Lattice Green's function of the hyperkagome lattice: modular uniformization at level 30 from an orthogonal differential Galois group
The lattice Green's functions of the cubic lattices are classical: each is a symmetric square of a second-order operator, with a closed form in complete elliptic integrals. The hyperkagome lattice, realized by the iridium sublattice of the spin-liquid candidate Na4Ir3O8, has resisted such a treatment: Varma and Monien reduced its density of states to a threefold integral they found no way to solve exactly. From exact lattice moments we obtain an irreducible third-order linear differential (Picard-Fuchs) operator annihilating the Green's function. It is not a literal symmetric square, yet its symmetric square carries a rational solution: the differential Galois group is orthogonal, and an order-two differential intertwiner makes the operator projectively a symmetric square. Our main result is that the underlying second-order operator is the uniformizing equation of the genus-zero modular curve of Gamma_0(30)+, the level-thirty group generated by Gamma_0(30) and all its Atkin-Lehner involutions, an explicit eta quotient parametrizing the natural variable. That uniformizing equation is itself not new - it is a row of the Lian-Yau table of genus-zero groups - so what is new here is the identification of a lattice Green's function with it, together with a proof: the Schwarzian identity is established by an a priori pole-degree bound rather than checked numerically, which also proves the tabulated entry. The hyperkagome Green's function is therefore modular at level thirty, supplying the closed form Varma and Monien sought. Its weight-two period is obtained explicitly: a depth-one quasimodular form which we prove is not a modular form times an algebraic function at any weight.