2. Matrix integrals

Let $H_{N}$ be the real vector space of self-adjoint $N\times N$ -matrices and $(E_1,\dots, E_{N})$ be not necessarily distinct positive real numbers. By the Bochner–Minlos theorem [Reference Bochner6], combined with the Schur product theorem [Reference Schur27, section 4], there is a unique probability measure $d\mu_0(\Phi)$ on the dual space $H^{\prime}_{N}$ with

(2.1) \begin{align}\exp\!\Bigg({-} \frac{1}{2N} \sum_{k,l=1}^{N}\frac{M_{kl}M_{lk}}{E_k+E_l}\Bigg)= \int_{H^{\prime}_{N}} d\mu_0(\Phi) \,e^{\textrm{i} \Phi(M)}\end{align}

for any $M=M^*=\sum_{k,l=1}^{N} M_{kl} e_{kl}\in H_{N}$ , where $(e_{kl})$ is the standard matrix basis. The linear forms extend via $\Phi(M_1+\textrm{i}M_2)\,:\!=\,\Phi(M_1)+\textrm{i}\Phi(M_2)$ to arbitrary complex $N\times N$ -matrices. This allows us to evaluate $\Phi(e_{jk})$ and to identify the covariance

\begin{equation*}\int_{H^{\prime}_{N}} d\mu_0(\Phi) \,\Phi(e_{jk})\Phi(e_{lm}) = \frac{\delta_{kl} \delta_{jm}}{N \!\left(E_j+E_k\right)}\;.\end{equation*}

We are going to deform the Gaußian measure (2.1) by a quartic potential,

(2.2) \begin{align}d\mu_\lambda(\Phi) &\,:\!=\,\frac{d\mu_0(\Phi)\; \mathcal{P}_4(\Phi,\lambda)}{\int_{H^{\prime}_{N}} d\mu_0(\Phi)\;\mathcal{P}_4(\Phi,\lambda)}\;,\qquad\\\mathcal{P}_4(\Phi,\lambda)&=\exp\!\Bigg({-}\,\frac{\lambda N}{4} \textrm{Tr}\big(\Phi^4\big)\Bigg)\,:\!=\,\exp\!\Bigg({-}\,\frac{\lambda N}{4} \!\!\sum_{j,k,l,m=1}^{N} \!\!\!\!\! \Phi(e_{jk})\Phi(e_{kl})\Phi(e_{lm})\Phi\big(e_{mj}\big)\Bigg)\;,\nonumber \end{align}

for some $\lambda>0$ . This matrix measure is the quartic analogue of the Kontsevich model [Reference Kontsevich20] in which the deformation is given by the cubic term

\begin{equation*}\mathcal{P}_3(\Phi,\lambda)=\exp\!\bigg({-}\,\frac{\lambda N}{3} \textrm{Tr}\big(\Phi^3\big)\bigg)\,:\!=\,\exp\!\Bigg({-}\frac{\lambda N}{3}\sum_{k,l,m=1}^{N} \Phi(e_{kl})\Phi(e_{lm})\Phi(e_{mk})\Bigg)\:.\end{equation*}

The cubic measure was designed to prove Witten’s conjecture [Reference Witten33] that intersection numbers of tautological characteristic classes on the moduli space of stable complex curves are related to the KdV hierarchy. Kontsevich proved that $\log \int_{H^{\prime}_N} d\mu_0(\Phi)\;\mathcal{P}_3(\Phi, {\textrm{i}}/{2})$ , viewed as function of $t_k=-(2k-1)!!({1}/{N})\sum_{j=1}^{N} E_j^{-(2k+1)}$ , is the generating function of these intersection numbers.

We are interested in moments of the measure (2.2),

(2.3) \begin{align}\langle e_{k_1l_1}\dots e_{k_nl_n}\rangle &\,:\!=\, \int_{H^{\prime}_{N}} \!d\mu_\lambda(\Phi)\;\Phi\big(e_{k_1l_1}\big) \ldots \Phi\big(e_{k_nl_n}\big)= \frac{1}{\textrm{i}^{n}}\frac{\partial^n\mathcal{Z}(M)}{\partial M_{k_1l_1}\ldots \partial M_{k_nl_n}} \Big|_{M=0}\;,\nonumber\\\mathcal{Z}(M)&= \int_{H^{\prime}_N}\! d\mu_\lambda(\Phi) \;e^{\textrm{i}\Phi(M)}\;.\end{align}

As explained in Appendix A (see also [Reference McCullagh23, Reference Speed30]), the moments (2.3) decompose into cumulants

(2.4) \begin{align}\Bigg\langle \prod_{i=1}^n e_{k_il_i}\Bigg\rangle=\sum_{\substack{\text{partitions} \\ \text{$\pi$ of $\{1,\dots, n\}$}}} \prod_{\text{blocks $\beta \in \pi$}}\Bigg\langle \prod_{i\in \beta} e_{k_i l_i} \Bigg\rangle_c\;.\end{align}

For a quartic potential (2.2), moments and cumulants are only non-zero if n is even and every block $\beta$ is of even length. The structure of the Gaußian measure (2.1) (together with the invariance of a trace under cyclic permutations) implies that $\big\langle e_{k_1l_1}\dots e_{k_nl_n}\big\rangle_c$ is only non-zero if $(l_1,\dots,l_n)=\big(k_{\sigma(1)},\dots,k_{\sigma(n)}\big)$ is a permutation of $(k_1,\dots,k_n)$ , and in this case the cumulant only depends on the cycle type of this permutation $\sigma$ in the symmetric group $\mathcal{S}_n$ (see Appendix A, with $b\geqslant 1$ the number of cycles of length $n_i>0$ , $n_1+\ldots +n_b=n$ ):

(2.5) \begin{align}N^{n_1+\dots+n_b}\Big\langle \Big(e_{k_1^1k_2^1}e_{k_2^1k_3^1} \ldots e_{k_{n_1}^1k_1^1}\Big) \ldots\Big(e_{k_1^bk_2^b} e_{k_2^bk_3^b} \ldots e_{k_{n_b}^bk_1^b}\Big) \Big\rangle_c\nonumber\\\,=\!:\,N^{2-b} G_{\big|k_1^1\dots k_{n_1}^1\big|\dots\big|k_1^b\dots k_{n_b}^b\big|} \;.\end{align}

To correctly identify the cycles of the permutation it is necessary that all $k^i_j$ are pairwise different in (2.5). These N-rescaled cumulants (2.5) are further expanded as formal power series $G_{\dots}=\sum_{g=0}^\infty N^{-2g} G_{\dots}^{(g)} $ in $N^{-2}$ so that

(2.6) \begin{align}N^{n_1+\dots+n_b} \Big\langle \Big(e_{k_1^1k_2^1}e_{k_2^1k_3^1} \ldots e_{k_{n_1}^1k_1^1}\Big) \ldots\Big(e_{k_1^bk_2^b} e_{k_2^bk_3^b} \ldots e_{k_{n_b}^bk_1^b}\Big) \Big\rangle_c \nonumber\\= \sum_{g=0}^\infty N^{2-b-2g} \cdot G^{(g)}_{\big|k_1^1\dots k_{n_1}^1\big|\dots\big|k_1^b\dots k_{n_b}^b\big|} \;.\end{align}

It turns out that this grading (g, b) of $G^{(g)}_{\big|k_1^1\dots k_{n_1}^1\big|\dots \big|k_1^b\dots k_{n_b}^b\big|}$ fits with the combinatorics of ribbon graphs (with 4-valent vertices) on a connected oriented compact topological surface of genus $g\geqslant 0$ with $b\geqslant 1$ boundary components (and $n_i$ labels on the $i{\text{th}}$ boundary component) and Euler characteristic $\chi=2-2g-b$ (see e.g. [Reference Grosse and Wulkenhaar14, section 3] for the particular case of 4-valent vertices, and compare also with [Reference Kontsevich20] or [Reference Eynard11, sections 2 and 6]). Note that the moments are related to ribbon graphs on possibly non-connected oriented compact topological surfaces (see e.g. [Reference Lando and Zvonkin22, section 3, proposition 3.8.3]).

