String Theory: The Past Ten Years

Gary Horowitz, UC Santa Barbara

Given a choice between summarizing the past decade of achievements in string theory or speculating about what string theory might look like a decade from now, I have chosen the first option. Indeed, given the rapid progress over the last decade, I find it hard to guess where string theory will be even a few years from now.

Ten years ago, it was common (and correct) to distinguish the two main approaches to quantum gravity by saying that string theory [1] was perturbative, and background dependent while the other approach [2] was non-perturbative and background independent. In light of this, it is not surprising that most relativists were not interested in string theory. Today, this distinction is no longer applicable. As we will discuss, there is now a complete, non-perturbative and background independent formulation of the theory, at least for space-times with certain asymptotic boundary conditions.

Let me begin by summarizing the situation ten years ago. At that time, there were five perturbatively consistent string theories. They were all based on the idea that particles are just different excitations of a one-dimensional extended object - the string. They all included gravity, supersymmetry, and required ten spacetime dimensions. These theories differed in the amount of supersymmetry and type of gauge groups that were included. In addition to perturbations about Minkowski spacetime, nontrivial classical solutions were known, including space-times in which six of the spatial dimensional are compactified. In some cases, the resulting four dimensional effective theories were in qualitative agreement with observations. It was also known that spacetime is seen differently in string theory than in general relativity or ordinary field theory. In particular, flat spacetime with one direction compactified into a circle of radius $R$ is completely equivalent to a spacetime with a circle of radius $\ell_s^2/R$ where $\ell_s$ is a new dimensional parameter related to the string tension. This is possible since the string is an extended object and has winding states in addition to the usual momentum states.

One of the main things that has changed over the past decade is that we now know that string theory does not just involve strings. Higher (and lower) dimensional objects (called branes) play an equally fundamental role. Using these branes, convincing evidence has been accumulated that all five of the perturbative string theories are just different limits of the same theory, called M theory. (There is no agreement about what the M stands for.) There is yet another limit in which M theory reduces to eleven dimensional supergravity.

Without a doubt, the main achievement of string theory over the past decade has been an explanation of black hole entropy [3] For a class of near extremal four and five dimensional charged black holes (with the extra spatial dimensions compactified on e.g. a torus) one can count the number of microstates of string theory associated with the black hole. One finds that in the limit of large black holes, the number is exactly the exponential of the Bekenstein-Hawking entropy. The black holes can include angular momentum, and several different types of charges, so the entropy is a function of several parameters. The string calculation reproduces this function exactly. Even more surprising, it was shown that the radiation calculated in string theory agrees exactly with the Hawking radiation from the black hole, including the distortions of the thermal spectrum arising from the greybody factors [4].

By exploring these black hole results, Maldacena was led to his famous ``AdS/CFT" conjecture [5]. This states that string theory (or M theory) on space-times which asymptotically approach anti de Sitter (AdS) space is completely described by a conformally invariant field theory (CFT) which lives on the boundary of this spacetime. This is a remarkable conjecture which states that an ordinary field theory in a fixed spacetime can describe all of string theory with asymptotically AdS boundary conditions. Since only the asymptotic boundary conditions on the metric are fixed, this constitutes a background independent formulation of the theory. Since the CFT can be defined non-perturbatively, this is also a non-perturbative formulation. The AdS/CFT conjecture is a concrete implementation of the idea that quantum gravity should be ``holographic" [6], i.e., the true degrees of freedom live on the boundary, but can describe all physical processes in the bulk. This was originally suggested by the fact that black hole entropy is given by the horizon area, but now applies to all quantum gravity processes, not just black holes. This conjecture has withstood a number of nontrivial checks. It can be used to derive new predictions about strongly coupled CFT, or learn about quantum gravity. For example, one immediate consequence is that the formation and evaporation of a small black hole in AdS can be described by the unitary evolution of a state in the CFT.

There is much that remains to be done. Major open questions include: (1) Develop a dictionary to translate spacetime concepts into field theory language and vice versa. (Only a few entries in this dictionary are currently known.) In particular, find a ``spacetime reconstruction theorem" which allows us to reconstruct a semiclassical spacetime from certain states in the CFT. (2) Extend the AdS/CFT conjecture to other boundary conditions including asymptotically flat space-times. (3) Calculate the entropy of all black holes (including Schwarzschild) exactly and understanding why it is always proportional to the horizon area.


[1] J. Polchinski, String Theory, in 2 vols., Cambridge Univ. Press (1998).

[2] C. Rovelli, ``Loop Quantum Gravity", Living Reviews 1 (1998), gr-qc/9710008

[3] A. Strominger and C. Vafa, ``Microscopic Origin of the Bekenstein-Hawking Entropy", Phys. Lett. B379 (1996) 99, hep-th/9601029; G. Horowitz, ``Quantum States of Black Holes", in Black Holes and Relativistic Stars, ed. R. Wald, U. of Chicago Press (1998), gr-qc/9704072; A. Peet, ``TASI Lectures on Black Holes in String Theory", hep-th/0008241.

[4] J. Maldacena and A. Strominger, ``Black Hole Greybody Factors and D-Brane Spectroscopy", Phys. Rev. D55 (1997) 861.

[5] J. Maldacena, ``The Large N Limit of Superconformal Field Theories and Supergravity", Adv. Theor. Phys. 2 (1998) 231, hep-th/9711200; O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, ``Large N Field Theories, String Theory and Gravity", Phys. Rept. 323 (2000) 183, hep-th/9905111.

[6] G. 't Hooft, ``Dimensional Reduction in Quantum Gravity", gr-qc/9310026; L. Susskind, ``The World as a Hologram", J. Math. Phys. 36 (1995) 6377, hep-th/9409089.

Jorge Pullin