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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.

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Acknowledgements

The author is grateful to Youjin Zhang, Boris Dubrovin, Don Zagier for their advising and to Jian Zhou and Si-Qi Liu for helpful discussions. He thanks the referee for valuable suggestions; in particular, Appendix A comes out from the suggestions. He also wishes to thank Boris Dubrovin for introducing GUE to him and for helpful suggestions and discussions on this article. The work is partially supported by a starting research grant from University of Science and Technology of China.

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Correspondence to Di Yang.

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Dedicated to the memory of Boris Anatol’evich Dubrovin, with gratitude and admiration.

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Appendix A: Pair of abstract pre-wave functions

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.

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Yang, D. On tau-functions for the Toda lattice hierarchy. Lett Math Phys 110, 555–583 (2020). https://doi.org/10.1007/s11005-019-01232-5

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Keywords

  • Toda lattice hierarchy
  • Tau-function
  • Pair of wave functions
  • Matrix resolvent
  • Generating series

Mathematics Subject Classification

  • 37K10
  • 53D45
  • 14N35
  • 05A15
  • 33E15