Some open questions in quiver gauge theory
DOI:
https://doi.org/10.22199/issn.0717-6279-5274Keywords:
quivers, gauge theory, sypersymmetry, branesAbstract
Quivers, gauge theories and singular geometries are of great interest in both mathematics and physics. In this note, we collect a few open questions which have arisen in various recent works at the intersection between gauge theories, representation theory, and algebraic geometry. The questions originate from the study of supersymmetric gauge theories in different dimensions with different supersymmetries. Although these constitute merely the tip of a vast iceberg, we hope this guide can give a hint of possible directions in future research.
References
H. Nakajima, “Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras”, Duke Mathematical Journal, vol. 76, no. 2, pp. 365-416, 1994.
M. R. Douglas and G. W. Moore, “D-branes, quivers, and ALE instantons”, 1996, arXiv: hep-th/9603167.
A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling, and Z. Zhong, “Branes, Quivers, and the Affine Grassmannian”, 2021, arXiv:2102.06190.
S. Cabrera, A. Hanany, and M. Sperling, “Magnetic quivers, Higgs branches, and 6d N = (1, 0) theories orthogonal and symplectic gauge groups”, 2020, arXiv:1912.02773.
S. Cremonesi, A. Hanany, and A. Zaffaroni, “Monopole operators and Hilbert series of Coulomb branches of 3dN = 4 gauge theories”, 2014, arXiv:1309.2657v3.
D. Martelli, J. Sparks, and S.-T. Yau, “The Geometric dual of amaximisation for Toric Sasaki-Einstein manifolds”, 2006, arXiv:hep-th/0503183v3.
D. Martelli, J. Sparks, and S.-T. Yau, “Sasaki-Einstein manifolds and volume minimization”, 2008, arXiv:hep-th/0603021v5.
L. E. Cowie, H.-C. Herbig, D. Herden, and C. Seaton, “The Hilbert series and a-invariant of circle invariants”, 2017, arXiv:1707.03128v1.
H.-C. Herbig, D. Herden, and C. Seaton, “The laurent coefficients of the hilbert series of a gorenstein algebra”, 2017, arXiv:1605.01572v3.
T. C. Collins and G. Szekelyhidi, “K-Semistability for irregular Sasakian manifolds”, 2012, arXiv:1204.2230v1.
T. Collins and G. Szekelyhidi, “Sasakieinstein metrics and K-stability”, 2017, arXiv:1512.07213v2.
S. Cecotti and M. Del Zotto, “4d N=2 Gauge Theories and Quivers: the Non-Simply Laced Case”, 2012, arXiv:1207.7205v1.
S. Cremonesi, G. Ferlito, A. Hanany, and N. Mekareeya, “Coulomb Branch and The Moduli Space of Instantons”, 2014, arXiv:1408.6835v2.
H.-Y. Chen and T. Kimura, “Quantum integrability from nonsimply laced quiver gauge theory”, 2018, arXiv:1805.01308v1.
A. Hanany and A. Zajac, “Ungauging Schemes and Coulomb Branches of Non-simply Laced Quiver Theories”, 2020, arXiv:2002.05716v3.
A. Bourget, J. F. Grimminger, A. Hanany, R. Kalveks, M. Sperling, and Z. Zhong, “Folding Orthosymplectic Quivers”, 2021, arXiv:2107.00754v1.
A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling, and Z. Zhong, “Magnetic Quivers from Brane Webs with O5 Planes”, 2020, arXiv:2004.04082v2.
S. Cremonesi, A. Hanany, N. Mekareeya, and A. Zaffaroni, “Coulomb branch Hilbert series and Hall-Littlewood polynomials”, 2014, arXiv:1403.0585v2.
