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Multi-pole extension of the elliptic models of interacting integrable tops

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Abstract

We review and give a detailed description of the \(gl_{NM}\) Gaudin models related to holomorphic vector bundles of rank \(NM\) and degree \(N\) over an elliptic curve with \(n\) punctures. We introduce their generalizations constructed by means of \(R\)-matrices satisfying the associative Yang–Baxter equation. A natural extension of the obtained models to the Schlesinger systems is also given.

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Notes

  1. See the Appendix for the definitions of elliptic functions.

  2. “Half” refers to the fact that taking the difference of two such equations does yield the classical Yang–Baxter equation.

References

  1. F. Calogero, “Exactly solvable one-dimensional many-body problems,” Lett. Nuovo Cimento, 13, 411–416 (1975); “On a functional equation connected with integrable many-body problems,” Lett. Nuovo Cimento, 16, 77–80 (1976); J. Moser, “Three integrable Hamiltonian systems connected with isospectral deformations,” Adv. Math., 16, 197–220 (1975); M. A. Olshanetsky and A. M. Perelomov, “Classical integrable finite-dimensional systems related to Lie algebras,” Phys. Rep., 71, 313–400 (1981).

    Article  MathSciNet  Google Scholar 

  2. I. M. Krichever, “Elliptic solutions of the Kadomtsev–Petviashvili equation and integrable systems of particles,” Funct. Anal. Appl., 14, 282–290 (1980).

    Article  Google Scholar 

  3. J. Gibbons and T. Hermsen, “A generalization of the Calogero-Moser systems,” Phys. D, 11, 337–348 (1984); S. Wojciechowski, “An integrable marriage of the Euler equations with the Calogero–Moser system,” Phys. Lett. A, 111, 101–103 (1985).

    Article  MathSciNet  Google Scholar 

  4. E. Billey, J. Avan, and O. Babelon, “The \(r\)-matrix structure of the Euler–Calogero–Moser model,” Phys. Lett. A, 186, 114–118 (1994); arXiv: hep-th/9312042.

    Article  ADS  MathSciNet  Google Scholar 

  5. N. Nekrasov, “Holomorphic Bundles and many-body systems,” Commun. Math. Phys., 180, 587–604 (1996); arXiv: hep-th/9503157.

    Article  ADS  MathSciNet  Google Scholar 

  6. R. J. Baxter, “Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. II. Equivalence to a generalized ice-type lattice model,” Ann. Phys., 76, 25–47 (1973); L. A. Takhtadzhyan and L. D. Faddeev, “The quantum method of the inverse problem and the Heisenberg \(XYZ\) model,” Russian Math. Surveys, 34, 11–68 (1979); A. A. Belavin, “Dynamical symmetry of integrable quantum systems,” Nucl. Phys. B, 180, 189–200 (1981); M. P. Richey and C. A. Tracy, “Baxter model: symmetries and the Belavin parametrization,” J. Stat. Phys., 42, 311–348 (1986).

    Article  ADS  Google Scholar 

  7. E. K. Sklyanin, “On complete integrability of the Landau–Lifshitz equation,” Preprint LOMI, E-3-79, LOMI, Leningrad (1979); E. K. Sklyanin, “Poisson structure of a periodic classical \(XYZ\)-chain,” J. Soviet Math., 46, 1664–1683 (1989); L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (1987); E. K. Sklyanin, “Some algebraic structures connected with the Yang–Baxter equation,” Funct. Anal. Appl., 16, 263–270 (1982).

  8. A. Levin, M. Olshanetsky, and A. Zotov, “Hitchin systems – symplectic Hecke correspondence and two-dimensional version,” Commun. Math. Phys., 236, 93–133 (2003); arXiv: nlin/0110045 [nlin.SI]; A. V. Zotov, “Classical integrable systems and their field-theoretical generalizations,” Phys. Part. Nucl., 37, 400–443 (2006); A. V. Zotov and A. V. Smirnov, “Modifications of bundles, elliptic integrable systems, and related problems,” Theoret. and Math. Phys., 177, 1281–1338 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  9. A. G. Reyman and M. A. Semenov-Tian-Shansky, “Lie algebras and Lax equations with spectral parameter on an elliptic curve,” J. Sov. Math., 46, 1631–1640 (1989).

