Baxter Q-operators from a Schwinger-Boson construction of the master T-operator and the mKP hierarchy
The master T-operator is a generating function for transfer matrices and a tau-function of the modified KP (mKP) hierarchy. It is conventionally introduced through a Schur-function expansion whose coefficients are fused transfer matrices satisfying the Cherednik--Bazhanov--Reshetikhin determinant formula. For rational inhomogeneous gl(M) spin chains, we give an alternative definition: we realize gl(M) on an auxiliary Fock space generated by finitely many families of Schwinger bosons, form a monodromy matrix from the resulting L-operator, and define the master T-operator as its trace over the Fock space. Howe duality then reproduces the Schur-function expansion. This realization relates two constructions of Baxter Q-operators. In the construction of [arXiv:1112.3310] (see also [arXiv:1010.4022]), the Q-operators are obtained from residues of the master T-operator with respect to selected Miwa variables. In the construction of [arXiv:1010.3699], they are defined using degenerate Yangian L-operators in oscillator form. When selected Miwa variables approach the inverses of boundary-twist eigenvalues, the defining trace develops poles. After a suitable rescaling of this L-operator, only the terms contributing to the highest-order pole survive in the normalized limit; before the trace is taken, they form the degenerate Yangian L-operator of the second construction. The trace then gives an explicit residue formula for the corresponding Q-operator. Independently, a Holstein--Primakoff-type large-occupation-number contraction of this L-operator on subspaces with fixed total occupation numbers yields the same degenerate Yangian L-operator.