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Papers by Mikhail Olshanetsky
It is proved that the Painlevé VI equation (P V I α,β,γ,δ) for the special values of constants (α... more It is proved that the Painlevé VI equation (P V I α,β,γ,δ) for the special values of constants (α = ν 2 4 , β = − ν 2 4 , γ = ν 2 4 , δ = 1 2 − ν 2 4) is a reduced hamiltonian system. Its phase space is the set of flat SL(2, C) connections over elliptic curves with a marked point and time of the system is given by the elliptic module. This equation can be derived via reduction procedure from the free infinite hamiltonian system. The phase space of later is the affine space of smooth connections and the "times are the Beltrami differentials. This approach allows to define the associate linear problem, whose isomonodromic deformations is provided by the Painlevé equation and the Lax pair. In addition, it leads to description of solutions by a linear procedure. This scheme can be generalized to G bundles over Riemann curves with marked points, where G is a simple complex Lie group. In some special limit such hamiltonian systems convert into the Hitchin systems. In particular, for SL(N, C) bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero N-body system. Relations to the classical limit of the Knizhnik-Zamolodchikov-Bernard equations is discussed.
Moscow Mathematical Journal
We consider the large N limits of Hitchin-type integrable systems. The first system is the ellipt... more We consider the large N limits of Hitchin-type integrable systems. The first system is the elliptic rotator on GLN that corresponds to the Higgs bundle of degree 1 over an elliptic curve with a marked point. This system is gauge equivalent to the N-body elliptic Calogero-Moser system, which is obtained from the Higgs bundle of degree zero over the same curve. The large N limit of the former system is the integrable rotator on the group of the non-commutative torus. Its classical limit leads to an integrable modification of 2D hydrodynamics on the two-dimensional torus. We also consider the elliptic Calogero-Moser system on the group of the non-commutative torus and consider the systems that arise after the reduction to the loop group.
Lettere Al Nuovo Cimento Series 2
The Calogero model describes the one-dimensional motion of n particles, with coordinates {qk} and... more The Calogero model describes the one-dimensional motion of n particles, with coordinates {qk} and momenta {Pk} characterized by the Hamiltonian n 1 2 (1) H =-2-~P~ + U(q), k=l where the potential 0)2 n (2) u(q)-~ ~, (q~-q,)-~ +-Z ~" q~ k~l k=[ This model was investigated in the quantum case by CALOGERO (1), and in the classical case (n = 3) by MARCmORO (~). MOS~R (3) by using the technique of isospectral deformation proved the complete integrability in the classical case with e)-0. In the papers (4,s) the Moser method was generalized to the case ~ 0. Actually in the last paper more general potentials than (2) were considered. Namely, if R = {a} is the root system (~) in the space 54f, i9+ is the subset of positive roots, qa= (q, or) is the scalar product of the coordinate vector ~ and the root ~, and g~, og~ are constants depending only on the length of the root ~, then the Hamiltonian system (1) with the potential (7)
Journal of High Energy Physics
We construct special rational gl N Knizhnik-Zamolodchikov-Bernard (KZB) equations withÑ punctures... more We construct special rational gl N Knizhnik-Zamolodchikov-Bernard (KZB) equations withÑ punctures by deformation of the corresponding quantum gl N rational R-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is τ. At the level of classical mechanics the deformation parameter τ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Next, we notice that the identities underlying generic (elliptic) KZB equations follow from some additional relations for the properly normalized R-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in R matrices) relations are generated by noncommutative Fay identities. In particular, one can derive the quantum Yang-Baxter equations from the Fay identities. The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical r-matrices which can be treated as halves of the classical Yang-Baxter equation. At last we discuss the R-matrix valued linear problems which provide glÑ CM models and Painlevé equations via the above mentioned identities. The role of the spectral parameter plays the Planck constant of the quantum R-matrix. When the quantum gl N R-matrix is scalar (N = 1) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the glÑ CM models. The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without it.
American Mathematical Society Translations: Series 2
ABSTRACT
Nuclear Physics B
In our recent paper we suggested a natural construction of the classical relativistic integrable ... more In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R-matrices. Here we study the simplest casethe 11-vertex R-matrix and related gl 2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to rewrite the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of n-particle integrable systems with 2n constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau-Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is also described.