    Article  MathSciNet  Google Scholar 

  10. M. Gaudin, “Diagonalisation d’une classe d’Hamiltoniens de spin,” J. Physique, 37, 1087–1098 (1976); La fonction d’onde de Bethe, Masson, Paris (1983).

    Article  MathSciNet  Google Scholar 

  11. N. Hitchin, “Stable bundles and integrable systems,” Duke Math. J., 54, 91–114 (1987).

    Article  MathSciNet  Google Scholar 

  12. A. Levin, M. Olshanetsky, A. Smirnov, and A. Zotov, “Characteristic classes and Hitchin systems. General construction,” Commun. Math. Phys., 316, 1–44 (2012), arXiv: 1006.0702; “Calogero–Moser systems for simple Lie groups and characteristic classes of bundles,” J. Geom. Phys., 62, 1810–1850 (2012), arXiv: 1004.3163; “Hecke transformations of conformal blocks in WZW theory. I. KZB equations for non-trivial bundles,” SIGMA, 8, 095, 37 pp. (2012), arXiv: math-ph/1207.4386.

    Article  ADS  MathSciNet  Google Scholar 

  13. A. P. Polychronakos, “Calogero–Moser models with noncommutative spin interactions,” Phys. Rev. Lett., 89, 126403, 4 pp. (2002), arXiv: hep-th/0112141; “Generalized Calogero models through reductions by discrete symmetries,” Nucl. Phys. B, 543, 485–498 (1999), arXiv: hep-th/9810211; “The physics and mathematics of Calogero particles,” J. Phys. A: Math. Gen., 39, 12793–12845 (2006), arXiv: /hep-th/0607033.

    Article  ADS  Google Scholar 

  14. A. V. Zotov and A. M. Levin, “Integrable model of interacting elliptic tops,” Theoret. and Math. Phys., 146, 45–52 (2006); A. Levin, M. Olshanetsky, A. Smirnov, and A. Zotov, “Characteristic classes of \(\mathrm{SL}(N,\mathbb C)\)-bundles and quantum dynamical elliptic \(R\)-matrices,” J. Phys. A: Math. Theor., 46, 035201, 25 pp. (2013), arXiv: math-ph/1208.5750.

    Article  ADS  MathSciNet  Google Scholar 

  15. M. F. Atiyah, “Vector bundles over an elliptic curve,” Proc. London Math. Soc., 7, 414–452 (1957).

    Article  MathSciNet  Google Scholar 

  16. V. Pasquier, “Etiology of IRF models,” Commun. Math. Phys., 118, 355–364 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  17. A. Levin, M. Olshanetsky, and A. Zotov, “Planck constant as spectral parameter in integrable systems and KZB equations,” JHEP, 10, 109, 29 pp. (2014), arXiv: hep-th/1408.6246; A. M. Levin, M. A. Olshanetsky, and A. V. Zotov, “Quantum Baxter–Belavin \(R\)-matrices and multidimensional Lax pairs for Painlevé VI,” Theoret. and Math. Phys., 184, 924–939 (2015), arXiv: 1501.07351.

    Article  ADS  MathSciNet  Google Scholar 

  18. A. Levin, M. Olshanetsky, and A. Zotov, “Relativistic classical integrable tops and quantum \(R\)-matrices,” JHEP, 07, 012, 39 pp. (2014), arXiv: 1405.7523; T. Krasnov and A. Zotov, “Trigonometric integrable tops from solutions of associative Yang–Baxter equation,” Ann. Henri Poincaré, 20, 2671–2697 (2019), arXiv: 1812.04209; G. Aminov, S. Arthamonov, A. Smirnov, and A. Zotov, “Rational top and its classical \(r\)-matrix,” J. Phys. A: Math. Theor., 47, 305207, 19 pp. (2014), arXiv: 1402.3189.