A. Hanany and R. Kalveks, “Highest Weight Generating Functions for Hilbert Series”, 2014, arXiv:1408.4690v2.
A. Hanany and N. Mekareeya, “The small E₈ instanton and the Kraft Procesi transition”, 2018, arXiv:1801.01129v3.
S. Cabrera and A. Hanany, “Branes and the Kraft-Procesi Transition”, 2016, arXiv:1609.07798v2.
S. Cabrera and A. Hanany, “Branes and the Kraft-Procesi transition: classical case”, 2018, arXiv:1711.02378v2.
P. Koroteev and A. M. Zeitlin, “3d mirror symmetry for instanton moduli spaces”, 2021, arXiv:2105.00588v2.
S. Cabrera and A. Hanany, “Quiver Subtractions”, 2018, arXiv:1803.11205v1.
A. Hanany and G. Zafrir, “Discrete Gauging in Six Dimensions”, 2018, arXiv:1804.08857v1.
A. Hanany and M. Sperling, “Discrete quotients of 3-dimensional N =4 Coulomb branches via the cycle index”, 2018, arXiv:1807.02784v2.
A. Hanany and A. Zajac, “Discrete Gauging in Coulomb branches of Three Dimensional N = 4 Supersymmetric Gauge Theories”, 2018, arXiv:1807.03221v2.
A. Bourget, A. Hanany, and D. Miketa, “Quiver origami: discrete gauging and folding”, 2020, arXiv:2005.05273v2.
S. Cabrera, A. Hanany, and M. Sperling, “Magnetic quivers, Higgs branches, and 6d N=(1,0) theories”, arXiv:1904.12293v2.
A. Bourget, S. Cabrera, J. F. Grimminger, A. Hanany, M. Sperling, A. Zajac, and Z. Zhong, “The Higgs mechanism Hasse diagrams for symplectic singularities”, arXiv:1908.04245v2.
A. Bourget, S. Cabrera, J. F. Grimminger, A. Hanany, and Z. Zhong, “Brane Webs and Magnetic Quivers for SQCD”, 2020, arXiv:1909.00667v2.
J. F. Grimminger and A. Hanany, “Hasse diagrams for 3d N = 4 quiver gauge theories Inversion and the full moduli space”, 2020, arXiv:2004.01675v2.
A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling, G. Zafrir, and Z. Zhong, “Magnetic quivers for rank 1 theories”, 2020, arXiv:2006.16994v1.
A. Bourget, S. Giacomelli, J. F. Grimminger, A. Hanany, M. Sperling, and Z. Zhong, “S-fold magnetic quivers”, 2020, arXiv:2010.05889v1.
T. Braden, A. Licata, N. Proudfoot, and B.Webster, “Quantizations of conical symplectic resolutions ii: category O and symplectic duality”, 2016, arXiv:1407.0964v4.
H. Nakajima, “Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, I”, 2016, arXiv:1503.03676v2.
H. Nakajima, “Questions on provisional coulomb branches of 3-dimensional N = 4 gauge theories”, arXiv:1510.03908v2.
A. Braverman, M. Finkelberg, and H. Nakajima, “Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II”, 2019, arXiv:1601.03586v7.
B. Webster, “Koszul duality between Higgs and Coulomb categories O”, 2019, arXiv:1611.06541v4.
M. Bullimore, T. Dimofte, D. Gaiotto, and J. Hilburn, “Boundaries, Mirror Symmetry, and Symplectic Duality in 3d N = 4 Gauge Theory”, 2016, arXiv:1603.08382v3.
A. Balasubramanian and J. Distler, “Masses, Sheets and Rigid SCFTs”, 2020, arXiv:1810.10652v2.
A. Dancer, A. Hanany, and F. Kirwan, “Symplectic duality and implosions”, arXiv:2004.09620v1.
A. Bourget, A. Dancer, J. F. Grimminger, A. Hanany, F. Kirwan, and Z. Zhong, “Orthosymplectic Implosions”, 2021, arXiv:2103.05458v1.
