Voros Coefficients for the Hypergeometric Differential Equations and Eynard–Orantin’s Topological Recursion: Part I—For the Weber Equation

Abstract

We develop the theory of quantization of spectral curves via the topological recursion. We formulate a quantization scheme of spectral curves which is not necessarily admissible in the sense of Bouchard and Eynard. The main result of this paper and the second part (Iwaki et al. in Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion, part II: for the confluent family of hypergeometric equations, preprint; arXiv:1810.02946) establishes a relation between the Voros coefficients for the quantum curves and the free energy for spectral curves associated with the confluent family of Gauss hypergeometric differential equations. We focus on the Weber equation in this article and generalize the result for the other members of the confluent family in the second part. We also find explicit formulas of free energy for those spectral curves in terms of the Bernoulli numbers.

Appendices

Meromorphic Multidifferentials

The correlation function $W_{g, n}(z_1, z_2, \ldots , z_n)$ is a meromorphic multidifferential , i.e., a meromorphic section of the line bundle $\pi _1^{*}(T^*\mathbb {P}^1) \otimes \pi _2^{*}(T^*\mathbb {P}^1) \otimes \cdots \otimes \pi _n^{*}(T^*\mathbb {P}^1)$ on $(\mathbb {P}^1)^n$, where $\pi _j: (\mathbb {P}^1)^n \rightarrow \mathbb {P}^1$ denotes the j-th projection [7]. Thus, a multidifferential is a meromorphic differential in $\mathbb {P}^1$ for each variable. If all of the residue with respect to each variable vanish, then we call it a multidifferential of the second kind. We summarize here some notations on multidifferential which we use in this paper.

Local coordinate representation.   In a local coordinate, we express a meromorphic multidifferential $\Omega$ on $(\mathbb {P}^1)^n$ as:

$$\begin{aligned} \Omega = \Omega (z_1, z_2, \ldots , z_n) = f(z_1, z_2, \ldots , z_n) \, dz_1 dz_2 \cdots dz_n, \end{aligned}$$

(A.1)

where f is a meromorphic function. (We omit the tensor product $\otimes$ in its expression.) If $\Omega$ is symmetric under the permutation of variables, i.e.,

$$\begin{aligned} \Omega (z_1, \ldots , z_j , \ldots , z_k , \ldots z_n) = \Omega (z_1, \cdots , z_k , \ldots , z_j , \ldots z_n) \end{aligned}$$

(A.2)

for any $j, k \in \{1, 2, \ldots , n\}$, $\Omega$ is said to be a symmetric multidifferential.

Integration.   Its integral with respect to j-th variable is denoted by

$$\begin{aligned} \int _{z_j = a}^{z_j = b} \Omega (z_1, z_2, \ldots , z_n) := \left( \int _a^b f(z_1, z_2, \ldots , z_n) \, dz_j \right) dz_1 \cdots dz_{j-1} dz_{j + 1} \cdots dz_n \end{aligned}$$

(A.3)

or

$$\begin{aligned} \int _{z_j \in \gamma } \Omega (z_1, z_2, \ldots , z_n) := \left( \int _{\gamma } f(z_1, z_2, \ldots , z_n) \, dz_j \right) dz_1 \cdots dz_{j-1} dz_{j + 1} \cdots dz_n \end{aligned}$$

(A.4)

for an integration path $\gamma$ in $\mathbb {P}^1$. If $\Omega$ is symmetric under the permutation of the variables, we write a multiple integral with a same integration contour $\gamma$ like

$$\begin{aligned} \underbrace{\int _{\gamma } \cdots \int _{\gamma }}_{n-\text {th}} \Omega (z_1, \ldots , z_n) := \int _{\gamma } dz_1 \cdots \int _{\gamma } dz_n \, f(z_1, \ldots , z_n). \end{aligned}$$

(A.5)

Further, we sometimes drop off the word “n-th” if it is clear from the context.

