Editing Edward Witten (section)

==Research==
===Fields medal work===
Witten was awarded the [[Fields Medal]] by the [[International Mathematical Union]] in 1990.<ref>{{Cite web|date=2011|title=Edward Witten|url=http://www.sns.ias.edu/~witten/CurrentCV.pdf|url-status=dead|archive-url=https://web.archive.org/web/20120204111241/http://www.sns.ias.edu/~witten/CurrentCV.pdf|archive-date=February 4, 2012|access-date=April 13, 2021}}</ref>

In a written address to the [[International Congress of Mathematicians|ICM]], [[Michael Atiyah]] said of Witten:<ref name="atiyah" />

{{blockquote|text=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&nbsp;... 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.<ref name="atiyah" />}}

{{stack|[[File:Widden Mori.jpg|thumb|Edward Witten (left) with mathematician [[Shigefumi Mori]], probably at the [[International Congress of Mathematicians|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 value]]s of observable quantities encode information about the [[topology]] of [[spacetime]].<ref name="tqft">{{Citation | last1=Witten | first1=Edward | year=1988 | title=Topological quantum field theory | url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104161738 | journal=[[Communications in Mathematical Physics]] | volume=117 | issue=3 | pages=353–386 |bibcode = 1988CMaPh.117..353W |doi = 10.1007/BF01223371 | s2cid=43230714 }}</ref> In particular, Witten realized that a physical theory now called [[Chern–Simons theory]] could provide a framework for understanding the mathematical theory of [[knot (mathematics)|knots]] and [[3-manifold]]s.<ref>{{Cite journal |last=Witten |first=Edward |year=1989 |title=Quantum Field Theory and the Jones Polynomial |url=http://www.maths.ed.ac.uk/~aar/papers/witten.pdf |journal=[[Communications in Mathematical Physics]] |volume=121 |issue=3 |pages=351–399 |bibcode = 1989CMaPh.121..351W |doi = 10.1007/BF01217730 |s2cid=14951363 }}</ref> Although Witten's work was based on the mathematically ill-defined notion of a [[Feynman path integral]] and therefore not [[mathematical rigor|mathematically rigorous]], mathematicians were able to systematically develop Witten's ideas, leading to the theory of [[Reshetikhin–Turaev invariant]]s.<ref>{{cite journal |last1=Reshetikhin |first1=Nicolai |last2=Turaev |first2=Vladimir |year=1991 |title=Invariants of 3-manifolds via link polynomials and quantum groups |journal=[[Inventiones Mathematicae]] |volume=103 |issue=1 |pages=547–597 |bibcode = 1991InMat.103..547R |doi = 10.1007/BF01239527 |s2cid=123376541 }}</ref>

Another result for which Witten was awarded the Fields Medal was his proof in 1981 of the [[positive energy theorem]] in [[general relativity]].<ref>{{cite journal |last1=Witten |first1=Edward |year=1981 |title=A new proof of the positive energy theorem |url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103919981 |journal=[[Communications in Mathematical Physics]] |volume=80 |issue=3 |pages=381–402 |bibcode = 1981CMaPh..80..381W |doi = 10.1007/BF01208277 |s2cid=1035111 }}</ref> 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]],<ref>{{cite journal |last1=Schoen |first1=Robert |last2=Yau |first2=Shing-Tung |year=1979 |title=On the proof of the positive mass conjecture in general relativity |journal=[[Communications in Mathematical Physics]] |volume=65 |issue=1 |page=45 |bibcode = 1979CMaPh..65...45S |doi = 10.1007/BF01940959 |s2cid=54217085 |url=http://projecteuclid.org/euclid.cmp/1103904790 }}</ref><ref>{{cite journal |last1=Schoen |first1=Robert |last2=Yau |first2=Shing-Tung |year=1981 |title=Proof of the positive mass theorem. II |journal=[[Communications in Mathematical Physics]] |volume=79 |issue=2 |page=231 |bibcode = 1981CMaPh..79..231S |doi = 10.1007/BF01942062 |s2cid=59473203 |url=http://projecteuclid.org/euclid.cmp/1103908964 }}</ref> Witten's proof used ideas from [[supergravity theory]] to simplify the argument.<ref name="Parker 1985 Gauge choice in Witten">{{cite journal | last=Parker | first=Thomas H. | title=Gauge choice in Witten's energy expression | journal=Communications in Mathematical Physics | volume=100 | issue=4 | date=1985 | issn=0010-3616 | doi=10.1007/BF01217725 | pages=471–480| bibcode=1985CMaPh.100..471P | url=http://projecteuclid.org/euclid.cmp/1104114001 }}</ref>

