Compactification of the moduli space in symplectization and hidden symmetries of its boundary

The purpose of this paper and the forth coming [L1], [L3] is to lay down a foundation for a sequence of papers concerning the moduli space of connecting pseudo-holomorphic maps in the symplectization of a compact contact manifold and their applications. In this paper, we will establish the comapctification of the moduli space of the pseudo-holomorphic maps in the symplectization and exhibit some new phenomenon concerning bubbling and the ”hidden” symmetries of the boundary of the comapctification. Combining with the index formula, which will be proved in [L3], we will show in [L3] that the virtual co-dimension of the boundary components of the moduli space with at least one bubble is at least two, while the virtual co-dimension of the boundary components of broken connecting maps of two elements is one. In [L1], we will show that these virtual co-dimensions can be realized in the corresponding virtual moduli cycles. In a sequence of forth coming papers, we will give some of possible applications. In particular, we will define various versions of index homology for a contact manifold, relative index homology for a symplectic manifold with contact type boundary, as well as their multiplicative structures in these holmologies. These multiplicative structures can be thought as analogies of the usual quantum product and pants product in quantum cohomology and Floer cohomology. We will also investigate the implication of these homologies to Weinstein conjecture. It is well-known that a family of pseudo-holomorphic maps in the symplectization of a compact contact manifold may develop bubbles. Since in the symplectization the symplectic form is exact, each top bubble necessarily has nonremovable singularity at infinity, and along the end at infinity, the bubble is convergent to some closed orbit of the Reeb field of the contact manifold. This makes the behavior of the boundary components of the compactification of the moduli space here very much look like the one of the broken connecting orbits in the usual Floer homology. In particular, it is believed that the co-dimension of the boundary components even coming from bubbling should be one in general. We will show in this paper and [L3] that in the case of the moduli space the