D-Brane Stability and the Hodge Conjecture: A Programme via Bridgeland Stability Conditions

Abstract We propose a research programme connecting the Hodge conjecture to stability conditions on derived categories of coherent sheaves, motivated by the physics of D-branes in B-model topological string theory. Rational (p, p)-classes are interpreted as topological RamondRamond charges, and algebraic cycles as supports of stable objects in Db(Coh(X)). We outline how Bridgeland stability conditions and the Kontsevich Soibelman wall-crossing formula provide a natural framework for this correspondence, prove the programme works completely for K3 surfaces and abelian varieties, and identify precisely where the argument requires genuinely new mathematics for general n > 2. We do not claim a proof of the Hodge conjecture; rather, we formulate three specic open problems whose resolution would constitute such a proof within this framework.