There is more than one way for geometry and physics to interact with each other

The strings ones

Looking to the past the great classical era began with Newton, took a giant stride with Maxwell and culminated in Einstein’s spectacular theory of general relativity. But then came quantum mechanics and quantum field theory with a totally new point of view, rather far removed from the geometric ideas of the past.  

The major problem of our time for physicists is how to combine the two great themes of GR, that governs the large scale universe, and QM that deals with the very small scale. 

At the present time we have string theory, or perhaps “M-theory”, which is a beautiful rich mathematical story and will certainly play an important role in the future of both mathematics and physics. Already the applications of these ideas to mathematics have been spectacular. To name just a few, we have 

1. Results on the moduli spaces of Riemann surfaces. 

2. The Jones polynomials of knots and their extension by Witten to “quantum invariants” of 3-manifolds. 

3. Donaldson theory of 4-manifolds and the subsequent emergence of Seiberg-Witten theory. 

4. Mirror symmetry between holomorphic and symplectic geometry.  

Although string theory or M-theory are thought by many to be the ultimate theory combining QM and GR no-one knows what M-theory really is. String theory is recognized only as a perturbative theory, but the full theory is still a mystery (one of the roles of the letter M). Some claim that the final theory is close at hand - we are almost there. But perhaps this is misplaced optimism and we await a new resolution based on radical new ideas. There are, after all, some major challenges posed by astronomical observations. 

• Dark Matter 

• Dark Energy (with a very small cosmological constant)  

Moreover, the direct linkage between the rarified mathematics of string theory and the world of experimental physics is, as yet, very slender. A friend of mine, a retired Professor of Physics, commentating on a string theory lecture, said it was ”pure poetry”! This can be taken both as criticism and as a tribute. In the same way science fiction cannot compete with the modern mysteries of the quantum vacuum.

A Shifted View of Fundamental Physics

Michael Atiyah, Gregory W. Moore

(Submitted on 16 Sep 2010)

The twistor and the noncommutative geometric paths

As is well known, Einstein dreamed of a unified geometric theory, extending GR, and he never accepted the philosophical foundations of QM, with its uncertainty principle. In the long debate on this controversy between Einstein and Bohr the general verdict of the physics community was that Einstein lost and that his idea of a unified field theory was a hopeless pipe dream.

With the remarkable success of the standard model of elementary particles, incorporating geometrically the electromagnetic, the weak force and the strong force, Einstein’s ideas were given new life. But the framework remained that of QM, and GR remained strictly outside the scope of the unification. Now, with string theory offering the hope of the ultimate unification it might appear that the old controversy between Einstein and Bohr has been resolved, with the honours more equally split. Unification is perhaps being achieved, but QM has persisted. 

This is the orthodox view of string theorists and they have impressive evidence in their favor. The only fly in the ointment is that no one yet has any real idea of what their ultimate M-theory is. Perhaps in the coming years this will be clarified and we will learn to live with the mysterious world of 11 dimensions and its hidden supersymmetries. Perhaps only a few technical obstacles remain to complete the structure. 

But it is at least worth exploring alternative scenarios. There are in particular two attractive ideas that have their devotees. The first (in historical precedence) is Roger Penrose’s twistor theory. On the one hand this has, as a technical mathematical tool, proved its worth in a number of problems. It is also related to supersymmetry and duality. Links with string theory are being explored. But beyond these mathematical technicalities there lies a deeper philosophical idea. Penrose is an Einsteinian who believes that in the hoped-for marriage between GR and QM it is the latter that must give the most, adapting itself to the beauty of GR. Twistors are thought of as a first step to achieving this goal. Moreover, Penrose speculates that the mysterious role of complex numbers in QM should ultimately have a geometric origin in the natural complex structure of the base of the light-cone in Minkowski space. So far, it has to be conceded that the weight of evidence is not in Penrose’s favor, but that does not mean that he may not ultimately be vindicated. 

A completely different scenario is offered by Alain Connes’ noncommutative geometry, a theory with a rich mathematical background and a promising future. Links with physics exist and new ones are being discovered. In a sense Connes takes off from the Heisenberg commutation relations, in a definitely non-Einsteinian direction. However, he tries to keep the geometric spirit by using the same concepts and terminology. It is certainly possible that the final version of M-theory may use Connes’ framework for its formulation. 

Perhaps I can end by indulging in some wild speculation of my own, not I hope totally unrelated to the other ideas above. 

I start, further back, by asking some philosophical or metaphysical questions. If we end up with a coherent and consistent unified theory of the universe, involving extremely complicated mathematics, do we believe that this represents “reality’’? Do we believe that the laws of nature are laid down using the elaborate algebraic machinery that is now emerging in string theory? Or is it possible that nature’s laws are much deeper, simple yet subtle, and that the mathematical description we use is simply the best we can do with the tools we have? In other words, perhaps we have not yet found the right language or framework to see the ultimate simplicity of nature. 

