Basic Kirwan injectivity and its applications

Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study Hamiltonian torus actions on transversely K\"ahler foliations. Among other things, we prove a foliated version of the Carrell-Liberman theorem. As an immediate consequence, this confirms a conjecture raised by Battaglia and Zaffran on the basic Hodge numbers of symplectic toric quasifolds. As an aside, we also present a symplectic approach to the calculation of basic Betti numbers of symplectic toric quasifolds.Comment: 18 pages, comments welcom

Nonequilibrium transport through a quantum dot weakly coupled to Luttinger liquids

We study the nonequlibrium transport through a quantum dot weakly coupled to Luttinger liquids (LL). A general current expression is derived by using nonequilibrium Green function method. Then a special case of the dot with only a single energy level is discussed. As a function of the dot's energy level, we find that the current as well as differential conductance is strongly renormalized by the interaction in the LL leads. In comparison with the system with Fermi liquid (FL) leads, the current is suppressed, consistent with the suppression of the electron tunneling density of states of the LL; and the outset of the resonant tunneling is shifted to higher bias voltages. Besides, the linear conductance obtained by Furusaki using master equation can be reproduced from our result.Comment: 8 pages, 3 figures, Late

Equivariant Formality of Transversely Symplectic Foliations and Frobenius Manifolds

Consider the Hamiltonian action of a compact connected Lie group on a transversely symplectic foliation whose basic cohomology satisfies the Hard Lefschetz property. We establish an equivariant formality theorem and an equivariant symplectic dδ-lemma in this setting. As an application, we show that there exists a natural Frobenius manifold structure on the equivariant basic cohomology of the given foliation. In particular, this result provides a class of new examples of dGBV-algebras whose cohomology carries a Frobenius manifold structure