On tau-functions for the Toda lattice hierarchy

Abstract

We extend a recent result of Dubrovin et al. in On tau-functions for the KdV hierarchy, arXiv:1812.08488 to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions and use them to give explicit formulae for the generating series of k-point correlation functions of the solution. Applications to computing GUE correlators and Gromov–Witten invariants of the Riemann sphere are under consideration.

Appendix A: Pair of abstract pre-wave functions

Here, we construct a ring that is suitable for defining abstract pre-wave functions. Recall that ${{\mathcal {A}}}$ is the ring of polynomials of $v_k,w_k$, $k \in {{\mathbb {Z}}}$. Instead of the ${{\mathbb {Z}}}$-coefficients, we will use in this appendix the ${{\mathbb {Q}}}$-coefficients, i.e., ${{\mathcal {A}}}={{\mathbb {Q}}}\bigl [ \{v_k,w_k \,|\, k\in {{\mathbb {Z}}}\}\bigr ]$, is now under consideration. For each monic monomial $\alpha \in {{\mathcal {A}}}\backslash {{\mathbb {Q}}}$, we associate a symbol $m_\alpha$. Denote by ${\mathcal {B}}$ the polynomial ring

$$\begin{aligned} {\mathcal {B}}\, := \, {{\mathbb {Q}}}\bigl [\{ \,m_\alpha \,|\,\alpha \text{ is } \text{ a } \text{ monic } \text{ monomial } \text{ in } {{\mathcal {A}}}\backslash {{\mathbb {Q}}}\} \bigr ]. \end{aligned}$$

(121)

Define the action of $\Lambda ^k$ on ${\mathcal {B}}$ with $k\in {{\mathbb {Z}}}$ by

$$\begin{aligned} \Lambda ^k \bigl (m_{\alpha _1} \ldots m_{\alpha _l}) \; = \; m_{ \Lambda ^k(\alpha _1)} \ldots m_{ \Lambda ^k (\alpha _l)} \end{aligned}$$

(122)

for $\alpha _1,\ldots ,\alpha _l$ being monic monomials in ${{\mathcal {A}}}\backslash {{\mathbb {Q}}}$, as well as by linearly extending it to other elements of ${\mathcal {B}}$. For a monic monomial $\alpha =v_{i_1} \ldots v_{i_r} w_{j_1} \ldots w_{j_s}\in {{\mathcal {A}}}\backslash {{\mathbb {Q}}}$ with $i_1\le \cdots \le i_r$, $j_1\le \cdots \le j_s$ and $r+s\ge 1$, let $k_\alpha :=-i_1$ (if $r\ge 1$), $k_\alpha :=-j_1$ (if $r=0$); the monomial $\Lambda ^{k_\alpha } (\alpha )\in {{\mathcal {A}}}$ is then called the (unique) reduced monomial (associated to $\alpha$). Denote by ${\mathcal {C}}$ the polynomial ring generated by $m_{\beta }$, $v_k$, $w_k$ with ${{\mathbb {Q}}}$-coefficients, where $\beta$ are reduced monic monomials, and $k\in {{\mathbb {Z}}}$. Let us also define an action of $\Lambda ^k$ on ${\mathcal {C}}$, $k\in {{\mathbb {Z}}}$. To this end, we introduce some notations: For $\beta$ a reduced monic monomial of ${{\mathcal {A}}}$, denote

$$\begin{aligned} n_{\Lambda ^k(\beta )} \, := \, \left\{ \begin{array}{cc} m_\beta + \sum _{i=0}^{k-1} \Lambda ^i(\beta ), &{} ~ k\ge 0,\\ m_\beta - \sum _{i=k}^{-1} \Lambda ^i(\beta ), &{} ~ k\le -1. \end{array}\right. \end{aligned}$$

(123)

Then, for a monomial $\alpha \cdot m_{\beta _1} \ldots m_{\beta _s}$ of ${\mathcal {C}}$ with $\alpha$ being a monomial in ${{\mathcal {A}}}$, define

$$\begin{aligned} \Lambda ^k (\alpha \cdot m_{\beta _1} \ldots m_{\beta _s}) = \Lambda ^k (\alpha ) \cdot \prod _{j=1}^s n_{\Lambda ^k(\beta _j)}, \quad k\in {{\mathbb {Z}}}. \end{aligned}$$

(124)

