Integrable Systems, Shuffle Algebras, and Bethe Equations

Feigin, Boris

doi:10.1090/mosc/259

Integrable Systems, Shuffle Algebras, and Bethe Equations
HTML articles powered by AMS MathViewer

by Boris L. Feigin
Translated by: V. E. Nazaikinskii

Trans. Moscow Math. Soc. 2016, 203-246

DOI: https://doi.org/10.1090/mosc/259

Published electronically: November 28, 2016

PDF | Request permission

Abstract:

We speak about the part of integrable system theory dealing with conformal theory and $W$-algebras (ordinary and deformed). Some new approaches to finding Bethe equations that describe the spectrum of Hamiltonians of these quantum integrable systems are developed. The derivation of the Bethe equations is based on the technique of shuffle algebras arising in quantum group theory.

References

I. M. Gel′fand and L. A. Dikiĭ, Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations, Uspehi Mat. Nauk 30 (1975), no. 5(185), 67–100 (Russian). MR 0508337
V. G. Drinfel′d, Quantum groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), no. Differentsial′naya Geometriya, Gruppy Li i Mekh. VIII, 18–49, 193 (Russian, with English summary); English transl., J. Soviet Math. 41 (1988), no. 2, 898–915. MR 869575, DOI 10.1007/BF01247086
V. G. Drinfel′d, Almost cocommutative Hopf algebras, Algebra i Analiz 1 (1989), no. 2, 30–46 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 2, 321–342. MR 1025154
V. G. Drinfel′d, A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13–17 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 2, 212–216. MR 914215
V. G. Drinfel′d and V. V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 81–180 (Russian). MR 760998
Michio Jimbo and Tetsuji Miwa, Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, vol. 85, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1995. MR 1308712
V. E. Zaharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskiĭ, Teoriya solitonov, “Nauka”, Moscow, 1980 (Russian). Metod obratnoĭ zadachi. [The method of the inverse problem]. MR 573607
A. A. Kirillov, The orbits of the group of diffeomorphisms of the circle, and local Lie superalgebras, Funktsional. Anal. i Prilozhen. 15 (1981), no. 2, 75–76 (Russian). MR 617476
S. L. Luk′yanov and V. A. Fateev, Conformally invariant models of two-dimensional quantum field theory with $Z_n$-symmetry, Zh. Èksper. Teoret. Fiz. 94 (1988), no. 3, 23–37 (Russian); English transl., Soviet Phys. JETP 67 (1988), no. 3, 447–454. MR 966184
S. V. Manakov, A remark on the integration of the Eulerian equations of the dynamics of an $n$-dimensional rigid body, Funkcional. Anal. i Priložen. 10 (1976), no. 4, 93–94 (Russian). MR 0455031
A. S. Miščenko and A. T. Fomenko, Integrability of Euler’s equations on semisimple Lie algebras, Trudy Sem. Vektor. Tenzor. Anal. 19 (1979), 3–94 (Russian). MR 549008
L. G. Rybnikov, The shift of invariants method and the Gaudin model, Funktsional. Anal. i Prilozhen. 40 (2006), no. 3, 30–43, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 40 (2006), no. 3, 188–199. MR 2265683, DOI 10.1007/s10688-006-0030-3
A. M. Perelomov, Integrable systems of classical mechanics and Lie algebras. Vol. I, Birkhäuser Verlag, Basel, 1990. Translated from the Russian by A. G. Reyman [A. G. Reĭman]. MR 1048350, DOI 10.1007/978-3-0348-9257-5
A. G. Reyman and M. A. Semenov-Tian-Shansky, Integrable systems, Inst. Komp’yut. Issled, Moscow–Izhevsk, 2003.
E. K. Skljanin, L. A. Tahtadžjan, and L. D. Faddeev, Quantum inverse problem method. I, Teoret. Mat. Fiz. 40 (1979), no. 2, 194–220 (Russian, with English summary). MR 549615
D. V. Talalaev, The quantum Gaudin system, Funktsional. Anal. i Prilozhen. 40 (2006), no. 1, 86–91 (Russian); English transl., Funct. Anal. Appl. 40 (2006), no. 1, 73–77. MR 2223256, DOI 10.1007/s10688-006-0012-5
