A new solvable two-matrix model and the BKP tau function

Abstract

We present exactly solvable modifications of the two-matrix Zinn-Justin–Zuber model and write it as a tau function. The grand partition function of these matrix integrals is written as the fermion expectation value. The perturbation theory series is written explicitly in terms of a series in strict partitions. The related string equations are presented.

Appendix A: Partitions. The Schur and projective Schur functions

We recall that a nonincreasing set of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_l)$, is called a partition ${\lambda_1\ge\cdots\ge\lambda_k\ge 0}$, and $\lambda_i$ are called the parts of $\lambda$. The sum of parts is called the weight $|\lambda|$ of $\lambda$. The number of nonzero parts of $\lambda$ is called the length of $\lambda$ and is denoted by $\ell(\lambda)$ (see [39] for details). Partitions are denoted by Greek letters: $\lambda,\mu,\ldots{}\,$. Partitions with distinct parts are called strict partitions, we prefer the letters $\alpha$ and $\beta$ to denote them. The set of all strict partitions is denoted by $\mathrm{DP}$.

To define the projective Schur function $Q_\alpha$, where $\alpha\in\mathrm{DP}$, at the first step, we define the set of functions $\{q_i,i\ge 0\}$ by

$$\exp\biggl\{\sum_{m\in\mathbb{Z}^{+}_{\text{odd}}}\frac{2}{m}p_m x^m\biggr\}=\sum_{m\ge 0}x^mq_m(\mathbf p_{\text{odd}}),$$

where now $\mathbf p_{\text{odd}}=(p_1,p_3,\ldots)$. Next, we define the skew-symmetric matrix

$$Q_{(i,j)}(\mathbf p_{\text{odd}})= q_i(\mathbf p_{\text{odd}})q_j(\mathbf p_{\text{odd}})+2\sum_{k=1}^j(-1)^kq_{i+k}(\mathbf p_{\text{odd}})q_{j-k}(\mathbf p_{\text{odd}}) \;\,\text{if}\;\,(i,j)\neq(0,0),$$

and $Q_{(i,j)}(\mathbf p_{\text{odd}})=0$ if $(i,j)=(0,0)$. In particular,

$$Q_{(j,0)}(\mathbf p_{\text{odd}})=-Q_{(0,j)}(\mathbf p_{\text{odd}})=q_j(\mathbf p_{\text{odd}})\quad\text{for}\quad j\ge 1.$$

For a strict partition $\alpha=(\alpha_1,\ldots,\alpha_{2r})$, where $\alpha_{2r}\ge 0$, the projective Schur function is defined as

$$ Q_\alpha(\mathbf p_{\text{odd}}):= \operatorname{Pf} [Q_{\alpha_i \alpha_j}(\mathbf p_{\text{odd}})]_{1\le i,j\le 2r},\qquad Q_{\varnothing}:=1.$$

(44)

Let $X$ be a matrix. We put $p_m=p_m(X)= \operatorname{tr} (X^m-(-X)^m)$, where $m$ is odd, and call these variables odd power sums. We write $Q_\alpha(\mathbf p_{\text{odd}}(X))=Q_\alpha(X)$.

Appendix B: Neutral fermions [46]. Scalar product of the projective Schur functions [13]

For $\alpha=(\alpha_1,\ldots,\alpha_k)\in\mathrm{DP}$ we introduce

$$\Phi_\alpha:=2^{k/2}\phi_{\alpha_1}\ldots\phi_{\alpha_k},\qquad \Phi_{-\alpha}:=(-1)^{\sum_{i=1}^k\alpha_i}\,2^{k/2}\phi_{-\alpha_k}\ldots\phi_{-\alpha_{1}}.$$

We have $\langle 0|\Phi_{-\alpha}\Phi_\beta|0\rangle=2^{\ell(\alpha)}\delta_{\alpha,\beta}$.

