Abstract
We propose an efficient grassmannian formalism for solution of bi-linear finite-difference Hirota equation (T-system) on T-shaped lattices related to the space of highest weight representations of gl(K 1 , K 2|M ) superalgebra. The formalism is inspired by the quantum fusion procedure known from the integrable spin chains and is based on exterior forms of Baxter-like Q-functions. We find a few new interesting relations among the exterior forms of Q-functions and reproduce, using our new formalism, the Wronskian determinant solutions of Hirota equations known in the literature. Then we generalize this construction to the twisted Q-functions and demonstrate the subtleties of untwisting procedure on the examples of rational quantum spin chains with twisted boundary conditions. Using these observations, we generalize the recently discovered, in our paper with N. Gromov, AdS/CFT Quantum Spectral Curve for exact planar spectrum of AdS/CFT duality to the case of arbitrary Cartan twisting of AdS5×S5 string sigma model. Finally, we successfully probe this formalism by reproducing the energy of gamma-twisted BMN vacuum at single-wrapping orders of weak coupling expansion.
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H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205 [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for planar 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, JHEP 09 (2015) 187 [arXiv:1405.4857] [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].
N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and Z. Tsuboi, Wronskian solution for AdS/CFT Y-system, JHEP 01 (2011) 155 [arXiv:1010.2720] [INSPIRE].
Z. Tsuboi, Wronskian solutions of the T , Q and Y -systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl(M |N ), Nucl. Phys. B 870 (2013) 92 [arXiv:1109.5524] [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].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Analytic solution of bremsstrahlung TBA II: turning on the sphere angle, JHEP 10 (2013) 036 [arXiv:1305.1944] [INSPIRE].
D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP 08 (2012) 134 [arXiv:1203.1913] [INSPIRE].
N. Drukker, Integrable Wilson loops, JHEP 10 (2013) 135 [arXiv:1203.1617] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve for a cusped Wilson line in N = 4 SYM, JHEP 04 (2016) 134 [arXiv:1510.02098] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, Y-system and β-deformed N = 4 super-Yang-Mills, J. Phys. A 44 (2011) 015402 [arXiv:1006.5438] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, Twisting the mirror TBA, JHEP 02 (2011) 025 [arXiv:1009.4118] [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, Finite-size effect for four-loop Konishi of the β-deformed N = 4 SYM, Phys. Lett. B 693 (2010) 380 [arXiv:1006.2209] [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, Twisted Bethe equations from a twisted S-matrix, JHEP 02 (2011) 027 [arXiv:1010.3229] [INSPIRE].
M. de Leeuw and S.J. van Tongeren, Orbifolded Konishi from the mirror TBA, J. Phys. A 44 (2011) 325404 [arXiv:1103.5853] [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lüscher correction and double wrapping in twisted AdS/CFT, JHEP 12 (2011) 059 [arXiv:1108.4914] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, A piece of cake: the ground-state energies in γ i -deformed N = 4 SYM theory at leading wrapping order, JHEP 09 (2014) 078 [arXiv:1405.6712] [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].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Pomeron eigenvalue at three loops in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 115 (2015) 251601 [arXiv:1507.04010] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Quantum spectral curve and the numerical solution of the spectral problem in AdS 5 /CFT 4, JHEP 06 (2016) 036 [arXiv:1504.06640] [INSPIRE].
R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Jpn. 50 (1981) 3785.
M. Jimbo and T. Miwa, Solitons and infinite dimentional Lie algebras, Publ. Res. Inst. Math. Sci. Kyoto 19 (1983) 943.
A. Kuniba, T. Nakanishi and J. Suzuki, T-systems and Y-systems in integrable systems, J. Phys. A 44 (2011) 103001 [arXiv:1010.1344] [INSPIRE].
P.P. Kulish and N.Y. Reshetikhin, On gl 3 -invariant solutions to the Yang-Baxter equation and the assosiated quantum systems, Zap. Nauchn. Sem. POMI 120 (1982) 92.
N.Y. Reshetikhin, The functional equation method in the theory of exactly soluble quantum systems, Sov. Phys. JETP 57 (1983) 691 [Zh. Eksp. Teor. Fiz 84 (1983) 1190].
A. Klumper and P.A. Pearce, Conformal weights of RSOS lattice models and their fusion hierarchies, Physica A 183 (1992) 304 [INSPIRE].
A. Kuniba, T. Nakanishi and J. Suzuki, Functional relations in solvable lattice models. 1: functional relations and representation theory, Int. J. Mod. Phys. A 9 (1994) 5215 [hep-th/9309137] [INSPIRE].
Z. Tsuboi, Analytic Bethe ansatz and functional equations for Lie superalgebra sl(r + 1|s + 1), J. Phys. A 30 (1997) 7975 [arXiv:0911.5386] [INSPIRE].