American Mathematical Society Translations: Series 2, 2007
Research Reports in Physics, 1991
Il Nuovo Cimento A, 1980
ABSTRACT
Nuclear Physics B, 1993
We propose to study a generalization of the Klebanov-Polyakov-Witten (KPW) construction for the a... more We propose to study a generalization of the Klebanov-Polyakov-Witten (KPW) construction for the algebra of observables in the c = 1 string model to theories with c > 1. We emphasize the algebraic meaning of the KPW construction for c = 1 related to occurrence of a model of SU(2) as original structure on the algebra of observables. The attempts to preserve this structure in generalizations naturally leads to consideration of W-gravities. As a first step in the study of these generalized KPW constructions we design explicitly the subsector of the space of observables in appropriate W G-string theory, which forms the model of G for any simply laced G. The model structure is confirmed by the fact that corresponding one-loop Kac-Rocha-Caridi W G-characters for c = r G sum into a chiral (open string) k = 1 G-WZW partition function.
Lettere Al Nuovo Cimento Series 2, 1979
Michael Marinov Memorial Volume, 2002
The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathem... more The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathematical tool in investigations of dynamical systems with the first class constraints. Here we consider canonical symmetries of Hamiltonian systems generated by a special class of Lie algebroids. The "coordinate part" of the Hamiltonian phase space is the Poisson manifold M and the Lie algebroid brackets are defined by means of the Poisson bivector. The Lie algebroid action defined on M can be lifted to the phase space. The main observation is that the classical BRST operator has the same form as in the case of the Lie groups action. Two examples are analyzed. In the first, M is the space of SL(3, C)-opers on Riemann curves with the Adler-Gelfand-Dikii brackets. The corresponding Hamiltonian system is the W 3-gravity. Its phase space is the base of the algebroid bundle. The sections of the bundle are the second order differential operators on Riemann curves. They are the gauge symmetries of the theory. The moduli space of W 3 geometry of Riemann curves is the symplectic quotient with respect to their action. It is demonstrated that the nonlinear brackets and the second order differential operators arise from the canonical brackets and the standard gauge transformations in the Chern-Simons field theory, as a result of the partial gauge fixing. The second example is M = C 4 endowed with the Sklyanin brackets. The symplectic reduction with respect to the algebroid action leads to a generalization of the rational Calogero-Moser model. As in the previous example the Sklyanin brackets can be derived from a "free theory." In this case it is a "relativistic deformation" of the SL(2, C) Higgs bundle over an elliptic curve.
Research Reports in Physics, 1990
Physics Letters A, 1980
... THE BOUSSINESQ EQUATION D. LEVI a, MA OLSHANETSKY b, AM PERELOMOV b and 0. RAGNISCO aa Istitu... more ... THE BOUSSINESQ EQUATION D. LEVI a, MA OLSHANETSKY b, AM PERELOMOV b and 0. RAGNISCO aa Istituto di Fisica, Universit di Roma, 00185 ... some direct calculations, we obtain that the time evolution of L given by the Euler Lax equations (2.4) im-plies the following ...
Encyclopaedia of Mathematical Sciences, 1994
Physics Reports, 1981
Hamil-1. General description 319 tonian systems 2. Completely integrable Hamiltonian systems 320 ... more Hamil-1. General description 319 tonian systems 2. Completely integrable Hamiltonian systems 320 14. Explicit integration of the equations of motion for the 3. Systems with additional integrals of motion 321 systems of type IV and VI' (periodic Toda lattice) 4. Proof of complete integrability of the systems of section 3 324 14.1. The systems of type VI' AN_i 5. Explicit integration of theequations of motion for potentials 14.2. The systems of type IV AN_I V(q) of type I and V 327 15. Miscellanea 371 6. Explicit integration of theequations of motion for potentials 15.1. Motion of the poles of nonlinear partial differential of type II and III 331 equations and related many-body problems 371 7. Integration ofthe equations ofmotion for a system with two 15.2. Motion of the zeros of the linear evolution equatypes of particles 333 tions and related integrable many-body problems 374 8. Explicit integration of theequations of motion for the Toda 15.3. Rotation of a many-dimensional rigid body around a lattice 335 fixed point 376 9. Reduction of Hamiltonian systems with symmetries 15.4. Concluding remarks 380 (Methods of orbits) 337 Appendix A. Solution to the functional equation (3.9) 382 10. Equilibrium configurations and small oscillations of some Appendix B. Groups generated by reflections and root sysdynamical systems 346 tems 385 11. Abstract Hamiltonian systems related to root systems 352 Appendix C. Symmetric spaces 390 12. Complete integrability of the abstract Hamiltonian systems 355 References 398
It is proved that the Painlevé VI equation (P V I α,β,γ,δ) for the special values of constants (α... more It is proved that the Painlevé VI equation (P V I α,β,γ,δ) for the special values of constants (α = ν 2 4 , β = − ν 2 4 , γ = ν 2 4 , δ = 1 2 − ν 2 4) is a reduced hamiltonian system. Its phase space is the set of flat SL(2, C) connections over elliptic curves with a marked point and time of the system is given by the elliptic module. This equation can be derived via reduction procedure from the free infinite hamiltonian system. The phase space of later is the affine space of smooth connections and the "times are the Beltrami differentials. This approach allows to define the associate linear problem, whose isomonodromic deformations is provided by the Painlevé equation and the Lax pair. In addition, it leads to description of solutions by a linear procedure. This scheme can be generalized to G bundles over Riemann curves with marked points, where G is a simple complex Lie group. In some special limit such hamiltonian systems convert into the Hitchin systems. In particular, for SL(N, C) bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero N-body system. Relations to the classical limit of the Knizhnik-Zamolodchikov-Bernard equations is discussed.
