The essence of biological evolution

Abstract

We give a careful definition of the open string propagator in Schnabl gauge and present its worldsheet interpretation. The propagator requires two Schwinger parameters and contains the BRST operator. It builds surfaces by gluing strips of variable width to the left and to the right of off-shell states with contracted or expanded local frames. We evaluate explicitly the four-point amplitude of off-shell tachyons. The computation involves a subtle boundary term, crucial to enforce the correct exchange symmetries. Interestingly, the familiar on-shell physics emerges even though string diagrams produce Riemann surfaces more than once. Off-shell, the amplitudes do not factorize over intermediate on-shell states.1. The amplitudes have permutation symmetry among scattering states.2. The amplitudes are integrals over sections of fiber bundles with base the moduli spaces of Riemann surfaces and fibers spanning the possible choices of local coordinates at the punctures where the scattering states are inserted. Ignoring coordinates at the punctures, each Riemann surface contributes only once to the amplitude.3. The amplitudes satisfy factorization: near poles, all of which must arise from the propagator, the amplitude is a product of the relevant off-shell vertices.The first property arises because the vertices in the string field theory action are symmetric and so is the propagator. 1 The second property implies that the string diagrams for a given amplitude give a construction of the appropriate moduli space of Riemann surfaces: they produce all surfaces of fixed genus and fixed number of punctures, each surfaced produced only once. The third property arises because string diagrams at factorization develop infinitely long strips (or tubes, in closed string theory) that if cut, result in two allowed subdiagrams 2 which provide the two off-shell factors.Siegel gauge provides amplitudes that obey the above properties, but not all gauges do. Off-shell light-cone amplitudes do not satisfy property 1 because the string diagrams break the symmetry among states by assigning to them values of the light-cone momentum p + all of which cannot be the same. They do not satisfy property 3 either, because the Schwinger parameters associated with propagators are sometimes not independent.In this paper we begin a detailed study of off-shell amplitudes in Schnabl gauge, the gauge in which it was possible to obtain an analytic form for the tachyon vacuum string field [7]. The string field that represents a finite marginal deformation by a regular marginal operator is another solution in Schnabl gauge [8,9]. Other analytic solutions [10 -13] use the wedge states [14 -18] that are natural in Schnabl gauge, but do not actually satisfy the gauge condition. Recent related work appears in [19]- [28].The simplest amplitude to consider is the Veneziano amplitude. Its off-shell version in Schnabl gauge is the central topic in this paper. In Siegel gauge the Veneziano amplitude was first discussed by Giddings [29], who found the conformal map from th...