Ivancevic_Applied-Diff-Geom

Introduction 41which have two distinct endpoints, and closed strings, where the endpointsare joined to make a complete loop. The two types of string behave inslightly different ways, yielding two spectra. Not all modern string theoriesuse both types; some incorporate only the closed variety. However,the bosonic theory has problems. Most importantly, the theory has a fundamentalinstability, believed to result in the decay of space-time itself.Additionally, as the name implies, the spectrum of particles contains onlybosons, particles like the photon which obey particular rules of behavior.While bosons are a critical ingredient of the Universe, they are not its onlyconstituents. Investigating how a string theory may include fermions in itsspectrum led to supersymmetry, a mathematical relation between bosonsand fermions which is now an independent area of study. String theorieswhich include fermionic vibrations are now known as superstring theories;several different kinds have been described.Roughly between 1984 and 1986, physicists realized that string theorycould describe all elementary particles and interactions between them, andhundreds of them started to work on string theory as the most promisingidea to unify theories of physics. This so–called first superstring revolutionwas started by a discovery of anomaly cancellation in type I string theory byMichael Green and John Schwarz in 1984. The anomaly is cancelled due tothe Green–Schwarz mechanism. Several other ground–breaking discoveries,such as the heterotic string, were made in 1985.Note that in the type IIA and type IIB string theories closed stringsare allowed to move everywhere throughout the 10D space-time (called thebulk), while open strings have their ends attached to D–branes, which aremembranes of lower dimensionality (their dimension is odd - 1,3,5,7 or 9– in type IIA and even – 0,2,4,6 or 8 – in type IIB, including the timedirection).While understanding the details of string and superstring theories requiresconsiderable geometrical sophistication, some qualitative propertiesof quantum strings can be understood in a fairly intuitive fashion. Forexample, quantum strings have tension, much like regular strings made oftwine; this tension is considered a fundamental parameter of the theory.The tension of a quantum string is closely related to its size. Consider aclosed loop of string, left to move through space without external forces.Its tension will tend to contract it into a smaller and smaller loop. Classicalintuition suggests that it might shrink to a single point, but this wouldviolate Heisenberg’s uncertainty principle. The characteristic size of thestring loop will be a balance between the tension force, acting to make it