Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices

Abstract

In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as $1/k_i$, $k_i\in {\mathbb {Z}}_{>0}$ ($i=1,\,2$), the average characteristic polynomials admit $(k_1+k_2+2)$-term recurrence relations, which can be interpreted as spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest solvable example is given.

Appendix A

This appendix is devoted to a statement about how the superpotential (1.3) relates to the extended affine group ${\tilde{W}}^{(k,k+1)}(A_\ell )$. To this end, firstly, we give some basic concepts about the extended affine Weyl group.

Let V be an $\ell$-dimensional Euclidean space with inner product $(\cdot ,\cdot )$ and ${\mathcal {R}}$ be an irreducible root system defined in V with simple roots $\alpha _1,\cdots ,\alpha _\ell$ and coroots $\alpha _1^\vee ,\cdots ,\alpha _\ell ^\vee$. Then, the Weyl group $W({\mathcal {R}})$ can be generated by the reflection

$$\begin{aligned} x\mapsto x-(\alpha _j^\vee ,x)\alpha _j,\quad \forall x\in V,\quad j=1,\cdots ,\ell . \end{aligned}$$

The affine Weyl group $W_a({\mathcal {R}})$ which acts on the Euclidean space V can be realised as the semi-product of $W({\mathcal {R}})$ by the lattice of coroots via the map

$$\begin{aligned} {x}\mapsto w(x)+\sum _{j=1}^\ell m_j\alpha _j^\vee , \quad w\in W({\mathcal {R}}),\quad m_j\in {\mathbb {Z}}. \end{aligned}$$

Dubrovin and Zhang then proposed the extended affine Weyl group ${\tilde{W}}^{(k)}({\mathcal {R}})$ acting on $V\oplus {\mathbb {R}}$ in the study of Frobenius manifold. It is generated by the transformations

$$\begin{aligned}&(x,x_{\ell +1})\mapsto (w(x)+\sum _{j=1}^\ell m_j\alpha _j^\vee , x_{\ell +1}),\quad w\in W({\mathcal {R}}), \quad m_j\in {\mathbb {Z}},\\&(x,x_{\ell +1})\mapsto (x+\gamma w_k, x_{\ell +1}-\gamma ),\quad 1\le k\le \ell \end{aligned}$$

with $w_k$ being the fundamental weights defined by the relations $(w_k,\alpha _j^\vee )=\delta _{k,j}$ for $k,j=1,\cdots ,\ell$ and $\gamma$ is a constant related the root system. With ${\mathcal {R}}$ chosen as $A_\ell$, a construction of Frobenius manifold structure on the orbit of ${\tilde{W}}^{(k)}(A_\ell )$ was given in Dubrovin and Zhang (1998). It was shown that ${\tilde{W}}^{(k)}(A_\ell )$ describes the monodromy of roots of trigonometric polynomials admits the form

$$\begin{aligned} \lambda (\phi )=e^{ik\phi }+\mathbf {a_1}e^{i(k-1)\phi }+\cdots +a_{\ell +1} e^{i(k-\ell -1)\phi },\quad a_{\ell +1}\ne 0. \end{aligned}$$

If one sets $e^{i\phi }$ as the shift operator $\Lambda$, then the above polynomials will become

$$\begin{aligned} L=\Lambda ^k+\mathbf {a_1}\Lambda ^{k-1}+\cdots +a_{\ell +1}\Lambda ^{k-\ell -1},\quad a_{\ell +1}\ne 0 \end{aligned}$$

which is regarded as the spectral operator of the bigraded Toda hierarchy (c.f. Eq. (1.2)).

In the recent work (Zuo 2020), the author studied another extended affine Weyl group ${\tilde{W}}^{(k,k+1)}(A_\ell )$ acting on the space $V\oplus {\mathbb {R}}^2$, generated by the transformation

$$\begin{aligned}&(x,x_{\ell +1},x_{\ell +2})\mapsto (w(x)+\sum _{j=1}^\ell m_j\alpha _j^\vee ,x_{\ell +1},x_{\ell +2}),\quad w\in W(A_\ell ),\quad m_j\in {\mathbb {Z}},\\&(x,x_{\ell +1},x_{\ell +2})\mapsto (x+\gamma w_k,x_{\ell +1}-\gamma ,x_{\ell +2}),\quad (x,x_{\ell +1},x_{\ell +2})\\&\quad \mapsto (x+\gamma w_{k+1},x_{\ell +1},x_{\ell +2}-\gamma ) \end{aligned}$$

for $1\le k\le \ell -1$. It was proven by Zuo (2020) that the orbit space of the extended affine Weyl group is locally isomorphic to a simple Hurwitz space ${\mathbb {M}}_{k,\ell -k+1,1}$ which contains a natural structure of Frobenius manifold. Moreover, this space consists of trigonometric Laurent series admitting the form

$$\begin{aligned} \lambda (\phi )=(e^{i\phi }-a_{\ell +2})^{-1}(e^{i(k+1)\phi }+\mathbf {a_1}e^{ik\phi } +\cdots +a_{\ell +1} e^{i(k-\ell )}\phi ),\quad a_{\ell +1}a_{\ell +2}\ne 0. \end{aligned}$$

Similarly, by setting $e^{i\phi }=\Lambda$, one gets the spectral operator (1.3)

$$\begin{aligned} L\!=\!(\Lambda \!-\!a_{\ell +2})^{-1}(\Lambda ^{k+1}\!+\!\mathbf {a_1}\Lambda ^{k}\!+\cdots +a_{\ell +1}\Lambda ^{k-\ell }), \quad 1\!\le \! k\!\le \! \ell -1,\, a_{\ell +1}a_{\ell +2}\ne 0. \end{aligned}$$