Abstract
Sigma models on coset superspaces, such as odd dimensional superspheres, play an important role in physics and in particular the AdS/CFT correspondence. In this work we apply recent general results on the spectrum of coset space models and on supergroup WZNW models to study the conformal sigma model with target space S 3|2. We construct its vertex operators and provide explicit formulas for their anomalous dimensions, at least to leading order in the sigma model coupling. The results are used to revisit a non-perturbative duality between the supersphere and the OSP(4|2) Gross-Neveu model that was conjectured by Candu and Saleur. With the help of powerful all-loop results for \( \frac{1}{2}\mathrm{B}\mathrm{P}\mathrm{S} \) operators in the Gross-Neveu model we are able to recover the entire zero mode spectrum of the sigma model at a certain finite value of the Gross-Neveu coupling. In addition, we argue that the sigma model constraints and equations of motion are implemented correctly in the dual Gross-Neveu description. On the other hand, high(er) gradient operators of the sigma model are not all accounted for. It is possible that this discrepancy is related to an instability from high gradient operators that has previously been observed in the context of Anderson localization.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.M. Polyakov, Interaction of Goldstone Particles in Two-Dimensions. Applications to Ferromagnets and Massive Yang-Mills Fields, Phys. Lett. B 59 (1975) 79 [INSPIRE].
E. Brézin and J. Zinn-Justin, Renormalization of the nonlinear σ-model in 2 + ϵ dimensions. Application to the Heisenberg ferromagnets, Phys. Rev. Lett. 36 (1976) 691 [INSPIRE].
D.H. Friedan, Nonlinear Models in 2 + ϵ Dimensions, Annals Phys. 163 (1985) 318 [INSPIRE].
A. D’Adda, M. Lüscher and P. Di Vecchia, A 1/n Expandable Series of Nonlinear σ-models with Instantons, Nucl. Phys. B 146 (1978) 63 [INSPIRE].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].
S.R. Coleman, The Quantum sine-Gordon Equation as the Massive Thirring Model, Phys. Rev. D 11 (1975) 2088 [INSPIRE].
S. Mandelstam, Soliton Operators for the Quantized sine-Gordon Equation, Phys. Rev. D 11 (1975) 3026 [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
C. Candu, V. Mitev and V. Schomerus, Spectra of Coset σ-models, Nucl. Phys. B 877 (2013) 900 [arXiv:1308.5981] [INSPIRE].
C. Candu and H. Saleur, A lattice approach to the conformal OSp(2S + 2|2S) supercoset σ-model. Part I: Algebraic structures in the spin chain. The Brauer algebra, Nucl. Phys. B 808 (2009) 441 [arXiv:0801.0430] [INSPIRE].
C. Candu and H. Saleur, A lattice approach to the conformal OSp(2S + 2|2S) supercoset σ-model. Part II: The boundary spectrum, Nucl. Phys. B 808 (2009) 487 [arXiv:0801.0444] [INSPIRE].
V. Mitev, T. Quella and V. Schomerus, Principal Chiral Model on Superspheres, JHEP 11 (2008) 086 [arXiv:0809.1046] [INSPIRE].
C. Candu, V. Mitev and V. Schomerus, Anomalous Dimensions in Deformed WZW Models on Supergroups, JHEP 03 (2013) 003 [arXiv:1211.2238] [INSPIRE].
A.M. Polyakov, Supermagnets and σ-models, hep-th/0512310 [INSPIRE].
T. Quella, V. Schomerus and T. Creutzig, Boundary Spectra in Superspace σ-models, JHEP 10 (2008) 024 [arXiv:0712.3549] [INSPIRE].
C. Candu, V. Mitev, T. Quella, H. Saleur and V. Schomerus, The σ-model on Complex Projective Superspaces, JHEP 02 (2010) 015 [arXiv:0908.0878] [INSPIRE].
M. Bershadsky, S. Zhukov and A. Vaintrob, P SL(n|n) σ-model as a conformal field theory, Nucl. Phys. B 559 (1999) 205 [hep-th/9902180] [INSPIRE].
N. Berkovits, C. Vafa and E. Witten, Conformal field theory of AdS background with Ramond-Ramond flux, JHEP 03 (1999) 018 [hep-th/9902098] [INSPIRE].
N. Read and H. Saleur, Exact spectra of conformal supersymmetric nonlinear σ-models in two-dimensions, Nucl. Phys. B 613 (2001) 409 [hep-th/0106124] [INSPIRE].
