The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy

Abstract

This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hierarchy with the help of orthogonal polynomial theory and Toda-type equations. Starting from the symmetric reduction in Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the $\tau$-function of the CKP hierarchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems.

Proofs of Propositions 3.1 and 3.3 by Pfaffians

In this appendix, we first give a brief review on some known facts about Pfaffians and then prove Propositions 3.1 and 3.3 by Pfaffian techniques. It turns out that the C-Toda lattice is nothing but Pfaffian identities with the $\tau$-functions given by determinants.

The term Pfaffian was introduced by Arthur Cayley in 1852, who named it after Johann Friedrich Pfaff. Pfaffian is generally used in theoretical physics nowadays. Let us first have a look at the definition of a Pfaffian. Let $A=(a_{i,j})_{1\le i,j\le 2N}$ be a $2N\times 2N$ skew-symmetric matrix. The Pfaffian of A, that is, Pf(A) is defined as

$$\begin{aligned} Pf(A)&=Pf(a_{i,j})_{1\le i,j\le 2N}=Pf\left[ \begin{array}{cccc} 0&{}a_{1,2}&{}\cdots &{}a_{1,{2N}}\\ -a_{1,2}&{}0&{}\cdots &{}a_{2,{2N}}\\ \vdots &{}\vdots &{}\ddots &{}\vdots \\ -a_{1,{2N}}&{}-a_{2,{2N}}&{}\cdots &{}0 \end{array} \right] \\&=\sum {'sgn}\left( \begin{array}{cccc} 1&{}2&{}\cdots &{}2N\\ j_1&{}j_2&{}\cdots &{}j_{2N} \end{array} \right) a_{j_1j_2}a_{j_3j_4}\cdots a_{j_{2N-1}j_{2N}}, \end{aligned}$$

where $\sum '$ means the sum over all possible combinations of pairs selected from $\{1,2,\ldots ,2N\}$ satisfying $j_1<j_2$, $\ldots$, $j_{2N-1}<j_{2N}$ and $j_1<j_3<\cdots <j_{2N-1}$. As an equivalent definition, an n-th order Pfaffian $pf(1,2,\ldots , 2N)$ can be expanded as

$$\begin{aligned} pf(1,2,\ldots , 2N)=\sum _{j=2}^{2N}(-1)^jpf(1,j)pf(2,3,\ldots ,{\hat{j}},\ldots ,2N) \end{aligned}$$

where ${\hat{j}}$ means that the index j is omitted. By this formula, the Pfaffian $pf(1,2,\ldots , 2N)$ may be recursively defined when Pfaffian entries pf(i, j) are given. It is obvious that $pf(1,2,\ldots , 2N)=Pf(A)$ when $(i,j)=a_{i,j}$. In this sense, the above-mentioned two definitions can be unified.

In the remaining part, we would like to adopt the notation below to denote a Pfaffian

$$\begin{aligned} pf(i_1,i_2,\ldots ,i_{2N})=Pf\left[ \begin{array}{cccc} 0&{}a_{i_1,i_2}&{}\cdots &{}a_{i_1,i_{2N}}\\ -a_{i_1,i_2}&{}0&{}\cdots &{}a_{i_2,i_{2N}}\\ \vdots &{}\vdots &{}\ddots &{}\vdots \\ -a_{i_1,i_{2N}}&{}-a_{i_2,i_{2N}}&{}\cdots &{}0 \end{array} \right] , \end{aligned}$$

where the Pfaffian elements $pf(i_j,i_k)=a_{i_j,i_k}$.

It is noted that any n-th order determinant can be expressed as an n-th order Pfaffian. Therefore, if we define Pfaffian entries by

$$\begin{aligned} pf(i,j)=pf(i^*,j^*)=0,\ \ pf(i,j^*)=I_{i,j}, \end{aligned}$$

then $\tau _n$ and ${\tilde{\tau }}_n$ given before can be also expressed by means of Pfaffians as

$$\begin{aligned}&\tau _n=pf(0,1,\ldots ,n-1,n-1^*,\ldots ,1^*,0^*),\\&{\tilde{\tau }}_n=pf(0,1,\ldots ,n-2,n,n-1^*,\ldots ,1^*,0^*). \end{aligned}$$

Proposition A.1

For the Pfaffian $\tau _n=pf(0,1,\ldots ,n-1,n-1^*,\ldots ,1^*,0^*)$ defined above, if its Pfaffian entries satisfy the relation

