Abstract
We perform a detailed investigation of Bipartite Field Theories (BFTs), a general class of 4d \( \mathcal{N} \) = 1 gauge theories which are defined by bipartite graphs. This class of theories is considerably expanded by identifying a new way of assigning gauge symmetries to graphs. A new procedure is introduced in order to determine the toric Calabi-Yau moduli spaces of BFTs. For graphs on a disk, we show that the matroid polytope for the corresponding cell in the Grassmannian coincides with the toric diagram of the BFT moduli space. A systematic BFT prescription for determining graph reductions is presented. We illustrate our ideas in infinite classes of BFTs and introduce various operations for generating new theories from existing ones. Particular emphasis is given to theories associated to non-planar graphs.
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A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [INSPIRE].
S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane dimers and quiver gauge theories, JHEP 01 (2006) 096 [hep-th/0504110] [INSPIRE].
D. Gaiotto, N=2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [INSPIRE].
I. Bah and B. Wecht, New N = 1 Superconformal Field Theories In Four Dimensions, arXiv:1111.3402 [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, arXiv:1108.4389 [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
S. Franco, Bipartite Field Theories: from D-brane Probes to Scattering Amplitudes, arXiv:1207.0807 [INSPIRE].
D. Xie and M. Yamazaki, Network and Seiberg Duality, JHEP 09 (2012) 036 [arXiv:1207.0811] [INSPIRE].
S. Franco, Dimer Models, Integrable Systems and Quantum Teichmüller Space, JHEP 09 (2011) 057 [arXiv:1105.1777] [INSPIRE].
R. Eager, S. Franco and K. Schaeffer, Dimer Models and Integrable Systems, JHEP 06 (2012) 106 [arXiv:1107.1244] [INSPIRE].
A. Amariti, D. Forcella and A. Mariotti, Integrability on the Master Space, JHEP 06 (2012) 053 [arXiv:1203.1616] [INSPIRE].
S. Franco, D. Galloni and Y.-H. He, Towards the Continuous Limit of Cluster Integrable Systems, JHEP 09 (2012) 020 [arXiv:1203.6067] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605 [INSPIRE].
S. Cecotti and C. Vafa, Classification of complete N = 2 supersymmetric theories in 4 dimensions, Surveys in differential geometry 18 (2013) [arXiv:1103.5832] [INSPIRE].
M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi and C. Vafa, BPS Quivers and Spectra of Complete N = 2 Quantum Field Theories, arXiv:1109.4941 [INSPIRE].
M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi and C. Vafa, N=2 Quantum Field Theories and Their BPS Quivers, arXiv:1112.3984 [INSPIRE].
D. Xie, Network, Cluster coordinates and N = 2 theory I, arXiv:1203.4573 [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
D. Xie, Network, cluster coordinates and N = 2 theory II: irregular singularity, arXiv:1207.6112 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Spectral Networks and Snakes, arXiv:1209.0866 [INSPIRE].
A. Postnikov, Total positivity, Grassmannians and networks, math/0609764.
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].
J.J. Heckman, C. Vafa, D. Xie and M. Yamazaki, String Theory Origin of Bipartite SCFTs, arXiv:1211.4587 [INSPIRE].
S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [INSPIRE].
K.D. Kennaway, Brane Tilings, Int. J. Mod. Phys. A 22 (2007) 2977 [arXiv:0706.1660] [INSPIRE].
A. Ishii and K. Ueda, On moduli spaces of quiver representations associated with dimer models, arXiv:0710.1898.
S. Benvenuti, S. Franco, A. Hanany, D. Martelli and J. Sparks, An Infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals, JHEP 06 (2005) 064 [hep-th/0411264] [INSPIRE].
A. Butti, D. Forcella and A. Zaffaroni, The Dual superconformal theory for L pqr manifolds, JHEP 09 (2005) 018 [hep-th/0505220] [INSPIRE].
D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni, The Master Space of N = 1 Gauge Theories, JHEP 08 (2008) 012 [arXiv:0801.1585] [INSPIRE].
D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni, Mastering the Master Space, Lett. Math. Phys. 85 (2008) 163 [arXiv:0801.3477] [INSPIRE].
A. Hanany and A. Zaffaroni, The master space of supersymmetric gauge theories, Adv. High Energy Phys. 2010 (2010) 427891.
