On the quantum $$\mathfrak{osp}(1|2\ell)$$ 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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The Toda chain as a reduced system

Theoretical and Mathematical Physics ◽

10.1007/bf01047139 ◽

1980 ◽

Vol 45 (1) ◽

pp. 843-854 ◽

Cited By ~ 12

Author(s):

M. A. Ol'shanetskii ◽

A. M. Perelomov

Keyword(s):

Toda Chain

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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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