Abstract
We propose a model for the dS/CFT correspondence. The model is constructed in terms of a “Yang–Baxter operator” R for unitary representations of the de Sitter group \({SO(d,1)}\). This R-operator is shown to satisfy the Yang–Baxter equation, unitarity, as well as certain analyticity relations, including in particular a crossing symmetry. With the aid of this operator we construct: (a) a chiral (light-ray) conformal quantum field theory whose internal degrees of freedom transform under the given unitary representation of \({SO(d,1)}\). By analogy with the O(N) non-linear sigma model, this chiral CFT can be viewed as propagating in a de Sitter spacetime. (b) A (non-unitary) Euclidean conformal quantum field theory on \({\mathbb{R}^{d-1}}\), where SO(d, 1) now acts by conformal transformations in (Euclidean) spacetime. These two theories can be viewed as dual to each other if we interpret \({\mathbb{R}^{d-1}}\) as conformal infinity of de Sitter spacetime. Our constructions use semi-local generator fields defined in terms of R and abstract methods from operator algebras.
Article PDF
References
Arutyunov G., Frolov S.: Foundations of the \({AdS_5 \times S^5}\) Superstring. Part I. J. Phys. A 42, 254003 (2009)
Abdalla E., Abdalla M., Rothe K.: Non-perturbative methods in two-dimensional quantum field theory. World Scientific, Singapore (2001)
Arinshtein A.E., Fateev V.A., Zamolodchikov A.B.: Quantum S-matrix of the (1 + 1)-dimensional toda chain. Phys. Lett. B 87, 389–392 (1979)
Alazzawi, S.: Deformations of quantum field theories and the construction of interacting models. Ph.D. Thesis, University of Vienna (2014). arXiv:1503.00897
Beisert N. et al.: Review of AdS/CFT integrability: an overview. Lett. Math. Phys. 99, 3 (2012)
Bombardelli D. et al.: An integrability primer for the gauge–gravity correspondence: an introduction. J. Phys. A 49(32), 320301 (2016)
Borchers H., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219, 125–140 (2001) arXiv:hep-th/0003243
Bostelmann H., Cadamuro D.: An operator expansion for integrable quantum field theories. J. Phys. A: Math. Theor. 46, 095401 (2012)
Babujian H.M., Foerster A., Karowski M.: The form factor program: a review and new results—the nested SU(N) off-shell bethe ansatz. SIGMA 2, 082 (2006) arXiv:hep-th/0609130
Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–786 (2002) arXiv:math-ph/0203021
Buchholz D., Lechner G.: Modular nuclearity and localization. Ann. Henri Poincaré 5, 1065–1080 (2004) arXiv:math-ph/0402072
Bostelmann H., Lechner G., Morsella G.: Scaling limits of integrable quantum field theories. Rev. Math. Phys. 23, 1115–1156 (2011) arXiv:1105.2781
Bros J., Moschella U.: Two-point functions and quantum fields in de Sitter universe. Rev. Math. Phys. 8, 327–392 (1996) arXiv:gr-qc/9511019
Bostelmann H.: Operator product expansions as a consequence of phase space properties. J. Math. Phys. 46, 082304 (2005) arXiv:math-ph/0502004v3
Bischoff M., Tanimoto Y.: Integrable QFT and Longo–Witten endomorphisms. Ann. Henri Poincaré 16, 569–608 (2015) arXiv:1305.2171
Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303–321 (1976)
Chicherin D., Derkachov S., Isaev A.P.: Conformal algebra: R-matrix and star-triangle relation. JHEP 1304, 20 (2013) arXiv:1206.4150v2
de Medeiros P., Hollands S.: Conformal symmetry superalgebras. Class. Quant. Grav. 30, 175016 (2013)
Derkachov S.E., Korchemsky G.P., Manashov A.N.: Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables. Nucl. Phys. B 617, 375–440 (2001) arXiv:hep-th/0107193v2
Derkachov S.E., Manashov A.N.: R-matrix and baxter Q-operators for the noncompact SL(N,C) invariant spin chain. SIGMA 2, 084 (2006) arXiv:nlin/0612003v1
Derkachov S.E., Manashov A.N.: Noncompact SL(N) spin chains: BGG-resolution, Q-operators and alternating sum representation for finite dimensional transfer matrices. Lett. Math. Phys. 97, 185–202 (2011) arXiv:1008.4734v2
Duren P.: Theory of H p Spaces. Dover Books on Mathematics. Dover Publications, Inc., New York (1970)
Epstein H., Moschella U.: de Sitter tachyons and related topics. Commun. Math. Phys. 336(1), 381–430 (2015) arXiv:1403.3319v2
