On the Work of Edward Witten
Michael Atiyah
Trinity College, Cambridge CB2 1TQ, England
1. General
The past decade has seen a remarkable renaissance in the interaction between
mathematics and physics. This has been mainly due to the increasingly sophisticated
mathematical models employed by elementary particle physicists, and the
consequent need to use the appropriate mathematical machinery. In particular,
because of the strongly non-linear nature of the theories involved, topological
ideas and methods have played a prominent part.
The mathematical community has benefited from this interaction in two
ways. First, and more conventionally, mathematicians have been spurred into
learning some of the relevant physics and collaborating with colleagues in theoretical
physics. Second, and more surprisingly, many of the ideas emanating from
physics have led to significant new insights in purely mathematical problems, and
remarkable discoveries have been made in consequence. The main input from
physics has come from quantum field theory. While the analytical foundations
of quantum field theory have been intensively studied by mathematicians for
many years the new stimulus has involved the more formal (algebraic, geometric,
topological) aspects.
In all this large and exciting field, which involves many of the leading physicists
and mathematicians in the world, Edward Witten stands out clearly as the most
influential and dominating figure. Although he is definitely a physicist (as his list
of publications clearly shows) his command of mathematics is rivalled 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.
Witten's output is remarkable both for its quantity and quality. His list of
over 120 publications indicates the scope of his research and it should be noted
that many.of these papers are substantial works indeed.
In what follows I shall ignore the bulk of his publications, which deal with
specifically physical topics. This will give a very one-sided view of his contribution,
but it is the side which is relevant for the Fields Medal. Witten's standing as a
physicist is for others to assess.
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
32 Michael Atiyah
Let me begin by trying to describe some of Witten's more influential ideas and
papers before moving on to describe three specific mathematical achievements.
2. Influential Papers
His paper [2] on supersymmetry and Morse theory is obligatory reading for
geometers interested in understanding modern quantum field theory. It also
contains a brilliant proof of the classic Morse inequalities, relating critical points
to homology. The main point is that homology is defined via Hodge's harmonic
forms and critical points enter via stationary phase approximation to quantum
mechanics. Witten explains that "supersymmetric quantum mechanics" is just
Hodge-de Rham theory. The real aim of the paper is however to prepare the
ground for supersymmetric quantum field theory as the Hodge-de Rham theory
of infinite-dimensional manifolds. It is a measure of Witten's mastery of the field
that he has been able to make intelligent and skilful use of this difficult point of
view in much of his subsequent work.
Even the purely classical part of this paper has been very influential and has
led to new results in parallel fields, such as complex analysis and number theory.
Many of Witten's papers deal with the topic of "Anomalies". This refers to
classical symmetries or conservation laws which are violated at thé quantum
level. Their investigation is of fundamental importance for physical models and
the mathematical aspects are also extremely interesting. The topic has been
extensively written about (mainly by physicists) but Witten's contributions have
been deep and incisive. For example, he pointed out and investigated "global"
anomalies [3], which cannot be studied in the traditional perturbativi manner.
He also made the important observation that the rç-invariant of Dirac operators
(introduced by Atiyah, Patodi and Singer) is related to the adiabatic limit1
of a
certain anomaly [4]. This was subsequently given a rigorous proof by Bisrriut and
Freed.
One of Witten's best known ideas is that the index theorem for the Dirac
operator on compact manifolds should emerge by a formally exact functional
integral on the loop space. This idea (very much in the spirit of his Morse theory
paper) stimulated an extensive development by Alvarez-Gaumé, Getzler, Bismut
and others which amply justified Witten's view-point.
Also concerned with the Direc operator is a beautiful joint paper with Vafa
[5] which is remarkable for the fact that it produces sharp uniform bounds for
eigenvalues by an essentially topological argument. For the Dirac operator on an
odd-dimensional compact manifold, coupled to a background gauge potential,
Witten and Vafa prove that there is a constant C (depending on the metric,
but independent of the potential) such that every interval of length C contains an
eigenvalue. This is not true for Laplace operators or in even dimensions, and is a
very refined and unusual result.
On the Work of Edward Witten 33
3. The Positive Mass Conjecture
In General Relativity the positive mass conjecture asserts that (under appropriate
hypotheses) the total energy of a gravitating system is positive and can only be
zero for flat Minkowski space. It implies that Minkowski space is a stable
ground state. The conjecture has attracted much attention over the years and
was established in various special cases before being finally proved by Schoen
and Yau in 1979. The proof involved non-linear P. D. E. through the use of
minimal surfaces and was a major achievement (leading in part to Yau's Fields
Medal at the Warsaw Congress). It was therefore a considerable surprise when
Witten outlined in [6] a much simpler proof of the positive mass conjecture based
on linear P. D. E. Specifically Witten introduced spinors and studied the Dirac
operator. His approach had its origin in some earlier ideas of supergravity and it
is typical of Witten's insight and technical skill that he eventually emerged with a
simple and quite classical proof. Witten's paper stimulated both mathematicians
and physicists in various directions, demonstrating the fruitfulness of his ideas.
