We are now ready to state our main theorem. , to be restricted to connect sums of S 1 × S 2 's, and finite quotients of homotopy spheres. A direct calculation shows that τ (X) is essentially trivial for the manifolds in this list. Thus, the triviality of τ (X) is a necessary condition for a closed manifold to have a positive scalar curvature metric. Our main theorem now gives an alternative proof of this conclusion. Theorem 1.1 (Main Theorem)There is an invariant of X, the Alexander invariant, denoted by Δ(X), which is related to the Milnor torsion, but easy to compute from a link presentation, or Heegard splitting of X. And, from The Seiberg-Witten invariant in dimension 3Let X be as described in Theorem 1.1. If ∂X = ∅, then glue the half cylinder ∂X × [0, ∞) to X along their common boundaries to get an open
The Kepler problem is a physical problem about two bodies which attract each other by a force proportional to the inverse square of the distance. The MICZ-Kepler problems are its natural cousins and have been previously generalized from dimension three to dimension five. In this paper, we construct and analyze the (quantum) MICZ-Kepler problems in all dimensions higher than two.
For a simple euclidean Jordan algebra, let co be its conformal algebra, P be the manifold consisting of its semi-positive rank-one elements, C ∞ (P) be the space of complex-valued smooth functions on P. An explicit action of co on C ∞ (P), referred to as the hidden action of co on P, is exhibited. This hidden action turns out to be mathematically responsible for the existence of the Kepler problem and its recently discovered vast generalizations, referred to as J-Kepler problems. The J-Kepler problems are then reconstructed and re-examined in terms of the unified language of euclidean Jordan algebras. As a result, for a simple euclidean Jordan algebra, the minimal representation of its conformal group can be realized either as the Hilbert space of bound states for its J-Kepler problem or as L 2 (P, 1 r vol), where vol is the volume form on P and r is the inner product of x ∈ P with the identity element of the Jordan algebra. C 2011 American Institute of Physics.Euclidean Jordan algebras and J-Kepler problems J. Math. Phys. 52, 112104 (2011) i.e., it is a reformulation of the Kepler problem as a dynamic problem on the open future light cone-the Kepler cone in this case. The J-Kepler problems all share the key features of the Kepler problem, for example, the Ith eigenvalue of the Hamiltonian ishere, ρ and δ are the rank and degree of the Jordan algebra, respectively. The more detailed results are given in Theorem 5 on p. 61.A key mathematical result here is Theorem 1 on p. 36, which gives an explicit action of the conformal algebra of the Jordan algebra on the space of complex-valued smooth functions on the Kepler cone. In this action, elements of the conformal algebra are realized as differential operators of degree zero, one, and two, so this action does not come from an underlying action on the Kepler cone; consequently such an action is referred to as a hidden action of the conformal algebra on the Kepler cone. In our view, this hidden action is the mathematical origin for the J-Kepler problems.One curious fact, though not presented here, is that the magnetized versions of a given J-Kepler problem also exist unless the Jordan algebra is the exceptional one. Another fact, which is not proved here either, is that a unitary lowest weight representation can be realized by the Hilbert space of bound states of a magnetized version of a J-Kepler problem if and only if it has the minimal positive Gelfand-Kirillov dimension.Here is the organization of this paper. In Sec. II, we give a review of the euclidean Jordan algebras, tailored to our needs. In Sec. III, we review the TKK (Tits-Kantor-Koecher) construction, 10 something that assigns a simple real Lie algebra (the conformal algebra) to each simple euclidean Jordan algebra. In Sec. IV, we do a bit of structural analysis for the conformal algebra. In Sec. V, we introduce the notion of Kepler cone for simple euclidean Jordan algebras. In Sec. VI, we introduce the hidden action of the conformal algebra on the Kepler cone, which amounts to the dynamic symmetry of the...
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