Abstract
We study the kinematics of correlation functions of local and extended operators in a conformal field theory. We present a new method for constructing the tensor structures associated to primary operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate various techniques to generate the bulk channel blocks either from a radial expansion or by acting with differential operators on simpler seed blocks. For the defect channel, we detail a method to compute the blocks in closed form, in terms of projectors into mixed symmetry representations of the orthogonal group.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys.B 240 (1984) 514 [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFT d, JHEP07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP03 (2014)100 [arXiv:1310.5078] [INSPIRE].
F. Gliozzi, P. Liendo, M. Meineri and A. Rago, Boundary and interface CFTs from the conformal bootstrap, JHEP05 (2015) 036 [arXiv:1502.07217] [INSPIRE].
F. Gliozzi, Truncatable bootstrap equations in algebraic form and critical surface exponents, JHEP10 (2016) 037 [arXiv:1605.04175] [INSPIRE].
P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping the half-BPS line defect, JHEP10 (2018) 077 [arXiv:1806.01862] [INSPIRE].
M. Hogervorst, Crossing kernels for boundary and crosscap CFTs, arXiv:1703.08159 [INSPIRE].
M. Lemos, P. Liendo, M. Meineri and S. Sarkar, Universality at large transverse spin in defect CFT, JHEP09 (2018) 091 [arXiv:1712.08185] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques and applications, Rev. Mod. Phys.91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, JHEP05 (2019) 098 [arXiv:1705.04278] [INSPIRE].
A. Dymarsky, On the four-point function of the stress-energy tensors in a CFT, JHEP10 (2015) 075 [arXiv:1311.4546] [INSPIRE].
A. Dymarsky, F. Kos, P. Kravchuk, D. Poland and D. Simmons-Duffin, The 3d stress-tensor bootstrap, JHEP02 (2018) 164 [arXiv:1708.05718] [INSPIRE].
S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A general proof of the quantum null energy condition, arXiv:1706.09432 [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, JHEP10 (2014) 178 [arXiv:1407.6429] [INSPIRE].
L. Bianchi, M. Meineri, R.C. Myers and M. Smolkin, Rényi entropy and conformal defects, JHEP07 (2016) 076 [arXiv:1511.06713] [INSPIRE].
A. Lewkowycz and J. Maldacena, Exact results for the entanglement entropy and the energy radiated by a quark, JHEP05 (2014) 025 [arXiv:1312.5682] [INSPIRE].
B. Fiol, E. Gerchkovitz and Z. Komargodski, Exact bremsstrahlung function in N = 2 superconformal field theories, Phys. Rev. Lett.116 (2016) 081601 [arXiv:1510.01332] [INSPIRE].
L. Bianchi, M. Lemos and M. Meineri, Line defects and radiation in N = 2 conformal theories, Phys. Rev. Lett.121 (2018) 141601 [arXiv:1805.04111] [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].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
S. Guha and B. Nagaraj, Correlators of mixed symmetry operators in defect CFTs, JHEP10 (2018) 198 [arXiv:1805.12341] [INSPIRE].
M. Fukuda, N. Kobayashi and T. Nishioka, Operator product expansion for conformal defects, JHEP01 (2018) 013 [arXiv:1710.11165] [INSPIRE].
P. Liendo and C. Meneghelli, Bootstrap equations for N = 4 SYM with defects, JHEP01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
L. Rastelli and X. Zhou, The Mellin formalism for boundary CFT d, JHEP10 (2017) 146 [arXiv:1705.05362] [INSPIRE].
V. Goncalves and G. Itsios, A note on defect Mellin amplitudes, arXiv:1803.06721 [INSPIRE].
N. Kobayashi and T. Nishioka, Spinning conformal defects, JHEP09 (2018) 134 [arXiv:1805.05967] [INSPIRE].
M. Billò, M. Caselle, D. Gaiotto, F. Gliozzi, M. Meineri and R. Pellegrini, Line defects in the 3d Ising model, JHEP07 (2013) 055 [arXiv:1304.4110] [INSPIRE].
E. Lauria, M. Meineri and E. Trevisani, Radial coordinates for defect CFTs, JHEP11 (2018) 148 [arXiv:1712.07668] [INSPIRE].
M. Isachenkov, P. Liendo, Y. Linke and V. Schomerus, Calogero-Sutherland approach to defect blocks, JHEP10 (2018) 204 [arXiv:1806.09703] [INSPIRE].
A. Gadde, Conformal constraints on defects, arXiv:1602.06354 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP02 (2015) 151 [arXiv:1411.7351] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP07 (2016) 018 [arXiv:1603.05551] [INSPIRE].
V.K. Dobrev, V.B. Petkova, S.G. Petrova and I.T. Todorov, Dynamical derivation of vacuum operator product expansion in Euclidean conformal quantum field theory, Phys. Rev. D13 (1976)887 [INSPIRE].
F. Rejon-Barrera and D. Robbins, Scalar-vector bootstrap, JHEP01 (2016) 139 [arXiv:1508.02676] [INSPIRE].
M.S. Costa and T. Hansen, AdS weight shifting operators, JHEP09 (2018) 040 [arXiv:1805.01492] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Deconstructing conformal blocks in 4D CFT, JHEP08 (2015) 101 [arXiv:1505.03750] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, JHEP02 (2018) 096 [arXiv:1612.08987] [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys.96 (1984) 419 [INSPIRE].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Radial expansion for spinning conformal blocks, JHEP07 (2016) 057 [arXiv:1603.05552] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion relations for conformal blocks, JHEP09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP11 (2018) 102 [arXiv:1805.00098] [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].
R. de Mello Koch, P. Rabambi, R. Rabe and S. Ramgoolam, Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds, JHEP08 (2017) 077 [arXiv:1705.06702] [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].
L. Bianchi, M. Preti and E. Vescovi, Exact bremsstrahlung functions in ABJM theory, JHEP07 (2018) 060 [arXiv:1802.07726] [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: 1807.02522
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lauria, E., Meineri, M. & Trevisani, E. Spinning operators and defects in conformal field theory. J. High Energ. Phys. 2019, 66 (2019). https://doi.org/10.1007/JHEP08(2019)066
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2019)066