Ivancevic_Applied-Diff-Geom

1142 Applied Differential Geometry: A Modern Introductionscribing the process in which M incoming strings interact and produce Noutgoing strings looks at the topological level like a closed surface withM + N = E boundary components and any number of handles (see Figure6.17). This picture is a kind of topological generalization of nonlinear controlMIMO–systems with M inputs, N outputs X states (see section 4.9.1below).The internal loops may arise when virtual particle pairs are produced,just as in quantum field theory. For example, a Feynman diagram in quantumfield theory that involves a loop is shown in Figure 6.18 together withthe corresponding string diagram.Fig. 6.18 A QFT Feynman diagram that involves an internal loop (left), with thecorresponding string diagram (right).Surfaces associated with closed oriented strings have two topologicalinvariants: (i) the number of boundary components E = M +N (which maybe shrunk to punctures, under certain conditions), and (ii) the number hof handles on the surface, which equals the surface genus.Fig. 6.19 Number h of handles on the surface of closed oriented strings, which equalsthe string–surface genus: (a) h = 0 for sphere S 2 ; (b) h = 1 for torus T 2 ; (c) h = 2 forstring–surfaces with higher genus, etc.When E = 0, we just have the topological classification of compact orientedsurfaces without boundary. Rendering E > 0 is achieved by removingE discs from the surface.Recall that in QFT, an expansion in powers of Planck’s constant yieldsan expansion in the number of loops of the associated Feynman diagram,