Courant algebroid relations define spinor and Dirac structure relations, with T-duality inducing spinor relations that generalize twisted cohomology isomorphisms and are compatible with Type II supergravity equations.
Lie Group Valued Moment Maps
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abstract
We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map. As an application we obtain moduli spaces of flat connections on an oriented compact 2-manifold with boundary as quasi-Hamiltonian quotients of the space G^2 x ... x G^2.
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A theorem gives sufficient conditions for an integrable system to descend through Poisson reduction, with explicit generalized action variables, applied to moduli spaces of flat connections and to the three doubles of compact Lie groups.
Multiplicative quiver varieties carry a pencil of dimension ℓ(ℓ-1)/2 of compatible Poisson structures obtained by reduction from a pencil of Hamiltonian quasi-Poisson structures.
Singular fibers in type (ii) compactified Ruijsenaars-Schneider systems are smooth connected isotropic submanifolds, diffeomorphic to S^3 over singular vertices of the action polytope in simple cases.
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Generalised Complex and Spinor Relations
Courant algebroid relations define spinor and Dirac structure relations, with T-duality inducing spinor relations that generalize twisted cohomology isomorphisms and are compatible with Type II supergravity equations.
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Integrable systems from Poisson reductions of generalized Hamiltonian torus actions
A theorem gives sufficient conditions for an integrable system to descend through Poisson reduction, with explicit generalized action variables, applied to moduli spaces of flat connections and to the three doubles of compact Lie groups.
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Compatible Poisson structures on multiplicative quiver varieties
Multiplicative quiver varieties carry a pencil of dimension ℓ(ℓ-1)/2 of compatible Poisson structures obtained by reduction from a pencil of Hamiltonian quasi-Poisson structures.
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Spherical singularities in compactified Ruijsenaars--Schneider systems
Singular fibers in type (ii) compactified Ruijsenaars-Schneider systems are smooth connected isotropic submanifolds, diffeomorphic to S^3 over singular vertices of the action polytope in simple cases.