Edward Witten

Witten was awarded the Fields Medal by the International Mathematical Union in 1990, becoming the first physicist to win the prize.

In a written address to the ICM, Michael Atiyah said of Witten,[5]

Although he is definitely a physicist (as his list of publications clearly shows) his command of mathematics is rivaled by few mathematicians, and his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems... He has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics.

Edward Witten (left) with mathematician Shigefumi Mori, probably at the ICM in 1990, where they received the Fields Medal

As an example of Witten's work in pure mathematics, Atiyah cites his application of techniques from quantum field theory to the mathematical subject of low-dimensional topology. In the late 1980s, Witten coined the term topological quantum field theory for a certain type of physical theory in which the expectation values of observable quantities encode information about the topology of spacetime.[16] In particular, Witten realized that a physical theory now called Chern–Simons theory could provide a framework for understanding the mathematical theory of knots and 3-manifolds.[17] Although Witten's work was based on the mathematically ill-defined notion of a Feynman path integral and was therefore not mathematically rigorous, mathematicians were able to systematically develop Witten's ideas, leading to the theory of Reshetikhin–Turaev invariants.[18]

Another result for which Witten was awarded the Fields Medal was his proof in 1981 of the positive energy theorem in general relativity.[19] This theorem asserts that (under appropriate assumptions) the total energy of a gravitating system is always positive and can be zero only if the geometry of spacetime is that of flat Minkowski space. It establishes Minkowski space as a stable ground state of the gravitational field. While the original proof of this result due to Richard Schoen and Shing-Tung Yau used variational methods,[20][21] Witten's proof used ideas from supergravity theory to simplify the argument.

A third area mentioned in Atiyah's address is Witten's work relating supersymmetry and Morse theory,[22] a branch of mathematics that studies the topology of manifolds using the concept of a differentiable function. Witten's work gave a physical proof of a classical result, the Morse inequalities, by interpreting the theory in terms of supersymmetric quantum mechanics.

By the mid 1990s, physicists working on string theory had developed five different consistent versions of the theory. These versions are known as type I, type IIA, type IIB, and the two flavors of heterotic string theory (SO(32) and E₈×E₈). The thinking was that out of these five candidate theories, only one was the actual correct theory of everything, and that theory was the one whose low-energy limit matched the physics observed in our world today.

Speaking at the string theory conference at University of Southern California in 1995, Witten made the surprising suggestion that these five string theories were in fact not distinct theories, but different limits of a single theory which he called M-theory.[23][24] Witten's proposal was based on the observation that the five string theories can be mapped to one another by certain rules called dualities and are identified by these dualities. Witten's announcement led to a flurry of work now known as the second superstring revolution.

Edward Witten (center) with David Gross and Stephen Hawking at Strings 2001 at TIFR in Mumbai, India

Another of his contributions to physics was to the result of gauge/gravity duality. In 1997, Juan Maldacena formulated a result known as the AdS/CFT correspondence, which establishes a relationship between certain quantum field theories and theories of quantum gravity.[25] Maldacena's discovery has dominated high energy theoretical physics for the past 15 years because of its applications to theoretical problems in quantum gravity and quantum field theory. Witten's foundational work following Maldacena's result has shed light on this relationship.[26]

In collaboration with Nathan Seiberg, Witten established several powerful results in quantum field theories. In their paper on string theory and noncommutative geometry, Seiberg and Witten studied certain noncommutative quantum field theories that arise as limits of string theory.[27] In another well-known paper, they studied aspects of supersymmetric gauge theory.[28] The latter paper, combined with Witten's earlier work on topological quantum field theory,[16] led to developments in the topology of smooth 4-manifolds, in particular the notion of Seiberg–Witten invariants.

With Anton Kapustin, he has made deep mathematical connections between S-duality of gauge theories and the geometric Langlands correspondence.[29] Partly in collaboration with Seiberg, one of his recent interests include aspects of field theoretical description of topological phases in condensed matter and non-supersymmetric dualities in field theories that, among other things, are of high relevance in condensed matter theory. From a generalization of SYK models from condensed matter and quantum chaos, he has also recently brought tensor models of Gurau to the relevance of holographic and quantum gravity theories.

In general, Witten has done influential and insightful works in many aspects of quantum field theories and mathematical physics, including the physics and mathematics of anomalies, integrability, dualities, localization, homologies and so on. Many of his results have deeply influenced many areas in theoretical physics (often well beyond the original context of his results), including string theory, quantum gravity and topological condensed matter.