The refined BPS index from stable pair invariants

Abstract

A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi-Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a C* action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M-theory reduced on these Calabi-Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local P^1. We explicitly compute refined invariants in low degree for local P^2 and local P^1 x P^1 and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula. The modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined PT invariants on appropriate geometries to Nekrasov's partition function and a refinement of Chern-Simons theory on a lens space. We also relate our product formula to wallcrossing.

arXiv comment60 pages, 1 eps figure; reference updated; minor typos corrected

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Cited by in corpus (11)

• Non-perturbative effects and the refined topological string
• Refined stable pair invariants for E-, M- and [p,q]-strings
• Instanton effects and quantum spectral curves
• Parabolic refined invariants and Macdonald polynomials
• Geometric engineering of (framed) BPS states
• Nekrasov's Partition Function and Refined Donaldson-Thomas Theory: the Rank One Case
• The moduli space of stable coherent sheaves via non-archimedean geometry
• Exact solutions to quantum spectral curves by topological string theory
• The geometry of the moduli space of one-dimensional sheaves
• Topological String on elliptic CY 3-folds and the ring of Jacobi forms
• Virtual Poincaré polynomial of the space of stable pairs supported on quintic curves