Abstract
We show that the 4d \( \mathcal{N} \) = 1 SU(3) Nf = 6 SQCD is the model obtained when compactifying the rank one E-string theory on a three punctured sphere (a trinion) with a particular value of flux. The SU(6) × SU(6) × U(1) global symmetry of the theory, when decomposed into the SU(2)3 × U(1)3 × SU(6) subgroup, corresponds to the three SU(2) symmetries associated to the three punctures and the U(1)3 × SU(6) subgroup of the E8 symmetry of the E-string theory. All the puncture symmetries are manifest in the UV and thus we can construct ordinary Lagrangians flowing in the IR to any compactification of the E-string theory. We generalize this claim and argue that the \( \mathcal{N} \) = 1 SU(N + 2) SQCD in the middle of the conformal window, Nf = 2N + 4, is the theory obtained by compactifying the 6d minimal (DN +3, DN +3) conformal matter SCFT on a sphere with two maximal SU(N + 1) punctures, one minimal SU(2) puncture, and with a particular value of flux. The SU(2N + 4) × SU(2N + 4) × U(1) symmetry of the UV Lagrangian decomposes into SU(N + 1)2 × SU(2) puncture symmetries and the U(1)3 × SU(2N + 4) subgroup of the SO(12 + 4N ) symmetry group of the 6d SCFT. The models constructed from the trinions exhibit a variety of interesting strong coupling effects. For example, one of the dualities arising geometrically from different pair-of-pants decompositions of a four punctured sphere is an SU(N + 2) generalization of the Intriligator-Pouliot duality of SU(2) SQCD with Nf = 4, which is a degenerate, N = 0, instance of our discussion.
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References
S.S. Razamat and E. Sabag, Sequences of 6d SCFTs on generic Riemann surfaces, JHEP 01 (2020) 086 [arXiv:1910.03603] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
C.S. Chan, O.J. Ganor and M. Krogh, Chiral compactifications of 6-D conformal theories, Nucl. Phys. B 597 (2001) 228 [hep-th/0002097] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, Prog. Math. 319 (2016) 155 [arXiv:1306.4320] [INSPIRE].
S. Gukov, D. Pei, P. Putrov and C. Vafa, 4-manifolds and topological modular forms, arXiv:1811.07884 [INSPIRE].
D. Gaiotto and S.S. Razamat, \( \mathcal{N} \) = 1 theories of class \( \mathcal{S} \)k , JHEP 07 (2015) 073 [arXiv:1503.05159] [INSPIRE].
S.S. Razamat, C. Vafa and G. Zafrir, 4d \( \mathcal{N} \) = 1 from 6d (1, 0), JHEP 04 (2017) 064 [arXiv:1610.09178] [INSPIRE].
H.-C. Kim, S.S. Razamat, C. Vafa and G. Zafrir, E-String Theory on Riemann Surfaces, Fortsch. Phys. 66 (2018) 1700074 [arXiv:1709.02496] [INSPIRE].
H.-C. Kim, S.S. Razamat, C. Vafa and G. Zafrir, D-type Conformal Matter and SU/USp Quivers, JHEP 06 (2018) 058 [arXiv:1802.00620] [INSPIRE].
S.S. Razamat and G. Zafrir, Compactification of 6d minimal SCFTs on Riemann surfaces, Phys. Rev. D 98 (2018) 066006 [arXiv:1806.09196] [INSPIRE].
S. Pasquetti, S.S. Razamat, M. Sacchi and G. Zafrir, Rank Q E-string on a torus with flux, SciPost Phys. 8 (2020) 014 [arXiv:1908.03278] [INSPIRE].
S.S. Razamat, O. Sela and G. Zafrir, Curious patterns of IR symmetry enhancement, JHEP 10 (2018) 163 [arXiv:1809.00541] [INSPIRE].
O. Sela and G. Zafrir, Symmetry enhancement in 4d Spin(n) gauge theories and compactification from 6d, JHEP 12 (2019) 052 [arXiv:1910.03629] [INSPIRE].
D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, JHEP 10 (2016) 012 [arXiv:1412.2781] [INSPIRE].
S.S. Razamat, E. Sabag and G. Zafrir, From 6d flows to 4d flows, JHEP 12 (2019) 108 [arXiv:1907.04870] [INSPIRE].
S.S. Razamat and G. Zafrir, N = 1 conformal dualities, JHEP 09 (2019) 046 [arXiv:1906.05088] [INSPIRE].
K. Ohmori, H. Shimizu and Y. Tachikawa, Anomaly polynomial of E-string theories, JHEP 08 (2014) 002 [arXiv:1404.3887] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, PTEP 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [INSPIRE].
R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 95 [hep-th/9503121] [INSPIRE].
