Analysis of contact Cauchy-Riemann maps II: canonical neighborhoods and exponential convergence for the Morse-Bott case

This is a sequel to the papers [OW1], [OW2]. In [OW1], the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\lambda,J)$. In [OW2], they used the connection to establish a priori $W^{k,p}$-coercive estimates for maps $w: \dot \Sigma \to M$ satisfying $\overline{\partial}^\pi w= 0, \, d(w^*\lambda \circ j) = 0$ \emph{without involving symplectization}. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse-Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\lambda,J)$, with $\lambda$ a Morse-Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes. We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse-Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois [Bou] (resp. by Bao [Ba]), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse-Bott case is an essential ingredient in the set-up of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).Comment: 69 pages, final version to appear in Nagoya Math J, improvement of overall presentatio

Top Hypercharge

We propose a top hypercharge model with gauge symmetry SU(3)_C x SU(2)_L x U(1)_1 x U(1)_2 where the first two families of the Standard Model (SM) fermions are charged under U(1)_1 while the third family is charged under U(1)_2. The U(1)_1 x U(1)_2 gauge symmetry is broken down to the U(1)_Y gauge symmetry, when a SM singlet Higgs field acquires a vacuum expectation value. We consider the electroweak constraints, and compare the fit to experimental observables to that of the SM. We study the quark CKM mixing between the first two families and the third family, the neutrino masses and mixing, the flavour changing neutral current effects in meson mixing and decays, the Z' discovery potential at the Large Hadron Collider, the dark matter with a gauged Z_2 symmetry, and the Higgs boson masses.Comment: 17 pages and 4 figure

Analysis of Contact Cauchy-Riemann maps I: a priori Ck estimates and asymptotic convergence

In the present article, we develop tensorial analysis for solutions w of the following nonlinear elliptic system ∂ π w = 0, d(w ∗ λ ◦ j) = 0, associated to a contact triad (M, λ, J). The novel aspect of this approach is that we work directly with this elliptic system on the contact manifold without involving the symplectization process. In particular, when restricted to the case where the one-form w∗λ◦ j is exact, all a priori estimates for w-component can be written in terms of the map w itself without involving the coordinate from the symplectization. We establish a priori Ck coercive pointwise estimates for all k ≥ 2 in terms of the energy density dw2 by means of tensorial calculations on the contact manifold itself. Further, for any solution w under the finite π-energy assumption and the derivative bound, we also establish the asymptotic subsequence convergence to ‘spiraling’ instantons along the ‘rotating’ Reeb orbit.11Nsciescopu