Arithmetic gauge theory: a brief introduction. (English) Zbl 1451.14113
Yang, Chen Ning (ed.) et al., Topology and physics. Hackensack, NJ: World Scientific. 109-134 (2019).
Summary: Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.
Reprinted from Modern Phys. Lett. A 33, No. 29, 1830012, 26 p. (2018; Zbl 1397.81001).
For the entire collection see [Zbl 1414.81013].
Reprinted from Modern Phys. Lett. A 33, No. 29, 1830012, 26 p. (2018; Zbl 1397.81001).
For the entire collection see [Zbl 1414.81013].
MSC:
14H81 | Relationships between algebraic curves and physics |
14H60 | Vector bundles on curves and their moduli |
11R37 | Class field theory |
11G05 | Elliptic curves over global fields |
11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |
14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |