Abstract
We consider the one-parameter generalization S 4 q of 4-sphere with a conical singularity due to identification τ = τ +2πq in one isometric angle. We compute the value of the spectral zeta-function at zero \( \widehat{\zeta}(q)=\zeta \left(0;q\right) \) that controls the coefficient of the logarithmic UV divergence of the one-loop partition function on S 4 q . While the value of the conformal anomaly a-coefficient is proportional to \( \widehat{\zeta}(1) \), we argue that in general the second c ∼ C T anomaly coefficient is related to a particular combination of the second and first derivatives of \( \widehat{\zeta}(q) \) at q = 1. The universality of this relation for C T is supported also by examples in 6 and 2 dimensions. We use it to compute the c-coefficient for conformal higher spins finding that it coincides with the “r = −1” value of the one-parameter Ansatz suggested in arXiv:1309.0785. Like the sums of a s and c s coefficients, the regularized sum of \( {\widehat{\zeta}}_s(q) \) over the whole tower of conformal higher spins s = 1, 2,… is found to vanish, implying UV finiteness on S 4 q and thus also the vanishing of the associated Rényi entropy. Similar conclusions are found to apply to the standard 2-derivative massless higher spin tower. We also present an independent computation of the full set of conformal anomaly coefficients of the 6d Weyl graviton theory defined by a particular combination of the three 6d Weyl invariants that has a (2, 0) supersymmetric extension.
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References
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 and A.A. Tseytlin, On induced action for conformal higher spins in curved background, Nucl. Phys. B 919 (2017) 359 [arXiv:1702.00222] [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
A.A. Tseytlin, On limits of superstring in AdS 5 × S 5, Theor. Math. Phys. 133 (2002) 1376 [hep-th/0201112] [INSPIRE].
A.Y. Segal, Conformal higher spin theory, Nucl. Phys. B 664 (2003) 59 [hep-th/0207212] [INSPIRE].
X. Bekaert, E. Joung and J. Mourad, Effective action in a higher-spin background, JHEP 02 (2011) 048 [arXiv:1012.2103] [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].
M. Beccaria, S. Nakach and A.A. Tseytlin, On triviality of S-matrix in conformal higher spin theory, JHEP 09 (2016) 034 [arXiv:1607.06379] [INSPIRE].
T. Nutma and M. Taronna, On conformal higher spin wave operators, JHEP 06 (2014) 066 [arXiv:1404.7452] [INSPIRE].
M. Grigoriev and A.A. Tseytlin, On conformal higher spins in curved background, J. Phys. A 50 (2017) 125401 [arXiv:1609.09381] [INSPIRE].
R.R. Metsaev, Ordinary-derivative formulation of conformal totally symmetric arbitrary spin bosonic fields, JHEP 06 (2012) 062 [arXiv:0709.4392] [INSPIRE].
R.R. Metsaev, Arbitrary spin conformal fields in (A)dS, Nucl. Phys. B 885 (2014) 734 [arXiv:1404.3712] [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, On higher spin partition functions, J. Phys. A 48 (2015) 275401 [arXiv:1503.08143] [INSPIRE].
M.J. Duff, Observations on Conformal Anomalies, Nucl. Phys. B 125 (1977) 334 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469 [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].
D.V. Fursaev, Spectral geometry and one loop divergences on manifolds with conical singularities, Phys. Lett. B 334 (1994) 53 [hep-th/9405143] [INSPIRE].
D.V. Fursaev and S.N. Solodukhin, On the description of the Riemannian geometry in the presence of conical defects, Phys. Rev. D 52 (1995) 2133 [hep-th/9501127] [INSPIRE].
J.S. Apps and J.S. Dowker, The C(2) heat kernel coefficient in the presence of boundary discontinuities, Class. Quant. Grav. 15 (1998) 1121 [hep-th/9712019] [INSPIRE].
E. Perlmutter, A universal feature of CFT Rényi entropy, JHEP 03 (2014) 117 [arXiv:1308.1083] [INSPIRE].
D.V. Fursaev and S.N. Solodukhin, On one loop renormalization of black hole entropy, Phys. Lett. B 365 (1996) 51 [hep-th/9412020] [INSPIRE].
S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].
A.F. Astaneh, A. Patrushev and S.N. Solodukhin, Entropy vs Gravitational Action: Do Total Derivatives Matter?, arXiv:1411.0926 [INSPIRE].
D.V. Fursaev, A. Patrushev and S.N. Solodukhin, Distributional Geometry of Squashed Cones, Phys. Rev. D 88 (2013) 044054 [arXiv:1306.4000] [INSPIRE].
D.V. Fursaev and G. Miele, Cones, spins and heat kernels, Nucl. Phys. B 484 (1997) 697 [hep-th/9605153] [INSPIRE].
L. De Nardo, D.V. Fursaev and G. Miele, Heat kernel coefficients and spectra of the vector Laplacians on spherical domains with conical singularities, Class. Quant. Grav. 14 (1997) 1059 [hep-th/9610011] [INSPIRE].
M.A. Rubin and C.R. Ordónez, Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics, J. Math. Phys. 25 (1984) 2888.