Faddeev, L.D.: Quantum completely integrable models in field theory, volume~1 of Mathematical Physics Reviews, pp. 107–155 (1984). In Novikov, S.P. (Ed.): Mathematical Physics Reviews, Vol. 1, 107–155
Fredenhagen K., Hertel J.: Local algebras of observables and point-like localized fields. Commun. Math. Phys. 80, 555 (1981)
Fröhlich J., Osterwalder K., Seiler E.: On Virtual representations of symmetric spaces and their analytic continuation. Ann. Math. 118, 461–489 (1983) http://www.jstor.org/stable/2006979
Guica M., Hartman T., Song W., Strominger A.: The Kerr/CFT correspondence. Phys. Rev. D 80, 124008 (2009). arXiv:0809.4266v1
Guido D., Longo R., Wiesbrock H.: Extensions of conformal nets and superselection structures. Commun. Math. Phys. 192, 217–244 (1998) arXiv:hep-th/9703129
Goodman R., Wallach N.R.: Symmetry, Representations, and Invariants. Springer, Berlin (2009)
Haag R.: Local Quantum Physics—Fields, Particles, Algebras, 2nd edn. Springer, Berlin (1996)
Hollands S.: Massless interacting quantum fields in de Sitter spacetime. Ann. Henri Poincaré 13, 1039–1081 (2012)
Hollands S.: Correlators, Feynman diagrams, and quantum no-hair in de Sitter spacetime. Commun. Math. Phys. 319, 1–68 (2013)
Hull C.M.: Timelike T duality, de Sitter space, large N gauge theories and topological field theory. JHEP 9807, 021 (1998)
Iagolnitzer D.: Scattering in Quantum Field Theories. Princeton University Press, Princeton (1993)
Ketov S.V.: Quantum Non-linear Sigma-Models. Springer, Berlin (2000)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras II—Advanced Theory (1986)
Lang S.: SL 2(R). Springer, (1975)
Lechner G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64, 137–154 (2003) arXiv:hep-th/0303062
Lechner, G.: On the construction of quantum field theories with factorizing S-matrices. Ph.D. Thesis, University of Göttingen (2006). arXiv:math-ph/0611050
Lechner, G.: Algebraic constructive quantum field theory: integrable models and deformation techniques. In: Brunetti, R. et al. (eds.) Advances in Algebraic Quantum Field Theory, pp. 397–449. Springer, Berlin (2015)
Liguori A., Mintchev M.: Fock spaces with generalized statistics. Lett. Math. Phys. 33, 283–295 (1995)
Lechner G., Schützenhofer C.: Towards an operator-algebraic construction of integrable global gauge theories. Ann. Henri Poincaré 15, 645–678 (2014) arXiv:1208.2366v1
Maldacena J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998)
Marolf D., Morrison I.A.: The IR stability of de Sitter QFT: results at all orders. Phys. Rev. 84, 044040 (2011) arXiv:1010.5327v2
Neeb K.-H., Olafsson G.: Reflection positivity and conformal symmetry. J. Funct. Anal. 266, 2174–2224 (2014) arXiv:1206.2039
Schwartz L.: Theorie des Distributions. Hermann, Paris (1966)
Schmüdgen K.: An operator-theoretic approach to a cocycle problem in the complex plane. Bull. Lond. Math. Soc. 27, 341–346 (1995)
Smirnov F.A.: Form Factors in Completely Integrable Models of Quantum Field Theory. World Scientific, Singapore (1992)
Strominger A.: The dS/CFT Correspondence. JHEP 0110, 341–346 (2001) arXiv:hep-th/0106113v2
Schroer B., Wiesbrock H.: Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12, 301–326 (2000) arXiv:hep-th/9812251
Vilenkin N., Klimyk A.: Representations of Lie Groups and Special Functions Vol. I. Kluwer, Dordrecht (1991)
Witten E.: Anti De Sitter Space And Holography. Adv. Theor. Math. Phys. 2, 253–291 (1998) arXiv:hep-th/9802150v2
Zamolodchikov A.: Relativistic factorized S-matrix in two dimensions having O(N) isotopic symmetry. Nucl. Phys. B 133, 525–535 (1978)
Zamolodchikov A.B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models. Ann. Phys. 120, 253–291 (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D. Buchholz, K. Fredenhagen, Y. Kawahigashi.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hollands, S., Lechner, G. \({{SO(d,1)}}\)-Invariant Yang–Baxter Operators and the dS/CFT Correspondence. Commun. Math. Phys. 357, 159–202 (2018). https://doi.org/10.1007/s00220-017-2942-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-2942-6