# Citation for the Award of the Poincaré Prize 2006 to Edward Witten

Read on Arthur Jaffe's behalf at the IAMP meeting in Rio on 9 August 2006

## Edward Witten is in the midst of an enormously productive career as a mathematical physicist. Born in 1951 in Baltimore, he began his undergraduate studies by majoring in history. Edward certainly had the opportunity for prior exposure to sophisticated physics as his father Louis is a noted expert on relativity and gravitation. After his undergraduate studies, Edward returned to physics, working with David Gross at Princeton, and receiving his doctorate in 1976.

## Edward’s early work left an immediate impression on experts. He discovered a new class of instanton solutions to the classical Yang-Mills equations, very much a central subject at the time. He pioneered work on field theories with N components and the associated “large-N limit” as N tends to infinity. Three years later as a Junior Fellow at Harvard he had already established a solid international reputation—both in research and as a spell-binding lecturer. That year several major physics departments took the unusual step, at the time an extraordinary one, to attempt to recruit a young post‑doctoral fellow to join their faculty as a full professor! At that point Edward returned to Princeton with Chiara Nappi, my post-doctoral fellow and Edward’s new wife. Edward has been in great demand ever since.

## Edward already became well-known in his early work for having keen mathematical insights. He re-interpreted Morse theory in an original way and related the Atiyah-Singer index theorem to the concept of super-symmetry in physics. These ideas revolved about the classical formula expressing the Laplace-Beltrami operator in terms of the de Rham exterior derivative, Δ as the square of d+d*. This insight was interesting in its own right. But it inspired his applying the same ideas to study the index of infinite-dimensional Dirac operators D and the self-adjoint operator Q = D+D*, known in physics as super-charges, related to the energy H by ita representation as the square of Q analogous to the formula for Δ. This led to the name “ Witten index” for the index of D, a terminology that many physicists still use.

## In 1981 Witten also discovered an elegant approach to the positive energy theorem in classical relativity, proved in 1979 by Schoen and Yau. What developed as Witten’s hallmark is the insight to relate a set of ideas in one field to an apparently unrelated set of ideas in a different field. In the case of the positive energy theorem, Witten again took inspiration from super-symmetry to relate the geometry of space-time to the theory of spin structures and to an identity due to Lichnerowicz. The paper by Witten framed the new proof in a conceptual structure that related it to old ideas and made the result immediately accessible to a wide variety of physicists and mathematicians.

## In 1986 Witten’s had a spectacular insight by giving a quantum-field theory interpretation to Vaughan Jones’ recently-discovered knot invariant. Witten showed that the Jones polynomial for a knot can be interpreted as the expectation of the parallel transport operator around the knot in a theory of quantum fields with a Chern-Simons action. This work set the stage for many other geometric invariants, including the Donaldson invariants, being regarded as partition functions or expectations in quantum field theory. In most of these cases, the mathematical foundations of the functional integral representations can still not be justified, but the insights and understanding of the picture will motivate work for many years in the future.

## With the resurgence of “super-string theory” in 1984, Witten quickly became one of its leading exponents and one of its most original contributors. His 1987 monograph with Green and Schwarz became the standard reference in that subject. Later Witten unified the approach to string theory by showing that many alternative string theories could be regarded as different aspects of one grand theory.

## Witten also pioneered the interpretation of symmetries related to the electromagnetic duality of Maxwell’s equations, and its generalization in field theory, gauge theory, and string theory. He pioneered the discovery of SL(2,Z) symmetry in physics, and brought concepts from number theory, as well as geometry, algebra, and representation theory centrally into physics.

## In understanding Donaldson theory in 1995 Seiberg and Witten formulated the equations named after them which have provided so much insight into modern geometry. With the advent of this point of view and fueled by its rapid dissemination over the internet, many geometers saw progress in their field proceed so rapidly that they could not hope to keep up.

## Not only is Witten’s own work in the field of super-symmetry, string theory, M-theory, dualities and other symmetries of physics legend, but he has trained numerous students and postdoctoral coworkers who have come to play leading roles in string theory and other aspects of theoretical physics.

## I could continue on and on about other insights and advances made or suggested by Edward Witten. But perhaps it is just as effective to mention that for all his mentioned and unmentioned work, Witten has already received many national and international honors and awards. These include the Alan Waterman award in 1986, the Field’s Medal in 1990, the CMI Research Award in 2001, the U.S. National Medal of Science in 2002, and an honorary degree from Harvard University in 2005. Witten is a member of many honorary organizations, including the American Philosophical Society and the Royal Society. While Witten may not need any additional recognition, it is an especially great personal pleasure and honor, as one of the original founders of IAMP, to present Edward Witten to receive the Poincaré prize in 2006.

## Arthur Jaffe

Harvard University

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