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The refined BPS index from stable pair invariants

Jinwon Choi, Sheldon Katz, Albrecht Klemm

TL;DR

This work introduces refined Pandharipande–Thomas invariants for non-compact Calabi–Yau spaces via a virtual Bialynicki-Birula decomposition or Nekrasov–Okounkov equivariant index, enabling refined 5d BPS counts from M-theory. It develops both a direct integration approach using holomorphic anomaly and modularity and a localization-based geometric method for toric local CYs, culminating in a product formula that extends motivic counterparts. The authors compute explicit refined invariants for local models such as ${ m O}(-3) ightarrow P^2$ and ${ m O}(-2,-2) ightarrow P^1 imesP^1$, extract the corresponding $ ext{SU}(2) imes ext{SU}(2)$ BPS spectra, and relate generating functions to Nekrasov’s partition function and refined Chern–Simons theory on lens spaces. They further connect refinements to wall-crossing and provide asymptotic behavior for large degree, offering a coherent framework linking PT theory, GV invariants, and refined topological/string theories with potential applications to matrix models and 3d/4d dualities. These results advance the mathematical understanding of refined BPS state counting and its interplay with modularity, wall-crossing, and physical dualities in string/M-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.

The refined BPS index from stable pair invariants

TL;DR

This work introduces refined Pandharipande–Thomas invariants for non-compact Calabi–Yau spaces via a virtual Bialynicki-Birula decomposition or Nekrasov–Okounkov equivariant index, enabling refined 5d BPS counts from M-theory. It develops both a direct integration approach using holomorphic anomaly and modularity and a localization-based geometric method for toric local CYs, culminating in a product formula that extends motivic counterparts. The authors compute explicit refined invariants for local models such as and , extract the corresponding BPS spectra, and relate generating functions to Nekrasov’s partition function and refined Chern–Simons theory on lens spaces. They further connect refinements to wall-crossing and provide asymptotic behavior for large degree, offering a coherent framework linking PT theory, GV invariants, and refined topological/string theories with potential applications to matrix models and 3d/4d dualities. These results advance the mathematical understanding of refined BPS state counting and its interplay with modularity, wall-crossing, and physical dualities in string/M-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.

Paper Structure

This paper contains 33 sections, 10 theorems, 242 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Physical expectations
  3. The refined BPS supertrace
  4. The Schwinger loop calculation
  5. The direct integration approach
  6. Elliptic curve mirrors and closed modular expressions
  7. The local Calabi-Yau geometries
  8. The local Calabi-Yau manifold ${\cal O}(-3)\rightarrow \mathbb{P}^2$
  9. The local Calabi-Yau manifold ${\cal O}(-2,-2) \rightarrow \mathbb{P}^1\times \mathbb{P}^1$
  10. Nekrasov's 4d partition function at weak and strong coupling
  11. BPS invariants for ${\cal O}(-2,-2) \rightarrow \mathbb{P}^1\times \mathbb{P}^1$ in the large volume limit
  12. The refinement of perturbative CS theory on the Lens space $L(2,1)$
  13. Enumerative invariants of Calabi-Yau threefolds
  14. Pandharipande-Thomas invariants
  15. Gopakumar-Vafa invariants
  16. ...and 18 more sections

Key Result

Proposition 1

ptbps$P_{1-p_a+n}(S,\beta)\simeq{\mathcal{C}}^{[n]}$ for any $n\ge0$.

Figures (1)

  • Figure 1: The figure shows boxes of type $\mathrm{I}^-$ in red, boxes of type $\mathrm{I\space I}$ in blue (light) and two uncolored positive cylinders corresponding to the partitions $(3,1,1)$ and $(1,1)$. In local toric Calabi-Yau manifolds there are only two cylinders and no boxes of type $\mathrm{I\space I\space I}$.

Theorems & Definitions (15)

  • Definition 1
  • Conjecture 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4: pt
  • Proposition 5
  • proof
  • Lemma 1
  • proof
  • ...and 5 more