    Article  ADS  Google Scholar 

  19. A. Grekov, I. Sechin, and A. Zotov, “Generalized model of interacting integrable tops,” JHEP, 2019, 81, 33 pp. (2019), arXiv: 1905.07820; I. A. Sechin and A. V. Zotov, “\({\rm GL}_{NM}\) quantum dynamical \(R\)-matrix based on solution of the associative Yang–Baxter equation,” Russian Math. Surveys, 74, 767–769 (2019), arXiv: 1905.08724; A. Grekov and A. Zotov, “On \(R\)-matrix valued Lax pairs for Calogero–Moser models,” J. Phys. A: Math. Theor., 51, 315202, 26 pp. (2018), arXiv: 1801.00245.

    Article  MathSciNet  Google Scholar 

  20. V. I. Inozemtsev, “The finite Toda lattices,” Commun. Math. Phys., 121, 629–638 (1989); A. V. Zotov and Yu. B. Chernyakov, “Integrable many-body systems via the Inosemtsev limit,” Theoret. and Math. Phys., 129, 1526–1542 (2001), arXiv: hep-th/0102069.

    Article  ADS  MathSciNet  Google Scholar 

  21. A. V. Zotov, “\(1+1\) Gaudin Model,” SIGMA, 7, 067, 26 pp. (2011), arXiv: 1012.1072.

    MathSciNet  MATH  Google Scholar 

  22. S. Fomin and A. N. Kirillov, “Quadratic algebras, Dunkl elements, and Schubert calculus” (Progress in Mathematics, Vol. 172, J.-L. Brylinski, R. Brylinski, V. Nistor, B. Tsygan, and P. Xu, eds.), Birkhäuser, Boston, MA (1999), pp. 147–182; A. Polishchuk, “Classical Yang–Baxter equation and the \(A_\infty\)-constraint,” Adv. Math., 168, 56–95 (2002).

  23. A. Levin, M. Olshanetsky, and A. Zotov, “Noncommutative extensions of elliptic integrable Euler-Arnold tops and Painleve VI equation,” J. Phys. A: Math. Theor., 49, 395202, 24 pp. (2016); arXiv: 1603.06101.

    Article  MathSciNet  Google Scholar 

  24. I. Sechin and A. Zotov, “\(R\)-matrix-valued Lax pairs and long-range spin chains,” Phys. Lett. B, 781, 1–7 (2018); arXiv: 1801.08908.

    Article  ADS  MathSciNet  Google Scholar 

  25. A. M. Levin and M. A. Olshanetsky, “Hierarchies of isomonodromic deformations and Hitchin systems” (Amer. Math. Soc. Transl. Ser. 2, Vol. 191, A. Yu. Morozov and M. A. Olshanetsky, eds.), AMS, Providence, RI (1999), pp. 223–262; K. Takasaki, “Gaudin model, KZ equation, and isomonodromic problem on torus,” Lett. Math. Phys., 44, 143–156 (1998), arXiv: hep-th/9711058; Yu. Chernyakov, A. M. Levin, M. Olshanetsky, and A. Zotov, “Elliptic Schlesinger system and Painlevé VI,” J. Phys. A: Math. Gen., 39, 12083–12101 (2006), arXiv: nlin/0602043.

    MathSciNet  MATH  Google Scholar 

  26. A. M. Levin and M. A. Olshanetsky, “Painlevé–Calogero correspondence” in: Calogero– Moser– Sutherland Models (Montréal, Quebec, Canada, 10–15 March, 1997, CRM Series in Mathematical Physics, J. F. van Diejen and L. Vinet, eds.), Springer, New York (2000), pp. 313–332; arXiv: alg-geom/9706010.

    Article  Google Scholar 

  27. A. M. Levin, M. A. Olshanetsky, and A. V. Zotov, “Classification of isomonodromy problems on elliptic curves,” Russian Math. Surveys, 69, 35–118 (2014); arXiv: 1311.4498.

    Article  ADS  MathSciNet  Google Scholar 

  28. A. Weil, Elliptic Functions According to Eisenstein and Kronecker (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 88), Springer, Berlin (1976); D. Mumford, Tata Lectures on Theta I, II (Progress in Mathematics, Vol. 43), Birkhäuser, Boston (1984).

    Book  Google Scholar 

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Funding

This work is supported by the Russian Science Foundation under grant 19-11-00062 and performed in Steklov Mathematical Institute of Russian Academy of Sciences.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    E. S. Trunina & A. V. Zotov

  2. Institute of Physics and Technology, Dolgoprudny, Moscow region, Russia

    E. S. Trunina

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  1. E. S. Trunina
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  2. A. V. Zotov
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Correspondence to A. V. Zotov.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 209, pp. 16–45 https://doi.org/10.4213/tmf10114.

Appendix Elliptic functions

The basic element for the construction of Lax pairs is the Kronecker elliptic function [28]

$$\phi(z,u)=\frac{\vartheta'(0)\vartheta(z+u)}{\vartheta(z)\vartheta(u)}, $$
(A.1)
defined in terms of the odd Riemann theta function
$$\vartheta(z)=\vartheta(z\mid\tau)=-\sum_{k\in\mathbb Z} \exp\biggl(\pi i\tau\biggl(k+\frac{1}{2}\biggr)^2 +2\pi i\biggl(z+\frac{1}{2}\biggr)\biggl(k+\frac{1}{2}\biggr)\!\biggr). $$
(A.2)

Function (A.1) has the obvious properties

$$\phi(z,u)=\phi(u,z),\qquad \phi(-z,-u)=-\phi(z,u). $$
(A.3)

