Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1141chosen to observe the process. In the Figure above, equal time slices areindicated from the point of view of two different Lorentz frames, schematicallyindicated by t and t ′ . The closed string interaction, as seen fromframes t and t ′ , occurs at times t 2 and t ′ 2 and at (distinct) points P and P ′respectively.Lorentz invariance of interaction forbids that any point on the world–sheet be singled out as interaction point. Instead, the interaction resultspurely from the joining and splitting of strings. While free closed strings arecharacterized by their topology being that of a cylinder, interacting stringsare characterized by the fact that their associated world–sheet is connectedto at least 3 strings, incoming and/or outgoing.As a result, the free string determines the nature of the interactionscompletely, leaving only the string coupling constant undetermined.The orientation is an additional structure of closed strings, dividingthem into two categories: (i) oriented strings, in which all world–sheets areassumed to be orientable; and (ii) non–oriented strings, in which world–sheets are non–orientable, such as the Möbius strip, Klein bottle, etc.Fig. 6.17 Boundary components and handles of closed oriented system of M incomingstrings, interacting through internal loops, to produce N outgoing strings. Notethe striking similarity with MIMO–systems of nonlinear control theory, with M inputprocesses and N output processes(see section 4.9.1 below).6.5.4.3 Loop Expansion – Topology of Closed SurfacesFor simplicity, here we consider closed oriented strings only, so that theassociated world–sheet is also oriented. A general string configuration de-