Following a suggestion by Vafa, we present a quantum-mechanical model for
S-duality symmetries observed in the quantum theories of fields, strings and
branes. Our formalism may be understood as the topological limit of Berezin's
metric quantisation of the upper half-plane H, in that the metric dependence
has been removed. Being metric-free, our prescription makes no use of global
quantum numbers. Quantum numbers arise only locally, after the choice of a
local vacuum to expand around. Our approach may be regarded as a manifestly non
perturbative formulation of quantum mechanics, in that we take no classical
phase space and no Poisson brackets as a starting point. The reparametrisation
invariance of H under SL(2,R) induces a natural SL(2,R) action on the quantum
mechanical operators that implements S-duality. We also link our approach with
the equivalence principle of quantum mechanics recently formulated by Faraggi
and Matone.Comment: 14 pages, JHEP styl