A third area mentioned in Atiyah's address is Witten's work relating [[supersymmetry]] and [[Morse theory]],<ref name="Witten 1982 Supersymmetry and Morse theory">{{cite journal | last1=Witten | first1=Edward | title=Super-symmetry and Morse Theory | year=1982 | journal=[[Journal of Differential Geometry]]| pages=661–692 | volume=17| issue=4 | doi=10.4310/jdg/1214437492 | doi-access=free }}</ref> 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 theory#Morse inequalities|Morse inequalities]], by interpreting the theory in terms of [[supersymmetric quantum mechanics]].<ref name="Witten 1982 Supersymmetry and Morse theory"/>

===M-theory===
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 string|type I]], [[type IIA string|type IIA]], [[type IIB string|type IIB]], and the two flavors of [[heterotic string]] theory ([[special orthogonal group|SO(32)]] and [[E8 (mathematics)|E<sub>8</sub>×E<sub>8</sub>]]). The thinking was that 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.<ref name="Rickles 2016">{{cite book | last=Rickles | first=Dean | title=A Brief History of String Theory | publisher=Springer | date=2016-08-23 | isbn=978-3-662-50183-2}}</ref>

Speaking at [[Strings (conference)|Strings]] '95 conference at [[University of Southern California]], 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]].<ref>{{Cite conference 
| publisher=University of Southern California
| place=Los Angeles
| series=Future Perspectives in String Theory
| date = 13–18 March 1995
| author= Witten, E.
| title= Some problems of strong and weak coupling
|url=http://physics.usc.edu/Strings95/program.html|access-date=2023-02-01|website=physics.usc.edu}}</ref><ref>{{cite journal |last1=Witten |first1=Edward |year=1995 |title=String theory dynamics in various dimensions |journal=[[Nuclear Physics B]] |volume=443 |issue=1 |pages=85–126 |doi=10.1016/0550-3213(95)00158-O|arxiv = hep-th/9503124 |bibcode = 1995NuPhB.443...85W |s2cid=16790997 }}</ref> Witten's proposal was based on the observation that the five string theories can be mapped to one another by certain rules called [[String duality|dualities]] and are identified by these dualities. It led to a flurry of work now known as the [[second superstring revolution]].<ref name="Rickles 2016"/>

===Other work===
{{stack|[[File:Gross Witten Hawking TIFR 2001.jpg|thumb|Edward Witten (center) with [[David Gross]] and [[Stephen Hawking]] at [[Strings (conference)|Strings 2001]] at TIFR in  Mumbai, India]]}}

Another of Witten's 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]].<ref name="largeN">{{cite journal | author=Juan M. Maldacena | title=The Large N limit of superconformal field theories and supergravity | journal=[[Advances in Theoretical and Mathematical Physics]] | volume=2 | year=1998 | issue=2 | pages=231–252 | arxiv=hep-th/9711200|bibcode = 1998AdTMP...2..231M | doi=10.4310/ATMP.1998.V2.N2.A1 }}</ref> 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.<ref>{{cite journal | author=Edward Witten | title=Anti-de Sitter space and holography | journal=[[Advances in Theoretical and Mathematical Physics]] | volume=2 | issue=2 | year=1998 | pages=253–291 | arxiv=hep-th/9802150|bibcode = 1998AdTMP...2..253W | doi=10.4310/ATMP.1998.v2.n2.a2 | s2cid=10882387 }}</ref>