Z. Tsuboi, Analytic Bethe ansatz and functional equations associated with any simple root systems of the Lie superalgebra sl(r + 1|s + 1), Physica A 252 (1998) 565 [arXiv:0911.5387] [INSPIRE].
V. Kazakov and P. Vieira, From characters to quantum (super)spin chains via fusion, JHEP 10 (2008) 050 [arXiv:0711.2470] [INSPIRE].
Z. Tsuboi, Solutions of the T-system and Baxter equations for supersymmetric spin chains, Nucl. Phys. B 826 (2010) 399 [arXiv:0906.2039] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [INSPIRE].
V. Kazakov and S. Leurent, Finite size spectrum of SU(N ) principal chiral field from discrete Hirota dynamics, Nucl. Phys. B 902 (2016) 354 [arXiv:1007.1770] [INSPIRE].
N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
A. Alexandrov, V. Kazakov, S. Leurent, Z. Tsuboi and A. Zabrodin, Classical tau-function for quantum spin chains, JHEP 09 (2013) 064 [arXiv:1112.3310] [INSPIRE].
V. Bazhanov and N. Reshetikhin, Restricted solid on solid models connected with simply based algebras and conformal field theory, J. Phys. A 23 (1990) 1477 [INSPIRE].
I.V. Cherednik, An analogue of the character formula for Hekke algebras, Funct. Anal. Appl. 21 (1987) 172 [Funkt. Anal. Pril. 21 (1987) 94].
E. Moens and J. Van der Jeugt, A determinantal formula for supersymmetric Schur polynomials, J. Alg. Combinat. 17 (2003) 283.
J.-H. Kwon, Rational semistandard tableaux and character formula for the Lie superalgebra, Adv. Math. 217 (2008) 713.
N. Gromov, V. Kazakov and Z. Tsuboi, PSU(2, 2|4) character of quasiclassical AdS/CFT, JHEP 07 (2010) 097 [arXiv:1002.3981] [INSPIRE].
G.P. Pronko and Yu. G. Stroganov, The complex of solutions of the nested Bethe ansatz. The A 2 spin chain, J. Phys. A 33 (2000) 8267 [hep-th/9902085] [INSPIRE].
V.V. Bazhanov, A.N. Hibberd and S.M. Khoroshkin, Integrable structure of W 3 conformal field theory, quantum Boussinesq theory and boundary affine Toda theory, Nucl. Phys. B 622 (2002) 475 [hep-th/0105177] [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].
A.V. Belitsky, S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Baxter Q-operator for graded SL(2|1) spin chain, J. Stat. Mech. 01 (2007) P01005 [hep-th/0610332] [INSPIRE].
N. Gromov and P. Vieira, Complete 1-loop test of AdS/CFT, JHEP 04 (2008) 046 [arXiv:0709.3487] [INSPIRE].
V.V. Bazhanov and Z. Tsuboi, Baxter’s Q-operators for supersymmetric spin chains, Nucl. Phys. B 805 (2008) 451 [arXiv:0805.4274] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Solving the AdS/CFT Y-system, JHEP 07 (2012) 023 [arXiv:1110.0562] [INSPIRE].
J. Caetano, Unified approach to the SU(2) principal chiral field model at finite volume, arXiv:1012.2600 [INSPIRE].
A. Zabrodin, Discrete Hirota’s equation in quantum integrable models, Int. J. Mod. Phys. B 11 (1997) 3125 [hep-th/9610039] [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].
A. Zabrodin, Backlund transformations for difference Hirota equation and supersymmetric Bethe ansatz, Theor. Math. Phys. 155 (2007) 567 [Teor. Mat. Fiz. 155 (2008) 74] [arXiv:0705.4006] [INSPIRE].
S. Leurent, Integrable systems and AdS/CFT duality, arXiv:1206.4061 [INSPIRE].
S.-J. Cheng and W. Wang, Dualities and representations of Lie superalgebras, American Mathematical Society, U.S.A. (2012).
F. Gohmann and A. Seel, A note on the Bethe ansatz solution of the supersymmetric t-J model, Czech. J. Phys. 53 (2003) 1041 [cond-mat/0309138].
N. Beisert, V.A. Kazakov, K. Sakai and K. Zarembo, Complete spectrum of long operators in N = 4 SYM at one loop, JHEP 07 (2005) 030 [hep-th/0503200] [INSPIRE].
M. Alfimov, N. Gromov and V. Kazakov, QCD pomeron from AdS/CFT quantum spectral curve, JHEP 07 (2015) 164 [arXiv:1408.2530] [INSPIRE].
D. Volin, String hypothesis for gl(n|m) spin chains: a particle/hole democracy, Lett. Math. Phys. 102 (2012) 1 [arXiv:1012.3454] [INSPIRE].
