String Theory and Math: Why This Marriage May Last. Mathematics and Dualities of Quantum Physics

String theory is changing the relationship between mathematics and physics. The central role is played by the phenomenon of duality, which is intrinsic to quantum physics and abundant in string theory. The relationship between mathematics and physics has a long history. Traditionally, mathematics provides the language physicists use to describe nature, while physics brings mathematics to life: To discover the laws of mechanics, Newton needed to develop calculus. The very idea of a precise physical law, and mathematics to describe it, were born together. To unify gravity and special relativity, Einstein needed the language of Riemannian geometry. He used known mathematics to discover new physics. General relativity has been inspiring developments in differential geometry ever since. Quantum physics impacted many branches of mathematics, from geometry and topology to representation theory and analysis, extending the pattern of beautiful and deep interactions between physics and mathematics throughout centuries. String theory brings something new to the table: the phenomenon of duality. Duality is the equivalence between two descriptions of the same quantum physics in different classical terms. Ordinarily, we start with a classical system and quantize it, treating quantum fluctuations as small. However, nature is intrinsically quantum. One can obtain the same quantum system from two distinct classical starting points. For every precise question in one description of the theory, there is a corresponding question in the dual description. Duality is similar to a change of charts on a manifold, except it also has the power to map large fluctuations in one description to small fluctuations in the dual, and to relate very hard mathematical problems in one area of mathematics to more manageable ones in another. Dualities are pervasive in string theory. Understanding dualities requires extracting their mathematical predictions and proving the huge set of mathematical conjectures that follow. The best understood duality is mirror symmetry. But mirror symmetry is but one example—many striking dualities have been discovered in quantum field theory (QFT) and many more in string theory over the last 20 years. Duality gives quantum physics, and especially string theory, the power to unify disparate areas of mathematics in surprising ways, and it provides a basis for a long lasting and profound relationship between physics and mathematics. Received by the editors May 1, 2015. 2010 Mathematics Subject Classification. Primary 00-XX, 81-XX, 51-XX. 1Reviews of dualities in string theory can be found in [1]–[3]. For another review of interaction between mathematics and physics, see [4]. c ©2015 American Mathematical Society