Geometry and physics (Q5894675)

The notice under review is based on Yau's talk given at Tsinghua University in 2016. Yau starts by reviewing Einstein's theory of general relativity, which is greatly influenced by the works of many mathematicians like Euler, Gauss, Riemann, Lie, Klein, Ricci, Christoffel, Bianchi, Minkowski, Hilbert, Levi-Civita, etc. After Einstein's success, people have great desire to unify all known forces by using ideas similar to general relativity. One way is by the so-called gauge theory. The next two sections reviews the development of gauge theory or Yang-Mills theory. Weyl notices gauge symmetry on Maxwell's theory and introduces the concept of gauge theory, which builds on Levi-Civita and Ehresmann's theory of connections on principal bundles. Chern introduces Chern classes, Chern-Weil and Chern-Simons theory which play important role in quantum physics. For example, Chern-Simons theory is used later in the work of Witten to link Jones polynomial of knots to quantum field theory. In 1954, \textit{C. N. Yang} and \textit{R. L. Mills} [Phys. Rev., II. Ser. 96, 191--195 (1954; Zbl 1378.81075)] developed non-abelian gauge theory where structure groups of bundles are replaced from \(U(1)\) by general Lie groups. Based on \textit{L. Faddeev}-\textit{V. Popov}'s work [Phys. Lett. B 25, No. 1, 29--30 (1967)], 't Hooft finally succeeds in quantizing YM theory. This is later used to build standard models for elementary particles. YM theory also has great influence in mathematics, notably Donaldson-Uhlenbeck-Yau's work on relating Hermitian-Yang-Mills connections to stability of holomorphic bundles, Donaldson's work on using Yang-Mills moduli spaces to give striking applications to differential topology of smooth 4-manifolds. In the next section, Yau reviews his work on proving the Calabi conjecture (which aims to classify higher dimensional manifolds by imposing Kähler-Einstein metrics) and the fundamental role of Calabi-Yau manifolds (i.e. Ricci-flat Kähler-Einstein manifolds) in superstring theory. In the next section, the history of Hodge theory and index theory is reviewed, typically the fundamental contributions from Weyl, deRham, Lefschetz, Hodge, Kodaira and Atiyah-Singer, etc. In the last section, several developments in mathematical physics since 1970 are reviewed, e.g. Witten's interpretation of Morse theory, Taubes' `SW = Gr' correspondence, Atiyah-Bott's study of Yang-Mills theory on Riemann surfaces, Gromov-Witten invariants and the genus zero mirror symmetry on Calabi-Yau 3-folds, etc.