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The Refined BPS Index from Stable Pair Invariants

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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 \({\mathbb{C}^{*}}\) 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 \({\mathbb{P}^1}\) . We explicitly compute refined invariants in low degree for local \({\mathbb{P}^2}\) and local \({\mathbb{P}^1\,\times\,\mathbb{P}^1}\) 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 wall crossing.

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References

  1. Aganagic, M., Cheng, M.C.N., Dijkgraaf, R., Krefl, D., Vafa, C.: Quantum geometry of refined topological strings. JHEP 1211, 19 (2012). arXiv:1105.0630v1[hep-th]

  2. Aganagic, M., Klemm, A., Marino, M., Vafa, C.: Matrix model as a mirror of Chern–Simons theory. JHEP 0402, 010 (2004). arXiv:hep-th/0211098v1

  3. Aganagic, M., Schaeffer, K.: Orientifolds and the Refined Topological String. JHEP 1209, 084 (2012). arXiv:1202.4456v1[hep-th]

  4. Aganagic, M., Shakirov, S.: Refined Chern–Simons theory and topological string, (2012). arXiv:1210.2733v1[hep-th]

  5. Behrend K.: Donaldson–Thomas type invariants via microlocal geometry. Ann. Math. 170, 1307–1338 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Behrend, K., Bryan, J., Szendroi, B.: Motivic degree zero Donaldson–Thomas invariants. Inv. Math. 192(1), 111–160 (2013). arXiv:0909.5088[math.AG]

  7. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311–428 (1994). arXiv:hep-th/9309140v1

  8. Choi, J., Chung, K.: Moduli spaces of α-stable pairs and wall-crossing on \({\mathbb{P}^2}\) , (2012). arXiv:1210.2499v1[math.AG]

  9. Dimca A., Szendröi B.: The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on \({\mathbb{C}^3}\) . Math. Res. Lett. 16, 1037–1055 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Drukker, N., Marino, M., Putrov, P.: From weak to strong coupling in ABJM theory. Commun. Math. Phys. 306, 511 (2011). arXiv:1007.3837 [hep-th]

  11. Fantechi, B., Göttsche, L.: Riemann–Roch theorems and elliptic genus for virtually smooth Schemes, (2007). arXiv:0706.0988v1[math.AG]

  12. Gaiotto, D., Strominger, A., Yin, X.: New connections between 4D and 5D black holes. JHEP 0602, 024 (2006). arXiv:hep-th/0503217v2

  13. Gopakumar, R., Vafa, C.: M theory and topological strings–II, (1998). arXiv:hep-th/9812127

  14. Göttsche, L., Shende, V.: Refined curve counting on complex surfaces, (2012). arXiv/1208.1973v1[math.AG]

  15. Haghighat, B., Klemm, A., Rauch, M.: Integrability of the holomorphic anomaly equations. JHEP 0810, 097 (2008). arXiv:0809.1674v2[hep-th]

  16. He M.: Espaces de Modules de systèmes cohérents. Internat. J. Math. 7, 545–598 (1998)

    Article  Google Scholar 

  17. Hori, K., Vafa, C.: Mirror symmetry, (2000). arXiv:hep-th/0002222v3

  18. Huang, M.-x., Kashani-Poor, A.-K., Klemm, A.: The Omega deformed B-model for rigid N = 2 theories, (2013). arXiv:1109.5728v2[hep-th]

  19. Huang, M.-x., Klemm, A.: Holomorphicity and modularity in Seiberg–Witten theories with matter. JHEP 1007, 083 (2010). arXiv:0902.1325v1[hep-th]

  20. Huang, M.-x., Klemm, A.: Direct integration for general Ω backgrounds, (2010). arXiv:1009.1126v1[hep-th]

  21. Huang, M.-x., Klemm, A.: Holomorphic anomaly in gauge theories and matrix models, JHEP 0709, 054 (2007). arXiv:hep-th/0605195v2

  22. Huang, M.-x., Klemm, A.: Refined BPS index on local del Pezzo surfaces and half K3. to appear

  23. Huang, M.-x., Klemm, A., Quackenbush, S.: Topological string theory on compact Calabi–Yau: Modularity and boundary conditions. Lect. Notes Phys. 757, 45–102 (2009). arXiv:hep-th/0612125v1

  24. Iqbal, A., Kozcaz, C., Vafa, C.: The Refined topological vertex. JHEP 0910, 069 (2009). arXiv:hep-th/0701156v2

  25. Iqbal, A., Vafa, C.: BPS degeneracies and superconformal index in diverse dimensions, (2012). arXiv:1210.3605v1[hep-th]

  26. Iqbal, A., Kozcaz, C.: Refined topological strings and toric Calabi–Yau threefolds, (2012). arXiv:1210.3016v1[hep-th]

