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http://www.math.columbia.edu/~woit/blog/archives/000099.html
1982:
Witten comes up with a beautiful reinterpretation of Morse theory
in terms of supersymmetric quantum mechanics, writing an extremely
influential paper on "Supersymmetry and Morse Theory", which is
published in a math journal, the Journal of Differential Geometry.
Spring 1987:
Atiyah conjectures that Andreas Floer's new homology groups
(inspired by Witten's supersymmetry and Morse theory paper)
are the Hilbert space of a QFT. There are two cases where Floer
theory works: 1+1 dimensions where the observables of the QFT
would count curves (later to be known as Gromov-Witten invariants),
and 3+1 dimensions where the observables count instantons
(Donaldson invariants). Atiyah conjectures the existence of
two corresponding QFTs, and also notes that the new knot
polynomials of Vaughan Jones might correspond to a QFT in 2+1 d.
He talks to Witten about this and gives an amazing lecture
at a conference at Duke explaining these ideas.
Witten tries to find a supersymmetric QFT that will do what
Atiyah wants, but initially doesn't succeed.
Late 1987:
Atiyah visits Witten again at the IAS and keeps after him about the
TQFT idea. Witten finally realizes that things work if he uses a
"twisted" version of N=2 supersymmetry.
February 1988:
Two papers by Witten appear, one "Topological Quantum Field Theory"
about the 3+1 d case, one "Topological Sigma Models" about the 1+1 d case.
The second paper contains some vague speculation at the beginning about the
relation of these "topological strings" to physical string theory,
perhaps in some kind of "unbroken phase". At the end it also contains
a sketch of an attempt to get Jones polynomials by using a 3+1d TQFT
that would couple together his 3+1 topological gauge theory with a
topological sigma model on the worldsheet swept out by a knot in
3 dimensions moving through time. This doesn't actually work.
Summer 1988:
At a conference in Swansea, talking to Atiyah and Segal about Segal's
ideas about conformal field theory and "modular functors", Witten realizes
that the right theory to get Jones polynomials is a 3d QFT
whose Hilbert space is the finite dimensional space of conformal
blocks of a 2d WZW theory. He also realizes that one can think
of the Lagrangian of this theory as being the Chern-Simons functional.
His paper "Quantum Field Theory and the Jones Polynomial"
appears in September. There's not a word about string theory anywhere
in it and he has completely abandoned the idea of relating
Jones polynomials to topological sigma models.
I was in Berkeley at MSRI for the academic year 1988-89.
In January there was a workshop there involving Atiyah, Bott, Witten,
and many other mathematicians and physicists. Initially many of the
mathematicians were a bit skeptical, but by the end Witten had convinced
the skeptics that what he had made complete sense, and they were very
impressed. In the summer of 1990 he was awarded the Fields Medal for this work.
New ideas about relations between branes, topological strings, and Chern-Simons
appeared about ten years later, and that's an ongoing story, one which Lubos
conflates with what was going on in 1988-9 that got Witten the Fields medal.
These are two completely different stories.
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