$N$-soliton $\tau$-functions admit several determinant representations, organised by the Sato Grassmannian
[Sato ’81; Date–Jimbo–Kashiwara–Miwa ’82; Kodama ’17].
⟷KP line solitons: an explicit one-to-one correspondence between the two [Chakravarty–Lewkow–Maruno ’10]
Motivation
Why the anti-self-dual Yang–Mills equations?
A matrix-valued integrable system at the heart of four-dimensional gauge theory — instantons
[Belavin–Polyakov–Schwartz–Tyupkin ’75; Atiyah–Hitchin–Drinfeld–Manin ’78].
A unifying framework: many lower-dimensional integrable equations arise as symmetry reductions
[Ward ’85; Ablowitz–Chakravarty–Takhtajan ’93; Mason–Woodhouse ’96]
KdVNLSsine-GordonPainlevé…
A rich toolbox of exact solutions — and hence the question of how the various constructions are related.
✓ Same action densities[Li–Hamanaka–Huang–Zhang ’25]
✓ Yang matrices of general $N$-solitons are asymptotically equivalent [Huang ’26]
? A direct algebraic identification of the two constructions — open
?
The problem
Do the quasi-Wronskian and quasi-Grammian constructions of ASDYM admit an exact algebraic correspondence — just as the Wronskian and Grammian do for KP?
Answer (this talk): Yes — for complex $2\times2$ fields ($G=\GL_2(\mathbb C)$) both are governed by one and the same uniquely solvable matrix-polynomial interpolation problem.
Main result
Main result
Input: $2M$ pairwise distinct nonzero spectral parameters $\xi_1,\dots,\xi_M,\gamma_1,\dots,\gamma_M$; vacuum wave functions $\phi_i,\psi_i\in\mathbb C^2$;
a nonempty local domain on which the interpolation matrix $\cV$ is invertible. Set $\;q_\xi(\lambda)=\prod_{i=1}^M(\lambda-\xi_i),\ \ \Xi=\diag(\xi_1,\dots,\xi_M)$.
⇒All discrepancies are constants ⇒ identical gauge potentials in the gauge $A_{\tz}=A_{\tw}=0$: the two constructions give the same ASDYM field.
Background · §2.1
ASDYM: Lax pair, Yang matrix and potential
Coordinates $(z,\tz,w,\tw)$ on $\mathcal M\cong\mathbb C^4$, metric $ds^2=2(dz\,d\tz-dw\,d\tw)$, gauge group $G=\GL_2(\mathbb C)$, $\mathfrak g=\Mat_2(\mathbb C)$. In the gauge $A_{\tz}=A_{\tw}=0$, ASDYM is the compatibility condition of
Normalization freedom. For constant $C\in\GL_2(\mathbb C)$, $C_0\in\Mat_2(\mathbb C)$:
$\;\cJ\mapsto\cJ C,\ \ \cK\mapsto\cK+C_0\;$ leave $A_z,A_w$ unchanged. ⇒ distinguish exact identities of solution matrices from equality of gauge fields.
Tool 1 · §2.2
Quasideterminants
A
B
C
D
Definition [Gelfand–Gelfand–Retakh–Wilson ’05] ($A$ invertible, blocks of compatible sizes):
$$\left|\begin{array}{cc}A & B\\ C & \boxed{D}\end{array}\right|:=D-C\,A^{-1}B$$
= the Schur complement of $A$; the box marks the expansion block.
Noncommutative analogue of a determinant ratio: if $D$ is $1\times1$, $\left|\begin{smallmatrix}A&B\\C&\boxed{D}\end{smallmatrix}\right|=\dfrac{\det H}{\det A}$.
Natural when the entries are themselves matrices — here $2\times2$ blocks. Both quasi-Wronskian and quasi-Grammian solutions are written this way.
i.e. arbitrary functions of two variables:
$$\W{\phi_i}=\phi_i\big(\xi_iz+\tw,\ \xi_iw+\tz\big),\qquad \G{\psi_i}=\psi_i\big(\gamma_iz+\tw,\ \gamma_iw+\tz\big).$$
Exponentials ⇒ solitons. The same data feed both constructions.
Tool 2 · §2.3
Matrix-polynomial interpolation
Find a monic $2\times2$ matrix polynomial [Gohberg–Lancaster–Rodman ’82; Ball–Gohberg–Rodman ’90]
$4M$ unknowns $p_{k,ab}$, $4M$ scalar equations. Row $a$ of $P$ solves $\;\cV\,\mathbf p_a=-\mathbf b_a$ with the $2M\times2M$ interpolation matrix $\cV$ →
Standing assumption. $\det\cV\neq0$ on a nonempty local domain ⇒ $P(\lambda)$ exists and is unique.
Remark: $\det\cV=\det\big(f^{(1)},\chi f^{(1)},\dots,\chi^{M-1}f^{(1)};\,f^{(2)},\dots,\chi^{M-1}f^{(2)}\big)$, $\chi=\diag(\xi,\gamma)$ — a double Wronskian as in Kakei–Ikeda–Takasaki ’02.
