The refined BPS index from stable pair invariants

TL;DR

This paper introduces a refined stable pair invariant for non-compact Calabi-Yau spaces using a virtual Bialynicki-Birula decomposition, providing explicit computations and connections to physical theories like M-theory and Chern-Simons theory.

Contribution

It proposes a new refined invariant based on a virtual Bialynicki-Birula decomposition and extends the motivic product formula for local P^1, with explicit low-degree computations.

Findings

Refined invariants computed for local P^2 and P^1 x P^1.

Agreement with predictions from holomorphic anomaly and product formula.

Relation of generating functions to Nekrasov's partition function and Chern-Simons theory.

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.