Starting point for the investigation of cumulants are equations of motion for $\mathcal{Z}(M)$ :

Lemma 2.1. The Fourier transform $\mathcal{Z}(M)$ of the measure (2.2) satisfies

(2.7) \begin{align}\frac{1}{\textrm{i}}\frac{\partial \mathcal{Z}(M)}{\partial M_{ab}}= \frac{\textrm{i} M_{ba}\mathcal{Z}(M)}{N(E_a+E_b)}-\frac{\lambda }{\textrm{i}^3(E_a+E_b)}\sum_{k,l=1}^{N} \frac{\partial^3\mathcal{Z}(M)}{\partial M_{ak}\partial M_{kl}\partial M_{lb}}\;.\end{align}

Proof. This follows from basic properties of the Gaußian measure (2.1). The derivative $({1}/{\textrm{i}})({\partial}/{\partial M_{ab}})$ applied to $\mathcal{Z}(M)$ produces a factor $\Phi(e_{ab})$ under the integral. Moments of $d\mu_0(\Phi)$ are by (2.2) a sum over pairings. This means that $\Phi(e_{ab})$ is paired in all possible ways with a $\Phi(e_{cd})$ contained in $\exp\!(\textrm{i} \Phi(M))$ or in $\mathcal{P}_4(\Phi,\lambda)$ . Every such pair contributes a factor $\delta_{ad}\delta_{bc}/\big(N\big(E_a+E_b\big)\big)$ , and summing over all pairings is the same as taking the derivative, thus producing a term

\begin{equation*}\frac{1}{N(E_a+E_b)}\Bigg(\textrm{i} M_{ba}-\lambda N\sum_{k,l=1}^{N} \Phi(e_{ak})\Phi(e_{kl})\Phi(e_{lb})\Bigg)\end{equation*}

under the integral. The triple product of $\Phi(e_{..})$ is written as a third derivative with respect to the corresponding entries of M.

The Kontsevich model [Reference Kontsevich20] with cubic deformation $\mathcal{P}_3(\Phi,\lambda)$ is governed by the equation of motion

\begin{equation*}\frac{1}{\textrm{i}}\frac{\partial \mathcal{Z}(M)}{\partial M_{ab}} = \frac{\textrm{i} M_{ba}\mathcal{Z}(M)}{N(E_a+E_b)} -\frac{\lambda }{\textrm{i}^2 (E_a+E_b)} \sum_{k=1}^{N} \frac{\partial^2 \mathcal{Z}(M)}{\partial M_{ak} \partial M_{kb}} \:.\end{equation*}

For $N=1$ this is essentially the ODE

\begin{equation*}f^{\prime\prime}(x)+2c f^{\prime}(x)=x f(x)\end{equation*}

solved by the Airy function $f(x)=e^{-cx} \textrm{Ai}(x+c^2)$ , hence the title of [Reference Kontsevich20]. Its quartic analogue is the matrix version of the ODE

\begin{equation*}f^{\prime\prime\prime}(x)+3c f^{\prime}(x)=x f(x)\:,\end{equation*}

which does not seem to have a name. The Airy function is the case $p=2$ of a larger class

\begin{equation*}\textrm{Ai}_p(x)=\frac{1}{2\pi} \int_{-\infty}^\infty dt \; \exp\left( \mathrm{i} \left(\frac{t^{p+1}}{p+1}+xt\right)\right)\end{equation*}

of higher Airy functions. As remarked in [Reference Kontsevich20, section 4.3], they also give rise to higher matrix Airy functions. In particular, there is also a ‘quartic analogue’ $p=3$ in this class, which was studied in [Reference Itzykson and Zuber19, Reference Kristjansen21]. This matrix model does not seem to be related to our ‘quartic analogue’ of the Kontsevich model.

Another equation of motion will be necessary for the subsequent work in [Reference Branahl, Hock and Wulkenhaar3]:

Lemma 2.2. The Fourier transform $\mathcal{Z}(M)$ of the measure (2.2) satisfies

(2.8) \begin{align}\frac{1}{N} \frac{\partial \mathcal{Z}(M)}{\partial E_{a}}= \Bigg(\sum_{k=1}^N\frac{\partial^2}{\partial M_{ak} \partial M_{ka}}+ \frac{1}{N}\sum_{k=1}^N G_{|ak|}+\frac{1}{N^2}G_{|a|a|}\Bigg)\mathcal{Z}(M)\;.\end{align}

Proof. Application of $(1/N)(\partial/\partial E_a)-\!\sum_{k=1}^N \partial^2/\big(\partial M_{ak} \partial M_{ka}\big)-(1/N)\sum_{k=1}^N\! 1/\big(E_a+E_k\big)$ to the left-hand side of (2.1) yields zero so that it gives

\begin{align*}\frac{1}{N}\frac{\partial}{\partial E_a}\big(d\mu_0(\Phi)\big)= d\mu_0(\Phi)\sum_{k=1}^N \bigg(\frac{1}{N(E_a+E_k)}-\Phi(e_{ka})\Phi(e_{ak})\bigg)\end{align*}

when applying it to the right-hand side. Apply this identity to (2.2) to get

\begin{align*}\frac{1}{N} \frac{\partial}{\partial E_a}\big(d\mu_\lambda(\Phi)\big) &= d\mu_\lambda(\Phi)\sum_{k=1}^N \bigg(\frac{1}{N(E_a+E_k)}-\Phi(e_{ka})\Phi(e_{ak})\bigg)\nonumber\\& \quad- d\mu_\lambda(\Phi) \!\! \int_{H^{\prime}_{N}} d\mu_\lambda(\Phi) \sum_{k=1}^N \bigg(\frac{1}{N(E_a+E_k)}-\Phi(e_{ka})\Phi(e_{ak})\bigg)\;.\end{align*}

Multiplying with $e^{\textrm{i}\Phi(M)}$ and integrating over $H^{\prime}_N$ gives with (A3) the assertion.

The equations of motion (2.7) and (2.8) induce identities between cumulants. Some of them are derived in Appendix B, for others see [Reference Grosse and Wulkenhaar17]. Taking also the grading by (g, b) into account, one can establish a partial order in the homogeneous building blocks $G^{(g)}_{\dots}$ . The least element is the planar two-point function $G_{|ab|}^{(0)}$ , which is the dominant part (at large N) of the cumulant of length 2 and cycle type (0, 1) (i.e. one cycle ab of length 2). It satisfies a closed non-linear equation for it alone, given in (3.1) below. Any other homogeneous building block of (2.6) satisfies an affine equation with inhomogeneity that depends only on functions of strictly larger topological Euler characteristic $\chi=2-2g-b$ , which are known by induction. Similar recursive systems have been identified in many areas of mathematics. Their common universal structure has been axiomatised under the name topological recursion [Reference Eynard and Orantin10], since the recursion is by the topological Euler characteristic. Starting from a few initial data called the spectral curve, topological recursion constructs a hierarchy of differential forms and understands them as spectral invariants of the curve. A prominent example is the Kontsevich model [Reference Kontsevich20] whose topological recursion is described e.g. in [Reference Eynard11, section 6]. Other classes of examples are the one- and two-matrix models [Reference Chekhov, Eynard and Orantin8], Mirzakhani’s recursions [Reference Mirzakhani24] for the volume of moduli spaces of Riemann surfaces, and recursions in Hurwitz theory [Reference Bouchard and Mariño5] and Gromov–Witten theory [Reference Bouchard, Klemm, Mariño and Pasquetti4].

3. The planar two-point function

The two-point function $G_{|ab|}$ is the cumulant of length 2 and cycle type (0, 1) (i.e. one cycle ab of length 2), see Appendix A. We reprove in Appendix B that the planar two-point function $G^{(0)}_{|ab|}$ (of degree or genus $g=0$ ) satisfies

(3.1) \begin{align}\Bigg(E_a+E_b+\frac{\lambda}{N}\sum_{k=1}^{N} G^{(0)}_{|ak|}\Bigg)G^{(0)}_{|ab|}= 1+ \frac{\lambda}{N} \sum_{k=1}^{N}\frac{G^{(0)}_{|kb|}-G^{(0)}_{|ab|}}{E_k-E_a}\;.\end{align}

This equation was first established in [Reference Grosse and Wulkenhaar16]; equation (B7) which involves all $G^{(g)}_{|ab|}$ was obtained in [Reference Grosse and Wulkenhaar17].