B. Allen, Phase Transitions in de Sitter Space, Nucl. Phys. B 226 (1983) 228 [INSPIRE].
S.M. Christensen and M.J. Duff, Quantizing Gravity with a Cosmological Constant, Nucl. Phys. B 170 (1980) 480 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, One Loop Effective Potential in Gauged O(4) Supergravity, Nucl. Phys. B 234 (1984) 472 [INSPIRE].
M.G. Eastwood, Higher symmetries of the Laplacian, Annals Math. 161 (2005) 1645 [hep-th/0206233] [INSPIRE].
D.V. Fursaev and G. Miele, Finite temperature scalar field theory in static de Sitter space, Phys. Rev. D 49 (1994) 987 [hep-th/9302078] [INSPIRE].
J.S. Dowker, Renyi entropies and C T for higher derivative free scalars and spinors on even spheres, arXiv:1706.01369 [INSPIRE].
J.S. Dowker, Renyi entropy and C T for p-forms on even spheres, arXiv:1706.04574 [INSPIRE].
E. Shaghoulian, Modular invariance on S 1 × S 3 and circle fibrations, arXiv:1612.05257 [INSPIRE].
I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Renyi Entropies for Free Field Theories, JHEP 04 (2012) 074 [arXiv:1111.6290] [INSPIRE].
M. Beccaria and A.A. Tseytlin, C T for higher derivative conformal fields and anomalies of (1, 0) superconformal 6d theories, JHEP 06 (2017) 002 [arXiv:1705.00305] [INSPIRE].
A. Cappelli and A. Coste, On the Stress Tensor of Conformal Field Theories in Higher Dimensions, Nucl. Phys. B 314 (1989) 707 [INSPIRE].
D. Kutasov and F. Larsen, Partition sums and entropy bounds in weakly coupled CFT, JHEP 01 (2001) 001 [hep-th/0009244] [INSPIRE].
K.-W. Huang, Central Charge and Entangled Gauge Fields, Phys. Rev. D 92 (2015) 025010 [arXiv:1412.2730] [INSPIRE].
W. Donnelly, B. Michel and A. Wall, Electromagnetic Duality and Entanglement Anomalies, Phys. Rev. D 96 (2017) 045008 [arXiv:1611.05920] [INSPIRE].
G. Wong, A note on entanglement edge modes in Chern Simons theory, arXiv:1706.04666 [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].
M. Beccaria and A.A. Tseytlin, Conformal a-anomaly of some non-unitary 6d superconformal theories, JHEP 09 (2015) 017 [arXiv:1506.08727] [INSPIRE].
M. Beccaria and A.A. Tseytlin, Conformal anomaly c-coefficients of superconformal 6d theories, JHEP 01 (2016) 001 [arXiv:1510.02685] [INSPIRE].
A.A. Tseytlin, Weyl anomaly of conformal higher spins on six-sphere, Nucl. Phys. B 877 (2013) 632 [arXiv:1310.1795] [INSPIRE].
H.W.J. Bloete, J.L. Cardy and M.P. Nightingale, Conformal Invariance, the Central Charge and Universal Finite Size Amplitudes at Criticality, Phys. Rev. Lett. 56 (1986) 742 [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].
L. Bonora, P. Pasti and M. Bregola, Weyl cocycles, Class. Quant. Grav. 3 (1986) 635 [INSPIRE].
D. Butter, J. Novak and G. Tartaglino-Mazzucchelli, The component structure of conformal supergravity invariants in six dimensions, JHEP 05 (2017) 133 [arXiv:1701.08163] [INSPIRE].
D. Butter, S.M. Kuzenko, J. Novak and S. Theisen, Invariants for minimal conformal supergravity in six dimensions, JHEP 12 (2016) 072 [arXiv:1606.02921] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
R.R. Metsaev, 6d conformal gravity, J. Phys. A 44 (2011) 175402 [arXiv:1012.2079] [INSPIRE].
P.B. Gilkey, The Spectral geometry of a Riemannian manifold, J. Diff. Geom. 10 (1975) 601 [INSPIRE].
Y. Pang, One-Loop Divergences in 6D Conformal Gravity, Phys. Rev. D 86 (2012) 084039 [arXiv:1208.0877] [INSPIRE].
S. Giombi and I.R. Klebanov, One Loop Tests of Higher Spin AdS/CFT, JHEP 12 (2013) 068 [arXiv:1308.2337] [INSPIRE].
S. Giombi, I.R. Klebanov and A.A. Tseytlin, Partition Functions and Casimir Energies in Higher Spin AdS d+1 /CFT d , Phys. Rev. D 90 (2014) 024048 [arXiv:1402.5396] [INSPIRE].
S. Giombi, Higher Spin — CFT Duality, arXiv:1607.02967 [INSPIRE].
D. Iellici, Aspects and applications of quantum field theory on spaces with conical singularities, gr-qc/9805058 [INSPIRE].
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Beccaria, M., Tseytlin, A.A. C T for conformal higher spin fields from partition function on conically deformed sphere. J. High Energ. Phys. 2017, 123 (2017). https://doi.org/10.1007/JHEP09(2017)123
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DOI: https://doi.org/10.1007/JHEP09(2017)123