On Fields over Fields

Details On Fields over Fields On Fields over Fields

2010-03-15

arxiv.org/abs/1003.2986v1

Physics

High Energy Physics

High Energy Physics - Theory

52 pages, 5 figures

Scientific paper

We investigate certain arithmetic properties of field theories. In particular, we study the vacuum structure of supersymmetric gauge theories as algebraic varieties over number fields of finite characteristic. Parallel to the Plethystic Programme of counting the spectrum of operators from the syzygies of the complex geometry, we construct, based on the zeros of the vacuum moduli space over finite fields, the local and global Hasse-Weil zeta functions, as well as develop the associated Dirichlet expansions. We find curious dualities wherein the geometrical properties and asymptotic behaviour of one gauge theory is governed by the number theoretic nature of another.

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