The Quest for Unification

One of the outstanding goals of theoretical physics is to formulate a quantum theory of gravity that unifies all fundamental interactions of nature. Perhaps our greatest achievement so far is quantum mechanics. Combined with Einstein's special relativity, it serves as the framework which underlies all descriptions of nature at the microscopic level, including quantum electrodynamics -- describing the interactions of electrons and photons, thus providing the foundation for large parts of modern technology -- and quantum chromodynamics -- capturing the dynamics of quarks within the atomic nuclei. However, despite its remarkable success as a framework for describing nature, there are regimes where we lack calculational techniques for applying the universal principles of quantum mechanics. In particular, we do not know how to use this framework when gravitational interactions are strong, like in the center of a black hole or close to the big bang, thereby motivating physicists' intense search for the elusive “quantum theory of gravity". But, in fact, there are also non-gravitational regimes where explicit methods for calculations are missing, for instance when describing interactions at strong coupling, such as high temperature superconductivity. This suggests that we should pursue the development of a new universal framework which accounts for all quantum interactions including gravity.

String theory is a promising candidate for such a framework. As a first approximation, one may characterize string theory as a theory of tiny one-dimensional objects, or “strings”, whose quantized vibrational modes give rise to the plethora of elementary particles we observe. This relatively simple generalization of the conventional point of view in fundamental physics, namely to replace the notion of static pointlike particles with the notion of vibrating strings, has far-reaching consequences and leads to a surprisingly rich structure.

The first major implication of studying the dynamics of quantized strings is that the dimension of spacetime is determined by the theory itself, and when fermionic degrees of freedom are properly taken into account string theory tells us that we live in a ten-dimensional world. In other words, we are lead to the daunting idea that the universe in its infancy was a tiny “ball” of ten tightly curled up dimensions, whereby some mechanism caused four of them to expand and evolve into the universe we now inhabit. This conclusion lies at the origin of the the notion of “compactification” of space, which plays a key role in modern day research, and to which we shall return below.

A second surprising implication of string theory is the “prediction” of general relativity. Similarly as the dimension of spacetime emerges from the quantization of strings, also Einstein's theory of gravity is an unavoidable consequence of the string dynamics. In fact, string theory gives rise to an infinite series of corrections to Einstein's theory in a way that makes it fully consistent with quantum mechanics. For this reason, string theory provides the first example of a consistent theory of quantum gravity. We can thereby use the theory to describe previously inaccessible corners of the universe, notably the interior of a black hole, or even the big bang itself. A remarkable result in this context is the microscopic derivation of the Bekenstein-Hawking black hole entropy due to Strominger and Vafa.

In addition to the aforementioned outcomes of string theory, yet another ramification is the profound connection with deep questions in modern mathematics. A striking example is the concept of “mirror symmetry”, originally put forward as the equivalence between two different physical string theories, but which has evolved into a vast field of research in mathematics, to a large extent sparked by Kontsevich's categorical version of mirror symmetry, dubbed “homological mirror symmetry”.

String theory appears to be a consistent physical theory; whether or not it describes our universe is yet to be verified. Despite its frustrating lack of experimental evidence, the theory exhibits a truly beautiful and intricate structure which may potentially answer some of the most difficult questions we can ask: How did the universe begin? What is the fundamental structure of spacetime? What are the microscopic constituents of a black hole?

Symmetries and Dualities

The development of the standard model of elementary particle physics throughout the second half of the 20th century, relied crucially on the basic principle of symmetry. The Lagrangian describing the standard model is required to be invariant under symmetry transformations described by the gauge group SU(3)x SU(2) x U(1), and the elementary particles fall into irreducible representations of this group. Also Einstein's theory of gravity is characterized by the invariance under general diffeomorphisms of the spacetime metric. The modern theory of theoretical high energy physics is in this way tightly interconnected with the representation theory of Lie algebras and Lie groups, which is the mathematical framework in which one characterizes the symmetries of an object, or an equation.

One of the outstanding problems in string theory is how to find a satisfactory fundamental formulation of the theory from first principles, analogous to the way Einstein formulated general relativity based on the principle of equivalence. At present, such a formulation is not known, instead we have a variety of different descriptions which are valid in certain limits and regimes. The remarkable property of the theory, as realized in the works of Hull & Townsend and Witten, is that all of these different descriptions are related in various ways to each other, creating an intricate web of dualities which ties the whole structure together into a robust framework. These dualities reveal an elaborate generalization of the principle of symmetry: although the various descriptions of the theory change under duality transformations, the complete structure is left invariant.

It is likely that these dualities hold the key for understanding the ultimate formulation of string theory. Such a formulation is then expected to unify its various dual descriptions in such a way that the fundamental symmetries are manifestly built in. One of the long-term goals of my present research is to uncover these fundamental symmetries, and to elucidate their role in uncovering a formulation of string theory from first principles.

