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Quantum line defects and refined BPS spectra

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Abstract

In this note, we study refined BPS invariants associated with certain quantum line defects in quantum field theories of class \({{\mathcal {S}}}\). Such defects can be specified via geometric engineering in the UV by assigning a path on a certain curve. In the IR, they are described by framed BPS quivers. We study the associated BPS spectral problem, including the spin content. The relevant BPS invariants arise from the K-theoretic enumerative geometry of the moduli spaces of quiver representations, adapting a construction by Nekrasov and Okounkov. In particular, refined framed BPS states are described via Euler characteristics of certain complexes of sheaves.

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Notes

  1. There is in principle the possibility of the corresponding particle state being described by a different statistics; it does not appear to be the case in all the examples we have studied, but we do not have a clear argument for this.

  2. This condition effectively tells us that we are only interested in representations for which the arrows going to the framing node are represented trivially. Therefore, these arrows can be set to zero in the F-term relations and do not contribute to the construction of the moduli spaces.

  3. For simplicity, we omit those terms associated with the framing arrows which can be neglected in the localization computation.

References

  1. Alim, M., Cecotti, S., Cordova, C., Espahbodi, S., Rastogi, A., Vafa, C.: BPS quivers and spectra of complete \(N=2\) quantum field theories. Commun. Math. Phys. 323, 1185 (2013). arXiv:1109.4941 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  2. Alim, M., Cecotti, S., Cordova, C., Espahbodi, S., Rastogi, A., Vafa, C.: \({\cal{N}} = 2\) quantum field theories and their BPS quivers. Adv. Theor. Math. Phys. 18(1), 27 (2014). arXiv:1112.3984 [hep-th]

    Article  MathSciNet  Google Scholar 

  3. Allegretti, D.G.L.: Categorified canonical bases and framed BPS states. arXiv:1806.10394 [math.RT]

  4. Behrend, K., Fantechi, B.: Symmetric obstruction theories and Hilbert schemes of points on threefolds. Ó Algebra Number Theory 2, 313–345 (2008). arXiv:math.AG/0512556

    Article  MathSciNet  Google Scholar 

  5. Benini, F., Bonelli, G., Poggi, M., Tanzini, A.: Elliptic non-Abelian Donaldson–Thomas invariants of \({\mathbb{C}}^3\). arXiv:1807.08482 [hep-th]

  6. Brennan, T.D., Dey, A., Moore, G.W.: On Õt Hooft defects, monopole bubbling and supersymmetric quantum mechanics. JHEP 1809, 014 (2018). arXiv:1801.01986 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  7. Cecotti, S.: Categorical Tinkertoys for \(N=2\) Gauge theories. Int. J. Mod. Phys. A 28, 1330006 (2013). arXiv:1203.6734 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  8. Cecotti, S., Neitzke, A., Vafa, C.: R-twisting and 4d/2d correspondences. arXiv:1006.3435 [hep-th]

  9. Chuang, W., Diaconescu, D.E., Manschot, J., Moore, G.W., Soibelman, Y.: Geometric engineering of (framed) BPS states. Adv. Theor. Math. Phys. 18(5), 1063 (2014). arXiv:1301.3065 [hep-th]

    Article  MathSciNet  Google Scholar 

  10. Cirafici, M.: On framed quivers, BPS invariants and defects. Conflu. Math. 9(2), 71–99 (2017). arXiv:1801.03778 [hep-th]

    Article  MathSciNet  Google Scholar 

  11. Cirafici, M.: Quivers, line defects and framed BPS invariants. Annales Henri Poincare 19(1), 1 (2018). arXiv:1703.06449 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  12. Cirafici, M., Del Zotto, M.: Discrete integrable systems, supersymmetric quantum mechanics, and framed BPS states-I. arXiv:1703.04786 [hep-th]

  13. Cirafici, M.: Line defects and (framed) BPS quivers. JHEP 1311, 141 (2013). arXiv:1307.7134 [hep-th]

    Article  ADS  Google Scholar 

  14. Cirafici, M., Sinkovics, A., Szabo, R.J.: Instanton counting and wall-crossing for orbifold quivers. Annales Henri Poincare 14, 1001–1041 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  15. Córdova, C., Neitzke, A.: Line defects, tropicalization, and multi-centered quiver quantum mechanics. JHEP 1409, 099 (2014). arXiv:1308.6829 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  16. Del Zotto, M., Sen, A.: Commun. Math. Phys. 357(3), 1113 (2018). arXiv:1409.5442 [hep-th]