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 theory|noncommutative quantum field theories]] that arise as limits of string theory.<ref>{{cite journal |last1=Seiberg |first1=Nathan |last2=Witten |first2=Edward |year=1999 |title=String Theory and Noncommutative Geometry |journal=[[Journal of High Energy Physics]] |volume=1999 | doi = 10.1088/1126-6708/1999/09/032 | page=032 | issue = 9|arxiv = hep-th/9908142 |bibcode = 1999JHEP...09..032S |s2cid=668885 }}</ref> In another well-known paper, they studied aspects of [[supersymmetric gauge theory]].<ref>{{cite journal |last1=Seiberg |first1=Nathan |last2=Witten |first2=Edward |year=1994 |title=Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory |journal=[[Nuclear Physics B]] |volume=426 |issue=1 |pages=19–52 |arxiv = hep-th/9407087 |bibcode = 1994NuPhB.426...19S |doi = 10.1016/0550-3213(94)90124-4 |s2cid=14361074 }}</ref> The latter paper, combined with Witten's earlier work on topological quantum field theory,<ref name="tqft" /> led to developments in the topology of [[smooth manifold|smooth]] [[4-manifold]]s, in particular the notion of [[Seiberg–Witten invariant]]s.<ref>{{citation|last=Donaldson|first= Simon K. |author-link=Simon Donaldson |title=The Seiberg-Witten equations and 4-manifold topology. |journal=[[Bulletin of the American Mathematical Society]]| series=(N.S.)  |volume=33  |year=1996|issue= 1|pages= 45–70|doi=10.1090/S0273-0979-96-00625-8 |mr=1339810|doi-access=free}}</ref>

With [[Anton Kapustin]], Witten has made deep mathematical connections between S-duality of gauge theories and the [[geometric Langlands correspondence]].<ref>{{cite journal|last1=Kapustin|first1=Anton|last2=Witten|first2=Edward|date=April 21, 2006|title=Electric-Magnetic Duality And The Geometric Langlands Program|journal=Communications in Number Theory and Physics|volume=1|pages=1–236|arxiv=hep-th/0604151|bibcode=2007CNTP....1....1K|doi=10.4310/CNTP.2007.v1.n1.a1|s2cid=30505126}}</ref> Partly in collaboration with Seiberg, one of his recent interests includes 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. In 2016, he has also brought tensor models to the relevance of holographic and quantum gravity theories, by using them as a generalization of the [[Sachdev–Ye–Kitaev model]].<ref>{{cite journal|last=Witten|first=Edward|date=October 31, 2016|title=An SYK-Like Model Without Disorder|journal=Journal of Physics A: Mathematical and Theoretical|volume=52|issue=47|pages=474002|arxiv=1610.09758
|doi=10.1088/1751-8121/ab3752|s2cid=118412962}}</ref>

Witten has published influential and insightful work in many aspects of quantum field theories and mathematical physics, including the physics and mathematics of anomalies, integrability, dualities, localization, and homologies. Many of his results have deeply influenced areas in theoretical physics (often well beyond the original context of his results), including string theory, quantum gravity and topological condensed matter.<ref name="Stiftung 2023 Witten">{{cite web | last=Stiftung | first=Joachim Herz | title=News | website=Joachim Herz Stiftung | date=2023-07-03 | url=https://www.joachim-herz-stiftung.de/en/about-us/news/edward-witten | access-date=2024-02-25}}</ref> In particular, Witten is known for collaborating with [[Ruth Britto]] on a method calculating scattering amplitudes known as the [[BCFW recursion|BCFW recursion relations]].