Fredholm Pfaffian $\tau$-Functions for Orthogonal Isospectral and Isomonodromic Systems

Abstract

We extend the approach to $\tau$-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld–Sokolov reductions and isomonodromic deformations systems. The combinatorial expansion of the $\tau$-function as a sum of correlators, each expressed as products of finite determinants, follows from using multicomponent fermionic vacuum expectation values of certain dressing operators encoding the initial conditions and dependence on the time parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the determinantal $\tau$-functions, both in the Drinfeld–Sokolov and isomonodromic case, become squares of $\tau$-functions of Pfaffian type. The results are illustrated by several examples, consisting of polynomial $\tau$-functions of orthogonal Drinfeld–Sokolov type and isomonodromic ones with four regular singular points.

Appendices

A Appendix. A Polynomial $\tau$-Function of $B^{(1)}_2$ Type

Recall equation (4.28) in the main text, representing the map between decreasing sequences of positive half-integers $\{p_{\alpha ,i}\}_{i=1}^{n_\alpha }$ and strict partitions

$$\begin{aligned} \lambda ^{(\alpha )}=\left( p_{\alpha ,1}-\frac{1}{2},\dots ,p_{\alpha ,n_{\alpha }}-\frac{1}{2}\right) . \end{aligned}$$

(A.1)

In the tables below we group the coefficients in the Pfaffian minor expansions by the total weight of the N-tuple of (positive) strict partitions obtained by adding 1 to each part:

$$\begin{aligned} |\vec {\lambda }|:=\sum _{\alpha =1}^N\sum _{i=1}^{n_\alpha }\left( p_{\alpha ,i}+\frac{1}{2}\right) , \end{aligned}$$

(A.2)

counting the overall power of z (resp. $z^{-1}$) in the Fourier expansion of $\mathsf{{a}}$ (respectively $\mathsf{{d}}$) that contribute to the Pfaffian minor. For example

$$\begin{aligned} \mathrm {Pf}\left( \begin{array}{cc} 0 &{} \quad \left( S\mathsf{{a}}^{\frac{1}{2}}_{-\frac{1}{2}} \right) _{12} \\ \left( S\mathsf{{a}}^{\frac{1}{2}}_{-\frac{1}{2}} \right) _{12} &{}\quad 0 \end{array} \right) \end{aligned}$$

(A.3)

has weight $|\vec {\lambda }|=2$, since it contains two Fourier coefficients of $\mathsf{{a}}$ of weight 0 that multiply $z^0w^0$, while

$$\begin{aligned} \mathrm {Pf}\left( \begin{array}{cc} 0 &{} \left( S\mathsf{{a}}^{\frac{3}{2}}_{-\frac{3}{2}} \right) _{12} \\ \left( S\mathsf{{a}}^{\frac{3}{2}}_{-\frac{3}{2}} \right) _{12} &{} 0 \end{array} \right) \end{aligned}$$

(A.4)

has weight $|\vec {\lambda }|=4$, since it contains two Fourier coefficients of $\mathsf{{a}}$ of weight 2, that multiply zw.