H. Nakajima, “Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories”, 2017, arXiv:1706.05154v1.
H. Nakajima and A. Weekes, “Coulomb branches of quiver gauge theories with symmetrizers”, 2020, arXiv:1907.06552v2.
D. Xie and S.-T. Yau, “Three dimensional canonical singularity and five dimensional N = 1 SCFT”, 2017, arXiv:1704.00799v1.
C. Closset, S. Schafer-Nameki, and Y.-N. Wang, “Coulomb and Higgs Branches from Canonical Singularities: Part 0”, 2021, arXiv:2007.15600v5.
M. van Beest, A. Bourget, J. Eckhard, and S. Schafer-Nameki, “(Sym- plectic) Leaves and (5d Higgs) Branches in the Poly(go)nesian Tropical Rain Forest”, 2020, arXiv:2008.05577v2.
M. Van Beest, A. Bourget, J. Eckhard, and S. Schafer-Nameki, “(5d RG-flow) Trees in the Tropical Rain Forest”, 2020, arXiv:2011.07033v2.
F. Benini, S. Benvenuti, and Y. Tachikawa, “Webs of five-branes and N = 2 superconformal field theories”, 2009, arXiv:0906.0359v3.
O. DeWolfe, T. Hauer, A. Iqbal, and B. Zwiebach, “Uncovering the symmetries on [p,q] seven-branes: Beyond the Kodaira classification”, 1999, arXiv:hep-th/9812028v2.
O. Bergman and D. Rodríguez-Gomez, “The Cats Cradle: deforming the higher rank E₁ and Ē₁ theories”, 2020, arXiv:2011.05125v1.
A. Braverman, M. Finkelberg, and H. Nakajima, “Coulomb branches of N = 4 quiver gauge theories and slices in the affine Grassmannian”, 2018, arXiv:1604.03625v5.
T. Richarz, “Affine grassmannians and geometric satake equivalences”, 2015, arXiv:1311.1008v2.
D. Gaiotto and E. Witten, “S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory”, 2008, arXiv:0807.3720v1.
A. Hanany and R. Kalveks, “Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits”, 2017, arXiv:1601.04020v2.
A. Hanany and R. Kalveks, “Quiver Theories and Formulae for Nilpotent Orbits of Exceptional Algebras”, 2017, arXiv:1709.05818v1.
R. Eager, I. Saberi, and J. Walcher, “Nilpotence varieties”, 2018, arXiv:1807.03766v1.
B. Feng, A. Hanany, and Y.-H. He, “D-brane gauge theories from toric singularities and toric duality”, 2000, arXiv:hep-th/0003085v2.
Y.-H. He, “D-brane gauge theory and toric singularities”, International Journal of Modern Physics A, vol. 16, supp. 01c, pp. 981-983, 2001.
B. Feng, Y.-H. He, and F. Lam, “On correspondences between toric singularities and (p,q) webs”, 2004, arXiv:hep-th/0403133v1.
B. Feng, A. Hanany, Y.-H. He, and A. M. Uranga, “Toric duality as Seiberg duality and brane diamonds”, 2002, arXiv:hep-th/0109063v3.
A. Hanany and K. D. Kennaway, “Dimer models and toric diagrams”, 2005, arXiv:hep-th/0503149v2.
S. Franco, A. Hanany, K. D. Kennaway, D. Vegh, and B. Wecht, “Brane dimers and quiver gauge theories”, 2005, arXiv:hep-th/0504110v2.
M. Bianchi, U. Bruzzo, P. Fre, and D. Martelli, “Resolution a la Kro- nheimer of ℂ³/Г singularities and the Monge-Ampere equation for Ricci-flat Kaehler metrics in view of D3-brane solutions of supergravity”, 2021, arXiv:2105.11704v1.
