Invariants of Toric Seiberg Duality
  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. 
Amihay Hanany, Yang-Hui He, Vishnu Jejjala, Jurgis Pasukonis, Sanjaye   Ramgoolam, Diego Rodriguez-Gomez
hep-th math-ph math.MP math.NT