Abstract
We count Higgs “phase” BPS states of general non-Abelian quiver, possibly with loops, by mapping the problem to its Abelian, or toric, counterpart and imposing Weyl invariance later. Precise Higgs index computation is particularly important for quivers with superpotentials; the Coulomb “phase” index is recently shown to miss important BPS states, dubbed intrinsic Higgs states or quiver invariants. We demonstrate how the refined Higgs index is naturally decomposed to a sum over partitions of the charge. We conjecture, and show in simple cases, that this decomposition expresses the Higgs index as a sum over a set of partition-induced Abelian quivers of the same total charge but generically of smaller rank. Unlike the previous approach inspired by a similar decomposition of the Coulomb index, our formulae compute the quiver invariants directly, and thus offer a self-complete routine for counting BPS states.
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References
F. Denef, Quantum quivers and Hall/hole halos, JHEP 10 (2002) 023 [hep-th/0206072] [INSPIRE].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].
A. Sen, Equivalence of three wall-crossing formulae, Commun. Num. Theor. Phys. 6 (2012) 601 [arXiv:1112.2515] [INSPIRE].
H. Kim, J. Park, Z. Wang and P. Yi, Ab Initio Wall-Crossing, JHEP 09 (2011) 079 [arXiv:1107.0723] [INSPIRE].
K.-M. Lee and P. Yi, Dyons in N = 4 supersymmetric theories and three pronged strings, Phys. Rev. D 58 (1998) 066005 [hep-th/9804174] [INSPIRE].
J.P. Gauntlett, N. Kim, J. Park and P. Yi, Monopole dynamics and BPS dyons N = 2 super Yang-Mills theories, Phys. Rev. D 61 (2000) 125012 [hep-th/9912082] [INSPIRE].
M. Stern and P. Yi, Counting Yang-Mills dyons with index theorems, Phys. Rev. D 62 (2000) 125006 [hep-th/0005275] [INSPIRE].
F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050 [hep-th/0005049] [INSPIRE].
A. Ritz, M.A. Shifman, A.I. Vainshtein and M.B. Voloshin, Marginal stability and the metamorphosis of BPS states, Phys. Rev. D 63 (2001) 065018 [hep-th/0006028] [INSPIRE].
P.C. Argyres and K. Narayan, String webs from field theory, JHEP 03 (2001) 047 [hep-th/0101114] [INSPIRE].
S. Lee and P. Yi, Framed BPS States, Moduli Dynamics and Wall-Crossing, JHEP 04 (2011) 098 [arXiv:1102.1729] [INSPIRE].
J. Manschot, B. Pioline and A. Sen, Wall Crossing from Boltzmann Black Hole Halos, JHEP 07 (2011) 059 [arXiv:1011.1258] [INSPIRE].
J. Manschot, B. Pioline and A. Sen, A Fixed point formula for the index of multi-centered N = 2 black holes, JHEP 05 (2011) 057 [arXiv:1103.1887] [INSPIRE].
M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, arXiv:1006.0146 [INSPIRE].
S.-J. Lee, Z.-L. Wang and P. Yi, BPS States, Refined Indices and Quiver Invariants, JHEP 10 (2012) 094 [arXiv:1207.0821] [INSPIRE].
S.-J. Lee, Z.-L. Wang and P. Yi, Quiver Invariants from Intrinsic Higgs States, JHEP 07 (2012) 169 [arXiv:1205.6511] [INSPIRE].
M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Inventiones Mathematicae 152 (2003) 349 [math/0204059].
M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi et al., N = 2 Quantum Field Theories and Their BPS Quivers, arXiv:1112.3984 [INSPIRE].
A. Bertram, I. Ciocan-Fontanine and B.-s. Kim, Two proofs of a conjecture of Hori and Vafa, math/0304403 [INSPIRE].
A. Bertram, I. Ciocan-Fontanine and B. Kim, Gromov-Witten Invariants for Abelian and Nonabelian Quotients, math/0407254.
I. Ciocan-Fontanine, B. Kim and C. Sabbah, The Abelian/Nonabelian correspondence and Frobenius manifolds, Inventiones Mathematicae 171 (2007) 301 [math/0610265].
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978.
S. Martin, Symplectic quotients by a nonAbelian group and by its maximal torus, Annals Math. (1999) [math/0001002] [INSPIRE].
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 1966.
L. Hille, Toric Quiver Varieties, Canadian Math. Soc. Proceedings 24 (1998) 311.
H. Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009) 1016 [arXiv:0903.1463].
J. Manschot, B. Pioline and A. Sen, From Black Holes to Quivers, JHEP 11 (2012) 023 [arXiv:1207.2230] [INSPIRE].
J. Manschot, B. Pioline and A. Sen, On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants, JHEP 05 (2013) 166 [arXiv:1302.5498] [INSPIRE].
I. Bena, M. Berkooz, J. de Boer, S. El-Showk and D. Van den Bleeken, Scaling BPS Solutions and pure-Higgs States, JHEP 11 (2012) 171 [arXiv:1205.5023] [INSPIRE].
W. Fulton, Introduction to Toric Varieties, Princeton University Press, 1993.
T. Oda, Convex Bodies and Algebraic Geometry, Springer-Verlag, 1988.
D. Cox, Recent Developments in Toric Geometry, arXiv:alg-geom/9606016v1.
D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, American Mathematical Society, 1999.
F. Denef, Les Houches Lectures on Constructing String Vacua, arXiv:0803.1194 [INSPIRE].
J. Manschot, B. Pioline and A. Sen, Generalized quiver mutations and single-centered indices, JHEP 01 (2014) 050 [arXiv:1309.7053] [INSPIRE].
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Lee, SJ., Wang, ZL. & Yi, P. Abelianization of BPS quivers and the refined Higgs index. J. High Energ. Phys. 2014, 47 (2014). https://doi.org/10.1007/JHEP02(2014)047
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DOI: https://doi.org/10.1007/JHEP02(2014)047