Integration with a divisor.   Let $\omega$ be a meromorphic differential on some domain U in $\mathbb {P}^1$ of the second kind. Following [5], for a divisor $D(z; \underline{\nu }) = [z] - \sum _{k = 1}^m \nu _k [p_k]$, where $z, p_1, p_2, \ldots , p_m \in \mathbb {P}^1 {\setminus } \{\text {poles of} \omega \}$, and $\underline{\nu } = (\nu _k)_{k=1}^{m}$ being a tuple of complex numbers satisfying $\sum _{k = 1}^m \nu _k = 1$, we define the integral of $\omega$ with $D(z; \underline{\nu })$ by

$$\begin{aligned} \int _{D(z; \nu )} \omega := \sum _{k = 1}^m \nu _k \int ^z_{p_k} \omega . \end{aligned}$$

(A.6)

Because $\omega$ is of the second kind, this integral does not depend on the choice of paths in U from $p_j$ to z ($j = 1, 2, \cdots , m$). For a multidifferential $\Omega = \Omega (z_1, z_2, \cdots , z_n)$ of the second kind, we define

$$\begin{aligned} \int _{z_j \in D(z; \underline{\nu })} \Omega (z_1, \ldots , z_j, \ldots z_n) := \sum _{k = 1}^m \nu _k \int ^{z_j = z}_{z_j = p_k} \Omega (z_1, \cdots , z_j, \ldots , z_n). \end{aligned}$$

(A.7)

As in (A.5), we write the multiple integral as:

$$\begin{aligned} \underbrace{ \int _{D(z; \underline{\nu })} \cdots \int _{D(z; \underline{\nu })} }_{n-\text {th}} \Omega (z_1, \ldots , z_n) := \int _{z_1 \in D(z; \underline{\nu })} dz_1 \cdots \int _{z_n \in D(z; \underline{\nu })} dz_n \, f(z_1, \ldots , z_n) \end{aligned}$$

(A.8)

for a symmetric meromorphic multidifferential of the second kind.

Pull-back.   Finally, for a holomorphic map $\phi$ from some domain in $\mathbb {P}^1$ to $\mathbb {P}^1$, we write the pullback of $\Omega$ by $\phi$ with respect to the j-th variable as:

$$\begin{aligned} \Omega (z_1, \ldots , \phi (z_j), \ldots , z_n) := f(z_1, \ldots , \phi (z_j), \ldots , z_n) dz_1 \cdots d\phi (z_j) \cdots dz_n. \end{aligned}$$

We frequently use this expression mainly when $\phi$ is conjugate map defined near a ramification point.

Ineffectiveness of Ramification Points

Following [5], we define R as a set of ramification points of x(z), not as a set of zeros of dx(z) as in [9]. However, this modification does not cause difference when the ramification points are ineffective (in the sense of Definition 2.7). Here, we give a criterion for the ineffectiveness of ramification points (cf. Proposition 2.8).

Proposition A.1

For a ramification point r, the followings are equivalent:

• (a) the correlation function $W_{g,n}(z_1, \ldots , z_n)$ with $(g,n) \ne (1,0)$ is holomorphic at $z_i = r$ for each $i = 1,\cdots , n$ (i.e., r is an ineffective ramification point).

• (b) The differential $(y(z) - y(\overline{z})) dx(z)$ has a pole at r.

Proof

First let us give a remark on the pole order of correlation functions at a ramification point satisfying the above condition.

Lemma A.2

If $(y(z) - y(\overline{z})) dx(z)$ has a pole at a ramification point r, then the pole order of $(y(z) - y(\overline{z})) dx(z)$ at $z=r$ is greater than or equal to two.

Proof of Lemma A.2

It is enough to prove that $(y(z) - y(\overline{z})) dx(z)$ never has a simple pole at ramification point. In other words, it suffices to prove that there is no $r \in R$ satisfying $\rho (x(r); P) = -2$ (cf. Proposition 3.3).

Suppose for contradiction that a point $r \in R$ satisfies $\rho (x(r); P) = -2$.

We also assume that $x(r) \ne \infty$ for simplicity. (The case $x(r) = \infty$ can be treated by a similar way.) Then, the function $Q_0(x)$ defined by (3.9) has an expression

$$\begin{aligned} Q_0(x) = \frac{c_0}{(x - x(r))^2} (1 + f(x)), \end{aligned}$$

where $c_0$ is a nonzero constant, and f(x) is a rational function of x which vanishes at $x=x(r)$. Taking a square root with an appropriate branch, we obtain

$$\begin{aligned} y(z) - y(\overline{z}) = \frac{2\sqrt{c_0}}{x(z) - x(r)} (1 + f(x(z)))^{1/2}. \end{aligned}$$

Since the left-hand side is anti-invariant under the involution $z \mapsto \overline{z}$, the above equality and the relation $x(z) = x(\overline{z})$ imply

$$\begin{aligned} \frac{1}{x(z) - x(r)} = - \frac{1}{x(z) - x(r)} \end{aligned}$$

which leads a contradiction. This proves that no $r \in R$ satisfying $\rho (x(r); P) = -2$. $\square$

Remark A.3

By a similar argument presented in the proof of Lemma A.2, we can also show that there is no $r \in R$ satisfying $\rho (x(r); P) = -2m$ for some $m \ge 1$.