D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, Exactly Marginal Deformations and Global Symmetries, JHEP 06 (2010) 106 [arXiv:1005.3546] [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].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
F.A. Dolan and H. Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N = 1 Dual Theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].
L. Rastelli and S.S. Razamat, The supersymmetric index in four dimensions, J. Phys. A 50 (2017) 443013 [arXiv:1608.02965] [INSPIRE].
D.R. Morrison and C. Vafa, F-theory and \( \mathcal{N} \) = 1 SCFTs in four dimensions, JHEP 08 (2016) 070 [arXiv:1604.03560] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
K.A. Intriligator and P. Pouliot, Exact superpotentials, quantum vacua and duality in supersymmetric SP (Nc ) gauge theories, Phys. Lett. B 353 (1995) 471 [hep-th/9505006] [INSPIRE].
I. Bah, C. Beem, N. Bobev and B. Wecht, Four-Dimensional SCFTs from M5-Branes, JHEP 06 (2012) 005 [arXiv:1203.0303] [INSPIRE].
A. Hanany and K. Maruyoshi, Chiral theories of class \( \mathcal{S} \) , JHEP 12 (2015) 080 [arXiv:1505.05053] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].
O. Aharony, S.S. Razamat and B. Willett, From 3d duality to 2d duality, JHEP 11 (2017) 090 [arXiv:1710.00926] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
H.-C. Kim, S.S. Razamat, C. Vafa and G. Zafrir, Compactifications of ADE conformal matter on a torus, JHEP 09 (2018) 110 [arXiv:1806.07620] [INSPIRE].
I. Bah, A. Hanany, K. Maruyoshi, S.S. Razamat, Y. Tachikawa and G. Zafrir, 4d \( \mathcal{N} \) = 1 from 6d \( \mathcal{N} \) = (1, 0) on a torus with fluxes, JHEP 06 (2017) 022 [arXiv:1702.04740] [INSPIRE].
M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and 6d(1,0) → 4d(N =2) , JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d \( \mathcal{N} \) = (1, 0) theories on S1 /T 2 and class S theories: part II, JHEP 12 (2015) 131 [arXiv:1508.00915] [INSPIRE].
K.A. Intriligator and B. Wecht, The Exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
D. Kutasov, A. Parnachev and D.A. Sahakyan, Central charges and U(1)R symmetries in N = 1 superYang-Mills, JHEP 11 (2003) 013 [hep-th/0308071] [INSPIRE].
S.S. Razamat, E. Sabag and G. Zafrir, Weakly coupled conformal manifolds in 4d, JHEP 06 (2020) 179 [arXiv:2004.07097] [INSPIRE].
S.S. Razamat and G. Zafrir, \( \mathcal{N} \) = 1 conformal duals of gauged En MN models, JHEP 06 (2020) 176 [arXiv:2003.01843] [INSPIRE].
E. Barnes, K.A. Intriligator, B. Wecht and J. Wright, Evidence for the strongest version of the 4d a-theorem, via a-maximization along RG flows, Nucl. Phys. B 702 (2004) 131 [hep-th/0408156] [INSPIRE].
S. Benvenuti and S. Giacomelli, Supersymmetric gauge theories with decoupled operators and chiral ring stability, Phys. Rev. Lett. 119 (2017) 251601 [arXiv:1706.02225] [INSPIRE].
P. Agarwal, K. Intriligator and J. Song, Infinitely many \( \mathcal{N} \) = 1 dualities from m + 1 − m = 1, JHEP 10 (2015) 035 [arXiv:1505.00255] [INSPIRE].
K. Maruyoshi and J. Song, Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index, Phys. Rev. Lett. 118 (2017) 151602 [arXiv:1606.05632] [INSPIRE].
C. Beem and A. Gadde, The \( \mathcal{N} \) = 1 superconformal index for class S fixed points, JHEP 04 (2014) 036 [arXiv:1212.1467] [INSPIRE].
C. Beem, S.S. Razamat and G. Zafrir, Universal Deformations from Six Dimensions, to appear.
S.S. Razamat, Geometrization of relevance, talk at Avant-garde methods for quantum field theory and gravity, Nazareth, Israel, 17–21 February 2019 [https://phsites.technion.ac.il/the-fifth-israeli-indian-conference-on-string-theory/program/].
S.S. Razamat, O. Sela and G. Zafrir, Between Symmetry and Duality in Supersymmetric Quantum Field Theories, Phys. Rev. Lett. 120 (2018) 071604 [arXiv:1711.02789] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, A new 5d description of 6d D-type minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6d SCFTs, 5d Dualities and Tao Web Diagrams, JHEP 05 (2019) 203 [arXiv:1509.03300] [INSPIRE].
O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Twisted D-Series, JHEP 04 (2015) 173 [arXiv:1309.2299] [INSPIRE].
O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Z3-twisted D4 Theory, arXiv:1601.02077 [INSPIRE].