  27. Kachru, S., Klemm, A., Lerche, W., Mayr, P., Vafa, C.: Nonperturbative results on the point particle limit of N = 2 heterotic string compactifications. Nucl. Phys. B 459, 537 (1996). arXiv:hep-th/9508155v1

  28. Katz, S.H.: Gromov–Witten, Gopakumar-Vafa, and Donaldson–Thomas invariants of Calabi–Yau threefolds. Cont. Math. 401, 43–52 (2006). arXiv:math/0408266v2[math-AG]

  29. Katz S.H.: Genus zero Gopakumar-Vafa invariants of contractible curves. J. Diff. Geom. 79, 185–195 (2008)

    MATH  Google Scholar 

  30. Katz, S.H., Klemm, A., Vafa, C.: Geometric engineering of quantum field theories. Nucl. Phys. B 497, 173–195 (1997). arXiv:hep-th/9609239v2

  31. Katz, S.H., Klemm, A., Vafa, C.: M theory, topological strings and spinning black holes. Adv. Theor. Math. Phys. 3, 1445–1537 (1999). arXiv:hep-th/9910181v2

  32. Klemm, A., Marino, M., Schiereck, M., Soroush, M.: ABJM Wilson loops in the Fermi gas approach, (2013). arXiv:1207.0611v2[hep-th]

  33. Krefl, D., Walcher, J.: Extended holomorphic anomaly in gauge theory. Lett. Math. Phys. 95, 67–88 (2011). arXiv:1007.0263v1[hep-th]

  34. Krefl, D., Walcher, J.: Shift versus extension in refined partition functions, (2010). arXiv:1010.2635v1[hep-th]

  35. Krefl, D., Walcher, J.: ABCD of beta ensembles and topological strings. JHEP 1211, 111 (2012). arXiv:1207.1438v1[hep-th]

  36. Marino, M., Putrov, P.: Exact results in ABJM theory from topological strings. JHEP 1006, 011 (2010). arXiv:0912.3074v4[hep-th]

  37. Marino, M.: Chern–Simons theory, matrix integrals, and perturbative three manifold invariants. Commun. Math. Phys. 253, 25 (2004). arXiv:hep-th/0207096v4

  38. Maulik D., Nekrasov N., Okounkov A., Pandharipande R.: Gromov–Witten theory and Donaldson–Thomas theory. I. Compos. Math. 142(5), 1263–1285 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Morrison, A., Mozgovoy, S., Nagao, K., Szendroi, B.: Motivic Donaldson–Thomas invariants of the conifold and the refined topological vertex, Adv. Math 230, 2065–2093 (2012). arXiv:1107.5017 [math.46]

  40. Nakayama, Y., Ooguri, H.: Comments on worldsheet description of the omega background. Nucl. Phys. B 856, 342 (2012). arXiv:1106.5503v1[hep-th]

  41. Nekrasov, N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004). arXiv:hep-th/0206161v1

  42. Nekrasov, N., Okounkov, A.: The index of M-theory. in preparation

  43. Nekrasov, N., Rosly, A., Shatashvili, S.: Darboux coordinates, Yang-Yang functional, and gauge theory. Nucl. Phys. Proc. Suppl. 216, 69 (2011). arXiv:1103.3919v1[hep-th]

  44. Nekrasov, N., Witten, E.: The Omega deformation, branes, integrability, and Liouville Theory. JHEP 1009, 092 (2010). arXiv:1002.0888v2[hep-th]

  45. Pandharipande, R., Thomas, R.P.: Curve counting via stable pairs in the derived category. Inv. Math. 178 407–447 (2009). arXiv:0707.2348v3[math.AG]

  46. Pandharipande, R., Thomas, R.P.: The 3-fold vertex via stable pairs. Geom. Top. 13, 1835–1876 (2009). arXiv:0709.3823v2[math.AG]

  47. Pandharipande, R., Thomas, R.P.: Stable Pairs and BPS Invariants. J. AMS 23, 267–297 (2010). arXiv:0711.3899v3[math.AG]

  48. Witten, E.: Phases of N = 2 theories in two-dimensions. Nucl. Phys. B 403, 159 (1993). arXiv:hep-th/9301042

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Authors and Affiliations

  1. School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Korea

    Jinwon Choi

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, IL, 61801, USA

    Jinwon Choi & Sheldon Katz

  3. Bethe Center for Theoretical Physics, Physikalisches Institut Universität Bonn, Nussallee 12, 53115, Bonn, Germany

    Albrecht Klemm

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  1. Jinwon Choi
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  2. Sheldon Katz
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  3. Albrecht Klemm
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Correspondence to Sheldon Katz.

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Communicated by N. A. Nekrasov

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Choi, J., Katz, S. & Klemm, A. The Refined BPS Index from Stable Pair Invariants. Commun. Math. Phys. 328, 903–954 (2014). https://doi.org/10.1007/s00220-014-1978-0

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