With cedram.org
Confluentes Mathematici
Search for an article
Search within the site
Table of contents for this issue | Previous article | Next article
Jan Manschot; Boris Pioline; Ashoke Sen
The Coulomb Branch Formula for Quiver Moduli Spaces
Confluentes Mathematici, 9 no. 2 (2017), p. 49-69, doi: 10.5802/cml.41
Article PDF
Class. Math.: 16G20, 37P45, 81T60, 83E50
Keywords: representations of quivers, moduli spaces, quiver quantum mechanics, bound states

Résumé - Abstract

In recent series of works, by translating properties of multi-centered supersymmetric black holes into the language of quiver representations, we proposed a formula that expresses the Hodge numbers of the moduli space of semi-stable representations of quivers with generic superpotential in terms of a set of invariants associated to ‘single-centered’ or ‘pure-Higgs’ states. The distinguishing feature of these invariants is that they are independent of the choice of stability condition. Furthermore they are uniquely determined by the $\chi _{y}$-genus of the moduli space. Here, we provide a self-contained summary of the Coulomb branch formula, spelling out mathematical details but leaving out proofs and physical motivations.


[1] M. R. Douglas and G. W. Moore, D-branes, quivers, and ALE instantons. arXiv:hep-th/9603167.
[2] F. Denef, Quantum quivers and Hall / hole halos, J. High En. Phys. 0210:023, 2002. arXiv:hep-th/0206072.
[3] H. Derksen and J. Weyman, Quiver representations, Not. Amer. Math. Soc. 52:200, 2005.
[4] M. Reineke, Moduli of representations of quivers, Proc. ICRA XII, Toruń, Poland, August 15–24, 2007. arXiv:0802.2147.
[5] A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxf. II. Ser. 45:515–530, 1994.
[6] S. J. Lee, Z. L. Wang and P. Yi, Abelianization of BPS Quivers and the Refined Higgs Index, J. High En. Phys. 1402:047, 2014. arXiv:1310.1265.
[7] D. Joyce, Configurations in Abelian categories. IV. Invariants and changing stability conditions, Adv. Math. 217:125-204, 2008. arXiv:math.AG/0410268.
[8] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. arXiv:0811.2435.
[9] J. Manschot, B. Pioline, and A. Sen, A Fixed point formula for the index of multi-centered N=2 black holes, J. High En. Phys. 1105:057, 2011. arXiv:1103.1887.
[10] J. Manschot, B. Pioline, and A. Sen, From Black Holes to Quivers, J. High En. Phys. 1211:023, 2012. arXiv:1207.2230.
[11] J. Manschot, B. Pioline, and A. Sen, On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants, J. High En. Phys. 1305:166, 2013. arXiv:1302.5498.
[12] B. Pioline, Corfu lectures on wall-crossing, multi-centered black holes, and quiver invariants, PoS Corfu 2012:085, 2013. arXiv:1304.7159.
[13] J. Manschot, Quivers and BPS bound states, Lectures at the Winter School in Mathematical Physics, Les Diablerets, January 12–17, 2014.
[14] J. Manschot, B. Pioline, and A. Sen, Wall Crossing from Boltzmann Black Hole Halos, J. High En. Phys. 1107:059, 2011. arXiv:1011.1258.
[15] J. Manschot, B. Pioline and A. Sen, unpublished.
[16] F. Denef and G. W. Moore, Split states, entropy enigmas, holes and halos, J. High En. Phys. 1111:129, 2011. arXiv:hep-th/0702146.
[17] S. Mozgovoy, M. Reineke, Abelian quiver invariants and marginal wall-crossing", Lett. Math. Phys. 104495-525, 2014. arXiv:1212.0410.
[18] M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli., Invent. Math. 152(2):349–368, 2003.
[19] B. Pioline, Four ways across the wall, J. Phys. Conf. Ser. 346:012017, 2012. arXiv:1103.0261.
[20] H. Kim, J. Park, Z. Wang and P. Yi, Ab Initio Wall-Crossing, J. High En. Phys. 1109:079, 2011. arXiv:1107.0723.
[21] M. Reineke, J. Stoppa, T. Weist, MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence", Geom. Topol. 16:2097–2134, 2012. arXiv:1011.1258.
[22] A. Sen, Equivalence of Three Wall Crossing Formulae, Comm. Numb. Phys. 6:601–659, 2012. arXiv:1112.2515.
[23] S.-J. Lee, Z.-L. Wang, and P. Yi, BPS States, Refined Indices, and Quiver Invariants, J. High En. Phys. 1210:094, 2012. arXiv:1207.0821.
[24] I. Bena, M. Berkooz, J. de Boer, S. El-Showk, and D. Van den Bleeken, Scaling BPS Solutions and pure-Higgs States, J. High En. Phys. 1211:171, 2012. arXiv:1205.5023.
[25] S.-J. Lee, Z.-L. Wang, and P. Yi, Quiver Invariants from Intrinsic Higgs States, J. High En. Phys. 1207:169, 2012. arXiv:1205.6511.
[26] J. Manschot, B. Pioline and A. Sen, Generalized quiver mutations and single-centered indices, J. High En. Phys. 1401:050, 2014. arXiv:1309.7053.
[27] H. Derksen, J. Weyman, and A. Zelevinsky, Quivers with potentials and their representations. I: Mutations., Sel. Math., New Ser. 14(1):59–119, 2008.
[28] B. Keller and D. Yang, Derived equivalences from mutations of quivers with potential., Adv. Math. 226(3):2118–2168, 2011.
[29] S. Mukhopadhyay and K. Ray, Seiberg duality as derived equivalence for some quiver gauge theories, J. High En. Phys. 0402:070, 2004. arXiv:hep-th/0309191.
[30] D. Gaiotto, G. W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17:241–397, 2013. arXiv:1006.0146.
[31] C. Córdova and A. Neitzke, Line Defects, Tropicalization, and Multi-Centered Quiver Quantum Mechanics, J. High En. Phys. 1409:099, 2014. arXiv:1308.6829.
[32] C. Córdova and S. H. Shao, An Index Formula for Supersymmetric Quantum Mechanics. arXiv:1406.7853.
[33] K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, J. High En. Phys. 1501:124, 2015. arXiv:1407.2567.
eISSN : 1793-7434