To give a meaning to the term $k=a$ in (3.1) we make the decisive assumption that $\Big\{G^{(0)}_{|ab|}\Big\}_{a,b=1,\dots N}$ arise by evaluation of a holomorphic function in two complex variables. Let $E_1,\dots,E_d$ be the distinct entries in the tuple $(E_k)$ , which occur with multiplicities $r_1,\dots,r_d$ , with $N=r_1+\dots+r_d$ . We assume that for some neighbourhoods $\mathcal{U}_k\subset \mathbb{C}$ of $E_k$ there is a holomorphic function $G^{(0)}\,:\, \bigcup_{k,l=1}^d \!(\mathcal{U}_k\times \mathcal{U}_l)\to \mathbb{C}$ which interpolates $G^{(0)}_{|ab|}=G^{(0)}(E_a,E_b)$ and satisfies the natural (but by no means unique) holomorphic extension

(3.2) \begin{align}\Bigg(\zeta+\eta+\frac {\lambda}{N}\sum_{k=1}^d r_k G^{(0)}(\zeta,E_k)\Bigg)G^{(0)}(\zeta,\eta)&=1+\frac{\lambda}{N} \sum_{k=1}^d r_k\frac{G^{(0)}(E_k,\eta)-G^{(0)}(\zeta,\eta)}{E_k-\zeta}\end{align}

of (3.1), for $(\zeta,\eta)\in \bigcup_{k,l=1}^d \!(\mathcal{U}_k\times \mathcal{U}_l)$ . Equation (3.1) is understood as the limit $\zeta\to E_a$ and $\eta\to E_b$ of (3.2) when taking multiplicities into account. It is not possible to deduce (3.2) from (3.1) alone. Justification of (3.2) comes from the fact that it gives rise to interesting mathematical structures:

Theorem 3.1. Construct 2d functions $\{\varepsilon_k(\lambda),\varrho_k(\lambda)\}_{k=1,\dots,d}$ , with $\lim_{\lambda\to 0} \big(\varepsilon_k,\varrho_k\big)=\big(E_k,r_k\big)$ , as implicitly defined solution of the system

(3.3) \begin{align}E_k=R(\varepsilon_k)\;,\qquad r_k=\varrho_k R^{\prime}(\varepsilon_k)\;,\qquad \text{where}\quad R(z)=z-\frac{\lambda}{N}\sum_{j=1}^d \frac{\varrho_j}{\varepsilon_j+z}\;.\end{align}

Then (3.2) is solved by $G^{(0)}(\zeta,\eta)=\mathcal{G}^{(0)}\big(R^{-1}(\zeta),R^{-1}(\eta)\big)$ , where $\mathcal{G}^{(0)}\,:\,\hat{\mathbb{C}}\times \hat{\mathbb{C}} \to\hat{\mathbb{C}}$ is the rational function

(3.4) \begin{align}\mathcal{G}^{(0)}(z,w)&=\frac{1}{(R(w)-R({-}z))(R(z)-R({-}w))}\Bigg\{R(z)+R(w)\nonumber\\& \quad +\frac{\lambda}{N}\sum_{k=1}^d \bigg(\frac{r_k}{R(\varepsilon_k)-R(z)}+\frac{r_k}{R(\varepsilon_k)-R(w)}\bigg)\nonumber\\& \quad +\frac{\lambda^2}{N^2}\sum_{k,l=1}^d \frac{r_k r_l \mathcal{G}_{kl}}{(R(\varepsilon_k)-R(z))(R(\varepsilon_l)-R(w))}\Bigg\}\end{align}

with

(3.5) \begin{align}\mathcal{G}_{kl}=\frac{\displaystyle\Bigg(\prod_{j,m=1}^d \frac{\big({-}\widehat{\varepsilon_k}^j-\widehat{\varepsilon_l}^m\big)}{\varepsilon_j+\varepsilon_m}\Bigg)\left(\prod_{\substack{j=1\\j\neq k}}^d \frac{\varepsilon_k-\varepsilon_j}{R(\varepsilon_k){-}R\big(\varepsilon_j\big)}\right)\left(\prod_{\substack{m=1\\m\neq l}}^d \frac{\varepsilon_l-\varepsilon_m}{R(\varepsilon_l){-}R(\varepsilon_m)}\right)}{R^{\prime}(\varepsilon_k)R^{\prime}(\varepsilon_l) (\varepsilon_k+\varepsilon_l)} \;.\end{align}

Here $z\in \big\{u,\hat{u}^1,\dots,\hat{u}^d\big\}$ is the list of the different roots of $R(z)=R(u)$ , and the correct branch of $R^{-1}$ is chosen by the implicitly defined solutions above (i.e. $\varepsilon_k \in R^{-1}(\mathcal{U}_k)$ for this branch). In particular, $G^{(0)}_{|kl|}=\mathcal{G}^{(0)}(\varepsilon_k,\varepsilon_l)\equiv\mathcal{G}_{kl}$ solves (3.1).

Existence of $(\varepsilon_k(\lambda),\varrho_k(\lambda))$ in a neighbourhood of $\lambda=0$ is guaranteed by the implicit function theorem. We will prove several equivalent formulae for $\mathcal{G}^{(0)}(z,w)$ : (4.5), (4.9), (4.14) and eventually (3.4). Some of them were already proved in [Reference Grosse, Hock and Wulkenhaar12]. There, inspired by the solution of a particular case [Reference Panzer and Wulkenhaar26], equation (3.2) was interpreted as an integral equation for a Dirac measure. Approximating the Dirac measure by a Hölder-continuous function allowed to employ boundary values techniques for sectionally holomorphic functions. Residue theorem and Lagrange–Bürmann resummation gave a solution formula whose limit back to Dirac measure was arranged into (4.14).

In this paper we provide a more elementary proof of these equations which solely needs properties of Cauchy matrices established by Schechter [Reference Schechter28]:

Proposition 3.2 ([Reference Schechter28]). For two d-tuples $(a_1,\dots,a_d)$ and $(b_1,\dots,b_d)$ , with all $a_i,b_j$ distinct, consider the $d\times d$ -matrix $H=\big(\frac{1}{a_k-b_l}\big)_{kl}$ . Let $A(x)\,:\!=\,\prod_{i=1}^d\! (x-a_i)$ and $B(y)\,:\!=\,\prod_{j=1}^d\! (y-b_j)$ . Then the inverse of H is given by

(3.6) \begin{align}\big(H^{-1}\big)_{kl}= (a_l-b_k) A_l(b_k)\,B_k(a_l)\;,\end{align}

where $A_l,B_k$ are the Lagrange interpolation polynomials

(3.7) \begin{align}A_l(x)=\frac{A(x)}{(x-a_l) \,A^{\prime}(a_l)}\;,\qquad B_k(y)=\frac{B(y)}{(y-b_k) \,B^{\prime}(b_k)}\;.\end{align}

The inverse of H satisfies

(3.8) \begin{align}\frac{B_k(x)A(b_k)}{A(x)}=\sum_{l=1}^d \frac{\big(H^{-1}\big)_{kl}}{a_l-x}\;,\qquad \frac{A_l(x)B(a_l)}{B(x)}=\sum_{k=1}^d \frac{\big(H^{-1}\big)_{kl}}{x-b_k}\;.\end{align}

Moreover, the row sums and column sums of $H^{-1}$ are given by

(3.9) \begin{align}\sum_{j=1}^d \!\big(H^{-1}\big)_{kj}=-\frac{A(b_k)}{B^{\prime}(b_k)}\;,\qquad \sum_{i=1}^d \!\big(H^{-1}\big)_{il}=\frac{B(a_l)}{A^{\prime}(a_l)}\;,\end{align}

and one has, for all $j=1,\dots, d$ ,

(3.10) \begin{align}\sum_{k=1}^d \frac{A(b_k)}{\big(b_k-a_j\big) B^{\prime}(b_k)}=1\quad \text{and}\quad \sum_{l=1}^d \frac{B(a_l)}{\big(a_l-b_j\big) A^{\prime}(a_l)}=1\;.\end{align}

4. Proof of Theorem 3.1

We are going to construct a non-constant rational function $R\in\mathbb{C}(z)$ viewed as a branched cover $R\,:\,\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\to\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ of Riemann surfaces (with $z=\textrm{id}_{\mathbb{C}}$ the standard coordinate on $\mathbb{C}$ ) via the following:

With the properties (iv) and (vii) in this Ansatz 4.1 we turn (3.2) into

(4.2) \begin{align}\big(R(w)-R({-}z)\big)\mathcal{G}^{(0)}(z,w) =1+\frac{\lambda}{N} \sum_{k=1}^d r_k\frac{\mathcal{G}^{(0)}(\varepsilon_k,w)}{R(\varepsilon_k)-R(z)}\;,\end{align}

where $(z,w)\in \bigcup_{k,l=1}^d \!(\mathcal{V}_k\times\mathcal{V}_l)$ . Next, setting $z=-\hat{w}^l$ in (4.2) for $l=1,\dots,d$ and a given w close to some $\varepsilon_k$ , requirements (v) and (vi) of Ansatz 4.1 give (by $\infty\neq R\big({-}\hat{w}^l\big)$ and $\mathcal{G}^{(0)}\big({-}\hat{w}^l,w\big)$ is defined and finite) the d equations

