String Theory and Mathematics

Why are mathematicians so excited about string theory?

To answer this question, consider the history of interaction between physics and mathematics. For 3000 years the two subjects evolved in tandem. Once their separate identities emerged, there were long periods when mathematicians and physicists worked largely apart on seemingly unrelated problems. But no matter how physical laws came to be generalized (e.g., in the theory of general relativity, quantum mechanics, or gauge theory), the still more abstract mathematical language needed to express them had usually already been developed for non-physical reasons (e.g., differential geometry, functional analysis, or bundle theory, respectively).

To physicists, this "Unreasonable Effectiveness of Mathematics in the Natural Sciences" (summarized in Eugene Wigner's 1960 essay) required explanation; Wigner viewed it as an inevitable consequence of the laws of invariance (i.e., the notion that physical laws are valid at every point of space-time), and the empirical law of epistemology (i.e., the accrued evidence that mathematics has worked well so far). He contrasts this with the "nightmare of the theorist," illustrated by the problem of quantum gravity – the quest to unify quantum mechanics and general relativity – wherein "The two theories operate with different mathematical concepts ... [and] no mathematical formulation exists to which both of these theories are approximations."

To mathematicians, physics is a rich source of both examples and conjectures, each a powerful motivation for mathematical research.

On the one hand, with its wealth of "coherent structures," physics can be viewed as the ur-example: Classical mechanics has motivated the qualitative theory of differential equations and symplectic geometry; quantum mechanics has inspired "q-analogues" throughout algebra and geometry; general relativity has led to much research in partial differential equations and differential geometry.

On the other hand, to a mathematician the very best conjectures are those whose statements or proofs relate very different types of mathematics. Wigner's illustration of the theorist's nightmare now becomes a great opportunity! Since any candidate for a consistent theory of quantum gravity must relate in a fundamentally new way the different branches of mathematics used to describe each component theory, such a physical theory should suggest important mathematical conjectures.

String theory is the leading candidate theory of quantum gravity. In fact, there are several consistent variants (Types I, IIA, IIB, and two Heterotic string theories), and they all have certain common features. The most obvious are the replacement of point particles by strings and the six extra "curled-up" space-time dimensions. The shape of these compactified dimensions is what mathematicians call a Calabi-Yau manifold, and the particle spectrum of the physical theory is determined by the topology and geometry of this manifold. Due to their "extra mathematical" origin, these distinct string theories can admit very different mathematical descriptions.

Physically, the five string theories may be viewed as limits of a single as yet unknown "M-theory," and hence are related one to another by "string dualities." Identifying these via a string duality sometimes results in an equivalence between an easy problem (computation is feasible) and a hard problem (undeveloped mathematics). This has lead to some very precise mathematical predictions from physics, and proven quite useful for mathematicians seeking to refine the conjectures they inspire. The very non-uniqueness of string theory has become a boon to mathematics, yielding new and deep mathematical conjectures.

String theory has produced "derivations" of mathematical theories like toric geometry and K-theory, and a host of string-motivated conjectures in virtually every field of mathematics. Whatever its eventual status as a physical theory of quantum gravity, the inevitability of string theory as a mathematical theory of the highest order is hard to dispute. In light of this, perhaps we should turn Wigner's remark around and marvel instead at the unreasonable effectiveness of string theory in mathematics!

Chuck Doran

This article was inspired by Professor Doran's lecture in the Science Forum Colloquium last May. Powerpoint slides are available for download online at http://www.math.washington.edu/~doran/StringTheoryandMathematics.ppt.