Abstract
We explain that the set of new integrable systems, generalizing the Calogero family and implied by the study of WLZZ models, which was described in arXiv:2303.05273, is only the tip of the iceberg. We provide its wide generalization and explain that it is related to commutative subalgebras (Hamiltonians) of the W1+∞ algebra. We construct many such subalgebras and explain how they look in various representations. We start from the even simpler w∞ contraction, then proceed to the one-body representation in terms of differential operators on a circle, further generalizing to matrices and in their eigenvalues, in finally to the bosonic representation in terms of time-variables. Moreover, we explain that some of the subalgebras survive the β-deformation, an intermediate step from W1+∞ to the affine Yangian. The very explicit formulas for the corresponding Hamiltonians in these cases are provided. Integrable many-body systems generalizing the rational Calogero model arise in the representation in terms of eigenvalues. Each element of W1+∞ algebra gives rise to KP/Toda τ-functions. The hidden symmetry given by the families of commuting Hamiltonians is in charge of the special, (skew) hypergeometric τ-functions among these.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Mironov and A. Morozov, Many-body integrable systems implied by WLZZ models, Phys. Lett. B 842 (2023) 137964 [arXiv:2303.05273] [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, The Complete Structure of W∞, Phys. Lett. B 236 (1990) 173 [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, W∞ and the Racah-wigner Algebra, Nucl. Phys. B 339 (1990) 191 [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, A New Higher Spin Algebra and the Lone Star Product, Phys. Lett. B 242 (1990) 401 [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, Ideals of Kac-Moody Algebras and Realizations of W∞, Phys. Lett. B 245 (1990) 72 [INSPIRE].
H. Awata, M. Fukuma, Y. Matsuo and S. Odake, Representation theory of the W1+∞ algebra, Prog. Theor. Phys. Suppl. 118 (1995) 343 [hep-th/9408158] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar, T. Hartman and S. Raju, Partition Functions of Holographic Minimal Models, JHEP 08 (2011) 077 [arXiv:1106.1897] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in Minimal Model Holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
V. Kac and A. Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Commun. Math. Phys. 157 (1993) 429 [hep-th/9308153] [INSPIRE].
J.-t. Ding and K. Iohara, Generalization and deformation of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997) 181 [INSPIRE].
K. Miki, A (q, γ) analog of the W1+∞ algebra, J. Math. Phys. 48 (2007) 123520.
M. Fukuma, H. Kawai and R. Nakayama, Infinite dimensional Grassmannian structure of two-dimensional quantum gravity, Commun. Math. Phys. 143 (1992) 371 [INSPIRE].
I. Bakas and E. Kiritsis, Beyond the large N limit: Nonlinear W(infinity) as symmetry of the SL(2, R)/U(1) coset model, Int. J. Mod. Phys. A 7S1A (1992) 55 [hep-th/9109029] [INSPIRE].
I. Bakas, B. Khesin and E. Kiritsis, The Logarithm of the derivative operator and higher spin algebras of W∞ type, Commun. Math. Phys. 151 (1993) 233 [INSPIRE].
E. Frenkel, V. Kac, A. Radul and W.-Q. Wang, W1+∞ and W (gl(N)) with central charge N, Commun. Math. Phys. 170 (1995) 337 [hep-th/9405121] [INSPIRE].
V. Kac and A. Radul, Representation theory of the vertex algebra W1+∞, hep-th/9512150 [INSPIRE].
A. Tsymbaliuk, The affine Yangian of \( \mathfrak{gl} \)1 revisited, Adv. Math. 304 (2017) 583 [arXiv:1404.5240] [INSPIRE].
T. Procházka, \( \mathcal{W} \)-symmetry, topological vertex and affine Yangian, JHEP 10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
A. Mironov, V. Mishnyakov, A. Morozov and A. Popolitov, Commutative subalgebras from Serre relations, Phys. Lett. B 845 (2023) 138122 [arXiv:2307.01048] [INSPIRE].
I. Bakas, The Large n Limit of Extended Conformal Symmetries, Phys. Lett. B 228 (1989) 57 [INSPIRE].
I.G. Macdonald, Symmetric functions and Hall polynomials, second edition, Oxford University Press (1995).
W. Fulton, Young tableaux: with applications to representation theory and geometry, Oxford University Press (1997).
A. Morozov, On the Concept of Universal W Algebra, Sov. J. Nucl. Phys. 51 (1990) 758 [INSPIRE].
R. Wang, C.-H. Zhang, F.-H. Zhang and W.-Z. Zhao, CFT approach to constraint operators for (β-deformed) hermitian one-matrix models, Nucl. Phys. B 985 (2022) 115989 [arXiv:2203.14578] [INSPIRE].
