The Coulomb Branch Formula for Quiver Moduli Spaces
Confluentes Mathematici, Volume 9 (2017) no. 2, pp. 49-69.

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 χ 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/cml.41
Classification: 16G20,  37P45,  81T60,  83E50
Keywords: representations of quivers, moduli spaces, quiver quantum mechanics, bound states
Jan Manschot 1; Boris Pioline 2, 3; Ashoke Sen 4

1 Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR5208, Institut Camille Jordan, F-69622 Villeurbanne Cedex, France Current address: School of Mathematics, Trinity College, College Green, Dublin 2, Ireland
2 CERN PH-TH, Case C01600, CERN, CH-1211 Geneva 23, Switzerland
3 Sorbonne Universités; CNRS; UPMC Univ. Paris 06, UMR 7589, LPTHE, F-75005, Paris, France
4 Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211019, India
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CML_2017__9_2_49_0,
     author = {Jan Manschot and Boris Pioline and Ashoke Sen},
     title = {The {Coulomb} {Branch} {Formula} for {Quiver} {Moduli} {Spaces}},
     journal = {Confluentes Mathematici},
     pages = {49--69},
     publisher = {Institut Camille Jordan},
     volume = {9},
     number = {2},
     year = {2017},
     doi = {10.5802/cml.41},
     zbl = {1392.16015},
     mrnumber = {3745161},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.5802/cml.41/}
}
TY  - JOUR
TI  - The Coulomb Branch Formula for Quiver Moduli Spaces
JO  - Confluentes Mathematici
PY  - 2017
DA  - 2017///
SP  - 49
EP  - 69
VL  - 9
IS  - 2
PB  - Institut Camille Jordan
UR  - https://cml.centre-mersenne.org/articles/10.5802/cml.41/
UR  - https://zbmath.org/?q=an%3A1392.16015
UR  - https://www.ams.org/mathscinet-getitem?mr=3745161
UR  - https://doi.org/10.5802/cml.41
DO  - 10.5802/cml.41
LA  - en
ID  - CML_2017__9_2_49_0
ER  - 
%0 Journal Article
%T The Coulomb Branch Formula for Quiver Moduli Spaces
%J Confluentes Mathematici
%D 2017
%P 49-69
%V 9
%N 2
%I Institut Camille Jordan
%U https://doi.org/10.5802/cml.41
%R 10.5802/cml.41
%G en
%F CML_2017__9_2_49_0
Jan Manschot; Boris Pioline; Ashoke Sen. The Coulomb Branch Formula for Quiver Moduli Spaces. Confluentes Mathematici, Volume 9 (2017) no. 2, pp. 49-69. doi : 10.5802/cml.41. https://cml.centre-mersenne.org/articles/10.5802/cml.41/

[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. | DOI | MR

[3] H. Derksen and J. Weyman, Quiver representations, Not. Amer. Math. Soc. 52:200, 2005. | Zbl

[4] M. Reineke, Moduli of representations of quivers, Proc. ICRA XII, Toruń, Poland, August 15–24, 2007. arXiv:0802.2147. | DOI | Zbl

[5] A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxf. II. Ser. 45:515–530, 1994. | DOI | MR | Zbl

[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. | DOI

[7] D. Joyce, Configurations in Abelian categories. IV. Invariants and changing stability conditions, Adv. Math. 217:125-204, 2008. | arXiv | DOI | MR | Zbl

[8] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. | arXiv | DOI

[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. | DOI | MR | Zbl

[10] J. Manschot, B. Pioline, and A. Sen, From Black Holes to Quivers, J. High En. Phys. 1211:023, 2012. arXiv:1207.2230. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[12] B. Pioline, Corfu lectures on wall-crossing, multi-centered black holes, and quiver invariants, PoS Corfu 2012:085, 2013. arXiv:1304.7159. | DOI

[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. | DOI | MR | Zbl

[15] J. Manschot, B. Pioline and A. Sen, unpublished. | DOI | MR

[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. | DOI | MR | Zbl

[17] S. Mozgovoy, M. Reineke, Abelian quiver invariants and marginal wall-crossing", Lett. Math. Phys. 104495-525, 2014. arXiv:1212.0410. | DOI | MR | Zbl

[18] M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli., Invent. Math. 152(2):349–368, 2003. | DOI | MR | Zbl

[19] B. Pioline, Four ways across the wall, J. Phys. Conf. Ser. 346:012017, 2012. arXiv:1103.0261. | DOI

[20] H. Kim, J. Park, Z. Wang and P. Yi, Ab Initio Wall-Crossing, J. High En. Phys. 1109:079, 2011. arXiv:1107.0723. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[22] A. Sen, Equivalence of Three Wall Crossing Formulae, Comm. Numb. Phys. 6:601–659, 2012. arXiv:1112.2515. | DOI | Zbl

[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. | DOI | MR

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI

[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. | DOI | MR | Zbl

[28] B. Keller and D. Yang, Derived equivalences from mutations of quivers with potential., Adv. Math. 226(3):2118–2168, 2011. | DOI | MR | Zbl

[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. | DOI

[30] D. Gaiotto, G. W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17:241–397, 2013. arXiv:1006.0146. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[32] C. Córdova and S. H. Shao, An Index Formula for Supersymmetric Quantum Mechanics. arXiv:1406.7853. | DOI | MR | Zbl

[33] K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, J. High En. Phys. 1501:124, 2015. arXiv:1407.2567. | DOI | MR | Zbl

Cited by Sources: