Abstract
We study the geometry of 4d \( \mathcal{N}=1 \) SCFT’s arising from compactification of 6d (1, 0) SCFT’s on a Riemann surface. We show that the conformal manifold of the resulting theory is characterized, in addition to moduli of complex structure of the Riemann surface, by the choice of a connection for a vector bundle on the surface arising from flavor symmetries in 6d. We exemplify this by considering the case of 4d \( \mathcal{N}=1 \) SCFT’s arising from M5 branes probing ℤ k singularity compactified on a Riemann surface. In particular, we study in detail the four dimensional theories arising in the case of two M5 branes on ℤ 2 singularity. We compute the conformal anomalies and indices of such theories in 4d and find that they are consistent with expectations based on anomaly and the moduli structure derived from the 6 dimensional perspective.
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References
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
J.J. Heckman, D.R. Morrison and C. Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 1506 (2015) 017] [arXiv:1312.5746] [INSPIRE].
J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic Classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].
L. Bhardwaj, Classification of 6d \( \mathcal{N}=\left(1,0\right) \) gauge theories, JHEP 11 (2015) 002 [arXiv:1502.06594] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d \( \mathcal{N}=\left(1,0\right) \) theories on T 2 and class S theories: Part I, JHEP 07 (2015) 014 [arXiv:1503.06217] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d \( \mathcal{N}=\left(1,0\right) \) theories on S 1 /T 2 and class S theories: part II, JHEP 12 (2015) 131 [arXiv:1508.00915] [INSPIRE].
M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and \( 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} \), JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
D. Gaiotto and S.S. Razamat, \( \mathcal{N}=1 \) theories of class \( {\mathcal{S}}_k \) , JHEP 07 (2015) 073 [arXiv:1503.05159] [INSPIRE].
A. Hanany and K. Maruyoshi, Chiral theories of class \( \mathcal{S} \), JHEP 12 (2015) 080 [arXiv:1505.05053] [INSPIRE].
S. Franco, H. Hayashi and A. Uranga, Charting Class \( {\mathcal{S}}_k \) Territory, Phys. Rev. D 92 (2015) 045004 [arXiv:1504.05988] [INSPIRE].
I. Coman, E. Pomoni, M. Taki and F. Yagi, Spectral curves of \( \mathcal{N}=1 \) theories of class \( {\mathcal{S}}_k \) , arXiv:1512.06079 [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. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4-D SYM to 2-D σ-models, Nucl. Phys. B 448 (1995) 166 [hep-th/9501096] [INSPIRE].
T.T. Dumitrescu, An introduction to supersymmetric field theories in curved space, arXiv:1608.02957 [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, C. Beem, N. Bobev and B. Wecht, Four-Dimensional SCFTs from M5-Branes, JHEP 06 (2012) 005 [arXiv:1203.0303] [INSPIRE].
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
J.J. Heckman, P. Jefferson, T. Rudelius and C. Vafa, Punctures for Theories of Class \( {\mathcal{S}}_{\varGamma } \), arXiv:1609.01281 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
C. Beem and A. Gadde, The N = 1 superconformal index for class S fixed points, JHEP 04 (2014) 036 [arXiv:1212.1467] [INSPIRE].
D. Xie, M5 brane and four dimensional N = 1 theories I, JHEP 04 (2014) 154 [arXiv:1307.5877] [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].
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].
T. Dimofte and D. Gaiotto, An E 7 Surprise, JHEP 10 (2012) 129 [arXiv:1209.1404] [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].
K. Maruyoshi and J. Song, The Full Superconformal Index of the Argyres-Douglas Theory, arXiv:1606.05632 [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
V.P. Spiridonov and S.O. Warnaar, Inversions of integral operators and elliptic beta integrals on root systems, Adv. Math. 207 (2006) 91 [math/0411044].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The Superconformal Index of the E 6 SCFT, JHEP 08 (2010) 107 [arXiv:1003.4244] [INSPIRE].
K.A. Intriligator and B. Wecht, The exact superconformal R-symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [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].
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, arXiv:1608.02965 [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, PTEP 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
K. Maruyoshi and J. Yagi, Surface defects as transfer matrices, PTEP 2016 (2016) 113B01 [arXiv:1606.01041] [INSPIRE].
Y. Ito and Y. Yoshida, Superconformal index with surface defects for class \( {\mathcal{S}}_k \) , arXiv:1606.01653 [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, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [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].
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions, JHEP 06 (2012) 016 [arXiv:1203.5677] [INSPIRE].
A. Arabi Ardehali, High-temperature asymptotics of supersymmetric partition functions, JHEP 07 (2016) 025 [arXiv:1512.03376] [INSPIRE].
N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir Energy and the Anomaly Polynomial, JHEP 09 (2015) 142 [arXiv:1507.08553] [INSPIRE].
L. Di Pietro and Z. Komargodski, Cardy formulae for SUSY theories in d = 4 and d = 6, JHEP 12 (2014) 031 [arXiv:1407.6061] [INSPIRE].
D. Gaiotto and A. Tomasiello, Holography for (1,0) theories in six dimensions, JHEP 12 (2014) 003 [arXiv:1404.0711] [INSPIRE].
F. Apruzzi, M. Fazzi, A. Passias and A. Tomasiello, Supersymmetric AdS 5 solutions of massive IIA supergravity, JHEP 06 (2015) 195 [arXiv:1502.06620] [INSPIRE].
F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev. D 86 (2012)065015 [arXiv:1109.0283] [INSPIRE].
C. Closset and I. Shamir, The \( \mathcal{N}=1 \) Chiral Multiplet on T 2 × S 2 and Supersymmetric Localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
L.F. Alday, M. Bullimore and M. Fluder, On S-duality of the Superconformal Index on Lens Spaces and 2d TQFT, JHEP 05 (2013) 122 [arXiv:1301.7486] [INSPIRE].
S.S. Razamat and M. Yamazaki, S-duality and the N = 2 Lens Space Index, JHEP 10 (2013) 048 [arXiv:1306.1543] [INSPIRE].
J.A. Harvey, TASI 2003 lectures on anomalies, hep-th/0509097 [INSPIRE].
L. Álvarez-Gaumé and E. Witten, Gravitational Anomalies, Nucl. Phys. B 234 (1984) 269 [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].
S.S. Razamat, On the \( \mathcal{N}=2 \) superconformal index and eigenfunctions of the elliptic RS model, Lett. Math. Phys. 104 (2014) 673 [arXiv:1309.0278] [INSPIRE].
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Razamat, S.S., Vafa, C. & Zafrir, G. 4d \( \mathcal{N}=1 \) from 6d (1, 0). J. High Energ. Phys. 2017, 64 (2017). https://doi.org/10.1007/JHEP04(2017)064
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DOI: https://doi.org/10.1007/JHEP04(2017)064