    Article  ADS  Google Scholar 

  17. Drukker, N., Morrison, D.R., Okuda, T.: Loop operators and S-duality from curves on Riemann surfaces. JHEP 0909, 031 (2009). arXiv:0907.2593 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  18. Eager, R., Selmani, S.A., Walcher, J.: Exponential networks and representations of quivers. JHEP 1708, 063 (2017). arXiv:1611.06177 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  19. Gaiotto, D., Moore, G.W., Neitzke, A.: Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163 (2010). arXiv:0807.4723 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  20. Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation. arXiv:0907.3987 [hep-th]

  21. Gaiotto, D., Moore, G.W., Neitzke, A.: Framed BPS states. Adv. Theor. Math. Phys. 17(2), 241 (2013). arXiv:1006.0146 [hep-th]

    Article  MathSciNet  Google Scholar 

  22. Gaiotto, D., Moore, G.W., Neitzke, A.: Spectral networks. Annales Henri Poincare 14, 1643 (2013). arXiv:1204.4824 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  23. Gabella, M., Longhi, P., Park, C.Y., Yamazaki, M.: BPS graphs: from spectral networks to BPS quivers. JHEP 1707, 032 (2017). arXiv:1704.04204 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  24. Gabella, M.: Quantum holonomies from spectral networks and framed BPS states. Commun. Math. Phys. 351(2), 563 (2017). arXiv:1603.05258 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  25. Galakhov, D.: BPS hall algebra of scattering hall states. arXiv:1812.05801 [hep-th]

  26. Gang, D., Longhi, P., Yamazaki, M.: \(S\) duality and framed BPS states via BPS graphs. arXiv:1711.04038 [hep-th]

  27. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson–Thomas invariants and cluster transformations. arXiv:0811.2435 [math.AG]

  28. Kontsevich, M., Soibelman, Y.: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants. Commun. Number Theor. Phys. 5, 231 (2011). arXiv:1006.2706 [math.AG]

    Article  MathSciNet  Google Scholar 

  29. Manschot, J., Pioline, B., Sen, A.: Wall crossing from Boltzmann black hole halos. JHEP 1107, 059 (2011). arXiv:1011.1258 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  30. Moore, G.W., Royston, A.B., Van den Bleeken, D.: Semiclassical framed BPS states. JHEP 1607, 071 (2016). arXiv:1512.08924 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  31. Nekrasov, N., Okounkov, A.: Membranes and sheaves. Algebraic Geom. 3(3), 320–369 (2016). arXiv:1404.2323 [math.AG]

    Article  MathSciNet  Google Scholar 

  32. Nekrasov, N.: Magnificent four. arXiv:1712.08128 [hep-th]

  33. Nekrasov, N., Piazzalunga, N.: Magnificent four with colors. arXiv:1808.05206 [hep-th]

  34. Okounkov, A.: Lectures on \(K\)-theoretic computations in enumerative geometry. arXiv:1512.07363 [math.AG]

  35. On membranes and quivers. to appear

  36. Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in \(N=2\) supersymmetric Yang–Mills theory. Nucl. Phys. B 426, 19 (1994). Erratum: [Nucl. Phys. B 430 (1994) 485] [hep-th/9407087]

    Article  ADS  MathSciNet  Google Scholar 

  37. Seiberg, N., Witten, E.: Gauge dynamics and compactification to three-dimensions. arXiv:hep-th/9607163

  38. Szendröi, B.: Noncommutative Donaldson–Thomas theory and the conifold. Ó Geom. Topol. 12, 1171–1202 (2008). arXiv:0705.3419 [math.AG]

    Article  MathSciNet  Google Scholar 

  39. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  40. Witten, E.: Fivebranes and knots. Quantum Topol. 3, 1–137 (2012). arXiv:1101.3216 [hep-th]

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I am grateful to Michele del Zotto, Vivek Shende, Yan Soibelman, Vasily Pestun, and Johannes Wälcher for discussions. I am thankful to the organizers of the program Symplectic Geometry and Representation Theory at the Hausdorff Institute for Mathematics in Bonn for the warm hospitality during the last stages of this project. These results were presented at the workshops Geometry and Topology inspired by Physics in 2018 in Ascona and Young Researchers in String Mathematics in 2017 in Bonn, and I am grateful to the organizers for the invitation to speak and for the warm hospitality. I am a member of INDAM-GNFM, I am supported by INFN via the Iniziativa Specifica GAST and by the FRA2018 project “K-theoretic Enumerative Geometry in Mathematical Physics”.

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Cirafici, M. Quantum line defects and refined BPS spectra. Lett Math Phys 110, 501–531 (2020). https://doi.org/10.1007/s11005-019-01226-3

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