We also need the derivative \(f(z,u)=\partial_u\varphi(z,u)\) given by

$$f(z,u)=\phi(z,u)(E_1(z+u)-E_1(u)),\qquad f(-z,-u)=f(z,u), $$
(A.4)
where (the first and the second) Eisenstein functions are
$$\begin{aligned} \, &E_1(z)=\partial_z\ln\vartheta(z),\qquad E_2(z)=-\partial_zE_1(z)=\wp(z)-\frac{\vartheta'''(0)}{3\vartheta'(0)}, \\ &E_1(-z)=E_1(z),\qquad E_2(-z)=E_2(z). \end{aligned}$$
For these functions, the following local expansions hold near \(z=0\):
$$\phi(z,u)=\frac{1}{z}+E_1(u)+z\rho(u)+O(z^2),$$
(A.5)
$$E_1(z)=\frac{1}{z}+\frac{z}{3}\frac{\vartheta'''(0)}{\vartheta'(0)}+O(z^3),$$
(A.6)
$$f(0,u)=-E_2(u),$$
(A.7)
where we use the notation
$$\rho(z)=\frac{E^2_1(z)-\wp(z)}{2}. $$
(A.8)
We have the quasiperiodic behavior the lattice of periods \(1\) and \(\tau\)
$$\begin{aligned} \, &E_1(z+1)=E_1(z),\qquad E_1(z+\tau)=E_1(z)-2\pi i, \\ &E_2(z+1)=E_2(z),\qquad E_2(z+\tau)=E_2(z), \\ &\phi(z+1,u)=\phi(z,u),\qquad \phi(z+\tau,u)=e^{-2\pi iu}\phi(z,u), \\ &f(z+1,u)=f(z,u),\qquad f(z+\tau,u)=e^{-2\pi iu}(f(z,u)-2\pi i\phi(z,u)). \end{aligned} $$
(A.9)
The addition formula and its degenerations are
$$\phi(z_1,u_1)\phi(z_2,u_2)=\phi(z_1,u_1+u_2)\phi(z_2-z_1,u_2) +\phi(z_2,u_1+u_2)\phi(z_1-z_2,u_1),$$
(A.10)
$$f(z_1,u_1)\phi(z_2,u_2)-\phi(z_1,u_1)f(z_2,u_2)= \phi(z_2,u_1+u_2)f(z_{12},u_1) -\phi(z_1,u_1+u_2)f(z_{21},u_2),$$
(A.11)
$$f(z,u_1)\phi(z,u_2)-\phi(z,u_1)f(z,u_2) =\phi(z,u_1+u_2)(E_2(u_2)-E_2(u_1)),$$
(A.12)
$$\phi(z,u)\phi(z,-u)=E_2(z)-E_2(u)=\wp(z)-\wp(u),$$
(A.13)
$$\phi(z,u_1)\phi(z,u_2) =\phi(z,u_1+u_2)(E_1(z)+E_1(u_1)+E_1(u_2)-E_1(z+u_1+u_2)),$$
(A.14)
$$\phi(z_1,u)\phi(z_2,u) =\phi(z_1+z_2,u)(E_1(z_1)+E_1(z_2))-f(z_1+z_2,u),$$
(A.15)
$$\phi(z_1,u)\rho(z_2)-E_1(z_2)f(z_1,u) +\phi(z_2,u)f(z_{12},u)-\phi(z_1,u)\rho(z_{21}) =\frac{1}{2}\,\partial_uf(z_1,u),$$
(A.16)
$$(E_1(u+v)-E_1(u)-E_1(v))^2=\wp(u+v)+\wp(u)+\wp(v),$$
(A.17)
$$\phi(z,u)\rho(z)-E_1(z)f(z,u)-\phi(z,u)\wp(u) =\frac{1}{2}\,\partial_uf(z,u).$$
(A.18)
Using the Kronecker elliptic function and its derivative, we define the functions
$$\varphi_\alpha(z,\omega_\alpha+u) =e^{2\pi i\alpha_2z/N}\phi(z,\omega_\alpha+u),\qquad \omega_\alpha=\frac{\alpha_1+\alpha_2\tau}{N},$$
(A.19)
$$f_\alpha(z,\omega_\alpha+u)=e^{2\pi i\alpha_2z/N}f(z,\omega_\alpha+u),$$
(A.20)
$$f_\alpha(z,\omega_\alpha+u)=\partial_u\varphi_\alpha(z,\omega_\alpha+u)= \varphi_\alpha(z,\omega_\alpha+u)(E_1(z+\omega_\alpha+u)-E_1(\omega_\alpha+u)).$$
(A.21)
Functions (A.19) are elements of a basis in the space of sections of the \(\operatorname{End}(V)\) for a holomorphic vector bundle \(V\) (over the elliptic curve) of degree \(1\).

The addition formulas for the basis functions take the form

$$\begin{aligned} \, \varphi_\alpha(z_1,\omega_\alpha+u_1)\varphi_\beta(z_2,\omega_\beta+u_2) ={}&\varphi_\alpha(z_1-z_2,\omega_\alpha+u_1)\varphi_{\alpha+\beta} (z_2,\omega_{\alpha+\beta}+u_1+u_2)+{} \nonumber \\ &+\varphi_\beta(z_2-z_1,\omega_\beta+u_1) \varphi_{\alpha+\beta}(z_1,\omega_{\alpha+\beta}+u_1+u_2), \end{aligned}$$
(A.22)
In particular,
$$\begin{aligned} \, &\varphi_\alpha(z-z_a,\omega_\alpha)\varphi_\beta(z-z_b,\omega_\beta)= \nonumber \\ &=\varphi_\alpha(z_{ba},\omega_\alpha)\varphi_{\alpha+\beta} (z-z_b,\omega_{\alpha+\beta})+\varphi_\beta(z_{ab},\omega_\beta) \varphi_{\alpha+\beta}(z-z_a,\omega_{\alpha+\beta}), \end{aligned}$$
(A.23)
$$\begin{aligned} \, &\varphi_\alpha(z,\omega_\alpha+u_1)\varphi_\beta(z,\omega_\beta+u_2) =\varphi_{\alpha+\beta}(z,\omega_{\alpha+\beta}+u_1+u_2)\times{} \nonumber \\ &\times(E_1(z)+E_1(\omega_\alpha+u_1)+E_1(\omega_\beta+u_2) -E_1(z+\omega_{\alpha+\beta}+u_1+u_2)), \end{aligned}$$
(A.24)
$$\begin{aligned} \, &\varphi_\alpha(z_1,\omega_\alpha+u)\varphi_\alpha(z_2,\omega_\alpha+u)= \nonumber \\ &=\varphi_\alpha(z_1+z_2,\omega_\alpha+u) (E_1(z_1)+E_1(z_2))-f_\alpha(z_1+z_2,\omega_\alpha+u). \end{aligned}$$
(A.25)