Now let us prove Proposition A.1.

Let us assume (b), that is, $(y(z) - y(\overline{z})) dx(z)$ has a pole at r, and look at the behavior of $W_{0,3}(z_0,z_1,z_2)$ and $W_{1,1}(z_0)$ when $z_0$ approaches to r. By deforming the residue contour around r, we can decompose the contribution of residue at $z=r$ to $W_{0,3}(z_0,z_1,z_2)$ as:

(B.1)

Here, $C_{r,z_0, \overline{z_0}}$ is a contour satisfying

$$\begin{aligned} \{ \text {the domain bounded by} \quad C_{r,z_0, \overline{z_0}} \} \cap R \cap \{z_0, \overline{z_0}, \ldots , z_n , \overline{z_n} \} = \{r, z_0, \overline{z_0} \}, \end{aligned}$$

and the other contours $C_{z_0}$ and $C_{\overline{z_0}}$ are defined similarly. Then, the first integral in the right-hand side of (B.1) is holomorphic at $z_0 = r$, while the other two integrals are evaluated as:

$$\begin{aligned}&- \frac{1}{2 \pi i} \left( \oint _{z \in C_{z_0}} + \oint _{z \in C_{\overline{z_0}}} \right) K_r(z_0,z) \bigl ( B(z,z_1)B(\overline{z},z_2) + B(\overline{z},z_1)B(z,z_2) \bigr ) \nonumber \\&\quad = \frac{1}{(y(z_0) - y(\overline{z_0})) dx(z_0)} \bigl ( B(z_0,z_1)B(\overline{z_0},z_2) + B(\overline{z_0},z_1)B(z_0,z_2) \bigr ). \end{aligned}$$

(B.2)

Therefore, $W_{0,3}(z_0,z_1,z_2)$ is holomorphic at $z_0 = r$ (and hence, it is holomorphic at $z_i = r$ for $i=1,2$ as well) under the assumption on the pole property of $( y(z) - y(\overline{z}) )dx(z)$ at r.

On the other hand, the behavior of $W_{1,1}(z_0)$ when $z_0$ approaches to r is described in a similar manner:

(B.3)

Then, we can conclude that $W_{1,1}(z_0)$ is holomorphic at $z_0 = r$ because $(y(z_0) - y(\overline{z_0})) dx(z_0)$ has a double or higher-order pole at r (cf. Lemma A.2) while $B(z_0,\overline{z_0})$ has a double pole there.

Using the induction on $2g-2+n$, we can conclude that the correlation functions $W_{g,n}$ are holomorphic at r with respect to each variable $z_i$, as follows. For general (g, n), the contribution of the residue at $z=r$ to $W_{g,n+1}(z_0, z_1, \ldots , z_n)$ is given by the following form:

(B.4)

Although we omit an explicit expression of $F_{g,n}(z,\overline{z},z_1,\cdots ,z_n)$ (which can be read off from (2.21)), we know it has at most double pole at $z=r$ under the induction hypothesis. Then, by the similar argument for $W_{1,1}(z_0)$ presented above, we can verify that the right-hand side of (B.4) is holomorphic at $z_0 = r$. Thus, we have verified that r is ineffective.

Conversely, let us assume (a). Then, it follows from the definition of ineffectiveness that the correlation functions $W_{0,3}(z_0,z_1,z_2)$ must be holomorphic at $z_0 = r$. Then, in view of (B.2), we can conclude that the differential $(y(z_0) - y(\overline{z_0})) dx(z_0)$ must have a pole at $z_0 = r$. (Otherwise, the right-hand side of (B.2) never becomes holomorphic at $z_0 = r$.)

Thus, we have proved the equivalence between the conditions (a) and (b). This also completes the proof of Proposition 2.8 (i). $\square$

The remaining task for a proof of Proposition 2.8 is to show

Proposition A.4

If r is an ineffective ramification point, then the residue at r in (2.21) becomes zero.

Proof

Since $\int ^{\zeta =z}_{\zeta =\overline{z}}B(z_0,\zeta )$ is holomorphic and vanishes at $z=r$, we can verify that $K_r(z_0,z)$ has a double (or more higher order) zero at $z = r$ if r is ineffective (cf. Proposition A.1). Therefore, $K_r(z_0,z) F_{g,n}(z,\overline{z},z_1,\cdots ,z_n)$ in (B.4) is holomorphic and has no residue at $z = r$. This completes the proof. $\square$