1.1 A.1 The Pfaffian $\mathsf{{d}}$ Coefficients

1.2 A.2 The Pfaffian $\mathsf{{a}}$ Coefficients

$$\begin{aligned} \begin{array}{c|c} \mathrm {Pf}(\mathsf{{a}}_{\vec {\lambda }}) &{} \vec {\lambda },\,|\vec {\lambda }|=2 \\ \hline -\frac{1^{}}{12} \left( t_1^3+6 t_3\right) =-\frac{1}{2}Q_{(3,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad (0) , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (0) \\ \end{array} \right) \\ t_3-\frac{t_1^{3^{}}}{12}=-\frac{1}{2}Q_{(2,1)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad (0) , &{}\quad \emptyset \\ \end{array} \right) \\ -\frac{t_1^{2^{}}}{4}=-\frac{1}{2}Q_{(2,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad \emptyset , &{}\quad (0) \\ \end{array} \right) \\ -\frac{t_1^{}}{2}=-\frac{1}{2}Q_{(1,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad (0) \\ \end{array} \right) \\ \hline &{} \vec {\lambda },\,|\vec {\lambda }|=3 \\ \hline \frac{1}{360} \left( -t_1^6-30 t_3 t_1^3+180 t_5 t_1+180 t_3^2\right) ^{}=-\frac{1}{2}Q_{(5,1)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (1,0) , &{}\quad \emptyset \\ \end{array} \right) \\ -\frac{1}{96} t_1 \left( t_1^3+24 t_3\right) =-\frac{1}{4}Q_{(4,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (1,0) \\ \end{array} \right) \\ -\frac{t_1^6}{1440}-\frac{1}{12} t_3 t_1^3-\frac{t_5 t_1}{2}-\frac{t_3^2}{4}=-\frac{1}{2}Q_{(6,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad (1) , &{}\quad \emptyset , &{}\quad (0) \\ \end{array} \right) \\ \frac{1}{576} \left( t_1^3-12 t_3\right) ^2=\frac{1}{4}Q_{(4,2)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad \emptyset , &{}\quad (1) \\ \end{array} \right) \\ -\frac{t_1^5}{240}-\frac{1}{4} t_3 t_1^2-\frac{t_5}{2}=-\frac{1}{2}Q_{(5,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (1) , &{}\quad (0) \\ \end{array} \right) \\ \frac{1}{160} \left( t_1^5-80 t_5\right) =\frac{1}{4}Q_{(4,1)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad (1) \\ \end{array} \right) \\ \hline &{} \vec {\lambda },\,|\vec {\lambda }|=4 \\ \hline -\frac{t_1^9}{725760}-\frac{t_3 t_1^6}{1440}-\frac{1}{48} t_5 t_1^4-\frac{1}{24} t_3^2 t_1^3-\frac{1}{2} t_3 t_5 t_1-\frac{t_3^3}{12}=-\frac{1}{2}Q_{(9,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (2) , &{}\quad (0) \\ \end{array} \right) \\ \frac{t_1^9}{207360}+\frac{1}{720} t_3 t_1^6+\frac{1}{48} t_5 t_1^4+\frac{1}{48} t_3^2 t_1^3-\frac{1}{4} t_3 t_5 t_1-\frac{t_3^3}{12}=\frac{1}{4}Q_{(8,1)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (0) , &{}\quad (2) \\ \end{array} \right) \\ -\frac{t_1^9}{103680}+\frac{t_3 t_1^6}{2880}+\frac{1}{96} t_5 t_1^4-\frac{1}{24} t_3^2 t_1^3+\frac{1}{4} t_3 t_5 t_1+\frac{t_3^3}{6}=-\frac{1}{4}Q_{(5,4)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad \emptyset , &{}\quad \emptyset , &{}\quad (1) , &{}\quad (1) \\ \end{array} \right) \\ \frac{t_1^6-60 t_3 t_1^3+720 t_5 t_1-720 t_3^2}{1440}=\frac{1}{4}Q_{(3,2,1,0)} &{} \left( \begin{array}{ccccc} \emptyset , &{}\quad (0) , &{}\quad (0) , &{}\quad (0) , &{}\quad (0) \\ \end{array} \right) \\ \end{array} \end{aligned}$$

B Appendix. Matrix Representation of Orthogonal Affine Lie Algebras

In the following, $e_{ij}$ is the elementary matrix $(e_{ij})_{\alpha \beta }:=\delta _{\alpha i}\delta _{\beta j}$.

1.1 B.1 Matrix Realization of $B_\ell ^{(1)}$

• Weyl generators:

  $$\begin{aligned}&E_0=\frac{1}{2}(e_{1,2\ell }+e_{2,2\ell +1}), \quad F_0=2\left( e_{2\ell ,1}+e_{2\ell +1,2} \right) , \nonumber \\&H_0 =e_{1,1}+e_{2,2}-e_{2l,2l}-e_{2l+1,2l+1}, \nonumber \\&E_i=e_{i+1,i}+e_{2l+2-i,2l+1-i}, \quad F_i=e_{i,i+1}+e_{2\ell +1-i,2\ell +2-i} \nonumber \\&H_i=-e_{i,i}+e_{i+1,i+1}-e_{2\ell +1-i,2\ell +1-i}+e_{2\ell +2-i,2\ell +2-i}, \quad i=1,\dots ,\ell -1, \nonumber \\&E_\ell =e_{\ell +1,\ell }+e_{\ell +2,\ell +1}, \quad F_{\ell }=e_{\ell ,\ell +1}+e_{\ell +1,\ell +2}, \quad H_\ell =-e_{\ell ,\ell }+e_{\ell +2,\ell +2}. \nonumber \\ \end{aligned}$$