(4.3) \begin{align}\frac{\lambda}{N}\sum_{k=1}^d r_k \frac{\mathcal{G}^{(0)}(\varepsilon_k,w)}{R\big({-}\hat{w}^l\big)-R(\varepsilon_k)}=1\;.\end{align}

This identifies $({\lambda}/{N}) r_k\mathcal{G}^{(0)}(\varepsilon_k,w)$ as row sums of the inverse of a Cauchy matrix. Setting $a_j=R\big({-}\hat{w}^j\big)$ and $b_i=R(\varepsilon_i)$ in the first identity (3.9) in Proposition 3.2 we conclude (since the $a_j,b_i$ for $j,i=1,\dots,d$ are pairwise distinct by requirement (v) of Ansatz 4.1):

Corollary 4.2. With Ansatz 4.1 one has

(4.4) \begin{align}\frac{\lambda}{N} r_k \mathcal{G}^{(0)}(\varepsilon_k,w)= -\frac{\prod_{j=1}^d \!\big(R(\varepsilon_k)-R\big({-}\hat{w}^j\big)\big)}{\prod_{j=1,j\neq k}^d \!\big(R(\varepsilon_k)-R\big(\varepsilon_j\big)\big)}\;.\end{align}

Inserted back into (4.2) expresses $\mathcal{G}^{(0)}(z,w)$ in terms of R. The result simplifies:

Lemma 4.3. With Ansatz 4.1 one has

(4.5) \begin{align}\mathcal{G}^{(0)}(z,w)=\frac{1}{(R(w)-R({-}z))} \prod_{j=1}^d\frac{R(z)-R\big({-}\hat{w}^j\big)}{R(z)-R\big(\varepsilon_j\big)}\;.\end{align}

Proof. This follows from (3.10) for $(d+1)$ -tuples with index 0 prepended. Setting $b_0=R(z)$ , $b_k=R(\varepsilon_k)$ , $a_l=R\big({-}\hat{w}^l\big)$ for $k,l=1,\dots,d$ , then the case $j=0$ of the first identity (3.10) reads (independent of $a_0$ )

(4.6) \begin{align}\frac{\prod_{j=1}^d \!\big(R(z)-R\big({-}\hat{w}^j\big)\big)}{\prod_{j=1}^d \!\big(R(z)-R\big(\varepsilon_j\big)\big)}+ \sum_{k=1}^d \frac{\prod_{j=1}^d \!\big(R(\varepsilon_k)-R\big({-}\hat{w}^j\big)\big)}{(R(\varepsilon_k)-R(z))\prod_{j=1,j\neq k}^d \!\big(R(\varepsilon_k)-R\big(\varepsilon_j\big)\big)}=1\;.\end{align}

Equation (4.5) results from this identity when inserting (4.4) into (4.2).

Lemma 4.4. With Ansatz 4.1 the rational function $R\in\mathbb{C}(z)$ is necessarily given by

(4.7) \begin{align}R(z)=z+c_0-\frac{\lambda}{N}\sum_{k=1}^d \frac{\varrho_k}{\varepsilon_k+z}\quad \text{for some } c_0\in \mathbb{C}\;,\qquad \varrho_k=\frac{r_k}{R^{\prime}(\varepsilon_k)}\;.\end{align}

Proof. Comparing the limit $z\to \varepsilon_k$ of (4.5) with (4.4) shows that R has a simple pole at every $-\varepsilon_k$ with

\begin{equation*}{R^\prime }({\varepsilon _k})\mathop {{\rm{RES}}}\limits_{z = - {\varepsilon _k}} \,R(z)dz = - \frac{{\lambda {r_k}}}{N} \ne 0\:.\end{equation*}

By construction, R has also a pole at $\infty$ . Since R has degree $d+1$ by (i) in Ansatz 4.1, $\{-\varepsilon_1,\dots, -\varepsilon_d,\infty\}$ is already the complete list of poles (i.e. preimages of $\infty$ ) of R. Moreover, the pole at $\infty$ has to be simple with $\lim_{z\to\infty} {R(z)}/{z}=1$ by (iii) in Ansatz 4.1. Therefore, $R(z)-z+ ({\lambda}/{N}) \sum_{k=1}^d {\varrho_k}/({\varepsilon_k+z})$ is a bounded holomorphic function on $\hat{\mathbb{C}}$ , which by Liouville’s theorem is a constant $c_0$ .

Corollary 4.5. For $u\in \bigcup_{j=1}^d\mathcal{V}_j$ one has an equality of rational functions in z:

(4.8) \begin{align}R(z)-R(u)=(z-u) \prod_{k=1}^d \frac{z-\hat{u}^k}{z+\varepsilon_k}\;,\end{align}

where $\hat{u}^k$ are the other preimages of R(u) under R.

Proof. Inserting (4.8) into (4.5) gives for $z,w\in \bigcup_{j=1}^d\mathcal{V}_j$ :

(4.9) \begin{align}\mathcal{G}^{(0)}(z,w)&=\frac{\prod_{k=1}^d\!(\varepsilon_k-z)}{(z+w) \prod_{k=1}^d \!\big({-}z-\hat{w}^k\big)}\prod_{k=1}^d \frac{\displaystyle\frac{\big(z+\hat{w}^k\big)\prod_{l=1}^d \!\big({-}\hat{w}^k-\hat{z}^l\big)}{\prod_{l=1}^d \!\big(\varepsilon_l-\hat{w}^k\big)} }{\displaystyle \frac{(z-\varepsilon_k)\prod_{l=1}^d \!\big(\varepsilon_k-\hat{z}^l\big)}{\prod_{l=1}^d(\varepsilon_k+\varepsilon_l)}}\nonumber\\[-1ex]&= \frac{1}{(z+w)} \prod_{k,l=1}^d\frac{(\varepsilon_k+\varepsilon_l)\big({-}\hat{w}^k-\hat{z}^l\big)}{\big(\varepsilon_k-\hat{z}^l\big)\big(\varepsilon_l-\hat{w}^k\big)}\nonumber\\&= \frac{1}{(z+w)} \prod_{k,l=1}^d\frac{\big({-}\hat{w}^k-\hat{z}^l\big)}{(\varepsilon_k+\varepsilon_l)}\frac{(\varepsilon_k-z)}{(R(\varepsilon_k)-R(z))}\frac{(\varepsilon_l-w)}{(R(\varepsilon_l)-R(w))}\;.\end{align}

The limit $z\to \varepsilon_k$ and $w\to \varepsilon_l$ gives $\mathcal{G}^{(0)}(\varepsilon_k,\varepsilon_l)=\mathcal{G}_{kl}$ .

We prove that (ii), (iv), (v), (vi) and (vii) of Ansatz 4.1 are automatic. We start with

Proof. With Lemma 4.4 and Lemma 4.3, both a consequence of Ansatz 4.1, each side of (4.1) is a rational function, and all poles are simple. For the term $ {r_k}/({R(\varepsilon_k)-R(z)})$ this follows from the assumption (ii) of Ansatz 4.1. We show that both sides of (4.1) have the same simple poles with the same residues. Then by Liouville’s theorem their difference is a constant, which is easy to control.

First, it follows from (4.7) and (4.5) that both sides of (4.1) approach z for $z\to \infty$ . Near $\infty$ the difference between both sides of (4.1) is $\pm 2c_0$ , which shows that $c_0=0$ in (4.7) is necessary.

Next, (4.7) shows that the only other poles of the right-hand side of (4.1) are simple and located at $z=\varepsilon_k$ with residue $-({\lambda}/{N}) \varrho_k$ . The same simple poles with the same residues are produced by $({\lambda}/{N}) \sum_{k=1}^d {r_k}/({R(\varepsilon_k)-R(z)})$ on the left-hand side, taking $r_k/R^{\prime}(\varepsilon_k)=\varrho_k$ into account.