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}pf(i,j^*)=pf(i,j+1^*)+pf(i+1,j^*), \end{aligned}$$

then we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\tau _n&= \frac{\mathrm{d}}{\mathrm{d}t}pf(0,\ldots ,n-1,n-1^*,\ldots ,0^*)\nonumber \\&=2pf(0,\ldots ,n-2,n,n-1^*,\ldots ,0^*)\nonumber \\&=2{\tilde{\tau }}_n. \end{aligned}$$

(A.1)

Proof

Let us first prove the following equality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}pf(i_1,\ldots ,i_N,j_1^*,\ldots ,j_N^*)=&\sum _{k=1}^Npf(i_1,\ldots ,i_k+1,\ldots ,i_N,j_1^*,\ldots ,j_N^*)\nonumber \\&+\sum _{k=1}^Npf(i_1,\ldots ,i_N,j_1^*,\ldots ,j_k+1^*,\ldots ,j_N^*) \end{aligned}$$

(A.2)

with the Pfaffian entries defined by

$$\begin{aligned}&pf(i_k,i_l)=pf(j_k^*,j_l^*)=0, \\&\frac{\mathrm{d}}{\mathrm{d}t}pf(i_k,j_l^*)=pf(i_k,j_l+1^*)+pf(i_k+1,j_j^*). \end{aligned}$$

In what follows, we are going to prove (A.2) by induction. It is obvious that (A.2) is true for $N=1$. Assume that (A.2) holds for N. For $N+1$, we have

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}pf(i_1,\ldots ,i_N,i_{N+1},j_1^*,\ldots ,j_N^*,j_{N+1}^*)\\&\quad =\frac{\mathrm{d}}{\mathrm{d}t}\left[ \sum _{l=1}^{N+1}(-1)^{N+l}pf(i_{N+1},j_l^*)pf(i_1,\ldots ,i_N,j_1^*,\ldots ,{\hat{j}}_l^*,\ldots ,j_{N+1}^*)\right] \\&\quad =\sum _{l=1}^{N+1}(-1)^{N+l}\left[ pf(i_{N+1}+1,j_l^*)\right. \\&\left. \qquad +\,pf(i_{N+1},j_{l}+1^*)\right] pf(i_1,\ldots ,i_N,j_1^*,\ldots ,{\hat{j}}_l^*,\ldots ,j_{N+1}^*)\\&\qquad +\sum _{l=1}^{N+1}(-1)^{N+l}pf(i_{N+1},j_l^*)\\&\qquad \times \left[ \sum _{k=1}^Npf(i_1,\ldots ,i_k+1,\ldots ,i_N,j_1^*,\ldots ,{\hat{j}}_l^*,\ldots ,j_{N+1}^*)\right. \\&\qquad \left. +\sum _{k=1,k\not =l}^{N+1}pf(i_1,\ldots ,i_N,j_1,\ldots ,j_k+1^*,\ldots ,{\hat{j}}_l^*,\ldots ,j_{N+1}^*)\right] \\&\quad =\sum _{k=1}^{N+1}pf(i_1,\ldots ,i_k+1,\ldots ,i_{N+1},j_1^*,\ldots ,j_{N+1}^*)\\&\qquad +\sum _{k=1}^{N+1}pf(i_1,\ldots ,i_{N+1},j_1^*,\ldots ,j_k+1^*,\ldots ,j_{N+1}^*), \end{aligned}$$

So far, we have completed the proof of (A.2). Notice that $pf(0,\ldots ,n-2,n,n-1^*,\ldots ,0^*)=pf(0,\ldots ,n-1,n^*,n-2^*,\ldots ,0^*)$ due to the symmetry $I_{i,j}=I_{j,i}$. Simply by taking $\{i_1,\ldots ,i_N\}$ as $\{0,\ldots ,n-1\}$ and $\{j_1^*,\ldots ,j_N^*\}$ as $\{n-1^*,\ldots ,0^*\}$ in (A.2), we have (A.1). $\square$