Moscow Mathematical Journal
We consider the large N limits of Hitchin-type integrable systems. The first system is the ellipt... more We consider the large N limits of Hitchin-type integrable systems. The first system is the elliptic rotator on GLN that corresponds to the Higgs bundle of degree 1 over an elliptic curve with a marked point. This system is gauge equivalent to the N-body elliptic Calogero-Moser system, which is obtained from the Higgs bundle of degree zero over the same curve. The large N limit of the former system is the integrable rotator on the group of the non-commutative torus. Its classical limit leads to an integrable modification of 2D hydrodynamics on the two-dimensional torus. We also consider the elliptic Calogero-Moser system on the group of the non-commutative torus and consider the systems that arise after the reduction to the loop group.
Lettere Al Nuovo Cimento Series 2
The Calogero model describes the one-dimensional motion of n particles, with coordinates {qk} and... more The Calogero model describes the one-dimensional motion of n particles, with coordinates {qk} and momenta {Pk} characterized by the Hamiltonian n 1 2 (1) H =-2-~P~ + U(q), k=l where the potential 0)2 n (2) u(q)-~ ~, (q~-q,)-~ +-Z ~" q~ k~l k=[ This model was investigated in the quantum case by CALOGERO (1), and in the classical case (n = 3) by MARCmORO (~). MOS~R (3) by using the technique of isospectral deformation proved the complete integrability in the classical case with e)-0. In the papers (4,s) the Moser method was generalized to the case ~ 0. Actually in the last paper more general potentials than (2) were considered. Namely, if R = {a} is the root system (~) in the space 54f, i9+ is the subset of positive roots, qa= (q, or) is the scalar product of the coordinate vector ~ and the root ~, and g~, og~ are constants depending only on the length of the root ~, then the Hamiltonian system (1) with the potential (7)
Journal of High Energy Physics
We construct special rational gl N Knizhnik-Zamolodchikov-Bernard (KZB) equations withÑ punctures... more We construct special rational gl N Knizhnik-Zamolodchikov-Bernard (KZB) equations withÑ punctures by deformation of the corresponding quantum gl N rational R-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is τ. At the level of classical mechanics the deformation parameter τ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Next, we notice that the identities underlying generic (elliptic) KZB equations follow from some additional relations for the properly normalized R-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in R matrices) relations are generated by noncommutative Fay identities. In particular, one can derive the quantum Yang-Baxter equations from the Fay identities. The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical r-matrices which can be treated as halves of the classical Yang-Baxter equation. At last we discuss the R-matrix valued linear problems which provide glÑ CM models and Painlevé equations via the above mentioned identities. The role of the spectral parameter plays the Planck constant of the quantum R-matrix. When the quantum gl N R-matrix is scalar (N = 1) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the glÑ CM models. The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without it.
American Mathematical Society Translations: Series 2
ABSTRACT
Nuclear Physics B
In our recent paper we suggested a natural construction of the classical relativistic integrable ... more In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R-matrices. Here we study the simplest casethe 11-vertex R-matrix and related gl 2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to rewrite the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of n-particle integrable systems with 2n constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau-Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is also described.
American Mathematical Society Translations: Series 2, 2007
Research Reports in Physics, 1991
Il Nuovo Cimento A, 1980
ABSTRACT
Nuclear Physics B, 1993
We propose to study a generalization of the Klebanov-Polyakov-Witten (KPW) construction for the a... more We propose to study a generalization of the Klebanov-Polyakov-Witten (KPW) construction for the algebra of observables in the c = 1 string model to theories with c > 1. We emphasize the algebraic meaning of the KPW construction for c = 1 related to occurrence of a model of SU(2) as original structure on the algebra of observables. The attempts to preserve this structure in generalizations naturally leads to consideration of W-gravities. As a first step in the study of these generalized KPW constructions we design explicitly the subsector of the space of observables in appropriate W G-string theory, which forms the model of G for any simply laced G. The model structure is confirmed by the fact that corresponding one-loop Kac-Rocha-Caridi W G-characters for c = r G sum into a chiral (open string) k = 1 G-WZW partition function.
Lettere Al Nuovo Cimento Series 2, 1979
Michael Marinov Memorial Volume, 2002
The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathem... more The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathematical tool in investigations of dynamical systems with the first class constraints. Here we consider canonical symmetries of Hamiltonian systems generated by a special class of Lie algebroids. The "coordinate part" of the Hamiltonian phase space is the Poisson manifold M and the Lie algebroid brackets are defined by means of the Poisson bivector. The Lie algebroid action defined on M can be lifted to the phase space. The main observation is that the classical BRST operator has the same form as in the case of the Lie groups action. Two examples are analyzed. In the first, M is the space of SL(3, C)-opers on Riemann curves with the Adler-Gelfand-Dikii brackets. The corresponding Hamiltonian system is the W 3-gravity. Its phase space is the base of the algebroid bundle. The sections of the bundle are the second order differential operators on Riemann curves. They are the gauge symmetries of the theory. The moduli space of W 3 geometry of Riemann curves is the symplectic quotient with respect to their action. It is demonstrated that the nonlinear brackets and the second order differential operators arise from the canonical brackets and the standard gauge transformations in the Chern-Simons field theory, as a result of the partial gauge fixing. The second example is M = C 4 endowed with the Sklyanin brackets. The symplectic reduction with respect to the algebroid action leads to a generalization of the rational Calogero-Moser model. As in the previous example the Sklyanin brackets can be derived from a "free theory." In this case it is a "relativistic deformation" of the SL(2, C) Higgs bundle over an elliptic curve.
Research Reports in Physics, 1990
Physics Letters A, 1980
... THE BOUSSINESQ EQUATION D. LEVI a, MA OLSHANETSKY b, AM PERELOMOV b and 0. RAGNISCO aa Istitu... more ... THE BOUSSINESQ EQUATION D. LEVI a, MA OLSHANETSKY b, AM PERELOMOV b and 0. RAGNISCO aa Istituto di Fisica, Universit di Roma, 00185 ... some direct calculations, we obtain that the time evolution of L given by the Euler Lax equations (2.4) im-plies the following ...
Encyclopaedia of Mathematical Sciences, 1994
Physics Reports, 1981
Hamil-1. General description 319 tonian systems 2. Completely integrable Hamiltonian systems 320 ... more Hamil-1. General description 319 tonian systems 2. Completely integrable Hamiltonian systems 320 14. Explicit integration of the equations of motion for the 3. Systems with additional integrals of motion 321 systems of type IV and VI' (periodic Toda lattice) 4. Proof of complete integrability of the systems of section 3 324 14.1. The systems of type VI' AN_i 5. Explicit integration of theequations of motion for potentials 14.2. The systems of type IV AN_I V(q) of type I and V 327 15. Miscellanea 371 6. Explicit integration of theequations of motion for potentials 15.1. Motion of the poles of nonlinear partial differential of type II and III 331 equations and related many-body problems 371 7. Integration ofthe equations ofmotion for a system with two 15.2. Motion of the zeros of the linear evolution equatypes of particles 333 tions and related integrable many-body problems 374 8. Explicit integration of theequations of motion for the Toda 15.3. Rotation of a many-dimensional rigid body around a lattice 335 fixed point 376 9. Reduction of Hamiltonian systems with symmetries 15.4. Concluding remarks 380 (Methods of orbits) 337 Appendix A. Solution to the functional equation (3.9) 382 10. Equilibrium configurations and small oscillations of some Appendix B. Groups generated by reflections and root sysdynamical systems 346 tems 385 11. Abstract Hamiltonian systems related to root systems 352 Appendix C. Symmetric spaces 390 12. Complete integrability of the abstract Hamiltonian systems 355 References 398