But the left-hand side of (4.1) could also have poles at $z=-\varepsilon_j$ and $z=\widehat{\varepsilon_m}^j$ (see (4.7) and (4.9)). We have $\textrm{Res}_{z= -\varepsilon_j}\ \ R(z)dz= - ({\lambda}/{N}) \varrho_j$ . Setting $w\mapsto \varepsilon_l$ in (4.5), then with $\lim_{z\to -\varepsilon_j} \big({R(z)-R\big({-}\widehat{\varepsilon_l}^k\big)}\big)/({ R(z)-R(\varepsilon_k)})=1$ for any k, l (here (v) of Ansatz 4.1 is used) one easily finds that $\mathcal{G}^{(0)}({-}\varepsilon_j,\varepsilon_l)$ is finite for $j\neq l$ and that $\textrm{Res}_{z= -\varepsilon_j} ({\lambda}/{N})r_j\mathcal{G}^{(0)}\big(z,\varepsilon_j\big)dz= ({\lambda}/{N})\big({r_j}/{R^{\prime}\big(\varepsilon_j\big)}\big)$ , which thus cancels $\textrm{Res}_{z= -\varepsilon_j} \ \ R(z)dz= -({\lambda}/{N}) \varrho_j$ .

Finally, from (4.5) we conclude

\begin{align*}\mathop{Res}\limits_{z= \widehat{\varepsilon_m}^j}\mathcal{G}^{(0)}\big(z,\varepsilon_k\big)dz&=\frac{1}{\big(R\big(\varepsilon_k\big)-R\big({-}\widehat{\varepsilon_m}^j\big)\big) R^{\prime}\big(\widehat{\varepsilon_m}^j\big)}\frac{\prod_{i=1}^d\!\big(R(\varepsilon_m)-R\big({-}\widehat{\varepsilon_k}^i\big)\big)}{\prod_{i=1,i\neq m }^d \!\big(R(\varepsilon_m)-R(\varepsilon_i)\big)}\\&=\frac{1}{\big(R\big(\varepsilon_k\big)-R\big({-}\widehat{\varepsilon_m}^j\big)\big) R^{\prime}\big(\widehat{\varepsilon_m}^j\big)}\Big( - \frac{\lambda}{N} r_m\mathcal{G}^{(0)}\big(\varepsilon_m,\varepsilon_k\big)\Big)\\&= \frac{r_m}{r_k R^{\prime}\big(\widehat{\varepsilon_m}^j\big)}\frac{1}{\big(R\big(\varepsilon_k\big)-R\big({-}\widehat{\varepsilon_m}^j\big)\big) }\Big( - \frac{\lambda}{N} r_k\mathcal{G}^{(0)}\big(\varepsilon_k,\varepsilon_m\big)\Big)\\&= \frac{r_m}{r_k R^{\prime}\big(\widehat{\varepsilon_m}^j\big)}\frac{1}{\big(R\big(\varepsilon_k\big)-R\big({-}\widehat{\varepsilon_m}^j\big)\big)}\frac{\prod_{i=1}^d\!\big(R\big(\varepsilon_k\big)-R\big({-}\widehat{\varepsilon_m}^i\big)\big)}{\prod_{i=1,i\neq k }^d \!\big(R\big(\varepsilon_k\big)-R(\varepsilon_i)\big)}\;,\end{align*}

where (4.4), the symmetry $\mathcal{G}^{(0)}(\varepsilon_m,\varepsilon_k)=\mathcal{G}^{(0)}(\varepsilon_k,\varepsilon_m)$ and again (4.4) have been used. The first identity (3.10) for $b_k=R(\varepsilon_k)$ and $a_j=R\big({-}\widehat{\varepsilon_m}^j\big)$ gives

\begin{equation*}\mathop{Res}\limits_{z= \widehat{\varepsilon_m}^j}\sum_{k=1}^d r_k \mathcal{G}^{(0)}(z,\varepsilon_k)dz=\frac{r_m}{R^{\prime}\big(\widehat{\varepsilon_m}^j\big)}\;,\end{equation*}

which precisely cancels ${Res}_{z= \widehat{\varepsilon_m}^j}\sum_{k=1}^d\frac{r_k}{R(\varepsilon_k)-R(z)}dz= -\frac{r_m}{R^{\prime}\big(\widehat{\varepsilon_m}^j\big)}$ .

Let us consider now the rational function

(4.10) \begin{equation}R(z)=z-\frac{\lambda}{N}\sum_{k=1}^d \frac{\varrho_k}{\varepsilon_k+z}\end{equation}

from equation (4.7) with $c_0=0$ . We are interested in the real and complex solutions $\{\varepsilon_k,\varrho_k\}_{k=1,\dots,d}$ (depending on $\lambda$ ) of the 2d equations

(4.11) \begin{align}0&=R(\varepsilon_l)-E_l=\varepsilon_l -E_l -\frac{\lambda}{N}\sum_{k=1}^d \frac{\varrho_k}{\varepsilon_k+\varepsilon_l}\,=\!:\,\,f_l(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d,\lambda)\;,\end{align}

(4.12) \begin{align}0&= R^{\prime}(\varepsilon_l)- \frac{r_l}{\varrho_l}=1-\frac{r_l}{\varrho_l}+ \frac{\lambda}{N}\sum_{k=1}^d \frac{\varrho_k}{(\varepsilon_k+\varepsilon_l)^2}\,=\!:\,g_l(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d,\lambda)\end{align}

for $l=1,\dots,d$ from Theorem 3.1 for the given positive real numbers $E_l>0$ and $r_l>0$ from Section 2. In the following we only use that all these $E_l,r_l$ are positive (but not that $r_l$ is an integer counting the multiplicity of $E_l$ ). So we consider for $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ the real or complex algebraic subset

\begin{equation*} Z(\mathbb{K})\,:\!=\,\{(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d,\lambda)\in U(\mathbb{K})|\: f_l=0,g_l=0,\: l=1,\dots,d\}\end{equation*}

in

\begin{equation*}U(\mathbb{K})\,:\!=\,\{(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d,\lambda)|\:\varepsilon_k+\varepsilon_l\neq 0, \: \varrho_l\neq 0,\:k,l=1,\dots,d\}\subset \mathbb{K}^{2d+1}\:,\end{equation*}

i.e. in the complement of the corresponding central hyperplane arrangement in $\mathbb{K}^{2d+1}$ . Note that $(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d,0)\in Z(\mathbb{K})$ iff $\varepsilon_l=E_l$ and $\varrho_l=r_l$ for all $l=1,\dots,d$ , and this real point belongs to the chamber

\begin{equation*}U_{+}(\mathbb{R})\,:\!=\,\{(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d,\lambda)|\:\varepsilon_l > 0, \: \varrho_l > 0,\: l=1,\dots,d\}\subset U(\mathbb{R})\subset \mathbb{R}^{2d+1}\:.\end{equation*}

Note that the complex dimension of any irreducible component of $Z(\mathbb{C})$ is at least $1=(2d+1)-2d$ , since we are considering 2d equations in a Zariski-open subset of $ \mathbb{C}^{2d+1}$ .

Lemma 4.8. $Z(\mathbb{K})$ is a one-dimensional (real or complex) algebraic submanifold of $U(\mathbb{K})$ near the reference point $(E_1,r_1,\ldots, E_d,r_d,0)$ , with the projection

\begin{equation*}pr\,:\, \mathbb{K}^{2d+1} \supset Z(\mathbb{K}) \longrightarrow \mathbb{K}\end{equation*}

onto the last $\lambda$ -coordinate a submersion near $(E_1,r_1,\ldots,E_d,r_d,0)$ in the Zariski topology. In particular, the reference point $(E_1,r_1,\ldots, E_d,r_d,0)$ only belongs to one irreducible component of $Z(\mathbb{C})$ , which is of dimension one. Moreover, the map (of pointed sets)

\begin{equation*}pr\,:\, Z(\mathbb{K}) \longrightarrow \mathbb{K} \quad \text{ with} \quad pr(E_1,r_1,\ldots, E_d,r_d,0)=0\end{equation*}

becomes locally near $(E_1,r_1,\ldots, E_d,r_d,0)$ a (real or complex) analytic isomorphism onto an open interval or disc around $\lambda=0\in \mathbb{K}$ , fitting with the description given in Theorem 3.1 in terms of the implicit function theorem.

