- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Macdonald index from a refined Kontsevich-Soibelman operator

George Andrews*

Anindya Banerjee†

Ranveer Kumar Singh‡ and Runkai Tao§

  • NHETC and Department of Physics and Astronomy, Rutgers University, 126 Frelinghuysen Road, Piscataway, New Jersey 08855, USA

  • *Contact author: gea1@psu.edu
  • †Contact author: ab1702@scarletmail.rutgers.edu
  • ‡Contact author: rks158@scarletmail.rutgers.edu
  • §Contact author: runkai.tao@physics.rutgers.edu
PDF

Phys. Rev. D 113, 105011 – Published 18 May, 2026

DOI: https://doi.org/10.1103/gv2h-8hs8

Abstract

We propose a refinement of the Kontsevich-Soibelman operator for a class of four-dimensional N=2 superconformal field theories characterized by the following conditions: (1) Their Coulomb branch admits a source/sink chamber, i.e., a chamber in which the Bogomol’nyi-Prasad-Sommerfield (BPS) quiver consists of only source and sink nodes. (2) The nodes with valency greater than 2 of the BPS quiver in a source/sink chamber are either all sources or all sinks. We present strong evidence that the trace of this refined operator is related to the Macdonald index of the theory. In particular, we conjecture new closed form expressions for the Macdonald indices of the (A1,g) Argyres-Douglas theories for any simply laced Lie algebra g.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (30)

  1. N. Seiberg and E. Witten, Nucl. Phys. B426, 19 (1994).
  2. N. Seiberg and E. Witten, Nucl. Phys. B431, 484 (1994).

  3. The original Kontsevich-Soibelman operator was introduced in context of Donaldson-Thomas invariants in [4] and used in context of 4D N=2 theories in [5] to study wall crossing.

  4. M. Kontsevich and Y. Soibelman, arXiv:0811.2435.
  5. D. Gaiotto, G. W. Moore, and A. Neitzke, Commun. Math. Phys. 299, 163 (2010).
  6. T. Dimofte and S. Gukov, Lett. Math. Phys. 91, 1 (2010).
  7. T. Dimofte, S. Gukov, and Y. Soibelman, Lett. Math. Phys. 95, 1 (2011).
  8. S. Cecotti, A. Neitzke, and C. Vafa, arXiv:1006.3435.
  9. S. Cecotti, J. Song, C. Vafa, and W. Yan, J. High Energy Phys. 11 (2017) 013.
  10. C. Cordova and S.-H. Shao, J. High Energy Phys. 01 (2016) 040.
  11. C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees, Commun. Math. Phys. 336, 1359 (2015).
  12. A. Gadde, L. Rastelli, S. S. Razamat, and W. Yan, Commun. Math. Phys. 319, 147 (2013).
  13. M. Buican and T. Nishinaka, J. High Energy Phys. 02 (2016) 159.
  14. C. Bhargava, M. Buican, and H. Jiang, Phys. Rev. Lett. 132, 031602 (2024).
  15. J. Song, J. High Energy Phys. 08 (2017) 044.
  16. G. Andrews, A. Banerjee, C. Bhargava, R. K. Singh, and R. Tao, arXiv:2507.06294.
  17. M. J. Kang, C. Lawrie, and J. Song, arXiv:2507.12510.
  18. D. Gaiotto, G. W. Moore, and A. Neitzke, Adv. Theor. Math. Phys. 17, 241 (2013).
  19. F. Denef, J. High Energy Phys. 10 (2002) 023.
  20. F. Denef and G. W. Moore, J. High Energy Phys. 11 (2011) 129.
  21. J. Manschot, B. Pioline, and A. Sen, J. High Energy Phys. 07 (2011) 059.
  22. J. Manschot, B. Pioline, and A. Sen, J. High Energy Phys. 11 (2012) 023.
  23. M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi, and C. Vafa, Commun. Math. Phys. 323, 1185 (2013).

  24. A node with all arrows emanating from it is called a source node and a node with all arrows ending on it is called a sink node.

  25. J. Song, J. High Energy Phys. 02 (2016) 045.
  26. Y. Wang and D. Xie, Phys. Rev. D 94, 065012 (2016).
  27. A. Watanabe and R.-D. Zhu, J. High Energy Phys. 02 (2020) 004.
  28. P. Agarwal, S. Lee, and J. Song, J. High Energy Phys. 06 (2019) 102.
  29. H. Kim, H. Kim, and J. Song, J. High Energy Phys. 03 (2026) 213.
  30. G. Andrews, The Theory of Partitions, Cambridge Mathematical Library (Cambridge University Press, Cambridge, England, 1998).

Outline

Information

Phys. Rev. D 113, 105011– Published 18 May, 2026
  • Received 20 November 2025
  • Accepted 10 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation