Abstract
We consider a special class of N=4 quiver quantum mechanics relevant in the description of BPS states of D4D0 branes in type II Calabi-Yau compactifications and the corresponding 4-dimensional black holes. These quivers have two abelian nodes in addition to an arbitrary number of non-abelian nodes and satisfy some simple but stringent conditions on the set of arrows, in particular closed oriented loops are always present. The Higgs branch can be described as the vanishing locus of a section of a vector bundle over a product of a projective space with a number of Grassmannians. The Lefschetz-Sommese theorem then allows to separate induced from intrinsic cohomology which leads to the notion of pure-Higgs states. We compute explicit formulae for an index counting these pure-Higgs states and prove — for this special class of quivers — some previously stated conjectures about them.
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References
F. Denef, Quantum quivers and Hall/hole halos, JHEP 10 (2002) 023 [hep-th/0206072] [INSPIRE].
S. Cecotti and M. Del Zotto, The BPS spectrum of the 4d N = 2 SCFT’s H1, H2, D4, E6, E7, E8, JHEP 06 (2013) 075 [arXiv:1304.0614] [INSPIRE].
B. Pioline, Corfu lectures on wall-crossing, multi-centered black holes, and quiver invariants, PoS(Corfu2012)085 (2013) [arXiv:1304.7159] [INSPIRE].
J. Manschot, B. Pioline and A. Sen, The Coulomb branch formula for quiver moduli spaces, arXiv:1404.7154 [INSPIRE].
G. Beaujard, S. Mondal and B. Pioline, Quiver indices and Abelianization from Jeffrey-Kirwan residues, JHEP 10 (2019) 184 [arXiv:1907.01354] [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten index and wall crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].
K. Ohta and Y. Sasai, Exact results in quiver quantum mechanics and BPS bound state counting, JHEP 11 (2014) 123 [arXiv:1408.0582] [INSPIRE].
C. Cordova and S.-H. Shao, An index formula for supersymmetric quantum mechanics, arXiv:1406.7853 [INSPIRE].
J. Manschot, B. Pioline and A. Sen, Generalized quiver mutations and single-centered indices, JHEP 01 (2014) 050 [arXiv:1309.7053] [INSPIRE].
H. Kim, S.-J. Lee and P. Yi, Mutation, Witten index, and quiver invariant, JHEP 07 (2015) 093 [arXiv:1504.00068] [INSPIRE].
S.-J. Lee, Z.-L. Wang and P. Yi, Abelianization of BPS quivers and the refined Higgs index, JHEP 02 (2014) 047 [arXiv:1310.1265] [INSPIRE].
S. Alexandrov and B. Pioline, Attractor flow trees, BPS indices and quivers, Adv. Theor. Math. Phys. 23 (2019) 627 [arXiv:1804.06928] [INSPIRE].
F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050 [hep-th/0005049] [INSPIRE].
B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole composites, JHEP 11 (2011) 127 [hep-th/0304094] [INSPIRE].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [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].
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].
A. Chowdhury, R.S. Garavuso, S. Mondal and A. Sen, BPS state counting in N = 8 supersymmetric string theory for pure D-brane configurations, JHEP 10 (2014) 186 [arXiv:1405.0412] [INSPIRE].
A. Chowdhury, R.S. Garavuso, S. Mondal and A. Sen, Do all BPS black hole microstates carry zero angular momentum?, JHEP 04 (2016) 082 [arXiv:1511.06978] [INSPIRE].
P. Heidmann and S. Mondal, The full space of BPS multicenter states with pure D-brane charges, JHEP 06 (2019) 011 [arXiv:1810.10019] [INSPIRE].
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].
R. Lazarsfeld, Positivity in algebraic geometry II, Springer, Berlin, Germany (2004).
I. Bena, E. Martinec, D. Turton and N.P. Warner, M-theory superstrata and the MSW string, JHEP 06 (2017) 137 [arXiv:1703.10171] [INSPIRE].
M. Shigemori, Superstrata, Gen. Rel. Grav. 52 (2020) 51 [arXiv:2002.01592] [INSPIRE].
J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].
J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, A bound on the entropy of supergravity?, JHEP 02 (2010) 062 [arXiv:0906.0011] [INSPIRE].
E.J. Martinec, S. Massai and D. Turton, Little strings, long strings, and fuzzballs, JHEP 11 (2019) 019 [arXiv:1906.11473] [INSPIRE].
J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, Quantizing N = 2 multicenter solutions, JHEP 05 (2009) 002 [arXiv:0807.4556] [INSPIRE].
J. de Boer, F. Denef, S. El-Showk, I. Messamah and D. Van den Bleeken, Black hole bound states in AdS3 × S2 , JHEP 11 (2008) 050 [arXiv:0802.2257] [INSPIRE].
B. Pioline, Mathematica package CoulombHiggs, https://www.lpthe.jussieu.fr/∼pioline/computing.html.
I. Bena, C.-W. Wang and N.P. Warner, Mergers and typical black hole microstates, JHEP 11 (2006) 042 [hep-th/0608217] [INSPIRE].
I. Bena, C.-W. Wang and N.P. Warner, Plumbing the abyss: black ring microstates, JHEP 07 (2008) 019 [arXiv:0706.3786] [INSPIRE].
J. de Boer and M. Shigemori, Exotic branes in string theory, Phys. Rept. 532 (2013) 65 [arXiv:1209.6056] [INSPIRE].
A. Borel and H. Friedrich, Characteristic classes and homogeneous spaces, I, Amer. J. Math. 80 (1958) 458.
P.A. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, NY, U.S.A. (1994).
F. Hirzebruch, R. Hirzebruch, R. Schwarzenberger and A. Borel, Topological methods in algebraic geometry, Springer, Berlin, Germany (1966).
I. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, U.K. (1998).
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ArXiv ePrint: 2008.05432
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Messamah, I., Van den Bleeken, D. Pure-Higgs states from the Lefschetz-Sommese theorem. J. High Energ. Phys. 2020, 161 (2020). https://doi.org/10.1007/JHEP11(2020)161
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DOI: https://doi.org/10.1007/JHEP11(2020)161