Response to 2006 CAP-CRM Prize in Theoretical and Mathematical Physics

J. Harnad

Department of Mathematics and Statistics, Concordia University,
Centre de recherches mathematiques, Universite de Mnntreal

Sometimes it is difficult for a mathematical physicist to steer a course between the Scylla of mathematical techniques that appear to most physicists as too abstract to be of genuine use, and the Charybdis of imprecise logic that may lurk behind the guesswork and analogies characteristic of imaginative work in theoretical physics. It is gratifying therefore to have work recognized in this award that is based on precise use of mathematical constructions, whether concrete or abstract, in the resolution of problems of real interest to physics.

When I began my doctoral studies in Oxford in 1968, relativistic quantum field theory, which had proved so successful in the context of quantum electrodynamics, seemed virtually cast aside as a viable approach to the subnuclear forces. This was mainly due to the seemingly insurmountable obstacles in implementing the renormalization scheme for short range interactions involving vector (and axial-vector) currents, and the apparent inconsistency of trying to use a perturbative approach in the case of strong interactions. Both these obstacles were dramatically overcome with two great developments that occurred at the beginning of the 1970's. The first of these was the discovery that a consistent, renormalizable quantum field theory existed, unifying the electro-weak interactions, based on a nonabelian extension of the gauge symmetry of quantum electrodynamics, combined with the mechanism of spontaneous symmetry breaking. The second was the demonstration that this framework could be consistently extended to include the strong interactions, since considerations valid to all orders in perturbation theory implied that the effective coupling strengths became weak in the short distance limit. Over a dozen Nobel prizes were awarded in the following years in recognition of the various theoretical and experimental developments confirming these discoveries.

My own thesis work, done at the time of this transition, under the supervision of one of the pioneers of the unified electro-weak theory, J.C. Taylor, was unfortunately still set in the mold of the analyticity-symmetry approach to hadronic scattering theory characteristic of the preceding period. The importance of nonabelian gauge theory however became quickly clear to everyone. Approaches based on nonperturbative techniques also gained prominence, and there was a period of activity in which energy minimizing classical configurations in pure gauge theory, such as instantons and monopoles, were intensely studied. I was able to make some contributions in this direction, together with my collaborators[1], concerning classical solutions to field equations having a high degree of symmetry. This led to a general geometric framework, somewhat analogous to the Kaluza-Klein approach to the unification of gravitation and electrodynamics, which became known as "dimensional reduction"[2]. We were able to show that the Higgs field, essential to spontaneous symmetry breaking, could be deduced naturally through the process of symmetry reduction from a higher dimensional space. Another result to which I contributed at this time, together with my students and colleagues[3,4] was a proof that the field equations of supersymmetric Yang-Mills theory could, as suggested by Edward Witten, be given a purely geometrical formulation as integrability conditions of an associated linear system, analogous to the twistor approach to the self-dual Yang-Mills equations which had been so successfully applied to the study of instantons and nonabelian monopoles.

The experience this work provided in the use of geometrical techniques, complex analyticity and symmetry in the resolution of relativistic field equations gave a very useful background when I subsequently changed orientation towards the study of completely integrable systems. This area, although much more modest in its goals than elementary particle theory, had some important successes of its own. A fairly general framework was developed, based on inverse spectral methods, that gave a complete understanding of the mechanism underlying long term coherent phenomena characteristic of such systems, such as solitons and nonlinear quasi-periodic modes. From previous familiarity with twistor transform methods, it was clear to me that the inverse spectral transform, based upon the Riemann-Hilbert factorization problem of analytic function theory, had an analogous rôle and, indeed, the two could be usefully related through dimensional reduction.

My work on soliton theory began with the recognition that linearization of flows could be achieved by casting the equations of inverse spectral theory into a special form, which geometrically describe flows induced by linear group actions on Grassmann manifolds[5] . This related the "dressing method" of Zakharov and Shabat, based on the Riemann-Hilbert problem, to the transformation group approach of the Kyoto school. It also became clear that the abstract notion of symmetry reduction could be viewed, within a Hamiltonian context, as the key structural feature underlying the factorization method.

In subsequent work (again together with a number of close collaborators), I developed a method for canonically embedding finite dimensional integrable systems into a sort of "universal phase space" provided by loop algebras and loop groups, through a "dual pair" of moment mappings[6]. The notion of "duality" arose naturally in this context, providing two alternative representations of the same dynamics as isospectral flows in loop algebras. These involved a pair of rational "Lax matrices" sharing the same invariant "spectral curve", which encoded the same conserved quantities. Our approach made clear that the linearization of flows on the complex tori associated to the spectral curves was simply an instance of classical canonical transformation theory[7].

