Abstract
Separation of variables (SoV) is a special property of integrable models which ensures that the wavefunction has a very simple factorised form in a specially designed basis. Even though the factorisation of the wavefunction was recently established for higher rank models by two of the authors and G. Sizov, the measure for the scalar product was not known beyond the case of rank one symmetry. In this paper we show how this measure can be found, bypassing an explicit SoV construction. A key new observation is that the measure for spin chains in a highest-weight infinite dimensional representation of 𝔰𝔩(N) couples Q-functions at different nesting levels in a non-symmetric fashion. We also managed to express a large number of form factors as ratios of determinants in our new approach. We expect our method to be applicable in a much wider setup including the problem of computing correlators in integrable CFTs such as the fishnet theory, \( \mathcal{N} \) = 4 SYM and the ABJM model.
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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 [Zap. Nauchn. Semin.164 (1987) 151] [INSPIRE].
E.K. Sklyanin, Quantum inverse scattering method. Selected topics, in Quantum Group and Quantum Integrable Systems: Nankai Lectures on Mathematical Physics, Nankai Institute of Mathematics, Tianjin China (1991), World Scientific, Singapore (1992), pg. 63 [hep-th/9211111] [INSPIRE].
E.K. Sklyanin, Separation of variables — new trends, Prog. Theor. Phys. Suppl.118 (1995) 35 [solv-int/9504001] [INSPIRE].
V.E. Korepin, Calculation Of Norms Of Bethe Wave Functions, Commun. Math. Phys.86 (1982) 391 [INSPIRE].
Y. Kazama, S. Komatsu and T. Nishimura, A new integral representation for the scalar products of Bethe states for the XXX spin chain, JHEP09 (2013) 013 [arXiv:1304.5011] [INSPIRE].
G. Niccoli, Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors, Nucl. Phys.B 870 (2013) 397 [arXiv:1205.4537] [INSPIRE].
D. Levy-Bencheton, G. Niccoli and V. Terras, Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors, J. Stat. Mech.1603 (2016) 033110 [arXiv:1507.03404] [INSPIRE].
G. Niccoli and V. Terras, Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate construction by functional equations in separation of variables, Lett. Math. Phys.105 (2015) 989 [arXiv:1411.6488] [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].
N. Kitanine, J.M. Maillet, G. Niccoli and V. Terras, The open XXX spin chain in the SoV framework: scalar product of separate states, J. Phys.A 50 (2017) 224001 [arXiv:1606.06917] [INSPIRE].
N. Kitanine, J.M. Maillet, G. Niccoli and V. Terras, On determinant representations of scalar products and form factors in the SoV approach: the XXX case, J. Phys.A 49 (2016) 104002 [arXiv:1506.02630] [INSPIRE].
N. Kitanine, J.M. Maillet and G. Niccoli, Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables, J. Stat. Mech.1405 (2014) P05015 [arXiv:1401.4901] [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].
S.E. Derkachov, G.P. Korchemsky, J. Kotanski and A.N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD. 2. Quantization conditions and energy spectrum, Nucl. Phys.B 645 (2002) 237 [hep-th/0204124] [INSPIRE].
G. Niccoli, Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables, J. Math. Phys.54 (2013) 053516 [arXiv:1206.2418] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains, JHEP09 (2017) 111 [arXiv:1610.08032] [INSPIRE].
D. Martin and F. Smirnov, Problems with using separated variables for computing expectation values for higher ranks, Lett. Math. Phys.106 (2016) 469 [arXiv:1506.08042] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys.99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
E.K. Sklyanin, Separation of variables in the classical integrable SL(3) magnetic chain, Commun. Math. Phys.150 (1992) 181 [hep-th/9211126] [INSPIRE].
E.K. Sklyanin, Separation of variables in the quantum integrable models related to the Yangian Y[sl(3)], J. Math. Sci.80 (1996) 1861 [hep-th/9212076] [INSPIRE].
F. Smirnov, Separation of variables for quantum integrable models related to U q(\( {\hat{sl}}_N \)), math-ph/0109013.
D.R.D. Scott, Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain, J. Math. Phys.35 (1994) 5831 [hep-th/9403030] [INSPIRE].
