Ivancevic_Applied-Diff-Geom

36 Applied Differential Geometry: A Modern Introductionmade up of two of these: the up quark and down quark, bound togetherby the strong nuclear force. Together with the electron (bound to thenucleus in atoms by the electromagnetic force), those fermions constitutethe vast majority of everyday matter. To date, almost all experimentaltests of the three forces described by the Standard Model have agreed withits predictions. However, the Standard Model is not a complete theoryof fundamental interactions, primarily because it does not describe thegravitational force.For this reason, string theories are able to avoid problems associatedwith the presence of point–like particles in a physical theory. The basicidea is that the fundamental constituents of Nature are strings of energyof the Planck length (around 10 −35 m), which vibrate at specific resonantfrequencies (modes). Another key claim of the theory is that no measurabledifferences can be detected between strings that wrap around dimensionssmaller than themselves and those that move along larger dimensions (i.e.,physical processes in a dimension of size R match those in a dimension ofsize 1/R). Singularities are avoided because the observed consequences of‘big crunches’ never reach zero size. In fact, should the universe begin a‘big crunch’ sort of process, string theory dictates that the universe couldnever be smaller than the size of a string, at which point it would actuallybegin expanding.Recently, physicists have been exploring the possibility that the stringsare actually membranes, that is strings with 2 or more dimensions (membranesare refereed to as p−branes, where p is the number of dimensions,see Figure 1.3). Every p−brane sweeps out a (p + 1)−dimensional world–volume as it propagates through space–time. A special class of p−branes arethe so–called D–branes, named for the mathematician Johann Dirichlet. 49D–branes are typically classified by their dimension, which is indicated bya number written after the D: a D0–brane is a single point, a D1–brane isa line (sometimes called a ‘D-string’), a D2–brane is a plane, and a D25–brane fills the highest–dimensional space considered in old bosonic string49 Recall that Dirichlet boundary conditions have long been used in the study of fluidsand potential theory, where they involve specifying some quantity all along a boundary.In fluid dynamics, fixing a Dirichlet boundary condition could mean assigning a knownfluid velocity to all points on a surface; when studying electrostatics, one may establishDirichlet boundary conditions by fixing the voltage to known values at particular locations,like the surfaces of conductors. In either case, the locations at which values arespecified is called a D–brane. These constructions take on special importance in stringtheory, because open strings must have their endpoints attached to D–branes.