GV-DT Invariants, McKay Correspondence, and Integer BPS Counts — E8 Intelligence Research

FINDING: Gopakumar-Vafa invariants are integer-valued and conjecturally equivalent to Donaldson-Thomas invariants via a combinatorial identity, with deep links to the McKay correspondence for finite subgroups of SL(2,C). MATH: - GV/DT correspondence: For a Calabi-Yau threefold X, the generating function of DT invariants (counting ideal sheaves) equals the product of GV invariants (counting BPS states) via the identity: \[ Z_{DT}(q) = \prod_{n \geq 1} \prod_{\beta \in H_2(X,\mathbb{Z})} \left(1 - (-q)^n\right)^{n \cdot n_{\beta}^{(n)}} \] where \(n_{\beta}^{(n)}\) are integer GV invariants. - McKay correspondence: For a finite subgroup \(G \subset SL(2,\mathbb{C})\), the resolution of \(\mathbb{C}^2/G\) yields a Dynkin diagram of ADE type, with the number of irreducible representations of G equal to the rank of the corresponding Lie algebra. - Combinatorial identity: The integrality of GV invariants is equivalent to the DT partition function being a product of ration Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com