Introduction to String Theory
Roughly one hundred years ago, at the turn of the last century, the newtonian paradigm started to collapse. Scientists who only years before had declaimed the imminent end of Physics, were now faced with empirical data for which there was no adequate theoretical explanation.
Out of this conflict emerged two scientific revolutions: Einstein's general theory of relativity, to account for the discrepancies in planetary motion; and quantum mechanics, and later quantum field theory, to explain atomic and subatomic phenomena. It took physicists the first three quarters of the twentieth century to develop these theories to the point that they can account, in principle, for most if not all of observed phenomena. Why then the need for something else?
Part of the appeal of the newtonian paradigm was its universality. Newtonian physics seemed to account for a vast range of phenomena, from the very small to the very large. As the empirical horizons widened, it became necessary to replace newtonian physics by not one but two new theories. Furthermore, these two theories happen to be incompatible. In other words, either theory loses its predictive power whenever it becomes impossible to ignore the other. Therefore besides the purely aesthetic need to have a single fundamental physical theory, there is a very real need for a theory which explains what happens at those tiny length scales at which neither quantum mechanics nor gravity can be ignored. String theory emerged in the mid-eighties as a likely candidate for such a theory.
The fundamental premise of string theory is that the basic objects in nature are not point-like, but rather string-like. Remarkably, out of this deceptively simple generalisation, one obtains a theory which does not just incorporate gauge theory, supersymmetry and gravitation in a natural and elegant way, but actually needs all three of them for its very consistency. It is precisely this fact which makes string theory such a compelling candidate for a unified theory.
Two 'revolutions' punctuate the history of string theory: the first happened in 1984 as a result of the work of Green and Schwarz on anomaly cancellation, the second was sparked in 1994 by the work of Seiberg and Witten on supersymmetric gauge theories and that of Hull and Townsend on string dualities. (The third revolution is due any day now!)
The panorama after the first superstring revolution consisted of five perturbatively consistent superstring theories, living in a ten-dimensional spacetime. The primary tool for the analysis of perturbative string theory is two-dimensional conformal field theory. This is a very rich algebro-geometric theory with connections to a wide range of mathematical topics like representation theory of infinite-dimensional Lie algebras and algebraic geometry, and also to physical topics like statistical mechanics and condensed matter. The need to relate the ten-dimensional string theory to the four-dimensional physics we observe initiated an intensive study of certain six-dimensional complex geometries known as Calabi-Yau manifolds. A consequence of these investigations is the mirror symmetry conjecture, which has occupied a growing number of mathematicians ever since.
As a result of the second revolution the picture has now been drastically altered. We now understand that the five ten-dimensional theories, and a newly discovered eleven-dimensional theory, are but different manifestations of an underlying theory (code-named M-theory). Moreover the different 'subtheories' are related by duality transformations, akin to electromagnetic duality but generalising it in nontrivial ways. A suggestive analogy is that of the description of a manifold in terms of coordinate charts and transition functions. Here the different subtheories play the role of the coordinate charts and the duality transformations play the role of the transition functions. Continuing with this analogy, we know that in the case of manifolds it is possible to define them without having to resort to local charts; and the search for such an intrinsic description of M-theory is one of the main problems in the field today and one to which a lot of effort continues to be devoted.
Instrumental in the second superstring revolution are the so-called branes. These appeared initially as classical solutions of the supergravity theories which are the field-theory limits of the string theories and of the new eleven-dimensional theory. Some of these branes, the so-called D-branes, can also be understood as Dirichlet-type boundary conditions for open strings and hence amenable to the methods of boundary conformal field theory.
D-branes have also been instrumental in the so-called gauge/gravity correspondence. Initially conjectured by Maldacena in 1997, this correspondence states that there is a weak/strong coupling duality between (the 't Hooft limit of) supersymmetric gauge theory and (the supergravity limit of) superstring theory. This correspondence hints that string theory and gauge theory are but two sides of the same coin and not the drastically different theories that they appear to be.
Much effort continues to be devoted to testing and exploiting this correspondence. Recently a novel large rank limit of the gauge theory, inspired by Penrose's plane-wave limit in gravity, allows for the first time a perturbative test of the correspondence simultaneously in the gravity and gauge theory sides, albeit for a special sub-sector of the theory.
The jury is still out on whether string theory will be relevant for Physics; although in the words of Edward Witten, if string theory would be wrong, "it would seem like a kind of cosmic conspiracy."
What seems beyond the shadow of a doubt is the impact that string theory has had and continues to have in certain areas of Mathematics; although to be fair, it is difficult to disentangle the impact of string theory proper from that of supersymmetric quantum field theory, since the two theories are so intimately linked. Indeed, supersymmetry is arguably the only testable prediction of string theory thus far
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