Abstract
We compute conformal anomalies for conformal field theories with free conformal scalars and massless spin 1/2 fields in hyperbolic space ℍd and in the ball \( {\mathbb{B}}^d \), for 2≤d≤7. These spaces are related by a conformal transformation. In even dimensional spaces, the conformal anomalies on ℍ2n and \( {\mathbb{B}}^{2n} \) are shown to be identical. In odd dimensional spaces, the conformal anomaly on \( {\mathbb{B}}^{2n+1} \) comes from a boundary contribution, which exactly coincides with that of ℍ2n + 1 provided one identifies the UV short-distance cutoff on \( {\mathbb{B}}^{2n+1} \) with the inverse large distance IR cutoff on ℍ2n + 1, just as prescribed by the conformal map. As an application, we determine, for the first time, the conformal anomaly coefficients multiplying the Euler characteristic of the boundary for scalars and half-spin fields with various boundary conditions in d = 5 and d = 7.
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References
J.S. Dowker and J.P. Schofield, Conformal Transformations and the Effective Action in the Presence of Boundaries, J. Math. Phys. 31 (1990) 808 [INSPIRE].
D.V. Fursaev, Quantum Entanglement on Boundaries, JHEP 07 (2013) 119 [arXiv:1305.2334] [INSPIRE].
K. Jensen and A. O’Bannon, Holography, Entanglement Entropy and Conformal Field Theories with Boundaries or Defects, Phys. Rev. D 88 (2013) 106006 [arXiv:1309.4523] [INSPIRE].
K. Jensen and A. O’Bannon, Constraint on Defect and Boundary Renormalization Group Flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [INSPIRE].
C.P. Herzog, K.-W. Huang and K. Jensen, Universal Entanglement and Boundary Geometry in Conformal Field Theory, JHEP 01 (2016) 162 [arXiv:1510.00021] [INSPIRE].
D. Fursaev, Conformal anomalies of CFT’s with boundaries, JHEP 12 (2015) 112 [arXiv:1510.01427] [INSPIRE].
S.N. Solodukhin, Boundary terms of conformal anomaly, Phys. Lett. B 752 (2016) 131 [arXiv:1510.04566] [INSPIRE].
D.V. Fursaev and S.N. Solodukhin, Anomalies, entropy and boundaries, Phys. Rev. D 93 (2016) 084021 [arXiv:1601.06418] [INSPIRE].
K.-W. Huang, Boundary Anomalies and Correlation Functions, JHEP 08 (2016) 013 [arXiv:1604.02138] [INSPIRE].
C.P. Herzog and K.-W. Huang, Boundary Conformal Field Theory and a Boundary Central Charge, JHEP 10 (2017) 189 [arXiv:1707.06224] [INSPIRE].
D. Rodriguez-Gomez and J.G. Russo, Free energy and boundary anomalies on \( {\mathbb{S}}^a \times {\mathrm{\mathbb{H}}}^b \) spaces, JHEP 10 (2017) 084 [arXiv:1708.00305] [INSPIRE].
C. Herzog, K.-W. Huang and K. Jensen, Displacement Operators and Constraints on Boundary Central Charges, arXiv:1709.07431 [INSPIRE].
A.A. Bytsenko, E. Elizalde and S.D. Odintsov, The Conformal anomaly in N-dimensional spaces having a hyperbolic spatial section, J. Math. Phys. 36 (1995) 5084 [gr-qc/9505047] [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].
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. Levitin, Dirichlet and Neumann heat invariants for Euclidean balls, Diff. Geom. Appl. 8 (1998) 35.
M. Bordag, E. Elizalde and K. Kirsten, Heat kernel coefficients of the Laplace operator on the D-dimensional ball, J. Math. Phys. 37 (1996) 895 [hep-th/9503023] [INSPIRE].
J.S. Dowker, J.S. Apps, K. Kirsten and M. Bordag, Spectral invariants for the Dirac equation on the d ball with various boundary conditions, Class. Quant. Grav. 13 (1996) 2911 [hep-th/9511060] [INSPIRE].
P.D. D’Eath and G. Esposito, Spectral boundary conditions in one loop quantum cosmology, Phys. Rev. D 44 (1991) 1713 [gr-qc/9507005] [INSPIRE].
M.J. Duff, Observations on Conformal Anomalies, Nucl. Phys. B 125 (1977) 334 [INSPIRE].
S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].
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ArXiv ePrint: 1710.09327
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Rodriguez-Gomez, D., Russo, J.G. Boundary conformal anomalies on hyperbolic spaces and Euclidean balls. J. High Energ. Phys. 2017, 66 (2017). https://doi.org/10.1007/JHEP12(2017)066
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DOI: https://doi.org/10.1007/JHEP12(2017)066