Abstract
We study the one-dimensional complex conformal manifold that controls the infrared dynamics of a three-dimensional \( \mathcal{N} \) = 2 supersymmetric theory of three chiral superfields with a cubic superpotential. Two special points on this conformal manifold are the well-known XYZ model and three decoupled copies of the critical Wess-Zumino model. The conformal manifold enjoys a discrete duality group isomorphic to S4 and can be thought of as an orbifold of CP1. We use the 4 − ε expansion and the numerical conformal bootstrap to calculate the spectrum of conformal dimensions of low-lying operators and their OPE coefficients, and find a very good quantitative agreement between the two approaches.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Behan, Conformal manifolds: ODEs from OPEs, arXiv:1709.03967 [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].
S. Hollands, Action principle for OPE, Nucl. Phys. B 926 (2018) 614 [arXiv:1710.05601] [INSPIRE].
K. Sen and Y. Tachikawa, First-order conformal perturbation theory by marginal operators, arXiv:1711.05947 [INSPIRE].
N. Seiberg, Observations on the moduli space of superconformal field theories, Nucl. Phys. B 303 (1988) 286 [INSPIRE].
D. Kutasov, Geometry on the space of conformal field theories and contact terms, Phys. Lett. B 220 (1989) 153 [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Deformations of superconformal theories, JHEP 11 (2016) 135 [arXiv:1602.01217] [INSPIRE].
W. Nahm, Supersymmetries and their representations, Nucl. Phys. B 135 (1978) 149 [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].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
M.J. Strassler, On renormalization group flows and exactly marginal operators in three-dimensions, hep-th/9810223 [INSPIRE].
S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, Springer Briefs in Physics, Springer, Germany (2016).
D. Simmons-Duffin, The conformal bootstrap, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), June 1-26, Boulder, U.S.A. (2015), arXiv:1602.07982 [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N ) archipelago, JHEP 11 (2015) 106 [arXiv:1504.07997] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N ) models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
S. El-Showk et al., Solving the 3d Ising model with the conformal bootstrap II. c-minimization and precise critical exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
L. Iliesiu, F. Kos, D. Poland, S.S. Pufu and D. Simmons-Duffin, Bootstrapping 3D fermions with global symmetries, JHEP 01 (2018) 036 [arXiv:1705.03484] [INSPIRE].
N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping SCFTs with four supercharges, JHEP 08 (2015) 142 [arXiv:1503.02081] [INSPIRE].
N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping the three-dimensional supersymmetric Ising model, Phys. Rev. Lett. 115 (2015) 051601 [arXiv:1502.04124] [INSPIRE].
S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, Exact correlators of BPS operators from the 3d superconformal bootstrap, JHEP 03 (2015) 130 [arXiv:1412.0334] [INSPIRE].
S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, The \( \mathcal{N} \) = 8 superconformal bootstrap in three dimensions, JHEP 09 (2014) 143 [arXiv:1406.4814] [INSPIRE].
N.B. Agmon, S.M. Chester and S.S. Pufu, Solving M-theory with the conformal bootstrap, arXiv:1711.07343 [INSPIRE].
S.-S. Lee, Emergence of supersymmetry at a critical point of a lattice model, Phys. Rev. B 76 (2007) 075103 [cond-mat/0611658] [INSPIRE].
Y. Yu and K. Yang, Simulating Wess-Zumino supersymmetry model in optical lattices, Phys. Rev. Lett. 105 (2010) 150605 [arXiv:1005.1399] [INSPIRE].
P. Ponte and S.-S. Lee, Emergence of supersymmetry on the surface of three dimensional topological insulators, New J. Phys. 16 (2014) 013044 [arXiv:1206.2340] [INSPIRE].
T. Grover, D.N. Sheng and A. Vishwanath, Emergent space-time supersymmetry at the boundary of a topological phase, Science 344 (2014) 280 [arXiv:1301.7449] [INSPIRE].
S.-K. Jian, C.-H. Lin, J. Maciejko and H. Yao, Emergence of supersymmetric quantum electrodynamics, Phys. Rev. Lett. 118 (2017) 166802 [arXiv:1609.02146] [INSPIRE].
Z.-X. Li, A. Vaezi, C.B. Mendl and H. Yao, Observation of emergent spacetime supersymmetry at superconducting quantum criticality, arXiv:1711.04772 [INSPIRE].
V. Asnin, On metric geometry of conformal moduli spaces of four-dimensional superconformal theories, JHEP 09 (2010) 012 [arXiv:0912.2529] [INSPIRE].
Y. Tachikawa, Five-dimensional supergravity dual of a-maximization, Nucl. Phys. B 733 (2006) 188 [hep-th/0507057] [INSPIRE].
S. de Alwis, J. Louis, L. McAllister, H. Triendl and A. Westphal, Moduli spaces in AdS 4 supergravity, JHEP 05 (2014) 102 [arXiv:1312.5659] [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, JHEP 09 (2002) 046 [hep-th/0205141] [INSPIRE].
B. Kol, On conformal deformations II, arXiv:1005.4408 [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.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].
F. Benini, S. Benvenuti and S. Pasquetti, SUSY monopole potentials in 2 + 1 dimensions, JHEP 08 (2017) 086 [arXiv:1703.08460] [INSPIRE].
