  Three-branes at a given toric Calabi-Yau singularity lead to different phases
of the conformal field theory related by toric (Seiberg) duality. Using the
dimer model/brane tiling description in terms of bipartite graphs on a torus,
we find a new invariant under Seiberg duality, namely the Klein j-invariant of
the complex structure parameter in the distinguished isoradial embedding of the
dimer, determined by the physical R-charges. Additional number theoretic
invariants are described in terms of the algebraic number field of the
R-charges. We also give a new compact description of the a-maximization
procedure by introducing a generalized incidence matrix.
