Abstract
We study the transition between phases at large R-charge on a conformal manifold. These phases are characterized by the behaviour of the lowest operator dimension ∆(QR) for fixed and large R-charge QR. We focus, as an example, on the D = 3, \( \mathcal{N} \) = 2 Wess-Zumino model with cubic superpotential \( W= XYZ+\frac{\tau }{6}\left({X}^3+{Y}^3+{Z}^3\right) \), and compute ∆(QR, τ) using the ϵ-expansion in three interesting limits. In two of these limits the (leading order) result turns out to be
which leads us to the double-scaling parameter, ϵ|τ|2QR, which interpolates between the “near-BPS phase” (∆(Q) ∼ Q) and the “superfluid phase” (∆(Q) ∼ QD/(D−1)) at large R-charge. This smooth transition, happening near τ = 0, is a large-R-charge manifestation of the existence of a moduli space and an infinite chiral ring at τ = 0. We also argue that this behavior can be extended to three dimensions with minimal modifications, and so we conclude that ∆(QR, τ) experiences a smooth transition around QR ∼ 1/|τ|2. Additionally, we find a first-order phase transition for ∆(QR, τ) as a function of τ, as a consequence of the duality of the model. We also comment on the applicability of our result down to small R-charge.
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S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT Operator Spectrum at Large Global Charge, JHEP 12 (2015) 071 [arXiv:1505.01537] [INSPIRE].
L. Álvarez-Gaumé, O. Loukas, D. Orlando and S. Reffert, Compensating strong coupling with large charge, JHEP 04 (2017) 059 [arXiv:1610.04495] [INSPIRE].
A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone Bosons and CFT data, JHEP 06 (2017) 011 [arXiv:1611.02912] [INSPIRE].
S. Hellerman, S. Maeda and M. Watanabe, Operator Dimensions from Moduli, JHEP 10 (2017) 089 [arXiv:1706.05743] [INSPIRE].
S. Hellerman and S. Maeda, On the Large R-charge Expansion in \( \mathcal{N} \) = 2 Superconformal Field Theories, JHEP 12 (2017) 135 [arXiv:1710.07336] [INSPIRE].
S. Favrod, D. Orlando and S. Reffert, The large-charge expansion for Schrödinger systems, JHEP 12 (2018) 052 [arXiv:1809.06371] [INSPIRE].
S.M. Kravec and S. Pal, Nonrelativistic Conformal Field Theories in the Large Charge Sector, JHEP 02 (2019) 008 [arXiv:1809.08188] [INSPIRE].
S. Hellerman, S. Maeda, D. Orlando, S. Reffert and M. Watanabe, Universal correlation functions in rank 1 SCFTs, JHEP 12 (2019) 047 [arXiv:1804.01535] [INSPIRE].
S. Hellerman, S. Maeda, D. Orlando, S. Reffert and M. Watanabe, S-duality and correlation functions at large R-charge, arXiv:2005.03021 [INSPIRE].
S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, A Note on Inhomogeneous Ground States at Large Global Charge, JHEP 10 (2019) 038 [arXiv:1705.05825] [INSPIRE].
S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, Observables in Inhomogeneous Ground States at Large Global Charge, arXiv:1804.06495 [INSPIRE].
O. Loukas, A matrix CFT at multiple large charges, JHEP 06 (2018) 164 [arXiv:1711.07990] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
L.F. Alday, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
L.F. Alday, Solving CFTs with Weakly Broken Higher Spin Symmetry, JHEP 10 (2017) 161 [arXiv:1612.00696] [INSPIRE].
S.M. Kravec and S. Pal, The Spinful Large Charge Sector of Non-Relativistic CFTs: From Phonons to Vortex Crystals, JHEP 05 (2019) 194 [arXiv:1904.05462] [INSPIRE].
G. Cuomo, A. de la Fuente, A. Monin, D. Pirtskhalava and R. Rattazzi, Rotating superfluids and spinning charged operators in conformal field theory, Phys. Rev. D 97 (2018) 045012 [arXiv:1711.02108] [INSPIRE].
G. Cuomo, Superfluids, vortices and spinning charged operators in 4d CFT, JHEP 02 (2020) 119 [arXiv:1906.07283] [INSPIRE].
D. Jafferis, B. Mukhametzhanov and A. Zhiboedov, Conformal Bootstrap At Large Charge, JHEP 05 (2018) 043 [arXiv:1710.11161] [INSPIRE].
G. Arias-Tamargo, D. Rodriguez-Gomez and J.G. Russo, The large charge limit of scalar field theories and the Wilson-Fisher fixed point at ϵ = 0, JHEP 10 (2019) 201 [arXiv:1908.11347] [INSPIRE].
G. Badel, G. Cuomo, A. Monin and R. Rattazzi, The ϵ-expansion Meets Semiclassics, JHEP 11 (2019) 110 [arXiv:1909.01269] [INSPIRE].
M. Watanabe, Accessing Large Global Charge via the ϵ-Expansion, arXiv:1909.01337 [INSPIRE].
G. Arias-Tamargo, D. Rodriguez-Gomez and J.G. Russo, Correlation functions in scalar field theory at large charge, JHEP 01 (2020) 171 [arXiv:1912.01623] [INSPIRE].
