Edward Witten is a theoretical physicist at the Institute for Advanced Study.
interview by Hirosi Ooguri, Notices Amer. Math. Soc, May 2015 p491 (pdf)
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.
In
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
Then
Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987)
on the elliptic genus/Witten genus as the partition function of the type II superstring/heterotic superstring
And various articles on the foundations of topological field theory such as
Topological quantum field theory, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386 (Euclid)
on topologically twisted D=4 super Yang-Mills theory and the corresponding twists of the superstring which yield the A-model and B-model topological string 2d TQFTs;
On the structure of the topological phase of two dimensional gravity Nuclear Phys, B 340 (1990) 281 (scans)
on the topological string limit of the worldsheet 2d quantum gravity theory of the string, including the A-model and B-model topological twists of the type II superstring;
and
Quantum field theory and the Jones polynomial, Comm, Math, Phys, 121 (1989) 351
on the quantization of 3d Chern-Simons theory, its Jones polynomial knot invariant quantum observables and its holographic relation to the quantization of the WZW model (the string propagating on a suitable Lie group manifold).
(…)
clearly, this list is supposed to be expanded, eventually…