O. Chacaltana, J. Distler and Y. Tachikawa, Gaiotto duality for the twisted A2N −1 series, JHEP 05 (2015) 075 [arXiv:1212.3952] [INSPIRE].
T. Dimofte and D. Gaiotto, An E7 Surprise, JHEP 10 (2012) 129 [arXiv:1209.1404] [INSPIRE].
S.S. Razamat and G. Zafrir, E8 orbits of IR dualities, JHEP 11 (2017) 115 [arXiv:1709.06106] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Superconformal indices for N = 1 theories with multiple duals, Nucl. Phys. B 824 (2010) 192 [arXiv:0811.1909] [INSPIRE].
A. Gadde, S.S. Razamat and B. Willett, “Lagrangian” for a Non-Lagrangian Field Theory with \( \mathcal{N} \) = 2 Supersymmetry, Phys. Rev. Lett. 115 (2015) 171604 [arXiv:1505.05834] [INSPIRE].
P. Agarwal, K. Maruyoshi and J. Song, A “Lagrangian” for the E7 superconformal theory, JHEP 05 (2018) 193 [arXiv:1802.05268] [INSPIRE].
A. Amariti and L. Cassia, USp(2Nc ) SQC D3 with antisymmetric: dualities and symmetry enhancements, JHEP 02 (2019) 013 [arXiv:1809.03796] [INSPIRE].
S. Benvenuti, A tale of exceptional 3d dualities, JHEP 03 (2019) 125 [arXiv:1809.03925] [INSPIRE].
A. Gadde, K. Maruyoshi, Y. Tachikawa and W. Yan, New N = 1 Dualities, JHEP 06 (2013) 056 [arXiv:1303.0836] [INSPIRE].
J. Chen, B. Haghighat, S. Liu and M. Sperling, 4d N = 1 from 6d D-type N = (1, 0), JHEP 01 (2020) 152 [arXiv:1907.00536] [INSPIRE].
G. Zafrir, On the torus compactifications of Z2 orbifolds of E-string theories, JHEP 10 (2019) 040 [arXiv:1809.04260] [INSPIRE].
G. Zafrir, An N = 1 Lagrangian for the rank 1 E6 superconformal theory, arXiv:1912.09348 [INSPIRE].
J.J. Heckman and T. Rudelius, Top Down Approach to 6D SCFTs, J. Phys. A 52 (2019) 093001 [arXiv:1805.06467] [INSPIRE].
S. Elitzur, A. Giveon and D. Kutasov, Branes and N = 1 duality in string theory, Phys. Lett. B 400 (1997) 269 [hep-th/9702014] [INSPIRE].
A. Giveon and D. Kutasov, Brane Dynamics and Gauge Theory, Rev. Mod. Phys. 71 (1999) 983 [hep-th/9802067] [INSPIRE].
M. Bershadsky, A. Johansen, T. Pantev, V. Sadov and C. Vafa, F theory, geometric engineering and N = 1 dualities, Nucl. Phys. B 505 (1997) 153 [hep-th/9612052] [INSPIRE].
F. Cachazo, B. Fiol, K.A. Intriligator, S. Katz and C. Vafa, A Geometric unification of dualities, Nucl. Phys. B 628 (2002) 3 [hep-th/0110028] [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].
I. García-Etxebarria and D. Regalado, \( \mathcal{N} \) = 3 four dimensional field theories, JHEP 03 (2016) 083 [arXiv:1512.06434] [INSPIRE].
F. Apruzzi, J.J. Heckman, D.R. Morrison and L. Tizzano, 4D Gauge Theories with Conformal Matter, JHEP 09 (2018) 088 [arXiv:1803.00582] [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].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d Topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
E. Rains and S. Ruijsenaars, Difference operators of Sklyanin and van Diejen type, Commun. Math. Phys. 320 (2013) 851.
B. Nazzal and S.S. Razamat, Surface Defects in E-String Compactifications and the van Diejen Model, SIGMA 14 (2018) 036 [arXiv:1801.00960] [INSPIRE].
S.N.M. Ruijsenaars, Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type IV. The relativistic Heun (van Diejen) case, SIGMA 11 (2015) 004.
J.F. van Diejen, Integrability of difference Calogero-Moser systems, J. Math. Phys. 35 (1994) 2983.
C. Csáki, M. Schmaltz, W. Skiba and J. Terning, Selfdual N = 1 SUSY gauge theories, Phys. Rev. D 56 (1997) 1228 [hep-th/9701191] [INSPIRE].
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Razamat, S.S., Sabag, E. SQCD and pairs of pants. J. High Energ. Phys. 2020, 28 (2020). https://doi.org/10.1007/JHEP09(2020)028
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DOI: https://doi.org/10.1007/JHEP09(2020)028