Quantum relativistic Toda chain

Investigated is the quantum relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl’sqwe are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six-vertexR-matrix. Both independent Baxter’sQ-operators are constructed explicitly as seria over local Weyl generators. The operator-valued Wronskian ofR-matrix. Both independent Baxter’sQsis also calculated.

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Lecture on SUSY Gauge Theories and Integrable Systems

International Journal of Modern Physics B ◽

10.1142/s0217979297001519 ◽

1997 ◽

Vol 11 (26n27) ◽

pp. 3093-3124

Author(s):

A. Marshakov

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Integrable Systems ◽

Gauge Theories ◽

Toda Chain ◽

Quantum Field ◽

Standard Quantum ◽

Integrable Equations ◽

Complex Curves ◽

Periodic Toda Chain

I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to [Formula: see text] SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.

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Quantum relativistic Toda chain at root of unity: isospectrality, modifiedQ-operator, and functional Bethe ansatz

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202105059 ◽

2002 ◽

Vol 31 (9) ◽

pp. 513-553 ◽

Cited By ~ 3

Author(s):

Stanislav Pakuliak ◽

Sergei Sergeev

Keyword(s):

Bethe Ansatz ◽

Weyl Algebra ◽

Separation Of Variables ◽

Toda Chain ◽

Special Choice ◽

Finite Dimensional Representation ◽

Classical Counterpart ◽

Discrete Integrable System ◽

Root Of Unity ◽

Finite Dimensional

We investigate anN-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra withqbeingNth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter'sQ-operators. The classical counterpart of the modifiedQ-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modifiedQ-operators.

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