Abstract
In this paper we consider a conformal invariant chain of L sites in the unitary irreducible representations of the group SO(1, 5). The k-th site of the chain is defined by a scaling dimension ∆k and spin numbers \( \frac{\ell_k}{2},\frac{\ell_k}{2} \) The model with open and fixed boundaries is shown to be integrable at the quantum level and its spectrum and eigenfunctions are obtained by separation of variables. The transfer matrices of the chain are graph-builder operators for the spinning and inhomogeneous generalization of squared-lattice “fishnet” integrals on the disk. As such, their eigenfunctions are used to diagonalize the mirror channel of the Feynman diagrams of Fishnet conformal field theories. The separated variables are interpreted as momentum and bound-state index of the mirror excitations of the lattice: particles with SO(4) internal symmetry that scatter according to an integrable factorized \( \mathcal{S} \)-matrix in (1 + 1) dimensions
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
O. Gürdoğan and V. Kazakov, New Integrable 4D Quantum Field Theories from Strongly Deformed Planar \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 117 (2016) 201602 [Addendum ibid. 117 (2016) 259903] [arXiv:1512.06704] [INSPIRE].
J. Caetano, O. Gürdoğan and V. Kazakov, Chiral limit of \( \mathcal{N} \) = 4 SYM and ABJM and integrable Feynman graphs, JHEP 03 (2018) 077 [arXiv:1612.05895] [INSPIRE].
O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].
S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].
N. Beisert and R. Roiban, Beauty and the twist: The Bethe ansatz for twisted N = 4 SYM, JHEP 08 (2005) 039 [hep-th/0505187] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, A piece of cake: the ground-state energies in γi -deformed \( \mathcal{N} \) = 4 SYM theory at leading wrapping order, JHEP 09 (2014) 078 [arXiv:1405.6712] [INSPIRE].
C. Sieg and M. Wilhelm, On a CFT limit of planar γi -deformed \( \mathcal{N} \) = 4 SYM theory, Phys. Lett. B 756 (2016) 118 [arXiv:1602.05817] [INSPIRE].
D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, Strongly γ-Deformed \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory, Phys. Rev. Lett. 120 (2018) 111601 [arXiv:1711.04786] [INSPIRE].
V. Kazakov, E. Olivucci and M. Preti, Generalized fishnets and exact four-point correlators in chiral CFT4 , JHEP 06 (2019) 078 [arXiv:1901.00011] [INSPIRE].
A. Pittelli and M. Preti, Integrable fishnet from gamma-deformed \( \mathcal{N} \) = 2 quivers, Phys. Lett. B 798 (2019) 134971.
N. Gromov, V. Kazakov, G. Korchemsky, S. Negro and G. Sizov, Integrability of Conformal Fishnet Theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [INSPIRE].
D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, to appear.
V. Kazakov and E. Olivucci, Biscalar Integrable Conformal Field Theories in Any Dimension, Phys. Rev. Lett. 121 (2018) 131601 [arXiv:1801.09844] [INSPIRE].
V. Kazakov, Quantum spectral curve of gamma-twisted \( \mathcal{N} \) = 4 sym theory and fishnet cft, Rev. Math. Phys. 30 (2018) 1840010.
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in \( \mathcal{N} \) = 4 SYM, JHEP 10 (2017) 098 [arXiv:1611.05436] [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions, JHEP 01 (2017) 130 [arXiv:1611.05577] [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions II: Two-Particle Contributions, JHEP 02 (2018) 177 [arXiv:1711.05327] [INSPIRE].
B. Basso, J. Caetano and T. Fleury, Hexagons and Correlators in the Fishnet Theory, JHEP 11 (2019) 172 [arXiv:1812.09794] [INSPIRE].
S. Derkachov and E. Olivucci, Conformal quantum mechanics & the integrable spinning Fishnet, JHEP 11 (2021) 060 [arXiv:2103.01940] [INSPIRE].
E. Olivucci, Hexagonalization of Fishnet integrals II: form factors, to appear.
D. Chicherin, S. Derkachov and A.P. Isaev, Conformal group: R-matrix and star-triangle relation, JHEP 04 (2013) 020 [arXiv:1206.4150] [INSPIRE].
