These direct mathematical arguments gave no understanding of the original physics insight that connected the formula with solutions to an ordinary differential equation. Indeed, finding such a connection is one of the central problems today in geometry. Many mathematicians work on aspects of this problem, and there are partial hints but no complete understanding, even conjecturally. This statement characterizes much of the current input of physics into mathematics. It seems clear that the physicists are on the track of a deeper level of mathematical understanding that goes far beyond our current knowledge, but we have only the smallest hints of what that might be. Understanding this phenomenon is a central problem both in high-energy theoretical physics and in geometry and topology.
Nowadays, sophisticated mathematics is essential for stating many of the laws of physics. As mentioned, the formulation of the standard model of particle physics involves “gauge theories,” or fiber bundles. These have a very rich topology. These topological structures are described by Chern-Simons theories, Index theory, and K-theory. These tools are also useful for condensed matter systems. They characterize topological phases of matter, which could offer an avenue for quantum computing.12 Here the q-bits are encoded into the subtle topology of the fiber bundle, described by Chern-Simons theory. Recently, K-theory has been applied to the classification of topological insulators,13 another active area of condensed matter physics.
String theory and mathematics are very closely connected, and research in these areas often straddles physics and mathematics. One recent development, the gauge gravity duality, or AdS/CFT, has connected general relativity with quantum field theories, the theories we use for particle physics.14 The gravity theory lives in hyperbolic space. Thus, many developments in hyperbolic geometry, and black holes, could be used to describe certain strongly interacting systems of particles. Thinking along these lines has connected a certain long-distance limit of gravity equations to the equations of hydrodynamics. One considers a particular black-hole, or black-brane, solution of Einstein’s equations with a negative cosmological constant. These black holes have long-distance excitations representing small fluctuations of the geometry. The fluctuations are damped since they end up being swallowed by the black hole. According to AdS/CFT, this system is described by a thermal system on the boundary, a thermal fluid of quantum interacting particles. In this formulation, the long-distance perturbations are described by hydrodynamics, namely by the Navier-Stokes equation
12 A. Yu. Kitaev, 2003, Fault-tolerant quantum computation by anyons. Annals of Physics 303:2-30
13 A.P. Schnyder, S. Ryu, A. Furusaki, and A.W.W. Ludwig, 2008, Classification of topological insulators and superconductors in three spatial dimensions. Physical Review Letters B 78:195125.
14 J. Maldacena, 2005, The illusion of gravity. Scientific American October 24.