Although Pfaffians may be obtained from antisymmetric determinants, their properties are more varied than those of determinants. Determinantal identities such as Plücker relations and Jacobi identities are extended and unified as Pfaffian identities which are very useful in integrable systems. Here, we list two most useful identities:

$$\begin{aligned}&pf(a_1,a_2,a_3,a_4,1,\ldots ,2n)pf(1,2,\ldots ,2n)\\&\quad =\sum _{j=2}^{4}(-1)^jpf(a_1,a_j,1,\ldots ,2n)pf(a_2,{\hat{a}}_j,a_{4},1,\ldots ,2n),\\&pf(a_1,a_2,a_3,1,\ldots ,2n-1)pf(1,2,\ldots ,2n)\\&\quad =\sum _{j=1}^{3}(-1)^{j-1}pf(a_j,1,\ldots ,2n-1)pf(a_1,{\hat{a}}_j,a_{3},1,\ldots ,2n). \end{aligned}$$

In the following, we demonstrate how to use Pfaffian techniques to prove that $\tau _n$ and $\sigma _n$ are solutions to the C-Toda lattice (3.9) as an application.

Proposition A.2

The C-Toda lattice (3.9) has solutions

$$\begin{aligned}&\tau _n=\begin{vmatrix}I_{0,0}&\cdots&I_{0,n-1}\\ \vdots&\vdots \\ I_{n-1,0}&\cdots&I_{n-1,n-1}\end{vmatrix}=pf(0,\ldots ,n-1,n-1^*,\ldots ,0^*),\\&\sigma _n=\begin{vmatrix}I_{0,0}&\cdots&I_{0,n}\\ \vdots&\vdots \\I_{n-1,0}&\cdots&I_{n-1,n}\\w_0&\cdots&w_n\end{vmatrix}=(-1)^npf(d_0,0,\ldots ,n-1,n^*,\ldots ,0^*) \end{aligned}$$

with Pfaffian entries defined by

$$\begin{aligned}&pf(d_0,i)=pf(d_0^*,i^*)=pf(d_0,d_0^*)=0,\\&pf(d_0,i^*)=pf(d_0^*,i)=w_i, \\&\frac{\mathrm{d}}{\mathrm{d}t}pf(i,j^*)=pf(i+1,j^*)+pf(i,j+1^*)=pf(d_0,d_0^*,i,j^*),\\&\frac{\mathrm{d}}{\mathrm{d}t}pf(d_0,i^*)=pf(d_0,i+1^*) \end{aligned}$$

which correspond to the conditions $\frac{\mathrm{d}}{\mathrm{d}t}I_{i,j}=I_{i+1,j}+I_{i.j+1}=\omega _i\omega _j$, $\frac{\mathrm{d}}{\mathrm{d}t}\omega _i=\omega _{i+1}$.

Proof

By using derivative formulae for Pfaffians repeatedly, it is easy to derive that

$$\begin{aligned} \tau _{n,t}&=pf(d_0,d_0^*,0,\ldots ,n-1,n-1^*,\ldots ,0^*)\\&=2pf(0,\ldots ,n-2,n,n-1^*,\ldots ,0^*),\\ \tau _{n,tt}&=2pf(d_0,d_0^*,0,\ldots ,n-2,n,n-1^*,\ldots ,0^*). \end{aligned}$$

Substituting the above results into the C-Toda lattice (3.9), we obtain the following two expressions

$$\begin{aligned}&pf(d_0,d_0^*,0,\ldots ,n,n^*,\ldots ,0^*)pf(0,\ldots ,n-1,n-1^*,\ldots ,0^*)\\&\qquad -pf(0,\ldots ,n,n^*,\ldots ,0^*)pf(d_0,d_0^*,0,\ldots ,n-1,n-1^*,\ldots ,0^*)\\&\qquad =pf(d_0,0,\ldots ,n-1,n^*,\ldots ,0^*)pf(d_0^*,0,n,n-1^*,\ldots ,0^*),\\&pf(d_0,d_0^*,0,\ldots ,n-1,n^*,n-2^*,\ldots ,0^*)pf(0,\ldots ,n-1,n-1^*,\ldots ,0^*)\\&\qquad -pf(d_0,d_0^*,0,\ldots ,n-1,n-1^*,\ldots ,0^*)pf(0,\ldots ,n-1,n^*,n-2^*,\ldots ,0^*)\\&\qquad =pf(d_0,0,\ldots ,n-1,n^*,\ldots ,0^*)pf(d_0^*,0,\ldots ,n-1,n-2^*,\ldots ,0^*), \end{aligned}$$

which are indeed two special cases of Pfaffian identities mentioned above. $\square$