V.K. Dobrev et al., Harmonic analysis on the n-dimensional lorentz group and its application to conformal quantum field theory, Lect.Notes Phys. 63 12 (1977) 059.
D. Chicherin, S. Derkachov and A.P. Isaev, The spinorial R-matrix, J. Phys. A 46 (2013) 485201 [arXiv:1303.4929] [INSPIRE].
P.P. Kulish, N.Y. Reshetikhin and E.K. Sklyanin, Yang-Baxter Equation and Representation Theory. 1, Lett. Math. Phys. 5 (1981) 393 [INSPIRE].
S. Derkachov and E. Olivucci, Exactly solvable single-trace four point correlators in χCFT4 , JHEP 02 (2021) 146 [arXiv:2007.15049] [INSPIRE].
G.M. Sotkov and R.P. Zaikov, Conformal Invariant Two Point and Three Point Functions for Fields with Arbitrary Spin, Rept. Math. Phys. 12 (1977) 375 [INSPIRE].
M. D’Eramo, G. Parisi and L. Peliti, Theoretical predictions for critical exponents at the λ-point of bose liquids, Lett. Nuovo Cim. 2 (1971) 878 [INSPIRE].
D.I. Kazakov, Calculation of Feynman diagrams by the “Uniqueness” method, Theor. Math. Phys. 58 (1984) 223 [INSPIRE].
D.I. Kazakov, The method of uniqueness, a new powerful technique for multiloop calculations, Phys. Lett. B 133 (1983) 406 [INSPIRE].
S. Derkachov and E. Olivucci, Exactly solvable magnet of conformal spins in four dimensions, Phys. Rev. Lett. 125 (2020) 031603 [arXiv:1912.07588] [INSPIRE].
L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Proceedings of the Les Houches summer school, Session LXIV, Les Houches, France, August 1 - September 8 1995, p. 149, Connes A. et al. ed., Quantum symmetries/ Symétries quantiques, Amsterdam, North-Holland (1998) [hep-th/9605187] [ISBN: 9780444828675].
B. Basso, G. Ferrando, V. Kazakov and D.-l. Zhong, Thermodynamic Bethe Ansatz for Biscalar Conformal Field Theories in any Dimension, Phys. Rev. Lett. 125 (2020) 091601 [arXiv:1911.10213] [INSPIRE].
S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD: 1. Baxter Q operator and separation of variables, Nucl. Phys. B 617 (2001) 375 [hep-th/0107193] [INSPIRE].
G. Arutyunov and S.J. van Tongeren, AdS5 × S5 mirror model as a string sigma model, Phys. Rev. Lett. 113 (2014) 261605 [arXiv:1406.2304] [INSPIRE].
B. Basso and L.J. Dixon, Gluing Ladder Feynman Diagrams into Fishnets, Phys. Rev. Lett. 119 (2017) 071601 [arXiv:1705.03545] [INSPIRE].
F. Gantmacher, Lectures in Analytical Mechanics, Beekman Books (1975) Incorporated [ISBN: 9780846405511].
H.M. Babujian, Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spin S, Phys. Lett. A 90 (1982) 479 [INSPIRE].
N. Andrei, Diagonalization of the Kondo Hamiltonian, Phys. Rev. Lett. 45 (1980) 379 [INSPIRE].
P. Schlottmann, Impurity-induced critical behaviour in antiferromagnetic heisenberg chains, J. Phys. Condens. Matter 3 (1991) 6617.
O.A. Castro-Alvaredo and J.M. Maillet, Form factors of integrable Heisenberg (higher) spin chains, J. Phys. A 40 (2007) 7451 [hep-th/0702186] [INSPIRE].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge Monogr. Math. Phys., Cambridge (1993), [DOI] [INSPIRE].
V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [INSPIRE].
B. Basso, L.J. Dixon, D.A. Kosower, A. Krajenbrink and D.-l. Zhong, Fishnet four-point integrals: integrable representations and thermodynamic limits, JHEP 07 (2021) 168 [arXiv:2105.10514] [INSPIRE].