R. Wang, F. Liu, C.-H. Zhang and W.-Z. Zhao, Superintegrability for (β-deformed) partition function hierarchies with W-representations, Eur. Phys. J. C 82 (2022) 902 [arXiv:2206.13038] [INSPIRE].
A. Mironov and A. Morozov, Spectral curves and W-representations of matrix models, JHEP 03 (2023) 116 [arXiv:2210.09993] [INSPIRE].
V.G. Kac and D.H. Peterson, Spin and Wedge Representations of Infinite Dimensional Lie Algebras and Groups, Proc. Nat. Acad. Sci. 78 (1981) 3308 [INSPIRE].
W.L. Li, 2-cocycles on the algebra of differential operators, J. Algebra 122 (1989) 64.
B. Feigin, The Lie algebras gl(λ) and the cohomology of the Lie algebra of differential operators, Usp. Math. Nauk 35 (1988) 157.
I.B. Frenkel and V.G. Kac, Basic Representations of Affine Lie Algebras and Dual Resonance Models, Invent. Math. 62 (1980) 23 [INSPIRE].
G. Segal, Unitarity Representations of Some Infinite Dimensional Groups, Commun. Math. Phys. 80 (1981) 301 [INSPIRE].
M. Wakimoto, Fock representations of the affine lie algebra \( {A}_1^{(1)} \), Commun. Math. Phys. 104 (1986) 605 [INSPIRE].
A. Morozov, Bosonization and Multiloop Calculations for Wess-Zumino-Witten Model, Phys. Lett. B 229 (1989) 239 [INSPIRE].
A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys. A 5 (1990) 2495 [INSPIRE].
B. Feigin and E. Frenkel, A family of representations of affine Lie algebras, Usp. Math. Nauk 43 (1988) 227.
A. Gerasimov, A. Marshakov and A. Morozov, Free Field Representation of Parafermions and Related Coset Models, Nucl. Phys. B 328 (1989) 664 [INSPIRE].
A. Gerasimov, A. Marshakov and A. Morozov, Hamiltonian Reduction of Wess-Zumino-Witten Theory From the Point of View of Bosonization, Phys. Lett. B 236 (1990) 269 [INSPIRE].
A.N. Sergeev and A.P. Veselov, Calogero-Moser operators in infinite dimension, arXiv e-prints (2009) arXiv:0910.1984 [arXiv:0910.1984].
S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, Generalized Kazakov-Migdal-Kontsevich model: Group theory aspects, Int. J. Mod. Phys. A 10 (1995) 2015 [hep-th/9312210] [INSPIRE].
A. Orlov and D.M. Shcherbin, Hypergeometric solutions of soliton equations, Theor. Math. Phys. 128 (2001) 906.
A. Alexandrov, A. Mironov, A. Morozov and S. Natanzon, Integrability of Hurwitz Partition Functions. I. Summary, J. Phys. A 45 (2012) 045209 [arXiv:1103.4100] [INSPIRE].
A. Alexandrov, A. Mironov, A. Morozov and S. Natanzon, On KP-integrable Hurwitz functions, JHEP 11 (2014) 080 [arXiv:1405.1395] [INSPIRE].
K. Takasaki, Initial Value Problem for the Toda Lattice Hierarchy, Adv. Stud. Pure Math. 4 (1984) 139.
A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov, R. Wang and W.-Z. Zhao, Interpolating matrix models for WLZZ series, Eur. Phys. J. C 83 (2023) 377 [arXiv:2301.04107] [INSPIRE].
A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov and W.-Z. Zhao, On KP-integrable skew Hurwitz τ-functions and their β-deformations, Phys. Lett. B 839 (2023) 137805 [arXiv:2301.11877] [INSPIRE].
A. Orlov, Vertex operator \( \overline{\delta} \)-problem, symmetries, variational identities and Hamiltonian formalism for (2 + 1)-integrable systems, in Plasma theory and nonlinear and turbulent processes in physics. Vol. 1, World Scientific (1988), pg. 13.
P. Winternitz and A.Yu. Orlov, P∞ algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators, Theor. Mat. Fiz. 113 (1997) 231 [Theor. Math. Phys. 113 (1997) 1393].
K. Takasaki and T. Takebe, Quasiclassical limit of KP hierarchy, W symmetries and free fermions, hep-th/9207081 [INSPIRE].
K. Takasaki and T. Takebe, Quasiclassical limit of Toda hierarchy and W(infinity) symmetries, Lett. Math. Phys. 28 (1993) 165 [hep-th/9301070] [INSPIRE].
A.D. Mironov and A. Morozov, Generalized Q-functions for GKM, Phys. Lett. B 819 (2021) 136474 [arXiv:2101.08759] [INSPIRE].
A. Mironov, A. Morozov and S. Natanzon, Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory, Theor. Math. Phys. 166 (2011) 1 [arXiv:0904.4227] [INSPIRE].
