Abstract
We study the ODE/IM correspondence for ODE associated to \({\widehat{\mathfrak{g}}}\)-valued connections, for a simply-laced Lie algebra \({\mathfrak{g}}\). We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called \({\Psi}\)-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.
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Adamopoulou P., Dunning C.: Bethe Ansatz equations for the classical \({A_{n}^{(1)}}\) affine Toda field theories. J. Phys. A 47, 205205 (2014)
Bazhanov V.V., Hibberd A., Khoroshkin S.: Integrable structure of W3 conformal field theory, quantum Boussinesq theory and boundary affine Toda theory. Nucl. Phys. B 622(3), 475–547 (2002)
Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B.: Spectral determinants for Schrodinger equation and Q operators of conformal field theory. J. Stat. Phys. 102, 567–576 (2001)
Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B.: Integrable structure of conformal field theory. II. Q-operator and DDV equation. Commun. Math. Phys. 190(2), 247–278 (1997)
Bazhanov V.V, Lukyanov S.L., Zamolodchikov A.B.: Higher-level eigenvalues of Q-operators and Schroedinger equation. Adv. Theor. Math. Phys. 7, 711 (2004)
Berman S., Lee T., Moody R.: The spectrum of a coxeter transformation, affine coxeter transformations, and the defect map. J. Algebra 121, 339–357 (1987)
Collingwood, D., McGovern, W.: Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematical Series. Van Nostrand Reinhold Co., New-York (1993)
Destri C., de Vega H.J.: New thermodynamic Bethe ansatz equations without strings. Phys. Rev. Lett. 69, 2313–2317 (1992)
Dorey P., Dunning C., Masoero D., Suzuki J., Tateo R.: Pseudo-differential equations, and the Bethe ansatz for the classical Lie algebras. Nucl. Phys. B 772(3), 249–289 (2007)
Dorey P., Dunning C., Tateo R.: Differential equations for general SU(n) Bethe ansatz systems. J. Phys. A 33(47), 8427–8441 (2000)
Dorey P., Faldella S., Negro S., Tateo R.: The Bethe Ansatz and the Tzitzeica-Bullough-Dodd equation. Phil. Trans. R. Soc. Lond. A 371, 20120052 (2013)
Dorey P., Negro S., Tateo R.: Affine toda field theories and the bethe ansatz (2015, in preparation)
Dorey P., Tateo R.: Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations. J. Phys. A 32, L419–L425 (1999)
Dorey P., Tateo R.: On the relation between Stokes multipliers and the T-Q systems of conformal field theory. Nucl. Phys. B. 563(3), 573–602 (1999)
Drinfeld V.G., Sokolov V.V.: Lie algebras and equations of kdv type. Sov. J. Math. 30, 1975–2036 (1985)
Fedoryuk M.: Asymptotic analysis. Springer, Berlin (1993)
Feigin, B., Frenkel, E.: Quantization of soliton systems and Langlands duality. In: Exploring new structures and natural constructions in mathematical physics, Adv. Stud. Pure Math., vol. 61, pp. 185–274. Math. Soc. Japan, Tokyo (2011)
Fring A., Liao H., Olive D.: The Mass spectrum and coupling in affine Toda theories. Phys. Lett. B 266, 82–86 (1991)
Fulton, W., Harris, J.: Representation theory, Graduate Texts in Mathematics, vol. 129. Springer-Verlag, New York, 1991. A first course, Readings in Mathematics (1991)
Gaiotto D., Moore G., Neitzke A.: Wall-crossing, hitchin systems, and the wkb approximation. Adv. Math. 234, 239–403 (2013)
The GAP Group.: GAP—groups, algorithms, and programming, Version 4.7.6 (2014)
Ilyashenko, Y., Yakovenko, S.: Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86. American Mathematical Society, Providence (2008)
Kac, V., Scwarz, A.: Geometric interpretation of the partition function of 2D gravity. Phys. Lett. B 3–4(257), 329–334 (1991)
Kac V.G.: Infinite-dimensional Lie algebras, 3rd edn. Cambridge University Press, Cambridge (1990)
Kostant B.: The principal three-dimensional subgroup and the betti numbers of a complex simple lie group. Am. J. Math. 81, 973–1032 (1959)
Levinson N.: The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948)
Lukyanov S.L., Zamolodchikov A.B.: Quantum sine(h)-Gordon model and classical integrable equations. JHEP 1007, 008 (2010)
Masoero D.: Y-System and deformed thermodynamic bethe ansatz. Lett. Math. Phys. 94(2), 151–164 (2010)
Masoero D., Raimondo A., Valeri D.: Bethe Ansatz and the Spectral Theory of affine Lie algebra–valued connections II. The non simply–laced case. arXiv:1511.00895
Negro, S.: ODE/IM correspondence in Toda field theories and fermionic basis in sin(h)-Gordon model. PhD thesis, Università degli Studi di Torino (2014)
Reshetikhin N.Yu., Wiegmann P.B.: Towards the Classification of completely integrable quantum field theories. Phys. Lett. B 189, 125–131 (1987)
Sun, J.: Polynomial relations for q-characters via the ODE/IM correspondence. SIGMA Symmetry Integrability Geom. Methods Appl. 8: Paper 028, 34 (2012)
Suzuki, J.: Private communication
Suzuki J.: Stokes multipliers, spectral determinants and T -Q relations. Sūrikaisekikenkyūsho Kōkyūroku, (1221):21–37, 2001. Development in discrete integrable systems—ultra-discretization, quantization (Japanese) (Kyoto, 2000).
Zamolodchikov A.B.: On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories. Phys. Lett. B 253(3-4), 391–394 (1991)
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Masoero, D., Raimondo, A. & Valeri, D. Bethe Ansatz and the Spectral Theory of Affine Lie Algebra-Valued Connections I. The simply-laced Case. Commun. Math. Phys. 344, 719–750 (2016). https://doi.org/10.1007/s00220-016-2643-6
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DOI: https://doi.org/10.1007/s00220-016-2643-6