L. A. Tahtadžjan and L. D. Faddeev, The quantum method for the inverse problem and the $XYZ$ Heisenberg model, Uspekhi Mat. Nauk 34 (1979), no. 5(209), 13–63, 256 (Russian). MR 562799
B. L. Feĭgin, Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras, Uspekhi Mat. Nauk 39 (1984), no. 2(236), 195–196 (Russian). MR 740035
Vladimir V. Bazhanov, Sergei L. Lukyanov, and Alexander B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Comm. Math. Phys. 177 (1996), no. 2, 381–398. MR 1384140
Vladimir V. Bazhanov, Sergei L. Lukyanov, and Alexander B. Zamolodchikov, Integrable structure of conformal field theory. II. $\textrm {Q}$-operator and DDV equation, Comm. Math. Phys. 190 (1997), no. 2, 247–278. MR 1489571, DOI 10.1007/s002200050240
A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint, http://www.math.uchicago.edu/\string mitya/langlands/QuantizationHitchin.pdf
A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975), no. 1, 85–87. MR 434183, DOI 10.1016/0370-2693(75)90163-X
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), no. 2, 333–380. MR 757857, DOI 10.1016/0550-3213(84)90052-X
Alexandr Buryak and Paolo Rossi, Double ramification cycles and quantum integrable systems, Lett. Math. Phys. 106 (2016), no. 3, 289–317. MR 3462029, DOI 10.1007/s11005-015-0814-6
Jintai Ding and Kenji Iohara, Generalization of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997), no. 2, 181–193. MR 1463869, DOI 10.1023/A:1007341410987
Pavel Etingof, Whittaker functions on quantum groups and $q$-deformed Toda operators, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 9–25. MR 1729357, DOI 10.1090/trans2/194/02
Pavel Etingof and David Kazhdan, Quantization of Lie bialgebras. I, Selecta Math. (N.S.) 2 (1996), no. 1, 1–41. MR 1403351, DOI 10.1007/BF01587938
Boris Feigin and Edward Frenkel, Quantization of the Drinfel′d-Sokolov reduction, Phys. Lett. B 246 (1990), no. 1-2, 75–81. MR 1071340, DOI 10.1016/0370-2693(90)91310-8
Boris Feigin and Edward Frenkel, Duality in $W$-algebras, Internat. Math. Res. Notices 6 (1991), 75–82. MR 1136408, DOI 10.1155/S1073792891000119
Boris Feigin and Edward Frenkel, Free field resolutions in affine Toda field theories, Phys. Lett. B 276 (1992), no. 1-2, 79–86. MR 1153194, DOI 10.1016/0370-2693(92)90544-E
Boris Feigin and Edward Frenkel, Affine Kac-Moody algebras at the critical level and Gel′fand-Dikiĭ algebras, Infinite analysis, Part A, B (Kyoto, 1991) Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 197–215. MR 1187549, DOI 10.1142/s0217751x92003781
Boris Feigin and Edward Frenkel, Integrals of motion and quantum groups, Integrable systems and quantum groups (Montecatini Terme, 1993) Lecture Notes in Math., vol. 1620, Springer, Berlin, 1996, pp. 349–418. MR 1397275, DOI 10.1007/BFb0094794
Boris Feigin and Edward Frenkel, Quantum $\scr W$-algebras and elliptic algebras, Comm. Math. Phys. 178 (1996), no. 3, 653–678. MR 1395209
Boris Feigin and Edward Frenkel, Quantization of soliton systems and Langlands duality, Exploring new structures and natural constructions in mathematical physics, Adv. Stud. Pure Math., vol. 61, Math. Soc. Japan, Tokyo, 2011, pp. 185–274. MR 2867148, DOI 10.2969/aspm/06110185
B. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, Representations of quantum toroidal ${\mathfrak {gl}}_n$, J. Algebra 380 (2013), 78–108. MR 3023228, DOI 10.1016/j.jalgebra.2012.12.029
B. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, Quantum toroidal $\mathfrak {gl}_1$ and Bethe ansatz, J. Phys. A 48 (2015), no. 24, 244001, 27. MR 3355243, DOI 10.1088/1751-8113/48/24/244001
B. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, Finite type modules and Bethe ansatz for quantum toroidal $\mathfrak {gl}(1)$, arXiv:1603.02765.
B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, and S. Yanagida, A commutative algebra on degenerate $\Bbb {CP}^1$ and Macdonald polynomials, J. Math. Phys. 50 (2009), no. 9, 095215, 42. MR 2566895, DOI 10.1063/1.3192773
