Abstract
In the context of boundary conformal field theory, we derive a sum rule that relates two and three point functions of the displacement operator. For four dimensional conformal field theory with a three dimensional boundary, this sum rule in turn relates the two boundary contributions to the anomaly in the trace of the stress tensor. We check our sum rule for a variety of free theories and also for a weakly interacting theory, where a free scalar in the bulk couples marginally to a generalized free field on the boundary.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H.W. Diehl, The Theory of boundary critical phenomena, Int. J. Mod. Phys. B 11 (1997) 3503 [cond-mat/9610143] [INSPIRE].
D. Carmi, L. Di Pietro and S. Komatsu, A Study of Quantum Field Theories in AdS at Finite Coupling, JHEP 01 (2019) 200 [arXiv:1810.04185] [INSPIRE].
C.P. Herzog and I. Shamir, On Marginal Operators in Boundary Conformal Field Theory, JHEP 10 (2019) 088 [arXiv:1906.11281] [INSPIRE].
L. Bianchi, M. Meineri, R.C. Myers and M. Smolkin, Rényi entropy and conformal defects, JHEP 07 (2016) 076 [arXiv:1511.06713] [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].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [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].
C. Herzog, K.-W. Huang and K. Jensen, Displacement Operators and Constraints on Boundary Central Charges, Phys. Rev. Lett. 120 (2018) 021601 [arXiv:1709.07431] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
T. Hartman, S. Kundu and A. Tajdini, Averaged Null Energy Condition from Causality, JHEP 07 (2017) 066 [arXiv:1610.05308] [INSPIRE].
D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A Proof of the Conformal Collider Bounds, JHEP 06 (2016) 111 [arXiv:1603.03771] [INSPIRE].
C. Behan, L. Di Pietro, E. Lauria and B.C. Van Rees, Bootstrapping boundary-localized interactions, JHEP 12 (2020) 182 [arXiv:2009.03336] [INSPIRE].
L. Di Pietro, E. Lauria and P. Niro, 3d large N vector models at the boundary, SciPost Phys. 11 (2021) 050 [arXiv:2012.07733] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The Bootstrap Program for Boundary CFTd, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
K. Sen and Y. Tachikawa, First-order conformal perturbation theory by marginal operators, arXiv:1711.05947 [INSPIRE].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
E. Lauria, M. Meineri and E. Trevisani, Spinning operators and defects in conformal field theory, JHEP 08 (2019) 066 [arXiv:1807.02522] [INSPIRE].
S. Guha and B. Nagaraj, Correlators of Mixed Symmetry Operators in Defect CFTs, JHEP 10 (2018) 198 [arXiv:1805.12341] [INSPIRE].
D. Simmons-Duffin, Projectors, Shadows, and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151 [arXiv:1411.7351] [INSPIRE].
D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
L. Di Pietro, D. Gaiotto, E. Lauria and J. Wu, 3d Abelian Gauge Theories at the Boundary, JHEP 05 (2019) 091 [arXiv:1902.09567] [INSPIRE].
D.M. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655 [hep-th/9302068] [INSPIRE].
J. Erdmenger and H. Osborn, Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys. B 483 (1997) 431 [hep-th/9605009] [INSPIRE].
J.L. Cardy, Anisotropic Corrections to Correlation Functions in Finite Size Systems, Nucl. Phys. B 290 (1987) 355 [INSPIRE].
O. Aharony, O. DeWolfe, D.Z. Freedman and A. Karch, Defect conformal field theory and locally localized gravity, JHEP 07 (2003) 030 [hep-th/0303249] [INSPIRE].
O. Aharony, A.B. Clark and A. Karch, The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture, Phys. Rev. D 74 (2006) 086006 [hep-th/0608089] [INSPIRE].
J.L. Cardy, Universal critical point amplitudes in parallel plate geometries, Phys. Rev. Lett. 65 (1990) 1443 [INSPIRE].
M. Fukuda, N. Kobayashi and T. Nishioka, Operator product expansion for conformal defects, JHEP 01 (2018) 013 [arXiv:1710.11165] [INSPIRE].
E. Lauria, P. Liendo, B.C. Van Rees and X. Zhao, Line and surface defects for the free scalar field, JHEP 01 (2021) 060 [arXiv:2005.02413] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions, and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight Shifting Operators and Conformal Blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products, Lett. Nuovo Cim. 4S2 (1972) 115 [INSPIRE].
N. Drukker, M. Probst and M. Trépanier, Defect CFT techniques in the 6d \( \mathcal{N} \) = (2, 0) theory, JHEP 03 (2021) 261 [arXiv:2009.10732] [INSPIRE].
L. Bianchi and M. Lemos, Superconformal surfaces in four dimensions, JHEP 06 (2020) 056 [arXiv:1911.05082] [INSPIRE].
N. Drukker, I. Shamir and C. Vergu, Defect multiplets of \( \mathcal{N} \) = 1 supersymmetry in 4d, JHEP 01 (2018) 034 [arXiv:1711.03455] [INSPIRE].
N. Drukker, D. Martelli and I. Shamir, The energy-momentum multiplet of supersymmetric defect field theories, JHEP 08 (2017) 010 [arXiv:1701.04323] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2107.11604
Rights and permissions
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.
About this article
Cite this article
Herzog, C.P., Schaub, V. A sum rule for boundary contributions to the trace anomaly. J. High Energ. Phys. 2022, 121 (2022). https://doi.org/10.1007/JHEP01(2022)121
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2022)121