R. Eager, “Brane Tilings and Non-Commutative Geometry”, 2010, arXiv:1003.2862v2.
A. Craw, “Quiver representations in toric geometry”, 2008, arXiv:0807.2191v1.
N. Broomhead, “Dimer models and calabi-yau algebras”, 2010, arXiv:0901.4662v2.
R. Bocklandt, “A dimer ABC”, 2015, arXiv:1510.04242v1.
F. Cachazo, S. Katz, and C. Vafa, “Geometric transitions and N=1 quiver theories”, 2001, arXiv:hep-th/0108120v2.
F. Cachazo, B. Fiol, K. A. Intriligator, S. Katz, and C. Vafa, “A Geometric unification of dualities”, 2002, arXiv:hep-th/0110028v2.
M. Yamazaki, “Brane Tilings and Their Applications”, 2008, arXiv:0803.4474v2.
S. Gukov and D. Tong, “D-brane probes of special holonomy manifolds, and dynamics of N = 1 three-dimensional gauge theories”, 2002, arXiv:hep-th/0202126v1.
M. Fluder and J. Sparks, “D2-brane Chern-Simons theories: F- maximization = a-maximization”, 2016, arXiv:1507.05817v2.
A. Hanany and A. Zaffaroni, “Tilings, Chern-Simons Theories and M2 Branes”, 2008, arXiv:0808.1244v2.
A. Hanany, D. Vegh, and A. Zaffaroni, “Brane Tilings and M2 Branes”, 2008, arXiv:0809.1440v1.
A. Hanany and Y.-H. He, “Chern-Simons: Fano and Calabi-Yau”, 2009, arXiv:0904.1847v1.
J. Davey, A. Hanany, and J. Pasukonis, “On the Classification of Brane Tilings”, 2009, arXiv:0909.2868v1.
K. A. Intriligator and B. Wecht, “The Exact superconformal R symmetry maximizes a”, 2003, arXiv:hep- th/0304128v3.
J. L. Cardy, “Is There a c Theorem in Four-Dimensions?”, Physics Letters B, vol. 215, no. 4, pp. 749-752, 1988.
Z. Komargodski and A. Schwimmer, “On Renormalization Group Flows in Four Dimensions”, 2011, arXiv:1107.3987v2.
S. S. Gubser, “Einstein manifolds and conformal field theories”, 1998, arXiv:hep-th/9807164v2.
R. Eager, “Equivalence of A-Maximization and Volume Minimization”, 2010, arXiv:1011.1809v1.
A. Butti and A. Zaffaroni, “R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization”, 2005, arXiv:hep-th/0506232v3.
Y.-H. He, R.-K. Seong, and S.-T. Yau, “Calabi-Yau Volumes and Reflexive Polytopes”, 2017, arXiv:1704.03462v1.
J. Bao, G. Beaney Colverd, and Y.-H. He, “Quiver Gauge Theories: Beyond Reflexivity”, 2020, arXiv:2004.05295v5.
V. V. Batyrev, “Toroidal fano 3-folds”, Mathematics of the USSR- Izvestiya, vol 19, no. 1, 1982, pp. 13-25.
B. Nill, “Gorenstein toric fano varieties”, 2004, arXiv:math/0405448v1.
D. R. Gulotta, “Properly ordered dimers, R-charges, and an efficient inverse algorithm”, 2008, arXiv:0807.3012v2.
T. C. Collins, D. Xie, and S.-T. Yau, “K stability and stability of chiral ring”, 2016, arXiv:1606.09260v1.
G. Szekelyhidi, An Introduction to Extremal Kahler Metrics. American Mathematical Society, 2014.
J. Bao, Y.-H. He, and Y. Xiao, “Chiral rings, Futaki invariants, plethystics, and Grobner bases”, 2020, arXiv:2009.02450v3.
N. Ilten and H. Süß, “K-stability for Fano manifolds with Torus action of complexity one”, 2015, arXiv:1507.04442v2.
M. Fazzi and A. Tomasiello, “Holography, Matrix Factorizations and K-stability”, 2020, arXiv:1906.08272v2.
S. Benvenuti and S. Giacomelli, “Supersymmetric gauge theories with decoupled operators and chiral ring stability”, 2017, arXiv:1706.02225v2.
