Abstract
We study the constraints of crossing symmetry and unitarity for conformal field theories in the presence of a boundary, with a focus on the Ising model in various dimensions. We show that an analytic approach to the bootstrap is feasible for free-field theory and at one loop in the epsilon expansion, but more generally one has to resort to numerical methods. Using the recently developed linear programming techniques we find several interesting bounds for operator dimensions and OPE coefficients and comment on their physical relevance. We also show that the “boundary bootstrap” can be easily applied to correlation functions of tensorial operators and study the stress tensor as an example. In the appendices we present conformal block decompositions of a variety of physically interesting correlation functions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
S. Ferrara, R. Gatto and A. Grillo, Conformal Algebra in Space-Time and Operator Product Expansion, Springer (1973).
I.T. Todorov, M.C. Mintchev and V.B. Petkova, Conformal invariance in quantum field theory, Edizioni della Normale (1978).
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
V.S. Rychkov and A. Vichi, Universal Constraints on Conformal Operator Dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
F. Caracciolo and V.S. Rychkov, Rigorous Limits on the Interaction Strength in Quantum Field Theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Central Charge Bounds in 4D Conformal Field Theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].
D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D Conformal Field Theories with Global Symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].
A. Vichi, Improved bounds for CFT’s with global symmetries, JHEP 01 (2012) 162 [arXiv:1106.4037] [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].
S. Rychkov, Conformal Bootstrap in Three Dimensions?, arXiv:1111.2115 [INSPIRE].
S. El-Showk et al., Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
J.L. Cardy, Operator content and modular properties of higher dimensional conformal field theories, Nucl. Phys. B 366 (1991) 403 [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
S. Hellerman and C. Schmidt-Colinet, Bounds for State Degeneracies in 2D Conformal Field Theory, JHEP 08 (2011) 127 [arXiv:1007.0756] [INSPIRE].
D. Friedan, A. Konechny and C. Schmidt-Colinet, Lower bound on the entropy of boundaries and junctions in 1 + 1d quantum critical systems, Phys. Rev. Lett. 109 (2012) 140401 [arXiv:1206.5395] [INSPIRE].
J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].
K. Binder, Phase transitions and critical phenomena, vol. 8, Academic Press (1983).
J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [INSPIRE].
H. Diehl and S. Dietrich, Field-theoretical approach to multicritical behavior near free surfaces, Phys. Rev. B 24 (1981) 2878 [INSPIRE].
D. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655 [hep-th/9302068] [INSPIRE].
P. Di Francesco, P. Mathieu, and D. Senechal, Conformal field theory, chapter 11, Springer (1997).
D. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
P.A. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429.
G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].
M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical tables, Dover, New York, U.S.A., ninth dover printing, tenth gpo printing ed. (1964), pg. 559.
J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theories with A Higher Spin Symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Class. Quant. Grav. 30 (2013) 104003 [arXiv:1204.3882] [INSPIRE].
W. Diehl, Phase transitions and critical phenomena, vol. 10, Academic Press (1986).
J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, Cambridge, U.K. (1996).
H. Diehl and M. Shpot, Massive field theory approach to surface critical behavior in three-dimensional systems, Nucl. Phys. B 528 (1998) 595 [cond-mat/9804083] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
F. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liendo, P., Rastelli, L. & van Rees, B.C. The bootstrap program for boundary CFT d . J. High Energ. Phys. 2013, 113 (2013). https://doi.org/10.1007/JHEP07(2013)113
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2013)113