Define the action of $\Lambda ^k$ on other elements in ${\mathcal {C}}$ by requiring it as a linear operator. Denote by $p:{\mathcal {B}}\rightarrow {\mathcal {C}}$ the linear map which maps $m_{\alpha _1} \ldots m_{\alpha _l}\in {\mathcal {B}}$ to $n_{\alpha _1} \ldots n_{\alpha _l}\in {\mathcal {C}}$, for $\alpha _i$, $i=1,\ldots ,l$ being monic monomials in ${{\mathcal {A}}}\backslash {{\mathbb {Q}}}$. Denote by ${\mathcal {B}}^0$ the image of p. Clearly, ${{\mathcal {A}}}\subset {\mathcal {B}}^0$. Indeed, the element $(\Lambda -1) \bigl (\sum _{i=1}^l \lambda _i \, m_{\alpha _i}\bigr ) \in {\mathcal {B}}$ becomes $\sum _{i=1}^l \lambda _i \alpha _i \in {{\mathcal {A}}}$ under the map p. Here, $\alpha _1,\ldots ,\alpha _l$ are distinct monic monomials in ${{\mathcal {A}}}\backslash {{\mathbb {Q}}}$. Finally, we define an operator ${\mathbb {S}}: {{\mathcal {A}}}\backslash {{\mathbb {Q}}}\rightarrow {\mathcal {B}}^0$ by

$$\begin{aligned} {\mathbb {S}} \bigl (\lambda _1 \alpha _1 \; + \; \cdots \; + \; \lambda _l \alpha _l\bigr ) \; = \; \lambda _1 n_{\alpha _1} \; + \; \cdots \; + \; \lambda _l n_{\alpha _l} \, \end{aligned}$$

(125)

for $\alpha _1,\ldots ,\alpha _l$ being distinct monic monomials and $\lambda _1,\ldots ,\lambda _l\in {{\mathbb {Q}}}$.

Motivated by (62) and (63), define two families of elements $y_i,z_i \in {{\mathcal {A}}}$, $i\ge 1$ by

$$\begin{aligned}&y_{k+1} \ = - \sum _{\begin{array}{c} m_1,\ldots ,m_k\ge 0\\ \sum _{i=1}^{k} im_i=k+1 \end{array}} \frac{\prod _{i=1}^k y_i^{m_i}}{\prod _{i=1}^k m_i!} -v_0\delta _{k,0} \\&\qquad \qquad \quad - w_0 \sum _{\begin{array}{c} m_1,\ldots ,m_{k-1}\ge 0 \sum _{i=1}^{k-1} im_i=k-1 \end{array}} \frac{\prod _{i=1}^{k-1} (-1)^{m_i} \bigl (\Lambda ^{-1}(y_i)\bigr )^{m_i}}{\prod _{i=1}^{k-1} m_i!}, \\&z_{k+1} \; = \; \sum _{\begin{array}{c} m_1,\ldots ,m_k\ge 0\\ \sum _{i=1}^{k} im_i=k+1 \end{array}} \frac{\prod _{i=1}^k (-1)^{m_i} z_i^{m_i}}{\prod _{i=1}^k m_i!} \; + \; v_1\delta _{k,0} \\&\quad \qquad \qquad + w_2 \sum _{\begin{array}{c} m_1,\ldots ,m_{k-1}\ge 0\\ \sum _{i=1}^{k-1} im_i=k-1 \end{array} }\frac{\prod _{i=1}^{k-1} \bigl (\Lambda (z_i)\bigr )^{m_i}}{\prod _{i=1}^{k-1} m_i!}. \end{aligned}$$

Equivalently, the generating series $y(\lambda ):=\sum _{i\ge 1} y_i/\lambda ^i$, $z(\lambda ):=\sum _{i\ge 1} z_i/\lambda ^i$ satisfy

$$\begin{aligned}&\lambda \, e^{y(\lambda )} \; + \; v_0 \,-\, \lambda \; + \; w_0 \, \lambda ^{-1} \Lambda ^{-1}\bigl (e^{-y(\lambda )}\bigr ) \; = \; 0, \\&\lambda \, \Lambda ^{-1} \bigl (e^{-z(\lambda )}\bigr ) \; + \; v_0 \,-\, \lambda \; + \; w_1 \, \lambda ^{-1} e^{z(\lambda )} \; = \; 0. \end{aligned}$$

Define

$$\begin{aligned} \psi _A\, := \, e^{{\mathbb {S}} (y(\lambda ))} \otimes \lambda ^n \otimes 1, \quad \psi _B \, := \, e^{{\mathbb {S}} (z(\lambda ))} \otimes \lambda ^{-n} \otimes e^{-\sigma } , \end{aligned}$$