M. Gunaydin and D. Volin, in preparation.
A. Hegedus, Discrete Hirota dynamics for AdS/CFT, Nucl. Phys. B 825 (2010) 341 [arXiv:0906.2546] [INSPIRE].
V.V. Bazhanov, T. Lukowski, C. Meneghelli and M. Staudacher, A shortcut to the Q-operator, J. Stat. Mech. 1011 (2010) P11002 [arXiv:1005.3261] [INSPIRE].
V.V. Bazhanov, R. Frassek, T. Lukowski, C. Meneghelli and M. Staudacher, Baxter Q-operators and representations of yangians, Nucl. Phys. B 850 (2011) 148 [arXiv:1010.3699] [INSPIRE].
V. Kazakov, S. Leurent and Z. Tsuboi, Baxter’s Q-operators and operatorial Backlund flow for quantum (super)-spin chains, Commun. Math. Phys. 311 (2012) 787 [arXiv:1010.4022] [INSPIRE].
P.P. Kulish and E.K. Sklyanin, On the solution of the Yang-Baxter equation, J. Sov. Math. 19 (1982) 1596 [Zap. Nauchn. Semin. 95 (1980) 129] [INSPIRE].
P.P. Kulish, Integrable graded magnets, J. Sov. Math. 35 (1986) 2648 [Zap. Nauchn. Semin. 145 (1985) 140] [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].
S.E. Derkachov and A.N. Manashov, Factorization of R-matrix and Baxter Q-operators for generic sl(N ) spin chains, J. Phys. A 42 (2009) 075204 [arXiv:0809.2050] [INSPIRE].
D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe ansatz for planar AdS/CFT: a proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].
G. Arutyunov and S. Frolov, Thermodynamic Bethe ansatz for the AdS 5 × S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].
A. Cavaglià, D. Fioravanti and R. Tateo, Extended Y-system for the AdS 5 /CFT 4 correspondence, Nucl. Phys. B 843 (2011) 302 [arXiv:1005.3016] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk, G. Sizov and S. Valatka, Quantum spectral curve at work: from small spin to strong coupling in N = 4 SYM, JHEP 07 (2014) 156 [arXiv:1402.0871] [INSPIRE].
A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, Quantum spectral curve of the N = 6 supersymmetric Chern-Simons theory, Phys. Rev. Lett. 113 (2014) 021601 [arXiv:1403.1859] [INSPIRE].
N. Gromov and G. Sizov, Exact slope and interpolating functions in N = 6 supersymmetric Chern-Simons theory, Phys. Rev. Lett. 113 (2014) 121601 [arXiv:1403.1894] [INSPIRE].
L. Anselmetti, D. Bombardelli, A. Cavaglià and R. Tateo, 12 loops and triple wrapping in ABJM theory from integrability, JHEP 10 (2015) 117 [arXiv:1506.09089] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, Non-conformality of γ i -deformed N = 4 SYM theory, J. Phys. A 47 (2014) 455401 [arXiv:1308.4420] [INSPIRE].
Q. Jin, The emergence of supersymmetry in γ i -deformed N = 4 super-Yang-Mills theory, arXiv:1311.7391 [INSPIRE].
S. Frolov and A.A. Tseytlin, Rotating string solutions: AdS/CFT duality in nonsupersymmetric sectors, Phys. Lett. B 570 (2003) 96 [hep-th/0306143] [INSPIRE].
R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional N =1 supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 95 [hep-th/9503121] [INSPIRE].
S.A. Frolov, R. Roiban and A.A. Tseytlin, Gauge-string duality for (non)supersymmetric deformations of N = 4 super Yang-Mills theory, Nucl. Phys. B 731 (2005) 1 [hep-th/0507021] [INSPIRE].
G. Arutyunov, S. Frolov and R. Suzuki, Five-loop Konishi from the mirror TBA, JHEP 04 (2010) 069 [arXiv:1002.1711] [INSPIRE].
S.J. van Tongeren, Integrability of the AdS 5 × S 5 superstring and its deformations, J. Phys. A 47 (2014) 433001 [arXiv:1310.4854] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, The complete one-loop dilatation operator of planar real β-deformed N = 4 SYM theory, JHEP 07 (2014) 150 [arXiv:1312.2959] [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].
É. Ragoucy and G. Satta, Analytical Bethe ansatz for closed and open gl(M |N ) super-spin chains in arbitrary representations and for any Dynkin diagrams, JHEP 09 (2007) 001 [arXiv:0706.3327] [INSPIRE].
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Kazakov, V., Leurent, S. & Volin, D. T-system on T-hook: Grassmannian solution and twisted Quantum Spectral Curve. J. High Energ. Phys. 2016, 44 (2016). https://doi.org/10.1007/JHEP12(2016)044
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DOI: https://doi.org/10.1007/JHEP12(2016)044