L. Faddeev, Quantum completely integrable models in field theory, in 40 Years in Mathematical Physics, (1995) p. 187 [DOI].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, Annals Phys. 120 (1979) 253 [INSPIRE].
F.A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory, World Scientific (1992) [DOI].
D. Bombardelli, S-matrices and integrability, J. Phys. A 49 (2016) 323003 [arXiv:1606.02949] [INSPIRE].
S.E. Derkachov and A.N. Manashov, Iterative construction of eigenfunctions of the monodromy matrix for SL(2, ℂ) magnet, J. Phys. A 47 (2014) 305204 [arXiv:1401.7477] [INSPIRE].
D. Chicherin, V. Kazakov, F. Loebbert, D. Müller and D.-l. Zhong, Yangian Symmetry for Fishnet Feynman Graphs, Phys. Rev. D 96 (2017) 121901 [arXiv:1708.00007] [INSPIRE].
D. Chicherin, V. Kazakov, F. Loebbert, D. Müller and D.-l. Zhong, Yangian Symmetry for Bi-Scalar Loop Amplitudes, JHEP 05 (2018) 003 [arXiv:1704.01967] [INSPIRE].
F. Loebbert, D. Müller and H. Münkler, Yangian Bootstrap for Conformal Feynman Integrals, Phys. Rev. D 101 (2020) 066006 [arXiv:1912.05561] [INSPIRE].
L. Corcoran, F. Loebbert, J. Miczajka and M. Staudacher, Minkowski Box from Yangian Bootstrap, JHEP 04 (2021) 160 [arXiv:2012.07852] [INSPIRE].
F. Coronado, Perturbative four-point functions in planar \( \mathcal{N} \) = 4 SYM from hexagonalization, JHEP 01 (2019) 056 [arXiv:1811.00467] [INSPIRE].
F. Coronado, Bootstrapping the Simplest Correlator in Planar \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory to All Loops, Phys. Rev. Lett. 124 (2020) 171601 [arXiv:1811.03282] [INSPIRE].
I. Kostov, V.B. Petkova and D. Serban, The Octagon as a Determinant, JHEP 11 (2019) 178 [arXiv:1905.11467] [INSPIRE].
E.K. Sklyanin, Quantum inverse scattering method. Selected topics, hep-th/9211111 [INSPIRE].
E.K. Sklyanin, Separation of variables - new trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].
P. Ryan and D. Volin, Separation of Variables for Rational \( \mathfrak{gl}\left(\mathfrak{n}\right) \) Spin Chains in Any Compact Representation, via Fusion, Embedding Morphism and Bäcklund Flow, Commun. Math. Phys. 383 (2021) 311 [arXiv:2002.12341] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and P. Ryan, Determinant form of correlators in high rank integrable spin chains via separation of variables, JHEP 05 (2021) 169 [arXiv:2011.08229] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Separation of variables and scalar products at any rank, JHEP 09 (2019) 052 [arXiv:1907.03788] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Separation of variables in AdS/CFT: functional approach for the fishnet CFT, JHEP 06 (2021) 131 [arXiv:2103.15800] [INSPIRE].
A. Cavaglià, N. Gromov, J. Julius and M. Preti, Integrability and Conformal Bootstrap: One Dimensional Defect CFT, arXiv:2107.08510 [INSPIRE].
S.E. Derkachov and P.A. Valinevich, Separation of variables for the quantum SL(3, ℂ) spin magnet: eigenfunctions of Sklyanin B-operator, Zap. Nauchn. Semin. 473 (2018) 110 [arXiv:1807.00302] [INSPIRE].
T. Fleury and V. Goncalves, Decagon at Two Loops, JHEP 07 (2020) 030 [arXiv:2004.10867] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
V. Kazakov, Discussion session about integrability, talk at ICTP-SAIFR Strings 2021, https://www.youtube.com/watch?v=TTOwkmdKSmU.
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: 2107.13035
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
Olivucci, E. Hexagonalization of Fishnet integrals. Part I. Mirror excitations. J. High Energ. Phys. 2021, 204 (2021). https://doi.org/10.1007/JHEP11(2021)204
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2021)204