Torus manifolds with non-abelian symmetries

Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus of (G). A torus manifold with (G)-action is defined to be a smooth connected closed oriented manifold of dimension (2\dim T) with an almost effective action of (G) such that (M^T\neq \emptyset). We show that if there is a torus manifold (M) with (G)-action then the action of a finite covering group of (G) factors through (\tilde{G}=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod SO(2l_i)\times T^{l_0}). The action of (\tilde{G}) on (M) restricts to an action of (\tilde{G}'=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod U(l_i)\times T^{l_0}) which has the same orbits as the (\tilde{G})-action. We define invariants of torus manifolds with (G)-action which determine their (\tilde{G}')-equivariant diffeomorphism type. We call these invariants admissible 5-tuples. A simply connected torus manifold with (G)-action is determined by its admissible 5-tuple up to (\tilde{G})-equivariant diffeomorphism. Furthermore we prove that all admissible 5-tuples may be realised by torus manifolds with (\tilde{G}")-action where (\tilde{G}") is a finite covering group of (\tilde{G}').Comment: 56 pages; a mistake in section 6 corrected; accepted for publication in Trans. Am. Math. So

Symplectic torus actions with coisotropic principal orbits

In this paper we completely classify symplectic actions of a torus $T$ on a compact connected symplectic manifold $(M, \sigma)$ when some, hence every, principal orbit is a coisotropic submanifold of $(M, \sigma)$. That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action and the symplectic form. The invariants are invariants of the topology of the manifold, of the torus action, or of the symplectic form. In order to deal with symplectic actions which are not Hamiltonian, we develop new techniques, extending the theory of Atiyah, Guillemin-Sternberg, Delzant, and Benoist. More specifically, we prove that there is a well-defined notion of constant vector fields on the orbit space $M/T$. Using a generalization of the Tietze-Nakajima theorem to what we call $V$-parallel spaces, we obtain that $M/T$ is isomorphic to the Cartesian product of a Delzant polytope with a torus. We then construct special lifts of the constant vector fields on $M/T$, in terms of which the model of the symplectic manifold with the torus action is defined

Spectral action on noncommutative torus

The spectral action on noncommutative torus is obtained, using a Chamseddine--Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.Comment: 57 page

Topological classification of torus manifolds which have codimension one extended actions

A toric manifold is a compact non-singular toric variety equipped with a natural half-dimensional compact torus action. A torus manifold is an oriented, closed, smooth manifold of dimension $2n$ with an effective action of a compact torus $T^{n}$ having a non-empty fixed point set. Hence, a torus manifold can be thought of as a generalization of a toric manifold. In the present paper, we focus on a certain class \mM in the family of torus manifolds with codimension one extended actions, and we give a topological classification of \mM. As a result, their topological types are completely determined by their cohomology rings and real characteristic classes. The problem whether the cohomology ring determines the topological type of a toric manifold or not is one of the most interesting open problems in toric topology. One can also ask this problem for the class of torus manifolds even if its orbit spaces are highly structured. Our results provide a negative answer to this problem for torus manifolds. However, we find a sub-class of torus manifolds with codimension one extended actions which is not in the class of toric manifolds but which is classified by their cohomology rings.Comment: 20 page

Poincar\'e polynomials for Abelian symplectic quotients of pure rr-qubits via wall-crossings

In this paper, we compute a recursive wall-crossing formula for the Poincar\'e polynomials and Euler characteristics of Abelian symplectic quotients of a complex projective manifold under a special effective action of a torus with non-trivial characters. An analogy can be made with the space of pure states of a composite quantum system containing $r$ quantum bits under action of the maximal torus of Local Unitary operations