M.I. Gekhtman, Separation of variables in the classical SL(N) magnetic chain, Commun. Math. Phys.167 (1995) 593.
A. Chervov and D. Talalaev, Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, hep-th/0604128 [INSPIRE].
A. Chervov and G. Falqui, Manin matrices and Talalaev’s formula, J. Phys.A 41 (2008) 194006 [arXiv:0711.2236] [INSPIRE].
P.P. Kulish and N. Yu. Reshetikhin, Generalized Heisenberg ferromagnet and the Gross-Neveu model, Sov. Phys. JETP53 (1981) 108 [INSPIRE].
P.P. Kulish, Integrable graded magnets, J. Sov. Math.35 (1986) 2648 [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].
P. Dorey, C. Dunning, D. Masoero, J. Suzuki and R. Tateo, Pseudo-differential equations and the Bethe ansatz for the classical Lie algebras, Nucl. Phys.B 772 (2007) 249 [hep-th/0612298] [INSPIRE].
V. Kazakov, A.S. Sorin and A. Zabrodin, Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics, Nucl. Phys.B 790 (2008) 345 [hep-th/0703147] [INSPIRE].
N. Gromov and P. Vieira, Complete 1-loop test of AdS/CFT, JHEP04 (2008) 046 [arXiv:0709.3487] [INSPIRE].
V. Kazakov, S. Leurent and D. Volin, T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve, JHEP12 (2016) 044 [arXiv:1510.02100] [INSPIRE].
N. Gromov, Introduction to the Spectrum of N = 4 SYM and the Quantum Spectral Curve, arXiv:1708.03648 [INSPIRE].
V. Kazakov, Quantum Spectral Curve of γ-twisted \( \mathcal{N} \) = 4 SYM theory and fishnet CFT, arXiv:1802.02160 [INSPIRE].
E.K. Sklyanin, New approach to the quantum nonlinear Schrödinger equation, J. Phys.A 22 (1989) 3551 [INSPIRE].
S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Separation of variables for the quantum SL(2, ℝ) spin chain, JHEP07 (2003) 047 [hep-th/0210216] [INSPIRE].
A. Liashyk and N.A. Slavnov, On Bethe vectors in 𝔤𝔩3-invariant integrable models, JHEP06 (2018) 018 [arXiv:1803.07628] [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, Zap. Nauchn. Semin.473 (2018) 110 [arXiv:1807.00302] [INSPIRE].
J.M. Maillet and G. Niccoli, On quantum separation of variables, J. Math. Phys.59 (2018) 091417 [arXiv:1807.11572] [INSPIRE].
J.M. Maillet and G. Niccoli, Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables, SciPost Phys.6 (2019) 071 [arXiv:1810.11885] [INSPIRE].
J.M. Maillet and G. Niccoli, Complete spectrum of quantum integrable lattice models associated to \( \mathcal{U} \) q (\( \hat{gl_n} \)) by separation of variables, J. Phys.A 52 (2019) 315203 [arXiv:1811.08405] [INSPIRE].
S.E. Derkachov and A.N. Manashov, Noncompact sl(N) spin chains: BGG-resolution, Q-operators and alternating sum representation for finite dimensional transfer matrices, Lett. Math. Phys.97 (2011) 185 [arXiv:1008.4734] [INSPIRE].
S.E. Derkachov and A.N. Manashov, Baxter operators for the quantum sl(3) invariant spin chain, J. Phys.A 39 (2006) 13171 [nlin/0604018].
S.E. Derkachov, Factorization of the R-matrix. I., math/0503396.
N. Gromov and F. Levkovich-Maslyuk, New Compact Construction of Eigenstates for Supersymmetric Spin Chains, JHEP09 (2018) 085 [arXiv:1805.03927] [INSPIRE].
Ö. 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].
M.S. Costa, R. Monteiro, J.E. Santos and D. Zoakos, On three-point correlation functions in the gauge/gravity duality, JHEP11 (2010) 141 [arXiv:1008.1070] [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, JHEP10 (2018) 060 [arXiv:1802.04237] [INSPIRE].
