Abstract
We compute the conformal anomaly a-coefficient for some non-unitary (higher derivative or non-gauge-invariant) 6d conformal fields and their supermultiplets. We use the method based on a connection between 6d determinants on S6 and 7d determinants on AdS7. We find, in particular, that (1,0) supermultiplet containing 4-derivative gauge-invariant conformal vector has precisely the value of a-anomaly as attributed in arXiv:1506.03807 (on the basis of R-symmetry and gravitational ’t Hooft anomaly matching) to the standard (1,0) vector multiplet. We also show that higher derivative (2,0) 6d conformal supergravity coupled to exactly 26 (2,0) tensor multiplets has vanishing a-anomaly (and also vanishing Casimir energy on 5-sphere). This is the 6d counterpart of the known fact of cancellation of the conformal anomaly in the 4d system of \( \mathcal{N}=4 \) conformal supergravity coupled to 4 vector \( \mathcal{N}=4 \) multiplets. In the case when 5 of tensor multiplets are chosen to be ghost-like and the conformal symmetry is spontaneously broken by a quadratic scalar constraint the resulting IR theory may be identified with (2,0) Poincaré supergravity coupled to 21 = 26 − 5 tensor multiplets. The latter theory is known to be special — it is gravitational anomaly free and results upon compactification of 10d type IIB supergravity on K3.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Cordova, T.T. Dumitrescu and X. Yin, Higher Derivative Terms, Toroidal Compactification and Weyl Anomalies in Six-Dimensional (2,0) Theories, arXiv:1505.03850 [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Anomalies, Renormalization Group Flows and the a-Theorem in Six-Dimensional (1,0) Theories, arXiv:1506.03807 [INSPIRE].
J. Heckman and C. Herzog, A Conformal Anomaly for 6d SCFT’s, private communication.
L. Bonora, P. Pasti and M. Bregola, Weyl cocycles, Class. Quant. Grav. 3 (1986) 635 [INSPIRE].
S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].
F. Bastianelli, S. Frolov and A.A. Tseytlin, Conformal anomaly of (2,0) tensor multiplet in six-dimensions and AdS/CFT correspondence, JHEP 02 (2000) 013 [hep-th/0001041] [INSPIRE].
A.A. Tseytlin, Weyl anomaly of conformal higher spins on six-sphere, Nucl. Phys. B 877 (2013) 632 [arXiv:1310.1795] [INSPIRE].
S. Giombi, I.R. Klebanov, S.S. Pufu, B.R. Safdi and G. Tarnopolsky, AdS Description of Induced Higher-Spin Gauge Theory, JHEP 10 (2013) 016 [arXiv:1306.5242] [INSPIRE].
S. Giombi, I.R. Klebanov and B.R. Safdi, Higher Spin AdS d+1 /CFT d at One Loop, Phys. Rev. D 89 (2014) 084004 [arXiv:1401.0825] [INSPIRE].
M. Beccaria and A.A. Tseytlin, Higher spins in AdS 5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT, JHEP 11 (2014) 114 [arXiv:1410.3273] [INSPIRE].
M. Beccaria, G. Macorini and A.A. Tseytlin, Supergravity one-loop corrections on AdS 7 and AdS 3 , higher spins and AdS/CFT, Nucl. Phys. B 892 (2015) 211 [arXiv:1412.0489] [INSPIRE].
E.A. Ivanov, A.V. Smilga and B.M. Zupnik, Renormalizable supersymmetric gauge theory in six dimensions, Nucl. Phys. B 726 (2005) 131 [hep-th/0505082] [INSPIRE].
E.A. Ivanov and A.V. Smilga, Conformal properties of hypermultiplet actions in six dimensions, Phys. Lett. B 637 (2006) 374 [hep-th/0510273] [INSPIRE].
A.V. Smilga, Chiral anomalies in higher-derivative supersymmetric 6D theories, Phys. Lett. B 647 (2007) 298 [hep-th/0606139] [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
J. Erdmenger and H. Osborn, Conformally covariant differential operators: Symmetric tensor fields, Class. Quant. Grav. 15 (1998) 273 [gr-qc/9708040] [INSPIRE].
S. El-Showk, Y. Nakayama and S. Rychkov, What Maxwell Theory in D ≠ 4 teaches us about scale and conformal invariance, Nucl. Phys. B 848 (2011) 578 [arXiv:1101.5385] [INSPIRE].
H. Samtleben, E. Sezgin and R. Wimmer, (1,0) superconformal models in six dimensions, JHEP 12 (2011) 062 [arXiv:1108.4060] [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal Anomaly in Weyl Theory and Anomaly Free Superconformal Theories, Phys. Lett. B 134 (1984) 187 [INSPIRE].
E. Bergshoeff, E. Sezgin and A. Van Proeyen, (2,0) tensor multiplets and conformal supergravity in D = 6, Class. Quant. Grav. 16 (1999) 3193 [hep-th/9904085] [INSPIRE].