B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, and S. Yanagida, Kernel function and quantum algebras, RIMS kokyuroku, 1689 (2010, 133–152. arXiv:1002.2485.
B. L. Feigin and A. V. Odesskii, Quantized moduli spaces of the bundles on the elliptic curve and their applications, Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000) NATO Sci. Ser. II Math. Phys. Chem., vol. 35, Kluwer Acad. Publ., Dordrecht, 2001, pp. 123–137. MR 1873568
Boris Feigin and Alexander Tsymbaliuk, Bethe subalgebras of $U_q(\widehat {\mathfrak {gl}}_n)$ via shuffle algebras, Selecta Math. (N.S.) 22 (2016), no. 2, 979–1011. MR 3477340, DOI 10.1007/s00029-015-0212-z
Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2nd ed., Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2004. MR 2082709, DOI 10.1090/surv/088
Edward Frenkel and Nikolai Reshetikhin, Quantum affine algebras and deformations of the Virasoro and ${\scr W}$-algebras, Comm. Math. Phys. 178 (1996), no. 1, 237–264. MR 1387950
Edward Frenkel and Nicolai Reshetikhin, Deformations of $\scr W$-algebras associated to simple Lie algebras, Comm. Math. Phys. 197 (1998), no. 1, 1–32. MR 1646483, DOI 10.1007/BF02099206
Edward Frenkel and Nicolai Reshetikhin, The $q$-characters of representations of quantum affine algebras and deformations of $\scr W$-algebras, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998) Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205. MR 1745260, DOI 10.1090/conm/248/03823
Michio Jimbo, A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001, DOI 10.1007/BF00704588
Boris Khesin and Ilya Zakharevich, Poisson-Lie group of pseudodifferential symbols, Comm. Math. Phys. 171 (1995), no. 3, 475–530. MR 1346170
A. V. Litvinov, On spectrum of ILW hierarchy in conformal field theory, J. High Energy Phys. 11 (2013), 155, front matter+13. MR 3132164, DOI 10.1007/JHEP11(2013)155
D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287.
Kei Miki, A $(q,\gamma )$ analog of the $W_{1+\infty }$ algebra, J. Math. Phys. 48 (2007), no. 12, 123520, 35. MR 2377852, DOI 10.1063/1.2823979
E. Mukhin, V. Tarasov, and A. Varchenko, Bethe algebra of homogeneous $XXX$ Heisenberg model has simple spectrum, Comm. Math. Phys. 288 (2009), no. 1, 1–42. MR 2491616, DOI 10.1007/s00220-009-0733-4
E. Mukhin, V. Tarasov, and A. Varchenko, On separation of variables and completeness of the Bethe ansatz for quantum ${\mathfrak {gl}}_N$ Gaudin model, Glasg. Math. J. 51 (2009), no. A, 137–145. MR 2481232, DOI 10.1017/S0017089508004850
Nikita A. Nekrasov and Samson L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nuclear Phys. B Proc. Suppl. 192/193 (2009), 91–112. MR 2570974, DOI 10.1016/j.nuclphysbps.2009.07.047
Nikita A. Nekrasov and Samson L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, pp. 265–289. MR 2730782, DOI 10.1142/9789814304634_{0}015
A. Okounkov and R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, Invent. Math. 179 (2010), no. 3, 523–557. MR 2587340, DOI 10.1007/s00222-009-0223-5
N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky, Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990), no. 2, 133–142. MR 1039522, DOI 10.1007/BF01045884
Nicolai Reshetikhin and Alexander Varchenko, Quasiclassical asymptotics of solutions to the KZ equations, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 293–322. MR 1358621
Jun’ichi Shiraishi, Harunobu Kubo, Hidetoshi Awata, and Satoru Odake, A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 (1996), no. 1, 33–51. MR 1401054, DOI 10.1007/BF00398297
Alexander Tsymbaliuk, The affine Yangian of $\mathfrak {gl}_1$ revisited, Adv. Math. 304 (2017), 583–645. MR 3558218, DOI 10.1016/j.aim.2016.08.041
A. Yu. Volkov, Quantum lattice KdV equation, Lett. Math. Phys. 39 (1997), no. 4, 313–329. MR 1449577, DOI 10.1023/A:1007390915590