(126)

where $e^{-\sigma }$ is a formal element satisfying $e^{(1-\Lambda ^{-1})(-\sigma )} = w_0$, and $\lambda ^n$, $\lambda ^{-n}$ are formal elements satisfying $\Lambda ^k (1\otimes \lambda ^n)=\lambda ^k \otimes \lambda ^n$, $\Lambda ^k (1\otimes \lambda ^{-n})=\lambda ^{-k} \otimes \lambda ^{-n}$, $k\in {{\mathbb {Z}}}$. We have

$$\begin{aligned}&L \bigl (\psi _A\bigr ) \; = \; \lambda \, \psi _A,\quad L \bigl (\psi _B\bigr ) \; = \; \lambda \, \psi _B,\nonumber \\&\psi _A(\lambda ) \; = \; \bigl (1+{\mathrm{O}}\bigl (\lambda ^{-1}\bigr )\bigr )\otimes \lambda ^n \;\in \; {\mathcal {C}} \left[ \left[ \lambda ^{-1}\right] \right] \otimes \lambda ^n, \end{aligned}$$

(127)

$$\begin{aligned}&\psi _B(\lambda ) \; = \; \bigl (1+{\mathrm{O}}\bigl (\lambda ^{-1}\bigr )\bigr ) \otimes \lambda ^{-n} \otimes e^{-\sigma } \;\in \; {\mathcal {C}} \left[ \left[ \lambda ^{-1}\right] \right] \otimes \lambda ^{-n} \otimes e^{-\sigma }, \end{aligned}$$

(128)

where $L=\Lambda +v_0 + w_0 \, \Lambda ^{-1}$. We call $\psi _A$ and $\psi _B$ the abstract pre-wave functions of type A and of type B, respectively, associated with $v_0,w_0$.

Denote

$$\begin{aligned} d_{{\mathrm{pre}}}(\lambda )\, := \, \psi _A(\lambda ) \, \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) \,-\, \psi _B(\lambda ) \, \Lambda ^{-1}\bigl (\psi _A(\lambda )\bigr ) \end{aligned}$$

(129)

and

$$\begin{aligned} \Psi (\lambda ) \, := \, \begin{pmatrix} \psi _A (\lambda ) &{}\quad \psi _B(\lambda ) \\ \Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) &{}\quad \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) \end{pmatrix}. \end{aligned}$$

(130)

Then, we have the following identity:

$$\begin{aligned} R(\lambda ) \; = \; \Psi (\lambda ) \, \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix} \, \Psi ^{-1}(\lambda ) \;=:\; M(\lambda ). \end{aligned}$$

(131)

The proof is similar to that of Proposition 3. (The main fact used in the proof is that from the definition, the coefficients of entries of $R(\lambda )$ are uniquely determined in an algebraic way.) We omit its details here. However, let us explain the equality (131) by an equivalent form. From definition, we have

$$\begin{aligned} M(\lambda ) \; = \; \frac{1}{d_{{\mathrm{pre}}}(\lambda )} \begin{pmatrix} \psi _A (\lambda ) \, \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) &{}\quad -\psi _A (\lambda ) \, \psi _B(\lambda ) \\ \Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) \, \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) &{}\quad -\Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) \, \psi _B(\lambda ) \end{pmatrix}. \end{aligned}$$

Then, from a straightforward calculation by using the definitions, we find

$$\begin{aligned} M_{11}(\lambda )&\; = \; \frac{1}{1 \,-\, \frac{w_0}{\lambda ^2} \, e^{ \Lambda ^{-1} (z(\lambda ) - y(\lambda ))}}, \end{aligned}$$

(132)

$$\begin{aligned} M_{12}(\lambda )&\; = \; \frac{1}{\lambda ^{-1} \, e^{-\Lambda ^{-1}(y(\lambda ))} \,-\, \frac{\lambda }{w_0} \, e^{-\Lambda ^{-1}(z(\lambda ))}}, \end{aligned}$$

(133)

$$\begin{aligned} M_{21}(\lambda )&\; = \; \frac{1}{\lambda \, e^{\Lambda ^{-1}(y(\lambda ))} \,-\, \frac{w_0}{\lambda } \, e^{\Lambda ^{-1}(z(\lambda ))}}, \end{aligned}$$

(134)

$$\begin{aligned} M_{22}(\lambda )&\; = \; \frac{1}{1 \,-\, \frac{\lambda ^2}{w_0} \, e^{ \Lambda ^{-1} (y(\lambda )- z(\lambda ))}}. \end{aligned}$$

(135)

Hence, the equality (131) means new expressions for the entries of the basic matrix resolvent $R(\lambda )$ explicitly in terms of $y(\lambda ),z(\lambda )$. Substituting the following expansions

$$\begin{aligned} y(\lambda ) \; = \; -\frac{v_0}{\lambda } \,-\, \frac{\frac{1}{2} v_0^2+w_0}{\lambda ^2} \; + \; \cdots , \qquad z(\lambda ) \; = \; \frac{v_1}{\lambda } \; + \; \frac{\frac{1}{2} v_1^2+w_2}{\lambda ^2} \; + \; \cdots \nonumber \\ \end{aligned}$$