The web of dualities of string theory -- encompassing most of the modern day concepts and phenomena of theoretical physics -- in many ways mirrors the so called Langlands program in mathematics, which connects many different areas of mathematics under a single roof. More specifically, it refers to a number of conjectures outlined by Robert Langlands in the late 1960's, relating group representation theory to number theory in far-reaching ways. Subsequently, a geometric version of the Langlands program has also been developed, following ideas of Drinfeld and Laumon, which instead deals with Riemann surfaces (or complex curves). For a recent survey of the geometric Langlands program and its relation with gauge theory, see Edward Frenkel’s Bourbaki seminar. Through the work of Kapustin and Witten the geometric Langlands program has already made it into the realm of theoretical physics; as we shall see below there are many indications that also the original “number-theoretic” Langlands program will prove to be of importance for foundational questions in string theory.

Cosmological Singularities and Hyperbolic Kac-Moody Algebras

The idea that 13,7 billion years ago our universe sprang out of nothing is breathtaking, and represents the modern physical explanation for the evolution of the universe. Although we now have a fairly good theoretical understanding of this evolution -- starting from a fraction of a second after the big bang up until our time -- an outstanding question is to determine exactly how this event occurred and what preceded it. The general consensus is that the singular behaviour of Einstein's theory close to the big bang reflects the fact that one has reached its limit of validity, and in a more complete theory the singularity will be “resolved” -- meaning that the theory will exhibit a smooth behavior as we approach, and perhaps even cross, the initial spacelike hypersurface corresponding to the “origin” of the universe.

My point of entry into this field of research has been to examine carefully the dynamical behavior of general relativity when approaching a big bang singularity. The purpose is to determine whether the theory just prior to breakdown exhibits some interesting structure that could lead us further into an understanding of the resolution of the singularity. Indeed, as was first discovered by Belinskii, Khalatnikov and Lifshitz in the context of pure gravity in four dimensions, in this limit the theory predicts a complete causal separation of spatial points (“ultralocality”) and displays a chaotic dynamical approach to the singularity. This behavior can in turn be shown to be equivalent to an auxiliary dynamical system, namely that of a particle moving freely in a finite volume region of hyperbolic space, known as a hyperbolic billiard. Remarkably, it turns out that this alternative description of the dynamics reveals a surprisingly rich underlying mathematical structure: the geometric reflections of the billiard ball generate a beautiful mathematical object known as a Coxeter group.

These surprising discoveries hint at the possibility that the ultralocal limit of gravity close to a big bang singularity unveils a huge hidden symmetry of gravity, which might prevail also in the quantum theory . This could provide the key for understanding how such singularities will ultimately be resolved in a complete quantum theory of gravity. For an extensive review of this subject see my paper with Marc Henneaux and Philippe Spindel.

Black Holes, Non-Perturbative Effects and Automorphic Forms

As mentioned above, another regime where quantum gravity effects dominate is in the vicinity of a black hole singularity. Within string theory, black holes arise from fundamental extended objects, known as branes, whose low-energy description is given in terms of solitonic (black hole) gravitational solutions. Due to their solitonic nature, these objects are inherently non-perturbative, with a tension that scales inversely with the coupling constant and therefore become infinitely heavy in the weak-coupling limit. Non-perturbative contributions to the effective action typically manifest themselves as instantons, which are completely localized in spacetime and originate from saddle-points in the path integral. As realized a long time ago by Polyakov, there is a close relation between solitons and instantons, namely soliton excitations in D dimensions become instanton excitations in D-1 dimensions. Concretely, this can be understood by compactifying the theory on a circle, in which case the solitonic worldline wraps around the circle and creates an instanton in one dimension lower.

String theory is originally formulated in ten spacetime dimensions, and in order to make contact with the four-dimensional world that we inhabit we must assume that six space dimensions are curled up into tiny circles. By an extension of the discussion above, it follows that the branes of string theory may wrap around the compact directions and give rise to instanton effects in four dimensions. These effects correspond to quantum corrections in the four-dimensional effective action, and calculating them exactly is of utmost importance, both for fundamental reasons and for phenomenological applications of string theory.

Remarkably, the various dual descriptions of string theory provide us with a powerful method of extracting these quantum corrections. For example, in many cases the couplings that determine the Lagrangian are constrained by duality to be functions invariant under certain arithmetic Lie groups G(Z). Mathematically, this has the far-reaching consequence that the couplings in the effective action must be automorphic forms. These ideas have been successfully exploited in my own research and continue to play a major role in my current and future projects.

The theory of arithmetic groups and automorphic forms fills an important and vast area of mathematics, as is in particular manifested through their ubiquitous appearance in the Langlands program. It is therefore intriguing that automorphic forms enter the physics of string theory in such a direct way. The G(Z)-invariant couplings in the four-dimensional effective action contain information about the complete set of perturbative and non-perturbative quantum corrections to the classical theory. Mathematically, these effects are revealed through the Fourier coefficients of the automorphic form. As was already mentioned, geometric Langlands duality has been shown to be related to electric-magnetic duality in gauge theory. Presently it is not known if a similar physical interpretation exists for Langlands duality in number theory, although the omnipresence of automorphic forms in string theory strongly suggests that this is the case. Understanding the role of Langlands duality in string theory will undoubtedly reveal novel deep relations between mathematics and physics, and lead to further beneficial cross-fertilization between the two subjects.