$\cV$ for $M=3$: rows ↔ data points, columns ↔ (component, power of $\lambda$).
Proposition 2.3
The determinant of the interpolant
Proposition. With $q_\xi(\lambda)=\prod_i(\lambda-\xi_i)$, $q_\gamma(\lambda)=\prod_i(\lambda-\gamma_i)$:
$$\det P(\lambda)=q_\xi(\lambda)\,q_\gamma(\lambda)=\prod_{i=1}^M(\lambda-\xi_i)(\lambda-\gamma_i).$$
$\det\cV\neq0$ ⇒ none of the vectors $\phi_i,\psi_i$ vanishes.
$P(\xi_i)\phi_i=0,\ P(\gamma_i)\psi_i=0$ ⇒ $\det P$ vanishes at the $2M$ distinct points $\xi_i,\gamma_i$.
$P$ monic of degree $M$ ⇒ $\det P$ is a monic scalar polynomial of degree $2M$.
In particular $\;\det P(0)=\prod_{i}\xi_i\gamma_i\neq0$ — the reason for nonzero spectral parameters.
Apply $\partial_z$ to the annihilation identity, subtract its $\partial_{\tw}$-derivative times $\Lambda_i$: the dispersion relations kill all derivatives of $\Theta_i$:
$\;\sum_k\big[(\partial_zT_k)\Theta_i\Lambda_i^k-(\partial_{\tw}T_k)\Theta_i\Lambda_i^{k+1}\big]=0.$
Eliminate $\Theta_i\Lambda_i^M$ (again by annihilation): $\;\sum_{k}R_k\Theta_i\Lambda_i^k=0$ with
$\;R_k=\partial_zT_k-\partial_{\tw}T_{k-1}+(\partial_{\tw}T_{M-1})T_k$, $\;T_{-1}:=0$.
$(R_0,\dots,R_{M-1})\,\Sigma=0$ and $\Sigma$ invertible ⇒ $R_k=0$ ⇒ $\;(\partial_z-\lambda\partial_{\tw}+\partial_{\tw}T_{M-1})T(\lambda)=\sum_kR_k\lambda^k=0$.
Put $\lambda=0$; $\;\det T_0=\det P(0)=\prod_i\xi_i\gamma_i\neq0$.
⇒Miura relations ⇒ $\W{\cJ_{\rW}:=T_0},\quad \W{\cK_{\rW}:=T_{M-1}},\quad \W{\tau_{\rW}:=\det\Sigma}$ — recovering the known quasi-Wronskian formulas [GHHN ’20; Hamanaka–Huang ’22].
Quasi-Grammian side · §4.1
Cauchy matrix approach and the dressing matrix $\G{W(\lambda)}$
Factor $\diag(V_\xi,V_\xi)$, Laplace-expand along the first $M$ columns:
$\;(\det V_\xi)^2\det\big[(V_\xi^{-1}V_\gamma)_{ij}\,(\phi_i^{(1)}\psi_j^{(2)}-\phi_i^{(2)}\psi_j^{(1)})\big].$
Lagrange interpolation:
$\;(V_\xi^{-1}V_\gamma)_{ij}=\dfrac{\prod_{k}(\gamma_j-\xi_k)}{(\gamma_j-\xi_i)\prod_{k\neq i}(\xi_i-\xi_k)}$ — pull out row/column factors, the Cauchy kernel $\Omega_{ij}$ appears.
⇒$\tau_{\rW}$ and $\tau_{\rG}$ vanish simultaneously: same nonvanishing domain; $\tau_{\rG}\neq0 \iff \det\cV\neq0$. cf. [Ishikawa–Okada–Tagawa–Zeng ’06, Eq. (1.7)]
Main theorem · Theorem 4.2
Proof of $\;\W{T(\lambda)}=\Au{q_\xi(\lambda)}\,\G{W(\lambda)}$
Set $\;\widehat P(\lambda):=q_\xi(\lambda)\,W(\lambda)$. It suffices to show that $\widehat P$ solves the interpolation problem.
(A) Shape
$q_\xi(\lambda)(\lambda I_M-\Xi)^{-1}=\diag_k\Big(\prod_{j\neq k}(\lambda-\xi_j)\Big)$ is polynomial, so
$$\widehat P(\lambda)=q_\xi(\lambda)I_2-\theta\Omega^{-1}\diag_k\Big(\prod_{j\neq k}(\lambda-\xi_j)\Big)\rho$$
and $W=I_2+O(\lambda^{-1})$ ⇒ monic of degree $M$.