P.K. Townsend, A New Anomaly Free Chiral Supergravity Theory From Compactification on K3, Phys. Lett. B 139 (1984) 283 [INSPIRE].
E. Witten, Five-branes and M-theory on an orbifold, Nucl. Phys. B 463 (1996) 383 [hep-th/9512219] [INSPIRE].
R.R. Metsaev, Massless mixed symmetry bosonic free fields in d-dimensional anti-de Sitter space-time, Phys. Lett. B 354 (1995) 78 [INSPIRE].
F.A. Dolan, Character formulae and partition functions in higher dimensional conformal field theory, J. Math. Phys. 47 (2006) 062303 [hep-th/0508031] [INSPIRE].
O.V. Shaynkman, I. Yu. Tipunin and M.A. Vasiliev, Unfolded form of conformal equations in M dimensions and o(M + 2) modules, Rev. Math. Phys. 18 (2006) 823 [hep-th/0401086] [INSPIRE].
X. Bekaert and M. Grigoriev, Higher order singletons, partially massless fields and their boundary values in the ambient approach, Nucl. Phys. B 876 (2013) 667 [arXiv:1305.0162] [INSPIRE].
G. Barnich, X. Bekaert and M. Grigoriev, Notes on conformal invariance of gauge fields, arXiv:1506.00595 [INSPIRE].
A.O. Barvinsky and D.V. Nesterov, Quantum effective action in spacetimes with branes and boundaries, Phys. Rev. D 73 (2006) 066012 [hep-th/0512291] [INSPIRE].
A.O. Barvinsky, Holography beyond conformal invariance and AdS isometry?, J. Exp. Theor. Phys. 120 (2015) 449 [arXiv:1410.6316] [INSPIRE].
S.S. Gubser and I.R. Klebanov, A Universal result on central charges in the presence of double trace deformations, Nucl. Phys. B 656 (2003) 23 [hep-th/0212138] [INSPIRE].
D.E. Diaz and H. Dorn, Partition functions and double-trace deformations in AdS/CFT, JHEP 05 (2007) 046 [hep-th/0702163] [INSPIRE].
D.E. Diaz, Polyakov formulas for GJMS operators from AdS/CFT, JHEP 07 (2008) 103 [arXiv:0803.0571] [INSPIRE].
A.A. Tseytlin, On partition function and Weyl anomaly of conformal higher spin fields, Nucl. Phys. B 877 (2013) 598 [arXiv:1309.0785] [INSPIRE].
M. Beccaria, X. Bekaert and A.A. Tseytlin, Partition function of free conformal higher spin theory, JHEP 08 (2014) 113 [arXiv:1406.3542] [INSPIRE].
S. Giombi, I.R. Klebanov and A.A. Tseytlin, Partition Functions and Casimir Energies in Higher Spin AdS d+1 /CF T d , Phys. Rev. D 90 (2014) 024048 [arXiv:1402.5396] [INSPIRE].
A. Cappelli and A. Coste, On the Stress Tensor of Conformal Field Theories in Higher Dimensions, Nucl. Phys. B 314 (1989) 707 [INSPIRE].
C.P. Herzog and K.-W. Huang, Stress Tensors from Trace Anomalies in Conformal Field Theories, Phys. Rev. D 87 (2013) 081901 [arXiv:1301.5002] [INSPIRE].
M.T. Grisaru, N.K. Nielsen, W. Siegel and D. Zanon, Energy Momentum Tensors, Supercurrents, (Super)traces and Quantum Equivalence, Nucl. Phys. B 247 (1984) 157 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Quantum Equivalence of Dual Field Theories, Annals Phys. 162 (1985) 31 [INSPIRE].
S. Ferrara and B. Zumino, Structure of Conformal Supergravity, Nucl. Phys. B 134 (1978) 301 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, One Loop β-function in Conformal Supergravities, Nucl. Phys. B 203 (1982) 157 [INSPIRE].
H. Liu and A.A. Tseytlin, D = 4 super Yang-Mills, D = 5 gauged supergravity and D = 4 conformal supergravity, Nucl. Phys. B 533 (1998) 88 [hep-th/9804083] [INSPIRE].
I.L. Buchbinder, N.G. Pletnev and A.A. Tseytlin, ’Induced’ \( \mathcal{N}=4 \) conformal supergravity, Phys. Lett. B 717 (2012) 274 [arXiv:1209.0416] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
M. Günaydin, P. van Nieuwenhuizen and N.P. Warner, General Construction of the Unitary Representations of Anti-de Sitter Superalgebras and the Spectrum of the S 4 Compactification of Eleven-dimensional Supergravity, Nucl. Phys. B 255 (1985) 63 [INSPIRE].
P. van Nieuwenhuizen, The Complete Mass Spectrum of d = 11 Supergravity Compactified on S 4 and a General Mass Formula for Arbitrary Cosets M 4, Class. Quant. Grav. 2 (1985) 1 [INSPIRE].