G. Badel, G. Cuomo, A. Monin and R. Rattazzi, Feynman diagrams and the large charge expansion in 3 − ε dimensions, Phys. Lett. B 802 (2020) 135202 [arXiv:1911.08505] [INSPIRE].
O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang and C. Zhang, Charging the O(N) model, Phys. Rev. D 102 (2020) 045011 [arXiv:2003.13121] [INSPIRE].
O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang and C. Zhang, Charging the Walking U(N) × U(N) Higgs Theory as a Complex CFT, Phys. Rev. D 102 (2020) 125033 [arXiv:2006.10078] [INSPIRE].
D. Banerjee, S. Chandrasekharan and D. Orlando, Conformal dimensions via large charge expansion, Phys. Rev. Lett. 120 (2018) 061603 [arXiv:1707.00711] [INSPIRE].
D. Banerjee, S. Chandrasekharan, D. Orlando and S. Reffert, Conformal dimensions in the large charge sectors at the O(4) Wilson-Fisher fixed point, Phys. Rev. Lett. 123 (2019) 051603 [arXiv:1902.09542] [INSPIRE].
A. De La Fuente, The large charge expansion at large N, JHEP 08 (2018) 041 [arXiv:1805.00501] [INSPIRE].
N. Karthik, Monopole scaling dimension using Monte-Carlo simulation, Phys. Rev. D 98 (2018) 074513 [arXiv:1808.08970] [INSPIRE].
A. Bourget, D. Rodriguez-Gomez and J.G. Russo, A limit for large R-charge correlators in \( \mathcal{N} \) = 2 theories, JHEP 05 (2018) 074 [arXiv:1803.00580] [INSPIRE].
M. Beccaria, On the large R-charge \( \mathcal{N} \) = 2 chiral correlators and the Toda equation, JHEP 02 (2019) 009 [arXiv:1809.06280] [INSPIRE].
M. Beccaria, Double scaling limit of N = 2 chiral correlators with Maldacena-Wilson loop, JHEP 02 (2019) 095 [arXiv:1810.10483] [INSPIRE].
A. Grassi, Z. Komargodski and L. Tizzano, Extremal Correlators and Random Matrix Theory, arXiv:1908.10306 [INSPIRE].
M. Beccaria, F. Galvagno and A. Hasan, \( \mathcal{N} \) = 2 conformal gauge theories at large R-charge: the SU(N) case, JHEP 03 (2020) 160 [arXiv:2001.06645] [INSPIRE].
Z. Komargodski, S.S. Razamat, O. Sela and A. Sharon, A Nilpotency Index of Conformal Manifolds, JHEP 10 (2020) 183 [arXiv:2003.04579] [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].
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].
M.J. Strassler, On renormalization group flows and exactly marginal operators in three-dimensions, hep-th/9810223 [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].
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].
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].
L. Álvarez-Gaumé, D. Orlando and S. Reffert, Large charge at large N, JHEP 12 (2019) 142 [arXiv:1909.02571] [INSPIRE].
E. Dyer, M. Mezei, S.S. Pufu and S. Sachdev, Scaling dimensions of monopole operators in the \( \mathrm{\mathbb{C}}{\mathrm{\mathbb{P}}}^{N_b-1} \) theory in 2 + 1 dimensions, JHEP 06 (2015) 037 [Erratum ibid. 03 (2016) 111] [arXiv:1504.00368] [INSPIRE].
W. Lerche, D. Lüst and N.P. Warner, Duality Symmetries in N = 2 Landau-Ginzburg Models, Phys. Lett. B 231 (1989) 417 [INSPIRE].
A. Sharon, Global Aspects of Spaces of Vacua, JHEP 11 (2020) 083 [arXiv:2004.11182] [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].
H. Watanabe, T. Brauner and H. Murayama, Massive Nambu-Goldstone Bosons, Phys. Rev. Lett. 111 (2013) 021601 [arXiv:1303.1527] [INSPIRE].
A. Trautner, CP and other Symmetries of Symmetries, Ph.D. Thesis, Munich, Tech. University, Universe (2016) [arXiv:1608.05240] [INSPIRE].
L.V. Delacretaz, Heavy Operators and Hydrodynamic Tails, SciPost Phys. 9 (2020) 034 [arXiv:2006.01139] [INSPIRE].
O. Loukas, D. Orlando, S. Reffert and D. Sarkar, An AdS/EFT correspondence at large charge, Nucl. Phys. B 934 (2018) 437 [arXiv:1804.04151] [INSPIRE].
A. de la Fuente and J. Zosso, The large charge expansion and AdS/CFT, JHEP 06 (2020) 178 [arXiv:2005.06169] [INSPIRE].
J. Polchinski, Effective field theory and the Fermi surface, in Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles, pp. 235–276, 1992 [hep-th/9210046] [INSPIRE].
R. Camporesi and A. Higuchi, On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1 [gr-qc/9505009] [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton, NJ, U.S.A. (1992).
S.M. Chester, M. Mezei, S.S. Pufu and I. Yaakov, Monopole operators from the 4 – ϵ expansion, JHEP 12 (2016) 015 [arXiv:1511.07108] [INSPIRE].
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Sharon, A., Watanabe, M. Transition of large R-charge operators on a conformal manifold. J. High Energ. Phys. 2021, 68 (2021). https://doi.org/10.1007/JHEP01(2021)068
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DOI: https://doi.org/10.1007/JHEP01(2021)068