Abstract
The major simplification in a number of quantum integrable systems is the existence of special coordinates in which the eigenstates take a factorised form. Despite many years of studies, the basis realising the separation of variables (SoV) remains unknown in \( \mathcal{N} \) = 4 SYM and similar models, even though it is widely believed they are integrable. In this paper we initiate the SoV approach for observables with nontrivial coupling dependence in a close cousin of \( \mathcal{N} \) = 4 SYM — the fishnet 4D CFT. We develop the functional SoV formalism in this theory, which allows us to compute non-perturbatively some nontrivial observables in a form suitable for numerical evaluation. We present some applications of these methods. In particular, we discuss the possible SoV structure of the one-point correlators in presence of a defect, and write down a SoV-type expression for diagonal OPE coefficients involving an arbitrary state and the Lagrangian density operator. We believe that many of the findings of this paper can be applied in the \( \mathcal{N} \) = 4 SYM case, as we speculate in the last part of the article.
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References
E.K. Sklyanin, The Quantum Toda Chain, Lect. Notes Phys. 226 (1985) 196 [INSPIRE].
E.K. Sklyanin, Separation of variables in the Gaudin model, J. Sov. Math. 47 (1989) 2473 [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].
S.E. Derkachov, K.K. Kozlowski and A.N. Manashov, Completeness of SoV representation for SL(2, ℝ) spin chains, arXiv:2102.13570 [INSPIRE].
T. Gombor and B. Pozsgay, On factorized overlaps: Algebraic Bethe Ansatz, twists, and Separation of Variables, Nucl. Phys. B 967 (2021) 115390 [arXiv:2101.10354] [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].
H. Pei and V. Terras, On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model, arXiv:2011.06109 [INSPIRE].
S. Derkachov and E. Olivucci, Exactly solvable single-trace four point correlators in χCFT4, JHEP 02 (2021) 146 [arXiv:2007.15049] [INSPIRE].
J.M. Maillet, G. Niccoli and L. Vignoli, On Scalar Products in Higher Rank Quantum Separation of Variables, SciPost Phys. 9 (2020) 086 [arXiv:2003.04281] [INSPIRE].
P. Ryan and D. Volin, Separation of Variables for Rational \( \mathfrak{gl}(n) \) Spin Chains in Any Compact Representation, via Fusion, Embedding Morphism and Bäcklund Flow, Commun. Math. Phys. 383 (2021) 311 [arXiv:2002.12341] [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].
N. Gromov, F. Levkovich-Maslyuk, P. Ryan and D. Volin, Dual Separated Variables and Scalar Products, Phys. Lett. B 806 (2020) 135494 [arXiv:1910.13442] [INSPIRE].
J.M. Maillet, G. Niccoli and L. Vignoli, Separation of variables bases for integrable \( {gl}_{\left.\mathrm{\mathcal{M}}\right|\mathcal{N}} \) and Hubbard models, SciPost Phys. 9 (2020) 060 [arXiv:1907.08124] [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].
P. Ryan and D. Volin, Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame, J. Math. Phys. 60 (2019) 032701 [arXiv:1810.10996] [INSPIRE].
S.E. Derkachov and P.A. Valinevich, Separation of variables for the quantum SL(3, ℂ) spin magnet: eigenfunctions of Sklyanin B-operator, J. Math. Sci. 242 (2019) 658 [arXiv:1807.00302] [INSPIRE].
A. Liashyk and N.A. Slavnov, On Bethe vectors in \( {\mathfrak{gl}}_3 \)-invariant integrable models, JHEP 06 (2018) 018 [arXiv:1803.07628] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, New Compact Construction of Eigenstates for Supersymmetric Spin Chains, JHEP 09 (2018) 085 [arXiv:1805.03927] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains, JHEP 09 (2017) 111 [arXiv:1610.08032] [INSPIRE].
F.A. Smirnov, Quasiclassical study of form-factors in finite volume, hep-th/9802132 [INSPIRE].