  (B.1)

• Cartan matrix:

  $$\begin{aligned} A=\left( \begin{array}{cccccc} 2 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 2 &{}\quad -1 &{}\quad &{}\quad &{}\quad \vdots \\ -1 &{}\quad -1 &{}\quad 2 &{}\quad \ddots &{}\quad &{}\quad 0 \\ 0 &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad -1 &{}\quad 0 \\ \vdots &{}\quad &{}\quad &{}\quad -1 &{}\quad 2 &{}\quad -1 \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad -2 &{}\quad 2 \end{array} \right) _{(\ell +1)\times (\ell +1)}, \end{aligned}$$

  (B.2)

• Chevalley involution:

  $$\begin{aligned} S= & {} \mathrm {antidiag}\left( 1,-1,1,\dots ,-1,1 \right) _{2\ell +1\times 2\ell +1} \nonumber \\= & {} \left( \begin{array}{cccccc} 0 &{}\quad 0 &{}\quad \cdots &{}\quad &{}\quad &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad &{}\quad -1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \ddots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -1 &{}\quad 0 &{}\quad &{}\quad \cdots &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad &{}\quad \cdots &{}\quad 0 \end{array} \right) _{2\ell +1\times 2\ell +1}. \end{aligned}$$

  (B.3)

1.2 B.2 Matrix Realization of $D_\ell ^{(1)}$

• Weyl generators:

  $$\begin{aligned}&E_0=\frac{1}{2}(e_{1,2\ell -1}+e_{2,2\ell }), \quad F_0=2\left( e_{2\ell -1,1}+e_{2\ell ,2}\right) , \nonumber \\&H_0 =e_{1,1}+e_{2,2}-e_{2l-1,2l-1}-e_{2l,2l}, \nonumber \\&E_i=e_{i+1,i}+e_{2l+1-i,2l-i}, \quad F_i=e_{i,i+1}+e_{2\ell -i,2\ell +1-i} \nonumber \\&H_i=-e_{i,i}+e_{i+1,i+1}-e_{2\ell -i,2\ell -i}+e_{2\ell +1-i,2\ell +1-i}, \quad i=1,\dots ,\ell -1, \nonumber \\&E_\ell =\frac{1}{2}\left( e_{\ell +1,\ell -1}+e_{\ell +2,\ell }\right) , \quad F_{\ell }=2\left( e_{\ell -1,\ell +1}+e_{\ell ,\ell +2}\right) , \nonumber \\&H_\ell =-e_{\ell -1,\ell -1}-e_{\ell ,\ell }+e_{\ell +1,\ell +1} +e_{\ell +2,\ell +2}. \end{aligned}$$

  (B.4)

• Cartan matrix:

  $$\begin{aligned} A=\left( \begin{array}{cccccc} 2 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 2 &{}\quad -1 &{}\quad &{}\quad &{}\quad \vdots \\ -1 &{}\quad -1 &{}\quad 2 &{}\quad \ddots &{}\quad &{}\quad 0 \\ 0 &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad -1 &{}\quad -1 \\ \vdots &{}\quad &{}\quad &{}\quad -1 &{}\quad 2 &{}\quad 0 \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 2 \end{array} \right) _{(\ell +1)\times (\ell +1)}. \end{aligned}$$

  (B.5)

• Chevalley involution:

  $$\begin{aligned} S=\mathrm {antidiag}\left( 1,-1,1,\dots ,(-1)^{\ell },(-1)^{\ell },(-1)^{\ell +1},\dots ,-1,1\right) _{2\ell \times 2\ell }. \end{aligned}$$

  (B.6)