B. Feng, S. Franco, A. Hanany and Y.-H. He, Symmetries of toric duality, JHEP 12 (2002) 076 [hep-th/0205144] [INSPIRE].
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
R. Kenyon, A. Okounkov and S. Sheffield, Dimers and amoebae, math-ph/0311005 [INSPIRE].
S. Franco and D. Vegh, Moduli spaces of gauge theories from dimer models: proof of the correspondence, JHEP 11 (2006) 054 [hep-th/0601063] [INSPIRE].
A. Hanany, Counting BPS operators in the chiral ring: the plethystic story, AIP Conf. Proc. 939 (2007) 165 [INSPIRE].
A. Butti, D. Forcella, A. Hanany, D. Vegh and A. Zaffaroni, Counting Chiral Operators in Quiver Gauge Theories, JHEP 11 (2007) 092 [arXiv:0705.2771] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the Plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].
A. Hanany and D. Vegh, Quivers, tilings, branes and rhombi, JHEP 10 (2007) 029 [hep-th/0511063] [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 [hep-th/0003085] [INSPIRE].
R. Kenyon and J.-M. Schlenker, Rhombic embeddings of planar graphs with faces of degree 4, math-ph/0305057.
B. Feng, A. Hanany and Y.-H. He, Phase structure of D-brane gauge theories and toric duality, JHEP 08 (2001) 040 [hep-th/0104259] [INSPIRE].
B. Feng, A. Hanany, Y.-H. He and A.M. Uranga, Toric duality as Seiberg duality and brane diamonds, JHEP 12 (2001) 035 [hep-th/0109063] [INSPIRE].
C.E. Beasley and M.R. Plesser, Toric duality is Seiberg duality, JHEP 12 (2001) 001 [hep-th/0109053] [INSPIRE].
S. Franco, A. Hanany and Y.-H. He, A Trio of dualities: walls, trees and cascades, Fortsch. Phys. 52 (2004) 540 [hep-th/0312222] [INSPIRE].
B. Feng, S. Franco, A. Hanany and Y.-H. He, UnHiggsing the del Pezzo, JHEP 08 (2003) 058 [hep-th/0209228] [INSPIRE].
A. Postnikov, D. Speyer and L. Williams, Matching polytopes, toric geometry and the non-negative part of the Grassmannian, arXiv:0706.2501.
B. Feng, Y.-H. He, K.D. Kennaway and C. Vafa, Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. 12 (2008) 489 [hep-th/0511287] [INSPIRE].
S. Franco, A. Hanany, D. Krefl, J. Park, A.M. Uranga and D. Vegh, Dimers and orientifolds, JHEP 09 (2007) 075 [arXiv:0707.0298] [INSPIRE].
A. Hanany and R.-K. Seong, Brane Tilings and Specular Duality, JHEP 08 (2012) 107 [arXiv:1206.2386] [INSPIRE].
O. Aharony, A. Hanany and B. Kol, Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].
A. Goncharov and R. Kenyon, Dimers and cluster integrable systems, arXiv:1107.5588 [INSPIRE].
A. Hanany and R.-K. Seong, Brane Tilings and Reflexive Polygons, Fortsch. Phys. 60 (2012) 695 [arXiv:1201.2614] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
A. Hanany, D. Orlando and S. Reffert, Sublattice Counting and Orbifolds, JHEP 06 (2010) 051 [arXiv:1002.2981] [INSPIRE].
J. Davey, A. Hanany and R.-K. Seong, An Introduction to Counting Orbifolds, Fortsch. Phys. 59 (2011) 677 [arXiv:1102.0015] [INSPIRE].
A. Hanany and R.-K. Seong, Symmetries of Abelian Orbifolds, JHEP 01 (2011) 027 [arXiv:1009.3017] [INSPIRE].
J. Davey, A. Hanany and R.-K. Seong, Counting Orbifolds, JHEP 06 (2010) 010 [arXiv:1002.3609] [INSPIRE].
A. Hanany, V. Jejjala, S. Ramgoolam and R.-K. Seong, Calabi-Yau Orbifolds and Torus Coverings, JHEP 09 (2011) 116 [arXiv:1105.3471] [INSPIRE].
M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
N. Broomhead, Dimer models and Calabi-Yau algebras, arXiv:0901.4662 [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
Y. Terashima and M. Yamazaki, Emergent 3-manifolds from 4d Superconformal Indices, Phys. Rev. Lett. 109 (2012) 091602 [arXiv:1203.5792] [INSPIRE].
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ArXiv ePrint: 1211.5139
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Franco, S., Galloni, D. & Seong, RK. New directions in bipartite field theories. J. High Energ. Phys. 2013, 32 (2013). https://doi.org/10.1007/JHEP06(2013)032
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DOI: https://doi.org/10.1007/JHEP06(2013)032