The refined BPS index from stable pair invariants
Refined BPS invariants can be defined in rigid \(N=2\) theories in \(4\) dimensions, the only known examples of which are type II string compactifications on non-compact Calabi-Yau. They record multiplicities of BPS particles in the lattice of \(K\)-theory charges of even \(D\)-branes with given spin quantum numbers labeling representations of the twisted off-shell Lorentz group \(\mathrm{SU}(2)\times \mathrm{SU}(2)\) . In this paper, the authors extend a B-model calculation from unrefined to refined invariants, and relate them conjecturally to refined stable pair invariants, confirming the correspondence by explicit evaluation for local \(\mathbb{P}^2\) and \(\mathbb{P}^1\times\mathbb{P}^1\). The refined stable pair invariants can be defined mathematically in terms of a virtual Bialynicki-Birula decomposition, and are expected to be related to BPS invariants via a \(5\)-dimensional BPS supertrace and the Schwinger loop calculation. The stable pair invariants are evaluated via the localization algorithm for the Pandharipande-Thomas invariants of toric Calabi-Yaus, and their partition function is expressed as an infinite product depending only on \(\mathrm{SU}(2)\times \mathrm{SU}(2)\) BPS invariants and related to wall-crossing.
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