Abstract
AbstractFor an arbitrary generalized quantum integrable spin chain we introduce a “masterT-operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the masterT-operator, which allows one to identify it withτ-function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rationalGL(N)-invariantR-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic)R-matrices and to supersymmetric spin chains.
Highlights
The aim of this paper is to make precise the long anticipated connection between transfer matrices of quantum integrable models and τ -functions of classical integrable hierarchies of non-linear partial differential equations
For an arbitrary generalized quantum integrable spin chain we introduce a “master T -operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space
We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the master T -operator, which allows one to identify it with τ -function of an integrable hierarchy of classical soliton equations
Summary
The aim of this paper is to make precise the long anticipated connection between transfer matrices of quantum integrable models and τ -functions of classical integrable hierarchies of non-linear partial differential equations. An important further step was recently made in [12], where an operator realization of the Backlund flow describing the “undressing” process was suggested and a commuting family of transfer matrices (T -operators on different levels of the nested Bethe ansatz) depending on a number of discrete variables (labels of representation in the auxiliary space) and on the spectral parameter was constructed These T -operators obey the discrete Hirota equation for usual and for supersymmetric integrable models, as it was directly demonstrated in [13] for GL(M |N )-invariant spin chains. T -operator is introduced and shown to satisfy the classical bilinear identities and Hirota equations
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