The Interaction between Geometry and Physics

Michael Atiyah (2003?)

In any case Spinors matter  

Finally, I should say a word about the role of spinors. Ever since they arose in physics with the work of Dirac they have played a fundamental part, providing the fermions of the theory. In mathematics spinors are well understood algebraically (going back to Hamilton and Clifford) and their role in the representation theory of the orthogonal group provides the link with physics. However, in global geometry, spinors are much less understood. The Dirac operator can be defined on spinor fields and its square is similar to the Hodge–Laplace operator. Its index is given by the topological formula referred to above in connection with anomalies. However, while the geometric significance of differential forms (as integrands) is clear, the geometric meaning of spinor fields is still mysterious. The only case where they can be interpreted geometrically is for complex Kähler manifolds where holomorphic function theory essentially extracts the “square root of the geometry’’ that is needed. Gauss is reputed to have said that the true metaphysics of √−1 is not simple. The same could be said for spinors, which are also a mysterious kind of square root. Perhaps this remains the deepest mystery on the geometry–physics frontier..  

The most striking application of quantum field theory in three dimensions was undoubtedly Witten’s interpretation of the polynomial knot invariants discovered by Vaughan Jones. It was already clear, from the work of Jones, that his invariants were essentially new and very powerful. Old conjectures were quickly disposed of. What Witten did was to show how the Jones invariants could be easily understood (and generalized) in terms of the quantum field theory defined by the Chern–Simons Lagrangian. One immediate benefit of this was that it worked for any oriented 3- manifold, not just S3. In particular, taking the empty knot one obtained numerical invariants for compact 3-manifolds. These developments have stimulated a great deal of work by geometers. In particular, there are combinatorial treatments which are fully rigorous and mimic much of the physics. In three dimensions, we are in the odd situation of having two completely different theories. One the one hand there are the quantum invariants just discussed, while on the other hand there is the deep work of Thurston on geometric structures, including the important special case of hyperbolic 3-manifolds. It has been a long-standing and embarrassing situation that there was little or no connection between these two theories... 

For any compact oriented 4-manifold X, any compact Lie group G and any positive integer k, Donaldson studies the moduli space M of k-instantons. These are anti-self-dual connections for the G-bundle (with topology fixed by k). For this he has first to choose a Riemannian metric (or rather a conformal structure), but he then computes some intersection numbers on M and shows these are independent of the metric. In this way, Donaldson defines invariants of X, which are just polynomials on the second homology of X. 

As is now well known, these Donaldson invariants proved spectacularly successful in distinguishing between 4-manifolds and they opened up the whole subject of smooth 4-manifolds just as Freedman had closed the subject of topological 4- manifolds. While the idea of using instantons came from physics, Donaldson was just using the classical equations of Yang–Mills theory. But Witten subsequently explained that Donaldson’s theory could be interpreted as a suitable quantum field theory in four dimensions. Moreover, this was just a slight variant on a standard theory known as N=2 supersymmetric Yang–Mills. 

This physical interpretation of the Donaldson theory was interesting for physicists but it was not clear what the mathematical benefit was. However, a few years later, the benefit became abundantly clear. As part of some very general ideas of duality in quantum fields theories, Seiberg and Witten produced a quite different theory which was expected to be equivalent to Donaldson theory. This has now been essentially confirmed by mathematicians, though a rigorous proof of the equivalence is not yet complete. Moreover, the Seiberg–Witten equations are technically easier to handle and so they have proved more powerful in many cases...

I should emphasize that the equivalence between the Donaldson and Seiberg– Witten theories is one between generating functions. Each theory has its instantons, but there is no simple relation between instantons of separate degrees, only between the total sums over all degrees. This should be compared with the classical Poisson summation formula which expresses a sum over one lattice in terms of the sum of Fourier transforms over the dual lattice. Thus these dualities of quantum field theories should be viewed as some kind of nonlinear analogues of the Fourier transfom... One surprising feature of the Seiberg–Witten theory, and the classical equations they lead to, is that they deal with a U(1) theory coupled nonlinearly to spinors. Thus spinors appear explicitly here, while they do not appear in the SU(2) Donaldson theory. This only increases the mystery of spinors and emphasizes my earlier remarks about our lack of any deep understanding of them. At present it is not clear whether Donaldson theory, with various refinements, will explain all geometric phenomena in four dimensions. It may do so, but it is also possible that it may take another 100 years to fully understand the geometry of four dimensions, just as it has taken a century to move from Riemann surfaces to an equivalent understanding of three dimensions. If so, this may accompany a similar period for a proper understanding of the physics of space–time...

The Interaction between Geometry and Physics Michael Atiyah (2003?)