(B) At $\lambda=\xi_i$
$$\widehat P(\xi_i)\phi_i=-\theta\Omega^{-1}D_i\,\rho\,\phi_i,$$
$D_i=\diag_k\big(\prod_{j\neq k}(\xi_i-\xi_j)\big)$: only $(D_i)_{ii}\neq0$. But
$$(\rho\,\phi_i)_k=\phi_k^{(1)}\phi_i^{(2)}-\phi_k^{(2)}\phi_i^{(1)}$$
vanishes for $k=i$ ⇒ $\widehat P(\xi_i)\phi_i=0$.
(C) At $\lambda=\gamma_i$
Sylvester ⇔ $(\lambda I_M-\Xi)\Omega-\rho\theta=\Omega(\lambda I_M-\Gamma)$, hence
$$W(\lambda)\,\theta=\theta\,\Omega^{-1}(\lambda I_M-\Xi)^{-1}\,\Omega\,(\lambda I_M-\Gamma).$$
Column $i$ carries $(\lambda-\gamma_i)$; the resolvent is regular at $\gamma_i\neq\xi_j$ ⇒ $W(\gamma_i)\psi_i=0$.
∎$\widehat P$ is monic of degree $M$ with the same interpolation data ⇒ by uniqueness $\;\widehat P=P=T$, i.e. $\;\W{T(\lambda)}=\Au{q_\xi(\lambda)}\G{W(\lambda)}$.
$$-(\partial_z\cJ_{\rW})\cJ_{\rW}^{-1}=-(\partial_z\cJ_{\rG})\cJ_{\rG}^{-1},\qquad \partial_{\tw}\cK_{\rW}=\partial_{\tw}\cK_{\rG}\quad(\text{same for }w,\tz).$$
⇒$\det\Xi$, $\tr\Xi$ are spacetime constants — exactly the normalization freedom $\cJ\mapsto\cJ C$, $\cK\mapsto\cK+C_0$. Same $A_z,A_w$, same ASDYM field.
Two quick checks · Remark 4.4
Sanity check $M=1$ & an $\SL_2(\mathbb C)$ normalization
Since $\Theta_1\Lambda_1\Theta_1^{-1}=\xi_1I_2+(\gamma_1-\xi_1)\,\psi_1\rho_1/\det\Theta_1$: $(\lambda-\xi_1)W(\lambda)=T(\lambda)$ ✓ and $\tau_{\rW}=\det\Theta_1=(\gamma_1-\xi_1)\,\tau_{\rG}$ ✓
$\det\widehat\cJ=1$, $\tr\widehat\cK=0$, gauge potentials unchanged ⇒ an $\mathfrak{sl}_2(\mathbb C)$-valued connection.
Summary
The big picture
Spectral data
$\{(\xi_i,\phi_i),\,(\gamma_i,\psi_i)\}_{i=1}^M$
Darboux transformation
$\W{T(\lambda)}$, $\tau_{\rW}=\det\Sigma$
Cauchy matrix / Sylvester
$\G{W(\lambda)}$, $\tau_{\rG}=\det\Omega$
Unique interpolant
$\W{T}=P=\Au{q_\xi}\G{W}$ $\det P=q_\xi q_\gamma$
One ASDYM gauge field
$\cJ_{\rW}=(-1)^M\det\Xi\,\cJ_{\rG},\quad \cK_{\rW}=\cK_{\rG}-\tr\Xi\,I_2\quad\Rightarrow\quad$ same $A_z,\ A_w$
Conclusions
Conclusions
Quasi-Wronskian and quasi-Grammian constructions of $2\times2$ ASDYM fields are governed by the same uniquely solvable matrix-polynomial interpolation problem.
Exact identification of the generating matrices: $\W{T(\lambda)}=\Au{q_\xi(\lambda)}\,\G{W(\lambda)}$.
$\tau$-functions differ by the nonzero constant $(-1)^{M(M-1)/2}\prod_{i,j}(\gamma_j-\xi_i)$ ⇒ same nonvanishing domain.
Yang / potential matrices differ only by constant normalizations ⇒ identical gauge potentials: the same ASDYM field.
The two formalisms are two realizations of the same spectral data — the ASDYM counterpart of the KP Wronskian ⟷ Grammian dictionary.
Future problems
Outlook
Other solution classes
ASDYM admits many other exact solutions — instantons / ADHM, Riemann–Hilbert, bilinear, … Can similar exact correspondences be established among these representations?
Symmetry reductions
Many lower-dimensional integrable equations are reductions of ASDYM. How does the correspondence behave under reduction? Does it induce equivalences between solution constructions of the reduced equations?
Further questions
Beyond $2\times2$: $\GL_N(\mathbb C)$ fields and more general (e.g. non-simple) spectral data?
A Grassmannian picture for ASDYM in the spirit of KP, via the Sato-theoretic dressing [Li–Zhang ’26; Takasaki ’26]?
References
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The author thanks Masashi Hamanaka for valuable comments. Supported by JSPS (Grant No. P25326) and NSFC (Grant No. 1261462). Shangshuai Li · [email protected]
S. Li · Quasi-Wronskian ⟷ quasi-Grammian solutions of ASDYM非線形波動から可積分系へ 2026 · 松江
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