Proof. The claim follows from

\begin{align*}\bigg(\frac{\partial f_l}{\partial \varepsilon_k}\big(E_1,r_1,\ldots, E_d,r_d,0\big), \frac{\partial f_l}{\partial \varrho_k}\big(E_1,r_1,\ldots, E_d,r_d,0\big)\bigg)&=(\delta_{lk},0)\\\text{and}\quad \bigg(\frac{\partial g_l}{\partial \varepsilon_k}\big(E_1,r_1,\ldots, E_d,r_d,0\big), \frac{\partial g_l}{\partial \varrho_k}\big(E_1,r_1,\ldots, E_d,r_d,0\big)\bigg)&=\bigg(0, \delta_{lk}\cdot \frac{1}{r_l}\bigg)\end{align*}

for $l,k=1,\dots,d$ .

Remark 4.9. Let us rewrite for a fixed $\lambda\in \mathbb{C}$ the equations (4.11) and (4.12) in terms of the d polynomials

\begin{equation*}F_l\big(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d\big) \,:\!=\,f_l\big(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d,\lambda\big)\cdot \prod_{k=1}^d(\varepsilon_k+\varepsilon_l)\end{equation*}

of degree $d+1$ and the d polynomials

\begin{equation*}G_l\big(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d\big) \,:\!=\,g_l(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d,\lambda)\cdot\varrho_l \prod_{k=1}^d(\varepsilon_k+\varepsilon_l)^2\end{equation*}

of degree $2d+1$ (for $l=1,\dots,d$ ), so that:

\begin{equation*}Z(\mathbb{C})\cap\{pr=\lambda\} \subset\big\{\big(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d\big)\in \mathbb{C}^{2d}|\: F_l=0,G_l=0,\: l=1,\dots,d\big\}\times\{\lambda\} \:.\end{equation*}

If for a given $\lambda\in \mathbb{C}$ the set

\begin{equation*}\big\{\big(\varepsilon_1,\varrho_1,\ldots,\varepsilon_d,\varrho_d\big)\in \mathbb{C}^{2d}|\: F_l=0,G_l=0,\: l=1,\dots,d\big\}\end{equation*}

is finite, then one gets by the affine Bezout inequality [Reference Schmid29, Thm 3.1] the upper estimate $(d+1)^d(2d+1)^d$ for the number of solutions of the equations (4.11) and (4.12) (for this $\lambda$ ).

Let us come back to the rational function R(z) from (4.10) for the case of positive real $E_l>0$ and $r_l>0$ for $ l=1,\dots,d$ related to the solution of Theorem 3.1 as discussed before. Then

(4.13) \begin{equation}R^{\prime}(z)=1+\frac{\lambda}{N}\sum_{k=1}^d \frac{\varrho_k}{(\varepsilon_k+z)^2}>0\end{equation}

for all $\lambda\geqslant 0$ and $z\in \mathbb{R}\backslash \{-\varepsilon_1,\dots,-\varepsilon_d\}$ .

Proof. If we order the numbering of the $\varepsilon_l$ as $\varepsilon_i<\varepsilon_{i+1}$ for $i=0,\dots,d$ with $\varepsilon_0\,:\!=\,-\infty$ and $\varepsilon_{d+1}\,:\!=\,+\infty$ , then

\begin{equation*}R\,:\, ({-}\varepsilon_{i+1},-\varepsilon_i) \longrightarrow \mathbb{R}\end{equation*}

is for all $i=0,\dots,d$ strictly monotone increasing and bijective by the estimate $R^{\prime}(z)>0$ above and the intermediate value theorem. This proves the first claim. Similarly, $R^{-1}(R(w))$ consists for any $w\in\mathcal{V}\cap (0,\infty)$ of $d+1$ real points which we can order as $\hat{w}^l\in ({-}\varepsilon_{l+1},-\varepsilon_l)$ . Therefore, $-\hat{w}^l\in (0,\infty)$ and all $R\big({-}\hat{w}^l\big)$ are distinct for $w\in\mathcal{V}\cap (0,\infty)$ , because R is injective on $({-}\varepsilon_1,\infty)$ . Moreover, R is an injective immersion in $({-}\varepsilon_1,\infty)$ , which is an open condition so that also the second claim follows. Finally the assumption (iv) with $\mathcal{G}^{(0)}$ as in (4.9) follows from $0\neq z+w$ for all $z,w\in \mathcal{V}$ , since $\textrm{Re}(z+w)=\textrm{Re}(z) +\textrm{Re}(w)>0$ .

We finish this section with the:

It remains to show (3.4). On the right-hand side of (4.2) we use the the symmetry $\mathcal{G}^{(0)}(\varepsilon_k,w)=\mathcal{G}^{(0)}(w,\varepsilon_k)$ from Proposition 4.6 and express $\mathcal{G}^{(0)}(w,\varepsilon_k)$ as (4.5) for $w\mapsto \varepsilon_k$ and $z\mapsto w$ . Dividing by $(R(w)-R(z))$ gives

(4.14) \begin{align}\mathcal{G}^{(0)}(z,w)=\frac{\displaystyle1 -\frac{\lambda}{N} \sum_{k=1}^d \frac{r_k}{(R(z)-R(\varepsilon_k))(R(\varepsilon_k)-R({-}w))}\prod_{j=1}^d \frac{R(w){-}R\big({-}\widehat{\varepsilon_k}^j\big)}{ R(w)-R\big(\varepsilon_j\big)}}{R(w)-R({-}z)}\;.\end{align}

This equation was obtained in [Reference Grosse, Hock and Wulkenhaar12] by another method. We rearrange it as

\begin{align*}\mathcal{G}^{(0)}(z,w)&=\frac{1}{(R(w)-R({-}z))(R(z)-R({-}w))}\Bigg\{R(z)-R({-}w)\\&\quad -\frac{\lambda}{N} \sum_{l=1}^d \frac{r_l}{(R(\varepsilon_l)-R({-}w))}\prod_{j=1}^d \frac{R(w)-R\big({-}\widehat{\varepsilon_l}^j\big)}{ R(w)-R\big(\varepsilon_j\big)}\\&\quad -\frac{\lambda}{N} \sum_{k=1}^d \frac{r_k}{(R(z)-R(\varepsilon_k))}\prod_{j=1}^d \frac{R(w)-R\big({-}\widehat{\varepsilon_k}^j\big)}{ R(w)-R\big(\varepsilon_j\big)}\Bigg\}\;.\end{align*}

The second line is $- ({\lambda}/{N}) \sum_{l=1}^d\mathcal{G}^{(0)}(w,\varepsilon_l)$ by (4.5). We combine it with the term $-R({-}w)$ inside $\{ \ \}$ according to our main algebraic relation (4.1). In the last line, the factor $\prod_{j=1}^d \!\big({R(w)-R\big({-}\widehat{\varepsilon_k}^j\big)}\big)/\big({ R(w)-R\big(\varepsilon_j\big)}\big)$ is rewitten via (4.6), with $w\mapsto \varepsilon_k$ and $z\mapsto w$ . We arrive at

\begin{align*}\mathcal{G}^{(0)}(z,w)&=\frac{1}{(R(w)-R({-}z))(R(z)-R({-}w))}\Bigg\{R(z)+R(w)\\[4pt]& \quad +\frac{\lambda}{N} \sum_{k=1}^d \frac{r_k}{(R(\varepsilon_k)-R(z))}+\frac{\lambda}{N} \sum_{l=1}^d \frac{r_l}{(R(\varepsilon_l)-R(w))}\\[4pt]& \quad +\frac{\lambda}{N} \sum_{k,l=1}^d \frac{r_k}{(R(\varepsilon_k)-R(z))(R(\varepsilon_l)-R(w))}\frac{\prod_{j=1}^d\!\big(R(\varepsilon_l)-R\big({-}\widehat{\varepsilon_k}^j\big)\big)}{\prod_{j\neq l}^d \!(R(\varepsilon_l)-R\big(\varepsilon_j\big)}\Bigg\}\;.\end{align*}

The result (3.4) follows from equation (4.4) for $\mathcal{G}^{(0)}(\varepsilon_k,\varepsilon_l)$ .