S.L. Lukyanov, Form-factors of exponential fields in the sine-Gordon model, Mod. Phys. Lett. A 12 (1997) 2543 [hep-th/9703190] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve and structure constants in \( \mathcal{N} \) = 4 SYM: cusps in the ladder limit, JHEP 10 (2018) 060 [arXiv:1802.04237] [INSPIRE].
S. Giombi and S. Komatsu, Exact Correlators on the Wilson Loop in \( \mathcal{N} \) = 4 SYM: Localization, Defect CFT, and Integrability, JHEP 05 (2018) 109 [Erratum ibid. 11 (2018) 123] [arXiv:1802.05201] [INSPIRE].
S. Giombi and S. Komatsu, More Exact Results in the Wilson Loop Defect CFT: Bulk-Defect OPE, Nonplanar Corrections and Quantum Spectral Curve, J. Phys. A 52 (2019) 125401 [arXiv:1811.02369] [INSPIRE].
J. McGovern, Scalar insertions in cusped Wilson loops in the ladders limit of planar \( \mathcal{N} \) = 4 SYM, JHEP 05 (2020) 062 [arXiv:1912.00499] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling Handles: Nonplanar Integrability in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 121 (2018) 231602 [arXiv:1711.05326] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar \( \mathcal{N} \) = 4 Super-Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for arbitrary state/operator in AdS5/CFT4, JHEP 09 (2015) 187 [arXiv:1405.4857] [INSPIRE].
A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, Quantum Spectral Curve of the \( \mathcal{N} \) = 6 Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 113 (2014) 021601 [arXiv:1403.1859] [INSPIRE].
D. Bombardelli, A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, The full Quantum Spectral Curve for AdS4/CFT3, JHEP 09 (2017) 140 [arXiv:1701.00473] [INSPIRE].
D. Bombardelli, A. Cavaglià, R. Conti and R. Tateo, Exploring the spectrum of planar AdS4/CFT3 at finite coupling, JHEP 04 (2018) 117 [arXiv:1803.04748] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, Quantum Spectral Curve for a cusped Wilson line in \( \mathcal{N} \) = 4 SYM, JHEP 04 (2016) 134 [arXiv:1510.02098] [INSPIRE].
A. Cavaglià, M. Cornagliotto, M. Mattelliano and R. Tateo, A Riemann-Hilbert formulation for the finite temperature Hubbard model, JHEP 06 (2015) 015 [arXiv:1501.04651] [INSPIRE].
Y. Jiang, S. Komatsu, I. Kostov and D. Serban, The hexagon in the mirror: the three-point function in the SoV representation, J. Phys. A 49 (2016) 174007 [arXiv:1506.09088] [INSPIRE].
S. Derkachov, V. Kazakov and E. Olivucci, Basso-Dixon Correlators in Two-Dimensional Fishnet CFT, JHEP 04 (2019) 032 [arXiv:1811.10623] [INSPIRE].
A.V. Belitsky, S.E. Derkachov and A.N. Manashov, Quantum mechanics of null polygonal Wilson loops, Nucl. Phys. B 882 (2014) 303 [arXiv:1401.7307] [INSPIRE].
A.V. Belitsky, Supersymmetric quantum mechanics of the flux tube, Nucl. Phys. B 913 (2016) 551 [arXiv:1604.00418] [INSPIRE].
A.V. Belitsky, Separation of Variables for a flux tube with an end, Nucl. Phys. B 957 (2020) 115093 [arXiv:1902.08596] [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].
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].
N. Gromov, V. Kazakov, G. Korchemsky, S. Negro and G. Sizov, Integrability of Conformal Fishnet Theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [INSPIRE].
A.B. Zamolodchikov, ‘Fishnet’ diagrams as a completely integrable system, Phys. Lett. B 97 (1980) 63 [INSPIRE].