R.R. Metsaev, 6d conformal gravity, J. Phys. A 44 (2011) 175402 [arXiv:1012.2079] [INSPIRE].
J. Maldacena, Einstein Gravity from Conformal Gravity, arXiv:1105.5632 [INSPIRE].
A. Chang, J. Qing and P. Yang, On the renormalized volumes for conformally compact Einstein manifolds, math/0512376 [INSPIRE].
M. Nishimura and Y. Tanii, Local symmetries in the AdS 7 /CF T 6 correspondence, Mod. Phys. Lett. A 14 (1999) 2709 [hep-th/9910192] [INSPIRE].
G.W. Gibbons, M.J. Perry and C.N. Pope, Partition functions, the Bekenstein bound and temperature inversion in anti-de Sitter space and its conformal boundary, Phys. Rev. D 74 (2006) 084009 [hep-th/0606186] [INSPIRE].
E. Bergshoeff, M. de Roo and B. de Wit, Extended Conformal Supergravity, Nucl. Phys. B 182 (1981) 173 [INSPIRE].
F. Coomans and A. Van Proeyen, Off-shell N=(1,0), D = 6 supergravity from superconformal methods, JHEP 02 (2011) 049 [Erratum ibid. 1201 (2012) 119] [arXiv:1101.2403] [INSPIRE].
H. Romer and P. van Nieuwenhuizen, Axial Anomalies in N = 4 Conformal Supergravity, Phys. Lett. B 162 (1985) 290 [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \) symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
E. Bergshoeff, E. Sezgin and A. Van Proeyen, Superconformal Tensor Calculus and Matter Couplings in Six-dimensions, Nucl. Phys. B 264 (1986) 653 [Erratum ibid. B 598 (2001) 667] [INSPIRE].
L.J. Romans, Selfduality for Interacting Fields: Covariant Field Equations for Six-dimensional Chiral Supergravities, Nucl. Phys. B 276 (1986) 71 [INSPIRE].
F. Riccioni, Tensor multiplets in six-dimensional (2,0) supergravity, Phys. Lett. B 422 (1998) 126 [hep-th/9712176] [INSPIRE].
M. de Roo, Matter Coupling in \( \mathcal{N}=4 \) Supergravity, Nucl. Phys. B 255 (1985) 515 [INSPIRE].
S. Ferrara, R. Kallosh and A. Van Proeyen, Conjecture on hidden superconformal symmetry of \( \mathcal{N}=4 \) Supergravity, Phys. Rev. D 87 (2013) 025004 [arXiv:1209.0418] [INSPIRE].
J.J.M. Carrasco, R. Kallosh, R. Roiban and A.A. Tseytlin, On the U(1) duality anomaly and the S-matrix of \( \mathcal{N}=4 \) supergravity, JHEP 07 (2013) 029 [arXiv:1303.6219] [INSPIRE].
M. Beccaria and A.A. Tseytlin, On higher spin partition functions, J. Phys. A 48 (2015) 275401 [arXiv:1503.08143] [INSPIRE].
J. Erdmenger, Conformally covariant differential operators: Properties and applications, Class. Quant. Grav. 14 (1997) 2061 [hep-th/9704108] [INSPIRE].
E. Elizalde, M. Lygren and D.V. Vassilevich, Antisymmetric tensor fields on spheres: Functional determinants and nonlocal counterterms, J. Math. Phys. 37 (1996) 3105 [hep-th/9602113] [INSPIRE].
H. Samtleben, E. Sezgin, R. Wimmer and L. Wulff, New superconformal models in six dimensions: Gauge group and representation structure, PoS(CORFU2011)071 [arXiv:1204.0542] [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Quantum Properties of Higher Dimensional and Dimensionally Reduced Supersymmetric Theories, Nucl. Phys. B 227 (1983) 252 [INSPIRE].
C. Graham, Conformal powers of the Laplacian via stereographic projection, SIGMA 3 (2007) 121 [arXiv:0711.4798].
R. Manvelyan and D.H. Tchrakian, Conformal coupling of the scalar field with gravity in higher dimensions and invariant powers of the Laplacian, Phys. Lett. B 644 (2007) 370 [hep-th/0611077] [INSPIRE].
A. Juhl, Explicit formulas for GJMS-operators and Q-curvatures, arXiv:1108.0273 [INSPIRE].
Y. Pang, One-Loop Divergences in 6D Conformal Gravity, Phys. Rev. D 86 (2012) 084039 [arXiv:1208.0877] [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: 1506.08727
Also at Lebedev Institute, Moscow. (Arkady A. Tseytlin)
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
Beccaria, M., Tseytlin, A.A. Conformal a-anomaly of some non-unitary 6d superconformal theories. J. High Energ. Phys. 2015, 17 (2015). https://doi.org/10.1007/JHEP09(2015)017
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2015)017