GV-DT Correspondence: Integer BPS Invariants via MacMahon Identity — E8 Intelligence Research

FINDING: Gopakumar-Vafa invariants are integer-valued counts of BPS states in M-theory on a Calabi-Yau threefold, conjectured to equal the Euler characteristic of the moduli space of stable pairs (Donaldson-Thomas invariants) via a combinatorial identity involving the MacMahon function. MATH: - GV-DT correspondence: \( ∑n,β Nn,β \, q^n y²ⁿ⁻² = ∑n,β DTβ(n) \, q^n y²ⁿ⁻² \) where \( Nn,β \) are integer Gopakumar-Vafa invariants and \( DTβ(n) \) are rational Donaldson-Thomas invariants. - Integrality: \( Nn,β ∈ Z \) (conjectured by GV, proven via DT theory). - Combinatorial identity: \( ∏ₖ₌₁^∞ (1 - q^k)⁻ᵏ = MacMahon function \) appears in the generating function for DT invariants of the resolved conifold. - McKay correspondence: For a finite subgroup \( G ⊂ SL(2,C) \), the McKay quiver encodes the representation theory of \( G \), and the resolution of the Kle Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com