N. Gromov and A. Sever, The holographic dual of strongly γ-deformed \( \mathcal{N} \) = 4 SYM theory: derivation, generalization, integrability and discrete reparametrization symmetry, JHEP 02 (2020) 035 [arXiv:1908.10379] [INSPIRE].
N. Gromov and A. Sever, Derivation of the Holographic Dual of a Planar Conformal Field Theory in 4D, Phys. Rev. Lett. 123 (2019) 081602 [arXiv:1903.10508] [INSPIRE].
N. Gromov and A. Sever, Quantum fishchain in AdS5, JHEP 10 (2019) 085 [arXiv:1907.01001] [INSPIRE].
A. Cavaglià, D. Grabner, N. Gromov and A. Sever, Colour-twist operators. Part I. Spectrum and wave functions, JHEP 06 (2020) 092 [arXiv:2001.07259] [INSPIRE].
A. Cavaglià, N. Gromov, F. Levkovich-Maslyuk and A. Sever, Colour-Twist Operators. Part II: Correlation Functions, to appear.
M.S. Costa, R. Monteiro, J.E. Santos and D. Zoakos, On three-point correlation functions in the gauge/gravity duality, JHEP 11 (2010) 141 [arXiv:1008.1070] [INSPIRE].
B. Basso, J. Caetano and T. Fleury, Hexagons and Correlators in the Fishnet Theory, JHEP 11 (2019) 172 [arXiv:1812.09794] [INSPIRE].
J. Caetano and S. Komatsu, Functional equations and separation of variables for exact g-function, JHEP 09 (2020) 180 [arXiv:2004.05071] [INSPIRE].
Y. Jiang, S. Komatsu and E. Vescovi, Structure constants in \( \mathcal{N} \) = 4 SYM at finite coupling as worldsheet g-function, JHEP 07 (2020) 037 [arXiv:1906.07733] [INSPIRE].
S. Komatsu and Y. Wang, Non-perturbative defect one-point functions in planar \( \mathcal{N} \) = 4 super-Yang-Mills, Nucl. Phys. B 958 (2020) 115120 [arXiv:2004.09514] [INSPIRE].
T. Gombor and Z. Bajnok, Boundary states, overlaps, nesting and bootstrapping AdS/dCFT, JHEP 10 (2020) 123 [arXiv:2004.11329] [INSPIRE].
S. Komatsu, Wilson Loops as Matrix Product States talk at London Integrability Journal Club.
P. Dorey, D. Fioravanti, C. Rim and R. Tateo, Integrable quantum field theory with boundaries: The Exact g function, Nucl. Phys. B 696 (2004) 445 [hep-th/0404014] [INSPIRE].
S. Derkachov and E. Olivucci, Conformal quantum mechanics & the integrable spinning Fishnet, arXiv:2103.01940 [INSPIRE].
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, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, Lect. Notes Phys. 63 (1977) 1.
V. Kazakov and E. Olivucci, Biscalar Integrable Conformal Field Theories in Any Dimension, Phys. Rev. Lett. 121 (2018) 131601 [arXiv:1801.09844] [INSPIRE].
I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations, Commun. Math. Phys. 188 (1997) 267 [hep-th/9604080] [INSPIRE].
A. Chervov and D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, hep-th/0604128 [INSPIRE].
D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, unpublished.
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4, JHEP 06 (2016) 036 [arXiv:1504.06640] [INSPIRE].
N. Gromov, V. Kazakov and G. Korchemsky, Exact Correlation Functions in Conformal Fishnet Theory, JHEP 08 (2019) 123 [arXiv:1808.02688] [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].
A.C. Ipsen, M. Staudacher and L. Zippelius, The one-loop spectral problem of strongly twisted \( \mathcal{N} \) = 4 Super Yang-Mills theory, JHEP 04 (2019) 044 [arXiv:1812.08794] [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
M. de Leeuw, C. Kristjansen and K. Zarembo, One-point Functions in Defect CFT and Integrability, JHEP 08 (2015) 098 [arXiv:1506.06958] [INSPIRE].
