions on C n. It follows from one of our main results that a holomorphic G-action on ℂⁿ, which is locally G-biholomorphic over a common quotient to a generic linear action, is linearisable.2015-01-01T00:00:00ZSpin-structures and proper group actionsHochs, P.Mathai, V.http://hdl.handle.net/2440/997362016-06-16T23:09:20Z2016-01-01T00:00:00ZTitle: Spin-structures and proper group actions
Author: Hochs, P.; Mathai, V.
Abstract: Abstract not available2016-01-01T00:00:00ZTwisted chiral de Rham complex, generalized geometry, and T-dualityLinshaw, A.Varghese, M.http://hdl.handle.net/2440/994472016-06-06T23:46:23Z2015-01-01T00:00:00ZTitle: Twisted chiral de Rham complex, generalized geometry, and T-duality
Author: Linshaw, A.; Varghese, M.
Abstract: The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold Z , and contains the ordinary de Rham complex at weight zero. Given a closed 3-form H on Z , we construct the twisted chiral de Rham differential D H , which coincides with the ordinary twisted differential in weight zero. We show that its cohomology vanishes in positive weight and coincides with the ordinary twisted cohomology in weight zero. As a consequence, we propose that in a background flux, Ramond–Ramond fields can be interpreted as D H -closed elements of the chiral de Rham complex. Given a T-dual pair of principal circle bundles Z , Zˆ with fluxes H , Hˆ , we establish a degree-shifting linear isomorphism between a central quotient of the i R [ t ] -invariant chiral de Rham complexes of Z and Zˆ . At weight zero, it restricts to the usual isomorphism of S 1 - invariant differential forms, and induces the usual isomorphism in twisted cohomology. This is interpreted as T-duality in type II string theory from a loop space perspective. A key ingredient in defining this isomorphism is the language of Courant algebroids, which clarifies the notion of functoriality of the chiral de Rham complex.2015-01-01T00:00:00Z