W. Lerche, D. Lüst and N.P. Warner, Duality symmetries in N = 2 Landau-Ginzburg models, Phys. Lett. B 231 (1989) 417 [INSPIRE].
E.P. Verlinde and N.P. Warner, Topological Landau-Ginzburg matter at c = 3, Phys. Lett. B 269 (1991) 96 [INSPIRE].
Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, (2, 2) superconformal bootstrap in two dimensions, JHEP 05 (2017) 112 [arXiv:1610.05371] [INSPIRE].
K.G. Wilson and M.E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972) 240 [INSPIRE].
K.G. Wilson and J.B. Kogut, The renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].
S.M. Chester, L.V. Iliesiu, S.S. Pufu and R. Yacoby, Bootstrapping O(N ) vector models with four supercharges in 3 ≤ d ≤ 4, JHEP 05 (2016) 103 [arXiv:1511.07552] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, The \( \mathcal{N} \) = 4 superconformal bootstrap, Phys. Rev. Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, More \( \mathcal{N} \) = 4 superconformal bootstrap, Phys. Rev. D 96 (2017) 046014 [arXiv:1612.02363] [INSPIRE].
W. Thurston, The geometry and topology of 3-manifolds, lecture notes (1978).
M. Baggio, V. Niarchos and K. Papadodimas, Aspects of Berry phase in QFT, JHEP 04 (2017) 062 [arXiv:1701.05587] [INSPIRE].
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
J. Gomis and S. Lee, Exact Kähler potential from gauge theory and mirror symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].
E. Gerchkovitz et al., Correlation functions of Coulomb branch operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].
W. Lerche, C. Vafa and N.P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
S. Cecotti, Geometry of N = 2 Landau-Ginzburg families, Nucl. Phys. B 355 (1991) 755 [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric field theories on three-manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].
T. Nishioka and K. Yonekura, On RG flow of τ RR for supersymmetric field theories in three-dimensions, JHEP 05 (2013) 165 [arXiv:1303.1522] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
W. Witczak-Krempa and J. Maciejko, Optical conductivity of topological surface states with emergent supersymmetry, Phys. Rev. Lett. 116 (2016) 100402 [arXiv:1510.06397] [INSPIRE].
S.M. Chester et al., Accidental symmetries and the conformal bootstrap, JHEP 01 (2016) 110 [arXiv:1507.04424] [INSPIRE].
P.M. Ferreira, I. Jack and D.R.T. Jones, The three loop SSM β-functions, Phys. Lett. B 387 (1996) 80 [hep-ph/9605440] [INSPIRE].
P.M. Ferreira, I. Jack and D.R.T. Jones, The quasiinfrared fixed point at higher loops, Phys. Lett. B 392 (1997) 376 [hep-ph/9610296] [INSPIRE].
I. Jack, D.R.T. Jones and A. Pickering, The soft scalar mass β-function, Phys. Lett. B 432 (1998) 114 [hep-ph/9803405] [INSPIRE].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Yukawa CFTs and emergent supersymmetry, PTEP 2016 (2016) 12C105 [arXiv:1607.05316] [INSPIRE].
N. Zerf, C.-H. Lin and J. Maciejko, Superconducting quantum criticality of topological surface states at three loops, Phys. Rev. B 94 (2016) 205106 [arXiv:1605.09423] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, tt ∗ equations, localization and exact chiral rings in 4d \( \mathcal{N} \) = 2 SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565] [INSPIRE].
H. Kleinert and V. Schulte-Frohlinde, Critical properties of ϕ 4 -theories, (2001).
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N ) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D conformal field theories with global symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].
D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].
K. Papadodimas, Topological anti-topological fusion in four-dimensional superconformal field theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, Exact correlation functions in SU(2) \( \mathcal{N} \) = 2 superconformal QCD, Phys. Rev. Lett. 113 (2014) 251601 [arXiv:1409.4217] [INSPIRE].
O. Aharony, B. Kol and S. Yankielowicz, On exactly marginal deformations of N = 4 SYM and type IIB supergravity on AdS 5 × S 5, JHEP 06 (2002) 039 [hep-th/0205090] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
S. Cecotti, C. Cordova and C. Vafa, Braids, walls and mirrors, arXiv:1110.2115 [INSPIRE].
G.D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton U.S.A. (1973).
L.F. Abbott and M.T. Grisaru, The three loop β-function for the Wess-Zumino model, Nucl. Phys. B 169 (1980) 415 [INSPIRE].
D. Binosi and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun. 161 (2004) 76 [hep-ph/0309015] [INSPIRE].
D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun. 180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].
B.R. Greene, C. Vafa and N.P. Warner, Calabi-Yau manifolds and renormalization group flows, Nucl. Phys. B 324 (1989) 371 [INSPIRE].
E.J. Martinec, Criticality, catastrophes and compactifications, PRINT-89-0373 (1989).
L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The conformal field theory of orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1712.02698
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Baggio, M., Bobev, N., Chester, S.M. et al. Decoding a three-dimensional conformal manifold. J. High Energ. Phys. 2018, 62 (2018). https://doi.org/10.1007/JHEP02(2018)062
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2018)062