A. Mironov, A. Morozov and S. Natanzon, Algebra of differential operators associated with Young diagrams, J. Geom. Phys. 62 (2012) 148 [arXiv:1012.0433] [INSPIRE].
H. Jack, A class of symmetric polynomials with a parameter, Proc. R. Soc. Edinb. A 69 (1970) 1.
H. Jack, A surface integral and symmetric functions, Proc. R. Soc. Edinb. A 69 (1972) 347.
R.P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989) 76.
W. Ruhl and A. Turbiner, Exact solvability of the Calogero and Sutherland models, Mod. Phys. Lett. A 10 (1995) 2213 [hep-th/9506105] [INSPIRE].
N. Gurappa and P.K. Panigrahi, Free harmonic oscillators, Jack polynomials and Calogero-Sutherland systems, Phys. Rev. B 62 (2000) 1943 [hep-th/9910123] [INSPIRE].
H. Ujino and M. Wadati, Rodrigues Formula for Hi-Jack Symmetric Polynomials Associated with the Quantum Calogero Model, J. Phys. Soc. Jpn. 65 (1996) 2423 [cond-mat/9609041].
M. Wadati and H. Ujino, The Calogero Model: Integrable Structures and Orthogonal Basis, cond-mat/9706156.
A.M. Perelomov, Algebraic approach to the solution of a one-dimensional model of N interacting particles, Teor. Mat. Fiz. 6 (1971) 364 [Theor. Math. Phys. 6 (1971) 263].
A. Mironov and A. Morozov, Superintegrability summary, Phys. Lett. B 835 (2022) 137573 [arXiv:2201.12917] [INSPIRE].
A. Mironov, A. Morozov and Z. Zakirova, New insights into superintegrability from unitary matrix models, Phys. Lett. B 831 (2022) 137178 [arXiv:2203.03869] [INSPIRE].
A. Morozov, Integrability and Matrix Models, arXiv:2212.02632 [INSPIRE].
A.B. Balantekin, Character Expansion for U(N) Groups and U(N/m) Supergroups, J. Math. Phys. 25 (1984) 2028 [INSPIRE].
A.B. Balantekin, Character expansions, Itzykson-Zuber integrals, and the QCD partition function, Phys. Rev. D 62 (2000) 085017 [hep-th/0007161] [INSPIRE].
V.A. Kazakov, Solvable matrix models, hep-th/0003064 [INSPIRE].
A.Y. Morozov, Unitary Integrals and Related Matrix Models, Teor. Mat. Fiz. 161 (2010) 3 Theor. Math. Phys. 162 (2010) 1 [arXiv:0906.3518] [INSPIRE].
Harish-Chandra, Spherical Functions on a Semisimple Lie Group. I, Am. J. Math. 80 (1958) 241.
C. Itzykson and J.B. Zuber, The Planar Approximation. 2., J. Math. Phys. 21 (1980) 411 [INSPIRE].
V. Mishnyakov and A. Oreshina, Superintegrability in β-deformed Gaussian Hermitian matrix model from W-operators, Eur. Phys. J. C 82 (2022) 548 [arXiv:2203.15675] [INSPIRE].
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press (2007).
F. Liu et al., (q, t)-deformed (skew) Hurwitz τ-functions, Nucl. Phys. B 993 (2023) 116283 [arXiv:2303.00552] [INSPIRE].
A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett. B 762 (2016) 196 [arXiv:1603.05467] [INSPIRE].
A. Gerasimov, D. Lebedev and A. Morozov, Possible implications of integrable systems for string theory, Int. J. Mod. Phys. A 06 (1991) 977.
A. Morozov, Integrable systems and double-loop algebras in string theory, Mod. Phys. Lett. A 06 (1991) 1525.
A. Smirnov, Quantum differential and difference equations for Hilbn(ℂ2), arXiv:2102.10726 [INSPIRE].
K. Miki, Toroidal braid group action and an automorphism of toroidal algebra Uq, Lett. Math. Phys. 47 (1999) 365.
A. Marshakov, A. Mironov and A. Morozov, On equivalence of topological and quantum 2-d gravity, Phys. Lett. B 274 (1992) 280 [hep-th/9201011] [INSPIRE].
A. Mikhailov, Ward identities and W constraints in generalized Kontsevich model, Int. J. Mod. Phys. A 9 (1994) 873 [hep-th/9303129] [INSPIRE].
Acknowledgments
A.Mir. is grateful to A. Grigoriev-Savrasov for kind hospitality at late stages of the project. This work was supported by the Russian Science Foundation (Grant No.20-12-00195).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2306.06623
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Mironov, A., Mishnyakov, V., Morozov, A. et al. Commutative families in W∞, integrable many-body systems and hypergeometric τ-functions. J. High Energ. Phys. 2023, 65 (2023). https://doi.org/10.1007/JHEP09(2023)065
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2023)065