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BPS Invariants of Semi-Stable Sheaves on Rational Surfaces


Bogomolnyi–Prasad–Sommerfield (BPS) invariants are computed, capturing topological invariants of moduli spaces of semi-stable sheaves on rational surfaces. For a suitable stability condition, it is proposed that the generating function of BPS invariants of a Hirzebruch surface \({\Sigma_\ell}\) takes the form of a product formula. BPS invariants for other stability conditions and other rational surfaces are obtained using Harder–Narasimhan filtrations and the blow-up formula. Explicit expressions are given for rank ≤  3 sheaves on \({\Sigma_\ell}\) and the projective plane \({\mathbb{P}^2}\). The applied techniques can be applied iteratively to compute invariants for higher rank.

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Manschot, J. BPS Invariants of Semi-Stable Sheaves on Rational Surfaces. Lett Math Phys 103, 895–918 (2013).

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Mathematics Subject Classification (2000)

  • 14J60
  • 14D21
  • 14N35


  • sheaves
  • moduli spaces