# nLab Edward Witten

Edward Witten is a theoretical physicist at the Institute for Advanced Study.

Witten’s work originates in theoretical quantum field theory and stands out as making numerous and deep connections between that and mathematical geometry and cohomology. In the course of the 1980s Witten became the central and leading figure in string theory.

Insight gained from the study of quantum field theories and specifically those relevant in string theory led Witten to mathematical results deep enough to gain him a Fields medal, see below. Indeed, a whole list of sub-fields in mathematics originate in aspects of Witten’s work in QFT/string theory and carry his name, such as Chern-Simons theory which many people call “Chern-Simons-Witten theory”, Wess-Zumino-Witten theory, the Witten genus, Gromov-Witten theory, Seiberg-Witten theory, Rozansky-Witten invariant, the Witten cojecture?. Other parts are still waiting to be absorbed into the mathematical literature such as Horava-Witten theory, Diaconescu-Moore-Witten anomaly etc..

Despite the deeply theoretical and abstract mathematical aspects of his work, Witten has visibly always been motivated by fundamental questions in the phenomenology of the standard model of particle physics and cosmology. (Indeed, some of his work on scattering amplitudes crucially enters into the experimental detection of the Higgs particle, for more on this see at string theory results applied elsewhere. ) He prominently argued that specifically heterotic string theory is a plausible candidate for a fundamental grand unified gauge field theory including quantum gravity.

Since about the turn of the millennium Witten has tended to more esoteric mathematical aspects of string theory, such as its relation to Khovanov homology and geometric Langlands duality which apparently the string theory community at large is following less enthusiastically than it was the case during the excited 1990s.

# Contents

## Fields medal work

In

• Michael Atiyah, On the work of Edward Witten, Proceedings of the International Congress of Mathematics, Kyoto 1990 (pdf)

the following “influential papers” are listed as relevant for Edward Witten receiving the Fields Medal in 1990.

First of all

• Supersymmetry and Morse theory, J. Differential Geom. Volume 17, Number 4 (1982), 661-692. (Euclid)

This discusses deformations of supersymmetric quantum mechanics on a Riemannian manifold and how its supersymmetric ground states are related to the Morse theory of a deformation function. The way this supersymmetric quantum mechanics appears as the point-particle limit of the type II superstring is explained at the end of

• Global anomalies in string theory, in W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, pp. 61–99. World Scientific, 1985

which otherwise is on quantum anomalies in string theory, such as the Green-Schwarz anomaly, etc.

Finally Atiyah’s section 2 mentions

• (with Cumrun Vafa), Eigenvalue inequalities for Fermions in gauge theories, Comm. Math. Phys. 95 (1984) 257 (Euclid)

Then

• Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987)

And various articles on the foundations of topological field theory such as

and

(…)

clearly, this list is supposed to be expanded, eventually…

category: people

Revised on May 1, 2015 20:28:47 by Urs Schreiber (195.113.30.252)