Abstract
We show that exactly marginal operators of Supersymmetric Conformal Field Theories (SCFTs) with four supercharges cannot obtain a vacuum expectation value at a generic point on the conformal manifold. Exactly marginal operators are therefore nilpotent in the chiral ring. This allows us to associate an integer to the conformal manifold, which we call the nilpotency index of the conformal manifold. We discuss several examples in diverse dimensions where we demonstrate these facts and compute the nilpotency index.
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J.L. Cardy, Continuously Varying Exponents and the Value of the Central Charge, J. Phys. A 20 (1987) L891 [INSPIRE].
M.R. Gaberdiel, A. Konechny and C. Schmidt-Colinet, Conformal perturbation theory beyond the leading order, J. Phys. A 42 (2009) 105402 [arXiv:0811.3149] [INSPIRE].
Z. Komargodski and D. Simmons-Duffin, The Random-Bond Ising Model in 2.01 and 3 Dimensions, J. Phys. A 50 (2017) 154001 [arXiv:1603.04444] [INSPIRE].
V. Bashmakov, M. Bertolini and H. Raj, On non-supersymmetric conformal manifolds: field theory and holography, JHEP 11 (2017) 167 [arXiv:1709.01749] [INSPIRE].
C. Behan, Conformal manifolds: ODEs from OPEs, JHEP 03 (2018) 127 [arXiv:1709.03967] [INSPIRE].
M.F. Sohnius and P.C. West, Conformal Invariance in N = 4 Supersymmetric Yang-Mills Theory, Phys. Lett. B 100 (1981) 245 [INSPIRE].
P.S. Howe, K.S. Stelle and P.C. West, A Class of Finite Four-Dimensional Supersymmetric Field Theories, Phys. Lett. B 124 (1983) 55 [INSPIRE].
A. Parkes and P.C. West, Finiteness in Rigid Supersymmetric Theories, Phys. Lett. B 138 (1984) 99 [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].
B. Kol, On Conformal Deformations II, arXiv:1005.4408 [INSPIRE].
D. Kutasov, Geometry on the Space of Conformal Field Theories and Contact Terms, Phys. Lett. B 220 (1989) 153 [INSPIRE].
N. Seiberg, Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys. B 303 (1988) 286 [INSPIRE].
V. Asnin, On metric geometry of conformal moduli spaces of four-dimensional superconformal theories, JHEP 09 (2010) 012 [arXiv:0912.2529] [INSPIRE].
S.S. Razamat and G. Zafrir, N = 1 conformal dualities, JHEP 09 (2019) 046 [arXiv:1906.05088] [INSPIRE].
S.S. Razamat and G. Zafrir, N = 1 conformal duals of gauged En MN models, JHEP 06 (2020) 176 [arXiv:2003.01843] [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].
M.A. Luty and W. Taylor, Varieties of vacua in classical supersymmetric gauge theories, Phys. Rev. D 53 (1996) 3399 [hep-th/9506098] [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].
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].
A.E. Nelson and N. Seiberg, R symmetry breaking versus supersymmetry breaking, Nucl. Phys. B 416 (1994) 46 [hep-ph/9309299] [INSPIRE].
M.T. Grisaru, W. Siegel and M. Roček, Improved Methods for Supergraphs, Nucl. Phys. B 159 (1979) 429 [INSPIRE].
N. Seiberg, Naturalness versus supersymmetric nonrenormalization theorems, Phys. Lett. B 318 (1993) 469 [hep-ph/9309335] [INSPIRE].
M. Dine, G. Festuccia and Z. Komargodski, A Bound on the Superpotential, JHEP 03 (2010) 011 [arXiv:0910.2527] [INSPIRE].
S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [INSPIRE].
W. Lerche, C. Vafa and N.P. Warner, Chiral Rings in N = 2 Superconformal Theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
E. Witten, On the Landau-Ginzburg description of N = 2 minimal models, Int. J. Mod. Phys. A 9 (1994) 4783 [hep-th/9304026] [INSPIRE].
M. Bertolini, I.V. Melnikov and M.R. Plesser, Accidents in (0, 2) Landau-Ginzburg theories, JHEP 12 (2014) 157 [arXiv:1405.4266] [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].
E. Witten, On S duality in Abelian gauge theory, Selecta Math. 1 (1995) 383 [hep-th/9505186] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N = 2 SCFTs. Part I: physical constraints on relevant deformations, JHEP 02 (2018) 001 [arXiv:1505.04814] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Deformations of Superconformal Theories, JHEP 11 (2016) 135 [arXiv:1602.01217] [INSPIRE].
J. Hughes, J. Liu and J. Polchinski, Supermembranes, Phys. Lett. B 180 (1986) 370 [INSPIRE].
S. Ferrara, L. Girardello and M. Porrati, Spontaneous breaking of N = 2 to N = 1 in rigid and local supersymmetric theories, Phys. Lett. B 376 (1996) 275 [hep-th/9512180] [INSPIRE].
I. Antoniadis, H. Partouche and T.R. Taylor, Spontaneous breaking of N = 2 global supersymmetry, Phys. Lett. B 372 (1996) 83 [hep-th/9512006] [INSPIRE].
M.J. Strassler, On renormalization group flows and exactly marginal operators in three-dimensions, hep-th/9810223 [INSPIRE].
M. Baggio, N. Bobev, S.M. Chester, E. Lauria and S.S. Pufu, Decoding a Three-Dimensional Conformal Manifold, JHEP 02 (2018) 062 [arXiv:1712.02698] [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
K. Konishi, Anomalous Supersymmetry Transformation of Some Composite Operators in SQCD, Phys. Lett. B 135 (1984) 439 [INSPIRE].
K.-i. Konishi and K.-i. Shizuya, Functional Integral Approach to Chiral Anomalies in Supersymmetric Gauge Theories, Nuovo Cim. A 90 (1985) 111 [INSPIRE].
F. Cachazo, M.R. Douglas, N. Seiberg and E. Witten, Chiral rings and anomalies in supersymmetric gauge theory, JHEP 12 (2002) 071 [hep-th/0211170] [INSPIRE].
A. Ceresole, G. Dall’Agata, R. D’Auria and S. Ferrara, Spectrum of type IIB supergravity on AdS5 × T 11: Predictions on N = 1 SCFT’s, Phys. Rev. D 61 (2000) 066001 [hep-th/9905226] [INSPIRE].
N. Seiberg, Adding fundamental matter to ‘Chiral rings and anomalies in supersymmetric gauge theory’, JHEP 01 (2003) 061 [hep-th/0212225] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, JHEP 03 (2019) 163 [arXiv:1612.00809] [INSPIRE].
F.A. Dolan and H. Osborn, On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41 [hep-th/0209056] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, On the Superconformal Index of N = 1 IR Fixed Points: A Holographic Check, JHEP 03 (2011) 041 [arXiv:1011.5278] [INSPIRE].
D. Berenstein, V. Jejjala and R.G. Leigh, Marginal and relevant deformations of N = 4 field theories and noncommutative moduli spaces of vacua, Nucl. Phys. B 589 (2000) 196 [hep-th/0005087] [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].
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Komargodski, Z., Razamat, S.S., Sela, O. et al. A nilpotency index of conformal manifolds. J. High Energ. Phys. 2020, 183 (2020). https://doi.org/10.1007/JHEP10(2020)183
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DOI: https://doi.org/10.1007/JHEP10(2020)183