5. The diagonal 2-point function

The diagonal planar cumulant $z\mapsto \mathcal{G}^{(0)}(z,z)$ of length 2 and cycle type (0, 1) admits a simpler formula due to properties of the rational function $\tilde{R}$ with $\tilde{R}(z)\,:\!=\,R(z)-R({-}z)$ . Let $z\in \big\{0,\pm \alpha_1,\dots, \pm \alpha_d\big\}$ be the list of roots of

\begin{equation*}0= R(z)-R({-}z)=2z -\frac{\lambda}{N}\sum_{k=1}^d \frac{\varrho_k}{\varepsilon_k+z} -\frac{\lambda}{N}\sum_{k=1}^d \frac{\varrho_k}{-\varepsilon_k+z} \:,\end{equation*}

with the convention $\alpha_k>0$ and $\alpha_k\neq \alpha_l$ for $k\neq l$ . Since here all $\varepsilon_k>0$ and $\varrho_k>0$ are positive real numbers, we can argue as for the rational function R that also the rational function $\tilde{R}$ maps each of the $2d+1$ connected components of $\mathbb{R}\backslash\{\pm\varepsilon_1, \dots, \pm \varepsilon_d\}$ bijectively onto $\mathbb{R}$ . So the odd function $\tilde{R}$ has indeed $2d+1$ different real roots $\big\{0,\pm \alpha_1,\dots, \pm \alpha_d\big\}$ of the equation $\tilde{R}=0$ . Taking its poles $\{\infty,\pm \varepsilon_k\}$ into account, we have

(5.1) \begin{align}(R(z)-R({-}z))&= 2z \prod_{k=1}^d \frac{\big(z^2-\alpha_k^2\big)}{\big(z^2-\varepsilon_k^2\big)}\;.\end{align}

Proposition 5.1. For any $z\in \hat{\mathbb{C}}$ , the diagonal planar cumulant of cycle type $(0, 1)$ can be simplified to

(5.2) \begin{align}\mathcal{G}^{(0)}(z,z)&=\frac{2 R(z)-2R(0)}{(R(z)-R({-}z))^2}\prod_{k=1}^d \frac{(R(z)-R(\alpha_k))^2}{(R(z)-R(\varepsilon_k))^2}\\[-2pt]&\equiv \frac{1}{2z} \prod_{k=1}^d\frac{\big(z-\hat{0}^k\big)(z+e_k)\prod_{j=2}^d \!\big(z-\widehat{\alpha_k}^j\big)^2}{\prod_{l=1}^d \!\big(z-\widehat{e_k}^l\big)^2}\;,\nonumber\end{align}

in the convention $\widehat{\alpha_k}^1\equiv -\alpha_k$ .

Proof. The $d+1$ fold product of (5.1) for $\big\{z,\hat{z}^1,\dots,\hat{z}^d\big\}$ is inserted into (4.5):

(5.3) \begin{align}&(R(z)-R({-}z))^2 \mathcal{G}^{(0)}(z,z) \prod_{k=1}^d(R(z)-R(\varepsilon_k))\nonumber\\[-2ex]& \quad = (R(z)-R({-}z))\prod_{l=1}^d\big(R(z)-R\big({-}\hat{z}^l\big)\big) \nonumber\\[-1ex]& \quad = 2z\Bigg(\prod_{l=1}^d 2\hat{z}^l\Bigg)\prod_{k=1}^d \frac{\big(z^2-\alpha_k^2\big)\prod_{l=1}^d \!\big(\big(\hat{z}^l\big)^2-\alpha_k^2\big)}{\big(z^2-\varepsilon_k^2\big)\prod_{l=1}^d \!\big(\big(\hat{z}^l\big)^2-\varepsilon_k^2\big)}\;.\end{align}

We use cases of (4.8); the third one takes $R(\alpha_k)=R({-}\alpha_k)$ into account:

\begin{align*}2(R(z)-R(0))&=\frac{2z\prod_{l=1}^d \!\big(2\hat{z}_l\big) }{\prod_{k=1}^d \!\big({-}2\varepsilon_k\big)}\;,\\[-0.5ex](R(z)-R(\varepsilon_k))&=(z-\varepsilon_k)\frac{\prod_{l=1}^d \!\big(\varepsilon_k-\hat{z}^l\big) }{\prod_{l=1}^d \!(\varepsilon_k+\varepsilon_l)}\;,\\(R(z)-R(\alpha_k))^2&=\big(z^2-\alpha_k^2\big)\frac{\prod_{l=1}^d \!\big(\big(\hat{z}^l\big)^2-\alpha_k^2\big) }{\prod_{l=1}^d \!\big(\varepsilon_l^2-\alpha_k^2\big)}\;.\end{align*}

We identify in (5.3) the first equation and the product over k of the second and third equations:

\begin{align*}&(R(z)-R({-}z))^2 \mathcal{G}^{(0)}(z,z) \prod_{k=1}^d(R(z)-R(\varepsilon_k))\nonumber\\&= \frac{2(R(z)-R(0))}{\prod_{k=1}^d \!\big((z+\varepsilon_k)\prod_{l=1}^d \!\big(\hat{z}^l+\varepsilon_k\big)\big)}\prod_{k=1}^d\frac{(R(z)-R(\alpha_k))^2}{(R(z)-R(\varepsilon_k))}\cdot \prod_{k=1}^d \frac{(2\varepsilon_k)\prod_{l=1}^d \!\big(\varepsilon_l^2-\alpha_k^2\big)}{\prod_{l=1}^d \!(\varepsilon_l+\varepsilon_k)}\;.\end{align*}

Now observe that the residue of (4.8) at $z=-\varepsilon_k$ is the identity

(5.4) \begin{align}\frac{(u+\varepsilon_k)\prod_{l=1}^d\!\big(\hat{u}^l+\varepsilon_k\big)}{\prod_{j\neq k}^d\!\big(\varepsilon_j-\varepsilon_k\big)} = -\frac{\lambda}{N} \varrho_k\;,\end{align}

for any $u\notin R^{-1}(\{\infty\})$ . Consequently,

\begin{align*}\frac{(R(z)-R({-}z))^2 \mathcal{G}^{(0)}(z,z)}{2(R(z)-R(0))}\prod_{k=1}^d\frac{(R(z)-R(\varepsilon_k))^2}{(R(z)-R(\alpha_k))^2}&= \prod_{k=1}^d \frac{ N\prod_{l=1}^d \!\big(\varepsilon_l^2-\alpha_k^2\big)}{({-}\lambda\varrho_k)\prod_{j\neq k}^d\!\big(\varepsilon_j^2-\varepsilon_k^2\big)}=C\end{align*}

is a constant independent of z, which for $z\to \infty$ is identified as $C=1$ .

The following result will be needed in the next section:

Lemma 5.2. For any $w\in \hat{\mathbb{C}}$ one has

(5.5) \begin{align}\frac{1}{R(w)-R({-}w)}+ \sum_{k=1}^d\frac{1}{R(w)-R\big({-}\hat{w}^k\big)}&=\frac{1}{2(R(w){-}R(0))}+\sum_{k=1}^d \frac{1}{R(w)-R(\alpha_k)}\;.\end{align}

Note that $R(w)=R\big(\hat{w}^k\big)$ implies $R^{\prime}(w) =R^{\prime}\big(\hat{w}^k\big) \big(\hat{w}^k\big)^{\prime}(w)$ or $\big(\hat{w}^k\big)^{\prime}(w)= {R^{\prime}(w)}/{R^{\prime}\big(\hat{w}^k\big)}$ . Consider the pole at $w=\pm \alpha_l$ . Then there is precisely one $k_l\in \{1,\dots, d\}$ with $\hat{w}^{k_l}=\mp \alpha_l$ . Therefore,

\begin{align*}&\mathop{Res}\limits_{w= \pm \alpha_l}\Bigg(\frac{dw }{R(w)-R({-}w)}+ \sum_{k=1}^d\frac{dw}{R(w)-R\big({-}\hat{w}^k\big)}\Bigg)\nonumber\\&= \left(\frac{1}{R^{\prime}(w)+R^{\prime}({-}w)}+\frac{1}{R^{\prime}(w)+R^{\prime}\big({-}\hat{w}^{k_l}\big)\frac{R^{\prime}(w)}{R^{\prime}(\hat{w}^{k_l})}}\right)\Bigg|_{w=\pm \alpha_l,\hat{w}^{k_l}=\mp\alpha_l}=\frac{1}{R^{\prime}(\!\pm \alpha_l)}\;.\end{align*}

Consider in case of $d\geqslant 2$ the pole at $w=\widehat{\alpha_l}^j$ . There are precisely two distinct $k_+,k_- \in \{1,\dots, d\}$ with $\hat{w}^{k_+}=\alpha_l$ and $\hat{w}^{k_-}=-\alpha_l$ . Therefore,

\begin{align*}&\mathop{Res}\limits_{w= \widehat{\alpha_l}^j}\Bigg(\frac{dw }{R(w)-R({-}w)}+ \sum_{k=1}^d\frac{dw}{R(w)-R\big({-}\hat{w}^k\big)}\Bigg)\nonumber\\&=\left(\frac{1}{R^{\prime}(w)+R^{\prime}\big({-}\hat{w}^{k_+}\big)\frac{R^{\prime}(w)}{R^{\prime}\big(\hat{w}^{k_+}\big)}}+\frac{1}{R^{\prime}(w)+R^{\prime}\big({-}\hat{w}^{k_-}\big)\frac{R^{\prime}(w)}{R^{\prime}\big(\hat{w}^{k_-}\big)}}\right)\Bigg|_{w=\widehat{\alpha_l}^j,\hat{w}^{k_\pm}=\pm\alpha_l}\\&=\frac{1}{R^{\prime}\big(\widehat{\alpha_l}^j\big)}\;.\end{align*}

The rhs of (5.5) has exactly the same residues.

