Lorentz Surfaces and Lorentzian CFT

The interest in string Hamiltonian system has recently been rekindled due to its application to target-space duality. In this article, we explore another direction it motivates. In Sec. 1, conformal symmetry and some algebraic structures of the system that are related to interacting strings are discussed. These lead one naturally to the study of Lorentz surfaces in Sec. 2. In contrast to the case of Riemann surfaces, we show in Sec. 3 that there are Lorentz surfaces that cannot be conformally deformed into Mandelstam diagrams. Lastly in Sec. 4, we discuss speculatively the prospect of Lorentzian conformal field theory. Additionally, to have a view of what quantum picture a string Hamiltonian system may lead to, we discuss independently in the Appendix a formal geometric quantization of the string phase space. MSC number 1991: 05C90, 53C50, 53Z05, 57M50, 81S10, 81T40. Acknowledgements. This work follows from numerous discussions with Orlando Alvarez, who helped me clarify all sorts of premature ideas and generously proofread and commented the draft. It is also under the shadow of Bill Thurston, whose insight on geometry greatly influences me. To both of them I am deeply indebted. I would also like to express my gratitude to Jφrgen E. Anderson, Martin Halpern, Duong Phong, and Alan Weinstein for their courses, from which I had my first contact with CFT and quantization. Besides, I want to thank Hung-Wen Chang, Thom Curtright, Marco Monti, Rafael Nepomechie, and Radu Tǎtar for discussions and assistance. e-mail: [email protected] 0. Introduction and Outline. Lorentz Surfaces and CFT — with appendix Introduction. The interest in string Hamiltonian system has recently been rekindled due to its application to target-space duality (e.g. [C-Z], [A-AG-B-L]). In this article, we explore another direction it provides. Interacting strings can be realized as collections of partially ordered integral filaments in the string Hamiltonian system LT M . They can be regarded as maps from Lorentzian world-sheets Σ into the target-space M . Together with the conformal symmetry in the system, one is motivated to the study of Lorentz surfaces. Depending on the role singularities on Lorentz surfaces play, there are coarse and fine Lorentz surfaces. We discuss only coarse ones due to technical reasons. Like pants decompositions for Riemann surfaces, one has rompers decompositions for Lorentz surfaces. Such decompositions provide a way to study their moduli spaces. In contrast to Riemannian case, we show that there are Lorentz surfaces that cannot be rectified into Mandelstam diagrams. As theory of Riemann surfaces to conformal field theory (CFT), theory of Lorentz surfaces should lead to a Lorentzian counterpart of CFT. We explore this prospect speculatively at the end. Additionally, to have a view of what quantum picture the string Hamiltonian system may lead to, we discuss independently in the Appendix a formal geometric quantization of the string phase space. Readers are referred to [AG-G-M-V], [At1-2], [F-S], [G-S-W], [Ka1-2], [Mo-S1-2], [L-T], [Se1-3], [Thor], and [Zw] for strings, CFT, and string fields; [B-E], [H-E] and [Pe] for Lorentzian manifolds; [C-B] and [St] for surface theory; [Bo], [Kö] for graph theory; [Br], [G-S], [Mi], [Śn] and [Wo] for geometric quantization.