S. Franco and G. Musiker, “Higher Cluster Categories and QFT Dualities”, 2017, arXiv:1711.01270v1.
C. Closset, S. Franco, J. Guo, and A. Hasan, “Graded quivers and B-branes at Calabi-Yau singularities”, 2018, arXiv:1811.07016v1.
S. Franco and A. Hasan, “Graded Quivers, Generalized Dimer Models and Toric Geometry”, 2019, arXiv:1904.07954v1.
S. Franco, A. Hasan, and X. Yu, “On the Classification of Duality Webs for Graded Quivers”, 2020, arXiv:2001.08776v1.
S. Franco and A. Hasan, “Calabi-Yau products: graded quivers for general toric Calabi-Yaus”, 2020, arXiv:2004.13765v1.
R. Eager and I. Saberi, “Holomorphic field theories and Calabi--Yau algebras”, 2018, arXiv:1805.02084v1.
C. Closset, J. Guo, and E. Sharpe, “B-branes and supersymmetric quivers in 2d”, 2018, arXiv:1711.10195v3.
A. Grothendieck, “Esquisse dun programme”, Institut de Mathématiques de Jussieu-Paris Rive Gauche, pp. 1-48 [Online]. Available: https://bit.ly/3IRuPkJ
G. V. Belyĭ, “On Galois extensions of a maximal cyclotomic field”, Mathematics of the USSR-Izvestiya, vol. 14, no. 2, pp. 247–256, 1980. https://doi.org/10.1070/im1980v014n02abeh001096
B. Feng, Y.-H. He, K. D. Kennaway, and C. Vafa, “Dimer models from mirror symmetry and quivering amoebae”, 2005, arXiv:hep-th/0511287v1.
V. Jejjala, S. Ramgoolam, and D. Rodriguez-Gomez, “Toric CFTs, Permutation Triples and Belyi Pairs”, 2011, arXiv:1012.2351v3.
J. J. Heckman, C. Vafa, D. Xie, and M. Yamazaki, “String Theory Origin of Bipartite SCFTs”, 2013, arXiv:1211.4587v2.
S. Franco, “Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes”, 2012, arXiv:1207.0807v5.
M. Bianchi, S. Cremonesi, A. Hanany, J. F. Morales, D. Ricci Pacifici, and R.-K. Seong, “Mass-deformed Brane Tilings”, 2014, arXiv:1408.1957v1.
O. Aharony, A. Hanany, and B. Kol, “Webs of (p,q) five-branes, five- dimensional field theories and grid diagrams”, 1997, arXiv:hep-th/9710116v3.
A. Hanany, Y.-H. He, V. Jejjala, J. Pasukonis, S. Ramgoolam, and D. Rodriguez-Gomez, “The Beta Ansatz: A Tale of Two Complex Structures”, 2011, arXiv:1104.5490v1.
A. Hanany and D. Vegh, “Quivers, tilings, branes and rhombi”, Journal of High Energy Physics, vol. 2007, no. 10, 2007. http://dx.doi.org/10.1088/1126-6708/2007/10/029.
S. Franco, Y.-H. He, C. Sun, and Y. Xiao, “A Comprehensive Survey of Brane Tilings”, 2017, arXiv:1702.03958v1.
A. Hanany, Y.-H. He, V. Jejjala, J. Pasukonis, S. Ramgoolam, and D. Rodriguez-Gomez, “Invariants of Toric Seiberg Duality”, 2011, arXiv:1107.4101v1.
A. Strominger, S.-T. Yau, and E. Zaslow, “Mirror symmetry is t- duality”, Nuclear Physics B, vol. 479, no. 1-2, pp. 243-259, 1996. https://doi.org/10.1016/0550-3213(96)00434-8.
Y.-H. He, V. Jejjala, and D. Rodriguez-Gomez, “Brane Geometry and Dimer Models”, 2012, arXiv:1204.1065v2.