4. Rigidity Theorems
The space of solutions of an elliptic differential equation on a compact manifold is
naturally acted on by any group of symmetries of the equation. All representations
of compact connected Lie groups occur this way. However, for very special
equations, these representations are trivial. Notably this happens for the spaces of
harmonic forms, since these represent cohomology (which is homotopy invariant).
A less obvious case arises from harmonic spinors (solutions of the Dirac equation),
although the relevant space here is the "index" (virtual difference of solutions of
D and D*). This was proved by Atiyah and Hirzebruch in 1970. Witten raised
the question whether such "rigidity theorems" might be true for other equations
of interest in mathematical physics, notably the Rarita-Schwinger equation. This
stimulated Landweber and Stong to investigate the question topologically and
eventually Witten [7] produced an infinite sequence of such equations which arise
naturally in the physics of string theories, for which the Feynman path integral
provides a heuristic explanation of rigidity. As usual Witten's work, which was
very precise and detailed in its formal aspects, stimulated great activity in this
area, culminating in rigorous proofs of these new rigidity theorems by Boti and
Taubes [1], A noteworthy aspect of these proofs is that they involve elliptic
function theory and deal with the infinite sequence of operators simultaneously
rather than term by term. This is entirely natural from Witten's view-point, based
on the Feynman integral.
5. Topological Quantum Field Theories
One of the remarkable aspects of the Geometry/Physics interaction of recent
years has been the impact of quantum field theory on low-dimensional geometry
(of 2, 3 and 4 dimensions). Witten has systematized this whole area by showing
34 Michael Atiyah
that there are, in these dimensions, interesting topological quantum field theories
[8], [9], [10]. These theories have all the formal structure of quantum field theories
but they are purely topological and have no dynamics (i.e. the Hamiltonian is
zero). Typically the Hilbert spaces are finite-dimensional and various traces give
well-defined invariants.'For example, the Donaldson theory in 4 dimensions fits
into this framework, showing how rich such structures can be.
A more recent example, and in some ways a more surprising one, is the
theory of Vaughan Jones related to knot invariants, which has just been reported
on by Joan Birman. Witten has shown that the Jones invariants of knots can
be interpreted as Feynman integrals for a 3-dimensional gauge theory [11]. As
Lagrangian, Witten uses the Chern-Simons function, which is well-known in this
subject but had previously been used as an addition to the standard YangMills
Lagrangian.
Witten's theory is a major breakthrough, since it is the only
intrinsically 3-dimensional interpretation of the Jones invariants: all previous
definitions employ a presentation of a knot by a plane diagram or by a braid.
Although the Feynman integral is at present only a heuristic tool it does
lead, in this case, to a rigorous development from the Hamiltonian point of view.
Moreover, Witten's approach immediately shows how to extend the Jones theory
from knots in the 3-sphere to knots in arbitrary 3-manifolds. This generalization
(which includes as a specially interesting case the empty knot) had previously
eluded all other efforts, and Witten's formulas have now been taken as a basis
for a rigorous algorithmic definition, on general 3-manifolds, by Reshetikin and
Turaev,
Moreover, Witten's approach is extremely powerful and flexible, suggesting
a number of important generalizations of the theory which are currently being
studied and may prove to be important.
One, of the most exciting recent developments in theoretical physics in the
past year has been the theory of 2-dimensional quantum gravity. Remarkably this
theory appears to have close relations with the topological quantum field theories
that have been developed by Wittert [12]. Detailed reports on these recent ideas
will probably be presented by various speakers at this congress.
6. Conclusion
From this very brief summary of Witten's achievements it should be clear that
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. Of course physical insight does not always lead to immediately
rigorous mathematical proofs but it frequently leads one in the right direction,
and technically correct proofs can then hopefully be found. This is the case with
Witten's work. So far his insight has never let him down and rigorous proofs, of
the standard we mathematicians rightly expect, have always been forthcoming.
There is therefore no doubt that contributions to mathematics of this order are
fully worthy of a Fields Medal.
On the Work of Edward Witten 35
References
1. R. Bott and C. H. Taubes: On the rigidity theorems of Witten. J. Amer, Math. Soc,
2 (1989) 137
2. E. Witten; Supersymmetry and Morse theory. J. Diff. Geom. 17 (1984) 661
3. E. Witten: An SU(2) anomaly. Phys. Lett. 117 B (1982) 324
4. E, Witten: Global anomalies in string theory. Proc, Argonne-Chicago Symposium on
Geometry, Anomalies and Topology (1985)
5. E, Witten and C. Vafa: Eigenvalue inequalities for Fermions in gauge theories, Comm.
Math. Phys. 95 (1984) 257
6. E, Witten: A new proof of the positive energy theorem, Comm. Math, Phys. 80 (1981)
381
7. E, Witten: Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987)
525
8. E. Witten: Topological quantum field theory, Comm. Math. Phys, 117 (1988) 353
9. E. Witten: Topological gravity. Phys. Lett. B 206 (1988) 601
10. E. Witten: Topological sigma models. Comm. Math. Phys, 118 (1988) 411
11. E. Witten: Quantum field theory and the Jones polynomial, Comm, Math, Phys, 121
(1989) 351
12. E, Witten: On the structure of the topological phase of two dimensional gravity.
Nuclear Phys, B 340 (1990) 281
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