# String Theory

The incompatibility between local quantum field theory and general relativity is one of the main open problems in theoretical physics. String theory is an attempt at its resolution. While it was originally formulated as a theory of the strong interactions - a method in the madness of the proliferation of observed hadronic resonances - it was soon realized that the theory always contains a massless spin two excitation which can be interpreted as the quantum of the gravitational interaction, the graviton.

But the graviton is not the only excitation of the (closed) string. There is an infinite number of them and the massless ones can be interpreted as non-abelian gauge bosons and massless matter fermions. In this way string theory provides a unification of all elementary particles and their interaction. Since it is formulated as a quantum theory, it achieves the unifications of general relativity with quantum theory, albeit not within the framework of local quantum field theory. The latter emerges as an effective low energy description of the massless excitation modes of the string.

While, to the best of our knowledge, being a consistent theory, the crucial question is whether Nature has chosen this possibility of reconciling quantum theory with gravity. String theory makes many postdictions, such as the existence of gravity, but it makes also many predictions, e.g. an infinite tower of massive excitations or corrections to Einstein's theory of gravity. Unfortuntately, to test these predictions requires energies which are unobtainable in experiments by many orders of magnitude. The characteristic energy scale is the Planck scale which is intrinsic to the gravitational interaction and every alternative theory of quantum gravity will have to face the same problem regarding its verifiability.

During the last 15 years it has been realized that while string theory might be the dream of a final theory come true, it can also be viewed, similar to quantum field theory, as a framework in which physical questions can be formulated and answered. This is known as the AdS/CFT correspondence, of more generally the gauge/gravity duality or even more generally, the holographic principle. It states, very roughly, that a field theory without gravity in d dimensional space-time, has an alternative (dual) description as a string theory in d+1 dimensions. This difference in dimensions makes this duality holographic. This can be made very explicit in certain examples. These developments have led to very beautiful results e.g. in hydrodynamics where the ratio of certain fluid parameters can be computed using a dual description as a higher dimensional classical gravity theory (viewed as the low-energy effective description of string theory). Other results which came out of this development is the analytic computation of the quark-anti-quark potential in QCD-like theories. It is, however, fair to say that so far string theory which is, in some sense, unique, dictates which field theories allow for such a dual or holographic description. For instance, the dual description of QCD, the theory of the strong interactions which governs the world of hadrons, has not been found yet.

A fundamental question which can be asked in the context of string theory is: What is the geometry of spacetime? On macroscopic scales Einstein’s theory of relativity gives a precise and well-tested answer to this question: spacetime geometry and background matter fields obey Einstein’s equations. The structure of spacetime on microscopic scales remains more mysterious, and most approaches to quantum gravity suggest that on scales near the Planck length of 10^{-35}cm a conventional notion of spacetime geometry based on manifolds of fixed topology and probe objects following geodesics will not yield a useful description. Does this mean that to describe the microscopic structure of spacetime we must abandon our well-honed geometric tools and hard-won insights? In the final reckoning it may well be so. However, the utility of these tools and dearth of substitutes suggest that we do so gradually, identifying which geometric aspects may be safely kept, which modified, and which must be entirely discarded. In string theory, the resulting set of ideas constitutes the notion of "stringy geometry".

When assessing string theory it should be kept in mind that it has led to a number of interesting developments in physics and mathematics (the latter were not addressed here). As such it played a prominent role in theoretical and mathematical physics of the past 40 years and is expected to do so in the future. Even if its role as a quantum theory of gravity cannot be conclusively, i.e. experimentally, verified, it enlarged our horizon within which we formulate and study physical systems.