Invariants of Toric Seiberg Duality

TL;DR

This paper identifies a new invariant under toric Seiberg duality for 4D conformal field theories, using dimer models and algebraic number theory, providing deeper insight into the structure of these theories.

Contribution

It introduces the Klein j-invariant of the dimer's complex structure as a Seiberg duality invariant and offers a new algebraic approach to a-maximization.

Findings

Klein j-invariant remains unchanged under Seiberg duality.

Number theoretic invariants are derived from R-charges.

A new compact formulation of a-maximization is proposed.

Abstract

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.