In 1991, I attended some lectures by Witten at the Institute for Advanced Study in Princeton relating the flows of the KP integrable hierarchy to a matrix integral shown by Maxim Kontsevich to be the generating function for intersection indices on moduli spaces of Riemann surfaces, a quantity that entered into the theory of 2D topological gravity. Subsequently, at the 1992 Scheveningen summer meeting, while giving some lectures on linearization of isospectral flows in loop algebras, I met Craig Tracy, who was lecturing about his new results, with Harold Widom, on edge spacing distributions in random matrix ensembles. He introduced a system of nonlinear Hamiltonian equations that underlay the deformation dynamics, and noted that these appeared almost exactly identical to the isospectral ones. On closer examination, it became clear that the key difference lay in the fact that parameters that were fixed in the isospectral equations became identified as times in the Tracy-Widom system, making them nonautonomous, and changing them from isospectral flows to deformations of associated rational covariant derivative operators that preserved their monodromy - so-called "isomonodromic deformations".

We wrote up a brief paper together explaining this fact[8], and this became the starting point of much of my subsequent research. I was able afterwards to develop a general Hamiltonian theory of such isomonodromic deformation equations[9], as well as their applications to the deformation approach to the spectral statistics of random matrices[10]. Amongst the useful results that emerged from this work was the recognition that the same canonical framework (R-matrix theory) that underlies integrable isospectral flows in loop algebras was applicable in this nonautonomous setting to isomonodromic systems. Furthermore, the notion of "duality" turned out to be at least as pertinent to isomonodromic systems, providing alternative implementations of the same deformation dynamics[9, 11].

This also turned out to have importance in applications of isomonodromic deformations to the study of two-matrix models, since the duality previously introduced for quite general isomonodromic systems had in this setting very immediate implications for the spectral correlation functions. A key development of recent years, obtained through collaborative work with my colleagues Marco Bertola (Concordia and CRM) and Bertrand Eynard (CEA, Saclay) was the derivation of a Riemann-Hilbert framework characterizing the integral kernels entering in the computation of the correlators[12]. The large N asymptotics of such correlators are the main objects of interest, and further development of the Riemann-Hilbert technique, applied to the computation of their asymptotic limits, is the goal of an ongoing program of research on multi-matrix models.

Amongst the many other colleagues and students with whom I have had the pleasure of working, and sharing ideas, I would like to specially mention: Steve Shnider, with whom I had a long and rewarding collaboration over several years, Jacques Hurtubise, who contributed enormously to our many joint projects, and remains a close friend and colleague, Luc Vinet, whom I had the immeasurable good luck to have had as my first, unofficial graduate student, and Alexander Its, from whom I learned a great deal, and with whom I have had the pleasure of an ongoing scientific collaboration together with a shared friendship.


[1] J. Harnad, S. Shnider and L. Vinet, "Group Actions on Principal Bundles and Invariance Conditions for Gauge Fields", J. Math. Phys. 21, 2719-2724 (1980).
[2] J. Harnad, S. Shnider and J. Tafel, "Group Actions on Principal Bundles and Dimensional Reduction", Lett. Math. Phys. 4, 107-113 (1980).
[3] J. Harnad, J. Hurtubise, M. Légaré and S. Shnider, "Constraint Equations and Field Equations for Supersymmetric Yang-Mills Theory", Nucl. Phys. B256, 609-620 (1985).
[4] J. Harnad and S. Shnider, "Constraint Equations and Field Equations for Ten Dimensional Super Yang-Mills Theory", Commun. Math. Phys. 106, 183-199 (1986).
[5] J. Harnad, Y. Saint-Aubin and S. Shnider, "The Soliton Correlation Matrix and the Reduction Problem for Integrable Systems", Commun. Math. Phys. 93, 33-56 (1984).
[6] M. Adams, J. Harnad and J. Hurtubise, "Dual Moment Maps into Loop Algebras", Lett. Math. Phys. 20, 299-308 (1990).
[7] M.R. Adams, J. Harnad and J. Hurtubise, "Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras", Commun. Math. Phys. 155, 385-413 (1993).
[8] J. Harnad, C.A. Tracy, and H. Widom, "Hamiltonian Structure of Equations Appearing in Random Matrices", in: Low Dimensional Topology and Quantum Field Theory, ed. H. Osborn, pp. 231-245. (Plenum, New York, 1993).
[9] J. Harnad, "Dual Isomonodromic Deformations and Moment Maps into Loop Algebras", Commun. Math. Phys. 166, 337-365 (1994).
[10] M. Bertola, B. Eynard, J. Harnad, "Partition functions for Matrix Models and Isomonodromic Tau functions", J. Phys. A : Math. Gen. 36, 3067-3983 (2003).
[11] J. Harnad and Alexander R. Its "Integrable Fredholm Operators and Dual Isomonodromic Deformations", Commun. Math. Phys. 226, 497-530 (2002).
[12] M. Bertola, B. Eynard, J. Harnad "Differential systems for biorthogonal polynomials appearing in 2-matrix models, and the associated Riemann-Hilbert problem", Commun. Math. Phys. 243, 193-240 (2003).