N. Seiberg, “Electric-magnetic duality in supersymmetric non-abelian gauge theories”, Nuclear Physics B, vol. 435, no. 1-2, pp. 129-146, 1995. https://doi.org/10.1016/0550-3213(94)00023-8.
B. Feng, A. Hanany, Y.-H. He, and A. M. Uranga, “Toric duality as seiberg duality and brane diamonds”, Journal of High Energy Physics, vol. 2001, no. 12, 2001, 035. https://doi.org/bhdbdx.
S. Franco, A. Hanany, and Y.-H. He, “A trio of dualities: walls, trees and cascades”, Fortschritte der Physik. Progress of Physica, vol. 52, no. 6-7, pp. 540-547, 2004. https://doi.org/10.1002/prop.200310142.
S. Franco, A. Hanany, Y.-H. He, and P. Kazakopoulos, “Duality walls, duality trees and fractional branes”, 2003, arXiv:hep-th/0306092v1.
F. Cachazo, N. Seiberg, and E. Witten, “Phases of n = 1 su- persymmetric gauge theories”, Journal of High Energy Physics, vol. 2003, no. 02, 2003, 42. https://doi.org/10.1088/1126-6708/2003/02/042.
S. K. Ashok, F. Cachazo, and E. Dell’Aquila, “Childrens drawings from Seiberg-Witten curves”, 2006, arXiv:hep-th/0611082v1.
Y.-H. He, J. McKay, and J. Read, “Modular subgroups, dessins denfants and elliptic k3 surfaces”, LMS Journal of Computation and Mathematics, vol. 16, pp. 271-318, 2013. https://doi.org/f5hdvt.
Y.-H. He and J. McKay, “N=2 gauge theories: Congru- ence subgroups, coset graphs, and modular surfaces”, Journal of Mathematical Physics, vol. 54, 2013, 012301. https://doi.org/10.1063/1.4772976.
Y.-H. He, “Deep-Learning the Landscape”, 2018, arXiv:1706.02714v3.
D. Krefl and R.-K. Seong, “Machine Learning of Calabi-Yau Volumes”, 2017, arXiv:1706.03346v1.
F. Ruehle, “Evolving neural networks with genetic algorithms to study the String Landscape”, 2017, arXiv:1706.07024v2.
J. Carifio, J. Halverson, D. Krioukov, and B. D. Nelson, “Machine Learning in the String Landscape”, 2017, arXiv:1707.00655v1.
J. Bao, Y.-H. He, E. Hirst, J. Hofscheier, A. Kasprzyk, and S. Majumder, “Hilbert Series, Machine Learning, and Applications to Physics”, 2022, arXiv:2103.13436v3.
J. Bao, Y.-H. He, E. Hirst, J. Hofscheier, A. Kasprzyk, and S. Majumder, “Polytopes and machine learning”, 2021, arXiv:2109.09602v1.
J. Bao, Y.-H. He, and E. Hirst, “Neurons on Amoebae”, 2021, arXiv:2106.03695v1.
J. Bao, S. Franco, Y.-H. He, E. Hirst, G. Musiker, and Y. Xiao, “Quiver Mutations, Seiberg Duality and Machine Learning”, 2020, arXiv:2006.10783v1.
Y.-H. He, E. Hirst, and T. Peterken, “Machine-learning dessins d’Enfants: explorations via modular and Seiberg-Witten curves”, 2021, arXiv:2004.05218v4.
L. Zhang, G. Naitzat, and L.-H. Lim, “Tropical geometry of deep neural networks”, 2018, arXiv:1805.07091v1
E. d. M. Koch, R. de Mello Koch, and L. Cheng, “Is Deep Learning a Renormalization Group Flow?”, 2020, arXiv:1906.05212v2.
J. Halverson, A. Maiti, and K. Stoner, “Neural Networks and Quantum Field Theory”, 2021, arXiv:2008.08601v2.
K. Hashimoto, “AdS/CFT correspondence as a deep Boltzmann machine”, 2019, arXiv:1903.04951v1.
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