The fermionic formula for projective Schur functions was obtained in [22]:

$$Q_\alpha(\mathbf x)\Delta^*(\mathbf x)= 2^{-N/2}\langle 0|\phi(-x_1^{-1})\ldots\phi(-x_N^{-1})\Phi_\alpha |0\rangle= 2^{-N/2}\langle 0|\Phi_{-\alpha} \phi(x_1)\ldots\phi(x_N) |0\rangle.$$

We have

$$ \begin{aligned} \, &\langle 0|\phi(-v_1^{-1})\ldots \phi(-v_N^{-1})=\Delta^*(v)\sum_{\alpha} 2^{-N/2-\ell(\alpha)} Q_\alpha(v) \langle 0|\Phi_{-\alpha}, \\ &\phi(u_1)\ldots\phi(u_N)|0\rangle=\Delta^*(u)\sum_{\alpha} 2^{-N/2-\ell(\alpha)}Q_\alpha(u) \Phi_{\alpha}|0\rangle. \end{aligned}$$

(45)

Following [13], we consider $f(x)=\sum_{i\ge 0}f_ix^i$ and write

$$\begin{aligned} \, &2^{-N}\sum_\alpha\Phi_\alpha|0\rangle \langle 0|\Phi_{-\alpha}\prod_{i=1}^{\ell(\alpha)} f_{\alpha_i}= \\ &=\frac{1}{(2\pi i)^N}\oint\ldots\oint\phi(u_1)\ldots\phi(u_N)|0\rangle\langle 0|\phi(-v_N^{-1})\ldots\phi(-v_1^{-1}) \prod_{i=1}^N\frac{du_i\,dv_i}{u_iv_i}f(u_i^{-1}v_i^{-1})= \\ &=\frac{1}{(2\pi i)^N}\oint\ldots\oint \Delta^*(u)\Delta^*(v)Q_\alpha(u)Q_\beta(v)\sum_{\alpha,\beta}\Phi_\alpha|0\rangle \langle 0|\Phi_{-\beta}2^{-N-\ell(\alpha)-\ell(\beta)} \prod_{i=1}^N\frac{du_i\,dv_i}{u_iv_i}f(u_i^{-1}v_i^{-1}). \end{aligned}$$

Equation (12) then follows:

$$\frac{1}{(2\pi i)^N}\oint\ldots\oint\Delta^*(u)\Delta^*(v)Q_\alpha(u)Q_\beta(v)2^{-\ell(\alpha)-\ell(\beta)} \prod_{i=1}^N\frac{du_idv_i}{u_iv_i}f(u_i^{-1}v_i^{-1})=\prod_{i=1}^N f_{\alpha_i}.$$

Appendix C: Fermionic representation and string equations for the modified $$I_N(\mathbf p^{(1)},\mathbf p^{(2)}; f)$$ integral

We consider

$$\begin{aligned} \, &I_N(\mathbf p^{(1)},\mathbf p^{(2)};f)=\langle N,-N| e^{F(\mathbf p^{(1)},\mathbf p^{(2)})}|0,0\rangle, \\ &F(\mathbf p^{(1)},\mathbf p^{(2)})= -\frac{1}{4\pi^2}\oint\!\oint f(u^{-1}v^{-1})\psi^{(1)}(u,\mathbf p^{(1)})\psi^{\dagger(2)}(v,\mathbf p^{(2)})\frac{du\,dv}{uv}= \sum_{j} f_j \psi^{(1)}_j\psi^{\dagger(2)}_{-j-1}, \end{aligned}$$

where

$$f(z)=\sum_j f_j z^j,\qquad\psi(z)=\sum_j z^j\psi_j,\qquad\psi^\dagger(z)=\sum_j z^j\psi^\dagger_{-j-1},$$

and for $j> N$, we have $\langle N|\psi_j=0$ and $\langle 0|\psi^\dagger_{-1-j}=0$. Then

$$ \psi(ze^{y/2})\psi^\dagger(ze^{-y/2})=\sum_{m,n}\Omega_{\mathrm F}(m,n)z^{-m}y^n,$$

(46)

where

$$ \Omega_{\mathrm F}^{}=\sum_{j}\biggl(m+\frac{1}{2}\biggr)^{\!n}\psi_j^{}\psi^\dagger_{j+m}.$$

(47)

We let $\hat{\mathbf p}$ denote the collections $(p_1,p_2,\ldots)$ and $(\partial_{p_1},\partial_{p_2},\ldots)$. The bosonic version is

$$ \Omega_{\mathrm B}(m,n,\hat{\mathbf p})=- \mathop{\rm res}\limits _z\; \,\mathopen{\vdots\kern1pt} (V(z,\hat{\mathbf p})z^{-1/2}(z^mD^nz^{1/2}V(z,-\hat{\mathbf p})) \mathclose{\kern1pt\vdots}\, \,dz.$$

(48)

