"This is a period in which mathematics and physics are closely linked," says Geometry Center visiting professor Dan Freed. "This has happened quite frequently in the past. For example, the Greeks developed trigonometry to try to describe the stars' placements in the sky. Newton invented calculus while trying explain Kepler's work about astronomy. Gauss studied the geometry of surfaces because he was trying to understand and describe the curvature of the earth. Einstein's theory of general relativity gave importance to the subject of Riemannian geometry. Now it is the work of physicists in quantum field theory which is giving mathematics a new push." Dan Freed is currently working in the mathematical discipline of topological quantum field theory which relates closely to the physical results.
Using quantum field theory, physicists have brought new insight to unsolved problems in mathematics by using a physical intuition for the problem not available to mathematicians. For example, here is a mathematical situation which became more clear by physical intuition: Four-dimensional objects are have some problems not seen in any other dimension. There are some four-dimensional manifolds which are indistinguishable by traditional means, such as classical homotopy theory. They are homeomorphic but not diffeomorphic. Therefore, the mathematician Donaldson came up with a series of computable quantities which are invariant for a given manifold. These quantities are called Donaldson invariants. They enable one to distinguish between the four-dimensional manifolds which were previously indistinguishable by any computable means. Physicist Ed Witten fit these Donaldson invariants into a general framework using quantum field theory. Just this week Freed heard that Witten used quantum field theory techniques to "prove" (in the physicist's sense) an interesting result about the Donaldson invariants, generalizing recent work of Kronheimer and Mrowka.
In another case, mathematicians were trying to compute the number of a certain kind of rational curves of a given degree. They had computed some of the numbers of curves, but it was quite difficult and cumbersome. Then the physicist Candelas and his collaborators came up with a generating function for the number of curves. The numbers did not agree with those that the mathematicians had calculated in one case. This resulted in a great deal of debate. However, it turned out that the mathematicians had made a mistake, and Candelas' function has been successful in predicting the answers.
After the results on the Donaldson invariants, Witten did some further work in which he linked quantum field theory to the Jones polynomials, an important tool for knot theorists. While creating this link, Witten discovered a new invariant in three-dimensional manifolds. Though this has not been earth-shattering for topology, it is a rare and thus exciting discovery.
The physicists' work is not what mathematicians would call rigorous. Many of the methods used are cannot be justified mathematically. Yet the results seem to work mathematically. Freed says, "I would not say that the physicists are doing things that are wrong, but certainly they are using tools which do not belong to the toolkits of mathematicians. Though to mathematicians certain steps do not make sense, the physical framework of quantum field theory serves as an excellent prediction and motivation for mathematical results. Over the past decade we have seen many instances where these predictions--often numerical and very concrete--have proved correct. The challenge for mathematicians is to create the new mathematics behind these physical insights."