(136)

into (132)–(135), we find that the new expressions agree with (24). Combining with (56), (57), we obtain

$$\begin{aligned}&\frac{1}{\frac{\lambda ^2}{w_0} \, e^{ \Lambda ^{-1} (y(\lambda )- z(\lambda ))} \,-\, 1} \; = \; \sum _{p\ge 0} \Omega _{p,0} \, \lambda ^{-p-2} \;=:\; A, \end{aligned}$$

(137)

$$\begin{aligned}&\frac{1}{\lambda \, e^{\Lambda ^{-1}(y(\lambda ))} \,-\, \frac{w_0}{\lambda } \, e^{\Lambda ^{-1}(z(\lambda ))}} \; = \; \lambda ^{-1}\; + \;\sum _{p\ge 0} \Lambda ^{-1} \bigl (S_p\bigr ) \, \lambda ^{-p-2} \; =: \; B. \end{aligned}$$

(138)

We therefore arrive at the following formulae:

$$\begin{aligned} e^{\Lambda ^{-1}(y(\lambda ))} \; = \; \frac{1}{\lambda }\, \frac{1+A}{B}, \qquad e^{\Lambda ^{-1}(z(\lambda ))} \; = \; \frac{\lambda }{w_0} \, \frac{A}{B}. \end{aligned}$$

(139)

Let us proceed to the generating series of multi-point correlation functions. Define

$$\begin{aligned} D_{{\mathrm{pre}}}(\lambda ,\mu ) \, := \, \frac{\psi _A(\lambda ) \, \Lambda ^{-1} \bigl (\psi _B(\mu )\bigr ) \,-\, \Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) \,\psi _B(\mu )}{\lambda -\mu }. \end{aligned}$$

(140)

Using (131), Proposition 1, and a similar argument to the proof of Theorem 2, we obtain

$$\begin{aligned} \sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}}&= \frac{(-1)^{k-1}}{\prod _{j=1}^k d_{{\mathrm{pre}}}(\lambda _j)} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k D_{{\mathrm{pre}}}(\lambda _{\pi (j)},\lambda _{\pi (j+1)}) \nonumber \\&\quad -\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2} . \end{aligned}$$

(141)

For the reader’s convenience, we give the first few terms of the abstract pre-wave functions $\psi _A(\lambda )$ and $\psi _B(\lambda )$ as follows:

$$\begin{aligned} \psi _A&= \biggl (1 \,-\, \frac{m_{v_0}}{\lambda } \; + \; \frac{m_{v_0}^2-m_{v_0^2}-2m_{w_0}}{2 \lambda ^2} \nonumber \\&\quad - \frac{1}{6\lambda ^3} \Bigl (m_{v_0}^3 + 2m_{v_0^3} - 3m_{v_0} m_{v_0^2} + 6m_{v_0 w_0} + 6m_{v_0 w_1} \nonumber \\&\quad - 6 m_{v_0} m_{w_0} - 6v_{-1} w_0\Bigr ) \; + \; {\mathrm{O}}\Bigl (\frac{1}{\lambda ^4}\Bigr )\biggr ) \, \lambda ^n,\end{aligned}$$

(142)

$$\begin{aligned} \psi _B&= \biggl (1 \; + \;\frac{m_{v_0}+v_0}{\lambda } \; + \; \frac{m_{v_0}^2+m_{v_0^2} +2v_0 m_{v_0}+2m_{w_0}+2v_0^2+2w_0+2w_1}{2\lambda ^2} \nonumber \\&\quad + \frac{1}{6 \lambda ^3} \Bigl (m_{v_0}^3+6m_{v_0} m_{w_0} +3 m_{v_0} m_{v_0^2} +2m_{v_0^3} +6m_{v_0 w_1} + 6m_{v_0 w_0} \nonumber \\&\quad +3 v_0 m_{v_0}^2+6v_0^2m_{v_0} +6w_0 m_{v_0} +6w_1 m_{v_0} +3v_0 m_{v_0^2}+6v_0 m_{w_0} \nonumber \\&\quad +6 v_0^3+12v_0 w_0+12 v_0 w_1 + 6 v_1 w_1 \Bigr ) \; + \; {\mathrm{O}}\Bigl (\frac{1}{\lambda ^4}\Bigr )\biggr ) \, \lambda ^{-n} e^{-\sigma }. \end{aligned}$$

(143)

It turns out that the above abstract pre-wave functions form a pair. Namely, $d_{{\mathrm{pre}}}(\lambda ) = \lambda \, e^{\Lambda ^{-1}(-\sigma )}$. Interestingly, for given arbitrary initial value (f(n), g(n)), based on this statement, one obtains a constructive method for a pair of wave functions associated with (f(n), g(n)) (cf. (28) in Sect. 1.3 for the definition of a pair). This is important considering Theorem 1. We hope to confirm the statement on the pair property of the abstract pre-wave functions in another publication.