Unlike string equations, typical of any “diagonal” series (7), string equations that select the prefactor $f$ in (7) are rather different from the case of a two-matrix model with two Hermitian matrices. We have

$$L=\sum_{i}x(i)\psi^{(1)}_i\psi^{\dagger(1)}_{i+m},\quad M=\sum_{i}y(i)\psi^{(2)}_i\psi^{\dagger(2)}_{i-m},\qquad m>0,$$

and require that $M|0,0\rangle=0$ for $m>0$. We have

$$\sum_{i} y(i)\psi^{(2)}_i\psi^{\dagger(2)}_{i-m}|0,0\rangle= y(0)\psi^{(2)}_{0}\psi^{\dagger(2)}_{-m}|0,0\rangle +\cdots+y(m-1)\psi^{(2)}_{m-1}\psi^{\dagger(2)}_{}{-1}|0,0\rangle=0$$

Thus, $y(i)=0$ for $i=0,\ldots,m-1$. Therefore, we can take $y(i)=i(i-1)\ldots(i-m+1)$. Then, the condition

$$\biggl[L+M,\sum_{i}f_i\psi^{(1)}_i\psi^{\dagger(2)}_{-i-1}\biggr]=0.$$

yields $x(i+m)f_{i+m}=y(i)f_i$. Setting $f_i=1/\Gamma(i+1)$ and $m=1$, we can take $x(i)=i(i-1)$ and $y(i)=i$ (which is the Virasoro generator denoted by $L_{-1}$).

The bosonized versions are as follows:

$$ L^{\mathrm B}(\hat{\mathbf t}^{(1)})I(\mathbf t^{(1)},\mathbf t^{(2)};f)=M^{\mathrm B}(\hat{\mathbf t}^{(2)})I(\mathbf t^{(1)},\mathbf t^{(2)};f).$$

(49)

The explicit examples will be written in a more detailed presentation.

Appendix D: Hirota equations for the two-component BKP hierarchy

We have the fermionic form of the two-component BKP tau function

$$ \tau(\mathbf p^{(1)},\mathbf p^{(2)})=\langle 0\bigg| \exp\biggl\{\sum_{i=1,2}\,\sum_{m\in\mathbb{Z}^{+}_{\text{odd}}}\frac{2}{m}p_m^{(i)} \sum_{j\in\mathbb{Z}^{}_{}}\phi^{(i)}_{j-m}\phi^{(i)}_{-j}\biggr\} h\bigg|0\rangle,$$

(50)

where

$$h=\exp\biggl\{\sum_{i,j\in\mathbb{Z}}(a_{i,j} \,\mathopen{:\kern1pt} \phi^{(1)}\phi^{(1)} \mathclose{\kern1pt :}\, + b_{i,j} \,\mathopen{:\kern1pt} \phi^{(2)}\phi^{(2)} \mathclose{\kern1pt :}\, +f_{i,j}\phi^{(1)}\phi^{(2)})\biggr\}.$$

The fermionic form of the two-component hierarchy is

$$ \sum_{i=1,2}\sum_{j\in \mathbb{Z}}(-1)^j\phi^{(i)}_jh\otimes\phi^{(i)}_{-j}h=\frac{1}{2} h\phi^{(i)}_0\otimes h\phi^{(i)}_0.$$

(51)

The bilinear Hirota equations are the bosonized version of (51). They can be found in [41]. In short, we have

$$\begin{aligned} \, & \mathop{\rm res}\limits _z\bigl(\tau(\mathbf p^{(1)}+\epsilon^{(1)}-[z],\mathbf p^{(2)}+\epsilon^{(2)}) \tau(\mathbf p^{(1)}+[z],\mathbf p^{(2)})\bigr)\frac{dz}{z}+{} \\ &{}+ \mathop{\rm res}\limits _z\bigl(\tau(\mathbf p^{(1)}+\epsilon^{(1)},\mathbf p^{(2)}+\epsilon^{(2)}-[z]) \tau(\mathbf p^{(1)},\mathbf p^{(2)})+[z]\bigr)\frac{dz}{z}-{} \\ &{}-\tau(\mathbf p^{(1)}+\epsilon^{(1)},\mathbf p^{(2)}+\epsilon^{(2)})\tau(\mathbf p^{(1)},\mathbf p^{(2)})=0, \end{aligned}$$

where the sets $\epsilon^{(i)}=(\epsilon^{(i)}_1,\epsilon^{(i)}_3,\ldots)$ are the sets of arbitrary parameters and where $[z]$ denotes the set $(2z^{-1},2z^{-3},2z^{-5},\ldots)$. The vanishing condition for all coefficients of the Taylor series in these parameters gives bilinear differential equations for the tau functions, which we call the Hirota differential equations.