arXiv subjects
Publications and source records attributed to Brett Parker.
In this paper, we describe a future electron-ion collider (EIC), based on the existing Relativistic Heavy Ion Collider (RHIC) hadron facility, with two intersecting superconducting rings, each 3.8 km in circumference. A new ERL accelerator, which provide 5-30 GeV electron beam, will ensure 10^33 to 10^34 cm^-2 s^-1 level luminosity.
This paper describes the structure of the moduli space of holomorphic curves and constructs Gromov Witten invariants in the category of exploded manifolds. This includes defining Gromov Witten invariants relative to normal crossing divisors and proving the associated gluing theorem which involves summing relative invariants over a count of tropical curves.
The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the limits considered in tropical geometry appear smooth, also degenerations corresponding to an algebraic family with normal crossing singularities are smooth.) In this paper we prove a compactness theorem for (pseudo)-holomorphic curves in exploded torus fibrations. In the case of smooth manifolds, this is just a version of Gromov's compactness theorem in a topology strong enough for gluing analysis.
Initiated by Gromov, the study of holomorphic curves in symplectic manifolds has been a powerfull tool in symplectic topology, however the moduli space of holomorphic curves is often very difficult to find. A common technique is to study the limiting behavior of holomorphic curves in a degenerating family of complex structures which corresponds to a kind of adiabatic limit. The category of exploded fibrations is an extension of the smooth category in which some of these degenerations can be described as smooth families. The first part of this paper is devoted to defining exploded fibrations and a slightly more specialized category of exploded torus fibrations. Later sections contain the transverse interesction theory for exploded fibrations and some examples of holomorphic curves in exploded torus fibrations, including a brief discussion of the relationship between tropical geometry and exploded torus fibrations. In the final section, the perturbation theory of holomorphic curves in exploded torus fibrations is sketched.