The Baxter–Belavin elliptic \(R\)-matrix [6]

$$R^{\mathrm{BB}}_{12}(z,x)=\sum_\alpha\varphi_\alpha(x,z+\omega_\alpha) T_\alpha\otimes T_{-\alpha}\in\operatorname{Mat}(N,\mathbb C)^{\otimes 2} $$
(A.26)
satisfies all the required properties (5.3)–(5.19), but with a different normalization. We use an \(R\)-matrix slightly different from (A.26) to ensure that all properties hold with the correct normalization coefficients:
$$R^z_{12}(x)=R^{\mathrm{BB}}_{12}\biggl(\frac{z}{N},x\biggr) =\frac{1}{N}\sum_\alpha\varphi_\alpha \biggl(x,\frac{z}{N}+\omega_\alpha\biggr)T_\alpha\otimes T_{-\alpha}. $$
(A.27)
Using (A.5) and (5.3), we obtain the corresponding classical \(r\)- and \(m\)-matrices
$$\begin{aligned} \, &r_{12}(z)=\frac{1}{N}E_1(z)1_N\otimes 1_N +\frac{1}{N}\sum_{\alpha\ne 0}\varphi_\alpha(z,\omega_\alpha) T_\alpha\otimes T_{-\alpha}, \\ &m_{12}(z)=\frac{1}{N^2}\rho(z)1_N\otimes 1_N +\frac{1}{N^2}\sum_{\alpha\ne 0}f_\alpha(z,\omega_\alpha) T_\alpha\otimes T_{-\alpha}. \end{aligned} $$
(A.28)
Taking the derivative of the \(r\)-matrix gives
$$\begin{aligned} \, F^0_{12}(z) ={}&\partial_zr_{12}(z)=-\frac{1}{N}E_2(z)1_N\otimes 1_N+{} \nonumber \\ &+\frac{1}{N^2}\sum_{\alpha\ne 0}\varphi_\alpha(z,\omega_\alpha) (E_1(z+\omega_\alpha) -E_1(z)+2\pi i\,\partial\tau\omega_\alpha)T_\alpha\otimes T_{-\alpha}. \end{aligned}$$
(A.29)

The following identity for elliptic functions (finite Fourier transformation) is useful in proving the Fourier symmetry in (5.10) and other identities:

$$\frac{1}{N}\sum_\alpha\kappa^2_{\alpha,\beta}\varphi_\alpha \biggl(Nx,\omega_\alpha+\frac{z}{N}\biggr) =\varphi_\beta(z,\omega_\beta+x)\qquad \forall\beta\in\mathbb Z_N\times\mathbb Z_N. $$
(A.30)
Its special cases are
$$\sum_\alpha E_2(\omega_\alpha+x)=N^2E_2(N x) $$
(A.31)
and
$$\sum_\alpha\kappa^2_{\alpha,\beta}\varphi_\alpha(x,\omega_\alpha) (E_1(z+\omega_\alpha)-E_1(z)+2\pi i\,\partial_\tau\omega_\alpha)-E_2(x) =-E_2\biggl(\omega_\beta+\frac{x}{N}\biggr). $$
(A.32)

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Trunina, E.S., Zotov, A.V. Multi-pole extension of the elliptic models of interacting integrable tops. Theor Math Phys 209, 1331–1356 (2021). https://doi.org/10.1134/S0040577921100020

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