I. Buhl-Mortensen, M. de Leeuw, C. Kristjansen and K. Zarembo, One-point Functions in AdS/dCFT from Matrix Product States, JHEP 02 (2016) 052 [arXiv:1512.02532] [INSPIRE].
E.K. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys. A 21 (1988) 2375 [INSPIRE].
N. Gromov, J. Julius and N. Primi, Open Fishchain in N = 4 Supersymmetric Yang-Mills Theory, arXiv:2101.01232 [INSPIRE].
B. Oertel, O. Shahpo and E. Vescovi. Determinant Operators and Exact Correlators in the Fishnet Theory, to appear.
S. Ghoshal and A.B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841 [Erratum ibid. 9 (1994) 4353] [hep-th/9306002] [INSPIRE].
L. Piroli, B. Pozsgay and E. Vernier, What is an integrable quench?, Nucl. Phys. B 925 (2017) 362 [arXiv:1709.04796] [INSPIRE].
A. LeClair, G. Mussardo, H. Saleur and S. Skorik, Boundary energy and boundary states in integrable quantum field theories, Nucl. Phys. B 453 (1995) 581 [hep-th/9503227] [INSPIRE].
P. Dorey, I. Runkel, R. Tateo and G. Watts, g function flow in perturbed boundary conformal field theories, Nucl. Phys. B 578 (2000) 85 [hep-th/9909216] [INSPIRE].
F. Woynarovich, O(1) contribution of saddle point fluctuations to the free energy of Bethe Ansatz systems, Nucl. Phys. B 700 (2004) 331 [cond-mat/0402129] [INSPIRE].
B. Pozsgay, On O(1) contributions to the free energy in Bethe Ansatz systems: The Exact g-function, JHEP 08 (2010) 090 [arXiv:1003.5542] [INSPIRE].
I. Kostov, D. Serban and D.-L. Vu, Boundary TBA, trees and loops, Nucl. Phys. B 949 (2019) 114817 [arXiv:1809.05705] [INSPIRE].
C. Marboe and D. Volin, Quantum spectral curve as a tool for a perturbative quantum field theory, Nucl. Phys. B 899 (2015) 810 [arXiv:1411.4758] [INSPIRE].
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].
B. Basso and D.-l. Zhong, Continuum limit of fishnet graphs and AdS sigma model, JHEP 01 (2019) 002 [arXiv:1806.04105] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Pomeron Eigenvalue at Three Loops in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 115 (2015) 251601 [arXiv:1507.04010] [INSPIRE].
M. Alfimov, N. Gromov and G. Sizov, BFKL spectrum of \( \mathcal{N} \) = 4: non-zero conformal spin, JHEP 07 (2018) 181 [arXiv:1802.06908] [INSPIRE].
B. Basso and L.J. Dixon, Gluing Ladder Feynman Diagrams into Fishnets, Phys. Rev. Lett. 119 (2017) 071601 [arXiv:1705.03545] [INSPIRE].
M. Alfimov, N. Gromov and V. Kazakov, QCD Pomeron from AdS/CFT Quantum Spectral Curve, JHEP 07 (2015) 164 [arXiv:1408.2530] [INSPIRE].
I. Buric, S. Lacroix, J.A. Mann, L. Quintavalle and V. Schomerus, From Gaudin Integrable Models to d-dimensional Multipoint Conformal Blocks, Phys. Rev. Lett. 126 (2021) 021602 [arXiv:2009.11882] [INSPIRE].
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Cavaglià, A., Gromov, N. & Levkovich-Maslyuk, F. Separation of variables in AdS/CFT: functional approach for the fishnet CFT. J. High Energ. Phys. 2021, 131 (2021). https://doi.org/10.1007/JHEP06(2021)131
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DOI: https://doi.org/10.1007/JHEP06(2021)131