D. Kagan and C.A.S. Young, Conformal σ-models on supercoset targets, Nucl. Phys. B 745 (2006) 109 [hep-th/0512250] [INSPIRE].
A. Babichenko, Conformal invariance and quantum integrability of σ-models on symmetric superspaces, Phys. Lett. B 648 (2007) 254 [hep-th/0611214] [INSPIRE].
C. Candu, Discrétisation des modèles sigma invariants conformes sur des supersphères et superespaces projectifs, Ph.D. Thesis, Université Pierre et Marie Curie — Paris VI, Paris, France (2008).
R. Bott, Homogeneous vector bundles, Ann. Math. 66 (1957) 203.
J. Germoni, Indecomposable representations of osp(3|2), d(2,1;α) and g(3), Bol. Acad. Nac. Cienc. (Córdoba) 65 (2000) 147.
T. Quella and V. Schomerus, Superspace conformal field theory, J. Phys. A 46 (2013) 494010 [arXiv:1307.7724] [INSPIRE].
K. Pilch and A.N. Schellekens, Formulae for the Eigenvalues of the Laplacian on Tensor Harmonics on Symmetric Coset Spaces, J. Math. Phys. 25 (1984) 3455 [INSPIRE].
V.E. Kravtsov, I.V. Lerner and V.I. Yudson, Anomalous Dimensions of High Gradient Operators in the Extended Nonlinear σ Model and Distribution of Mesoscopic Fluctuations, Phys. Lett. A 134 (1989) 245 [INSPIRE].
F. Wegner, Anomalous dimensions of high-gradient operators in the n-vector model in 2 + ϵ dimensions, Z. Phys. B 78 (1990) 33.
F. Wegner, Anomalous dimensions of high gradient operators in the unitary matrix model, Nucl. Phys. B 354 (1991) 441 [INSPIRE].
H. Mall and F. Wegner, Anomalous dimensions of high gradient operators in the orthogonal matrix model, Nucl. Phys. B 393 (1993) 495 [INSPIRE].
A.N. Vasiliev and A.S. Stepanenko, A method of calculating the critical dimensions of composite operators in the massless nonlinear σ-model, Theor. Math. Phys. 94 (1993) 471 [INSPIRE].
K. Lang and W. Rühl, Critical nonlinear O(N) σ-models at 2 < d < 4: The degeneracy of quasiprimary fields and it resolution, Z. Phys. C 61 (1994) 495 [INSPIRE].
S.E. Derkachov, S.K. Kehrein and A.N. Manashov, High-gradient operators in the N-vector model, Nucl. Phys. B 493 (1997) 660 [INSPIRE].
S. Ryu, C. Mudry, A.W.W. Ludwig and A. Furusaki, High-gradient operators in perturbed Wess-Zumino-Witten field theories in two dimensions, Nucl. Phys. B 839 (2010) 341 [arXiv:1002.0118] [INSPIRE].
G.E. Castilla and S. Chakravarty, Is the phase transition in the Heisenberg model described by the (2+epsilon) expansion of the nonlinear σ-model?, Nucl. Phys. B 485 (1997) 613 [cond-mat/9605088] [INSPIRE].
E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
J. Van der Jeugt, Irreducible representations of the exceptional lie superalgebras D(2,1α), J. Math. Phys. 26 (1985) 913 [INSPIRE].
J. Van der Jeugt, Finite and infinite-dimensional representations of the orthosymplectic superalgebra OSP(3,2), J. Math. Phys. 25 (1984) 3334 [INSPIRE].
C. Candu, T. Creutzig, V. Mitev and V. Schomerus, Cohomological Reduction of σ-models, JHEP 05 (2010) 047 [arXiv:1001.1344] [INSPIRE].
F. Wegner, Anomalous Dimensions for the Nonlinear σ Model in (2 + ϵ)-dimensions. 2, Nucl. Phys. B 280 (1987) 210 [INSPIRE].
F. Wegner, Anomalous Dimensions for the Nonlinear σ Model in (2 + ϵ)-dimensions. 1, Nucl. Phys. B 280 (1987) 193 [INSPIRE].
C. Candu and V. Schomerus, Exactly marginal parafermions, Phys. Rev. D 84 (2011) 051704 [arXiv:1104.5028] [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: 1408.6838
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
Cagnazzo, A., Schomerus, V. & Tlapak, V. On the spectrum of superspheres. J. High Energ. Phys. 2015, 13 (2015). https://doi.org/10.1007/JHEP03(2015)013
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2015)013