A. Cavaglià, N. Gromov, F. Levkovich-Maslyuk and A. Sever, to appear.
F. Smirnov and V. Zeitlin, On The Quantization of Affine Jacobi Varieties of Spectral Curves, Statistical Field Theories, Springer, Dordrecht The Netherlands (2002), pg. 79.
F. Smirnov and V. Zeitlin, Affine Jacobians of spectral curves and integrable models, math-ph/0203037.
N. Gromov and A. Sever, The Holographic Fishchain, Phys. Rev. Lett.123 (2019) 081602 [arXiv:1903.10508] [INSPIRE].
O. Lipan, P.B. Wiegmann and A. Zabrodin, Fusion rules for quantum transfer matrices as a dynamical system on Grassmann manifolds, Mod. Phys. Lett.A 12 (1997) 1369 [solv-int/9704015][INSPIRE].
B. Sutherland, A General Model for Multicomponent Quantum Systems, Phys. Rev.B 12 (1975) 3795 [INSPIRE].
P.P. Kulish and N. Yu. Reshetikhin, Diagonalization of GL(N) invariant transfer matrices and quantum n wave system (Lee model), J. Phys.A 16 (1983) L591 [INSPIRE].
S.L. Lukyanov, Finite temperature expectation values of local fields in the sinh-Gordon model, Nucl. Phys.B 612 (2001) 391 [hep-th/0005027] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory. 2. Q operator and DDV equation, Commun. Math. Phys.190 (1997) 247 [hep-th/9604044] [INSPIRE].
G.P. Pronko and Yu. G. Stroganov, The Complex of solutions of the nested Bethe ansatz. The A 2spin chain, J. Phys.A 33 (2000) 8267 [hep-th/9902085] [INSPIRE].
N. Gromov and V. Kazakov, Review of AdS/CFT Integrability, Chapter III.7: Hirota Dynamics for Quantum Integrability, Lett. Math. Phys.99 (2012) 321 [arXiv:1012.3996] [INSPIRE].
A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral Duality Between Heisenberg Chain and Gaudin Model, Lett. Math. Phys.103 (2013) 299 [arXiv:1206.6349] [INSPIRE].
A. Gorsky, A. Zabrodin and A. Zotov, Spectrum of Quantum Transfer Matrices via Classical Many-Body Systems, JHEP01 (2014) 070 [arXiv:1310.6958] [INSPIRE].
E. Mukhin, V. Tarasovand A. Varchenko, Bispectral and (gl N, gl M) dualities, math/0510364.
E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and (gl N, gl M) dualities, discrete versus differential, Adv. Math.218 (2008) 216 [math/0605172].
A. Cavaglià, N. Grabner, N. Gromov and A. Sever, Twisting and Fishing, in preparation.
B. Basso, J. Caetano and T. Fleury, Hexagons and Correlators in the Fishnet Theory, arXiv:1812.09794 [INSPIRE].
S. Giombi and S. Komatsu, Exact Correlators on the Wilson Loop in \( \mathcal{N} \) = 4 SYM: Localization, Defect CFT and Integrability, JHEP05 (2018) 109 [Erratum ibid.1811 (2018) 123][arXiv:1802.05201] [INSPIRE].
S. Derkachov, V. Kazakov and E. Olivucci, Basso-Dixon Correlators in Two-Dimensional Fishnet CFT, JHEP04 (2019) 032 [arXiv:1811.10623] [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].
Y. Jiang, S. Komatsu and E. Vescovi, Structure Constants in \( \mathcal{N} \) = 4 SYM at Finite Coupling as Worldsheet g-Function, arXiv:1906.07733 [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 AdS 5/CFT 4, JHEP09 (2015) 187 [arXiv:1405.4857] [INSPIRE].
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Cavaglià, A., Gromov, N. & Levkovich-Maslyuk, F. Separation of variables and scalar products at any rank. J. High Energ. Phys. 2019, 52 (2019). https://doi.org/10.1007/JHEP09(2019)052
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DOI: https://doi.org/10.1007/JHEP09(2019)052