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: \( \sum_{n,\beta} N_{n,\beta} \, q^n y^{2n-2} = \sum_{n,\beta} \text{DT}_{\beta}(n) \, q^n y^{2n-2} \) where \( N_{n,\beta} \) are integer Gopakumar-Vafa invariants and \( \text{DT}_{\beta}(n) \) are rational Donaldson-Thomas invariants. - Integrality: \( N_{n,\beta} \in \mathbb{Z} \) (conjectured by GV, proven via DT theory). - Combinatorial identity: \( \prod_{k=1}^\infty (1 - q^k)^{-k} = \text{MacMahon function} \) appears in the generating function for DT invariants of the resolved conifold. - McKay correspondence: For a finite subgroup \( G \subset SL(2,\mathbb{C}) \), 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