Abstract
The coefficient τ RR of the two-point function of the superconformal U(1) R currents of \( \mathcal{N}=2 \) SCFTs in three-dimensions is recently shown to be obtained by differentiating the partition function on a squashed three-sphere with respect to the squashing parameter. With this method, we compute the τ RR for \( \mathcal{N}=2 \) Wess-Zumino models and SQCD numerically for small number of flavors and analytically in the large number limit. We study the behavior of τ RR under an RG flow by adding superpotentials to the theories. While the τ RR decreases for the gauge theories, we find an \( \mathcal{N}=2 \) Wess-Zumino model whose τ RR increases along the RG flow. Since τ RR is proportional to the coefficient C T of the two-point correlation function of the stress-energy tensors for \( \mathcal{N}=2 \) superconformal field theories, this rules out the possibility of C T being a measure of the degrees of freedom which monotonically decreases along RG flows in three-dimensions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-Theorem: \( \mathcal{N}=2 \) Field Theories on the Three-Sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact Terms, Unitarity and F-Maximization in Three-Dimensional Superconformal Theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
K.A. Intriligator and B. Wecht, The Exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without Supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
H. Liu and M. Mezei, A Refinement of entanglement entropy and the number of degrees of freedom, arXiv:1202.2070 [INSPIRE].
I.R. Klebanov, T. Nishioka, S.S. Pufu and B.R. Safdi, Is Renormalized Entanglement Entropy Stationary at RG Fixed Points?, JHEP 10 (2012) 058 [arXiv:1207.3360] [INSPIRE].
A. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
J.L. Cardy, Is there a c-theorem in four-dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
S. Sachdev, Polylogarithm identities in a conformal field theory in three-dimensions, Phys. Lett. B 309 (1993) 285 [hep-th/9305131] [INSPIRE].
H. Osborn and A. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
A. Cappelli, D. Friedan and J.I. Latorre, C theorem and spectral representation, Nucl. Phys. B 352 (1991) 616 [INSPIRE].
D. Anselmi, D. Freedman, M.T. Grisaru and A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526 (1998) 543 [hep-th/9708042] [INSPIRE].
A. Petkou, Conserved currents, consistency relations and operator product expansions in the conformally invariant O(N) vector model, Annals Phys. 249 (1996) 180 [hep-th/9410093] [INSPIRE].
A.C. Petkou, C T and C J up to next-to-leading order in 1/N in the conformally invariant 0(N) vector model for 2 < D < 4, Phys. Lett. B 359 (1995) 101 [hep-th/9506116] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
E. Barnes, E. Gorbatov, K.A. Intriligator, M. Sudano and J. Wright, The Exact superconformal R-symmetry minimizes τ RR , Nucl. Phys. B 730 (2005) 210 [hep-th/0507137] [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].
I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Entanglement Entropy of 3-D Conformal Gauge Theories with Many Flavors, JHEP 05 (2012) 036 [arXiv:1112.5342] [INSPIRE].
B.R. Safdi, I.R. Klebanov and J. Lee, A Crack in the Conformal Window, arXiv:1212.4502 [INSPIRE].
Y. Imamura, Relation between the 4D superconformal index and the S 3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].
Y. Imamura and D. Yokoyama, \( \mathcal{N}=2 \) supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
B. Willett and I. Yaakov, \( \mathcal{N}=2 \) Dualities and Z Extremization in Three Dimensions, arXiv:1104.0487 [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3D Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].
F. van de Bult, Hyperbolic hypergeometric functions, http://www.its.caltech.edu/˜vdbult/Thesis.pdf.
D. Martelli and J. Sparks, The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere, Nucl. Phys. B 866 (2013) 72 [arXiv:1111.6930] [INSPIRE].
E. Barnes, E. Gorbatov, K.A. Intriligator and J. Wright, Current correlators and AdS/CFT geometry, Nucl. Phys. B 732 (2006) 89 [hep-th/0507146] [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M. Strassler, Aspects of \( \mathcal{N}=2 \) supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of Boundary Conditions In \( \mathcal{N}=4 \) Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) [arXiv:0807.3720] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1303.1522
Rights and permissions
About this article
Cite this article
Nishioka, T., Yonekura, K. On RG flow of τ RR for supersymmetric field theories in three-dimensions. J. High Energ. Phys. 2013, 165 (2013). https://doi.org/10.1007/JHEP05(2013)165
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2013)165