Finally, it is also clear that both sides of (5.5) have the same residue ${1}/{2R^{\prime}(0)}$ at $w=0$ . For $w=\hat{0}^l$ there is a unique $k_l \in \{1,\dots, d\}$ with $\hat{w}^{k_l}=0$ . Then

\begin{align*}&\mathop{Res}\limits_{w= \widehat{0}^l}\Bigg(\frac{dw }{R(w)-R({-}w)}+ \sum_{k=1}^d\frac{dw}{R(w)-R\big({-}\hat{w}^k\big)}\Bigg)\nonumber\\& \quad = \frac{1}{R^{\prime}(w)+R^{\prime}\big({-}\hat{w}^{k_l}\big)\frac{R^{\prime}(w)}{R^{\prime}\big(\hat{w}^{k_l}\big)}}\Bigg|_{w=\hat{0}^l,\hat{w}^{k_l}=0}=\frac{1}{2R^{\prime}\big(\hat{0}^l\big)}\;,\end{align*}

which agrees with the residue of the rhs of (5.5). Therefore, the difference between lhs and rhs of (5.5) is a bounded entire function, i.e. a constant by Liouville’s theorem, which is zero when considering $w\to \infty$ . This completes the proof.

6. The planar $1+1$ -point function

The $1+1$ -point function $G_{|a|b|}$ is the cumulant of length 2 and cycle type (2, 0) (i.e. two cycles a and b of length 1), see Appendix A. We derive in Appendix B its equation of motion (B8) whose restriction to the planar sector (of degree or genus $g=0$ ) reads

(6.1) \begin{align}(E_a+E_a)G_{|a|b|}^{(0)} &=-\frac{\lambda}{N}\sum_{k=1}^{N} G^{(0)}_{|ak|}G^{(0)}_{|a|b|}+\frac{\lambda}{N}\sum_{k=1}^{N}\frac{G^{(0)}_{|k|b|}-G^{(0)}_{|a|b|}}{E_k-E_a}+\lambda\frac{G^{(0)}_{|bb|}-G^{(0)}_{|ab|}}{E_b-E_a}\;.\end{align}

We interpret this equation as evaluation $G_{|a|b|}^{(0)}=\mathcal{G}^{(0)}(\varepsilon_a|\varepsilon_b)$ of a functionFootnote 2 $\mathcal{G}^{(0)}(z|w)$ which satisfies

(6.2) \begin{align}\big( R(z)-R({-}z)\big) \mathcal{G}^{(0)}(z|w)- \frac{\lambda}{N} \sum_{k=1}^d\frac{r_k\mathcal{G}^{(0)}(\varepsilon_k|w) }{R(\varepsilon_k)-R(z)}&=\lambda \frac{\mathcal{G}^{(0)}(z,w)-\mathcal{G}^{(0)}(w,w)}{R(z)-R(w)}\;.\end{align}

The identity (4.1) was decisive here, and multiplicities $r_k$ of the $E_k=R(\varepsilon_k)$ were admitted.

Since $\mathcal{G}^{(0)}(\alpha_k|w)$ must be regularFootnote 3 for any $w>0$ , evaluation at $z=\alpha_k$ produces d equations

(6.3) \begin{align}\frac{\lambda}{N} \sum_{l=1}^d\frac{r_l\mathcal{G}^{(0)}(\varepsilon_l|w) }{R(\alpha_k)-R(\varepsilon_l)}&=\lambda \frac{\mathcal{G}^{(0)}(\alpha_k,w)-\mathcal{G}^{(0)}(w,w)}{R(\alpha_k)-R(w)}\;.\end{align}

Equation (6.3) is with Proposition 3.2 solved by

\begin{align*}\frac{r_k}{N} \mathcal{G}^{(0)}(\varepsilon_k|w)=\sum_{l=1}^{d}(R(\alpha_l)-R(\varepsilon_k)) \textbf{A}_l(R(\varepsilon_k))\textbf{E}_k(R(\alpha_l))\frac{\mathcal{G}^{(0)}(\alpha_l,w)-\mathcal{G}^{(0)}(w,w)}{R(\alpha_l)-R(w)}\;,\end{align*}

where

\begin{align*}\textbf{A}_i(x)&=\frac{\textbf{A}(x)}{(x-R(\alpha_i))\textbf{A}^{\prime}(R(\alpha_i))}&\text{and} \quad \textbf{E}_j(y)&=\frac{\textbf{E}(y)}{\big(y-R\big(\varepsilon_j\big)\big)\textbf{E}^{\prime}\big(R\big(\varepsilon_j\big)\big)}\:,\\\text{with}\quad \textbf{A}(x)&\,:\!=\,\prod_{k=1}^d \!(x-R(\alpha_k)) & \text{and} \quad \textbf{E}(y)&=\prod_{k=1}^d \!(y-R(\varepsilon_k)) \:.\end{align*}

Here, the $R(\alpha_k)$ and $R(\varepsilon_l)$ are pairwise distinct, since R is injective on $({-}\varepsilon_1,\infty)$ . Inserting this back into (6.2) gives the $1+1$ -point function

(6.4) \begin{align}\mathcal{G}^{(0)}(z|w)&= \frac{\lambda}{R(z)-R({-}z)}\Bigg\{\frac{\mathcal{G}^{(0)}(z,w)-\mathcal{G}^{(0)}(w,w)}{R(z)-R(w)}\nonumber\\&\quad -\sum_{k,l=1}^{d}\frac{(R(\alpha_l)-R(\varepsilon_k)) \textbf{A}_l(R(\varepsilon_k))\textbf{E}_k(R(\alpha_l)) }{R(z)-R(\varepsilon_k)}\frac{\mathcal{G}^{(0)}(\alpha_l,w)-\mathcal{G}^{(0)}(w,w)}{R(\alpha_l)-R(w)}\Bigg\}\end{align}

in terms of the 2-point function $\mathcal{G}^{(0)}(z,w)$ known from Theorem 3.1. We convert the solution (6.4) into a manifestly symmetric form:

Proposition 6.1. The planar cumulant of length 2 and cycle type (2, 0) has the solution

(6.5) \begin{align}\mathcal{G}^{(0)}(z|w)&= \frac{\lambda}{(R(z)-R(w))^2} \Bigg(\mathcal{G}^{(0)}(z,w)\\&\quad -\frac{R(z)+R(w)-2R(0)}{(R(z){-}R({-}z))(R(w){-}R({-}w))}\prod_{k=1}^d \frac{(R(z)-R(\alpha_k))(R(w)-R(\alpha_k))}{(R(z)-R(\varepsilon_k))(R(w)-R(\varepsilon_k))}\Bigg)\,.\nonumber\end{align}

Proof. Using (3.8) we evaluate the k-sum in (6.4) to

(6.6) \begin{align}\mathcal{G}^{(0)}(z|w)&= \frac{\lambda}{R(z)-R({-}z)}\Bigg\{\frac{\mathcal{G}^{(0)}(z,w)-\mathcal{G}^{(0)}(w,w)}{R(z)-R(w)}\nonumber\\&\quad -\sum_{l=1}^{d}\frac{\textbf{A}_l(R(z))\textbf{E}(R(\alpha_l))}{\textbf{E}(R(z))}\frac{\mathcal{G}^{(0)}(\alpha_l,w)-\mathcal{G}^{(0)}(w,w)}{R(\alpha_l)-R(w)}\Bigg\}\;.\end{align}

In the second line we have

\begin{equation*}\frac{\textbf{A}_l(R(z))\textbf{E}(R(\alpha_l))}{\textbf{E}(R(z))}=-\frac{\textbf{A}(R(z))}{\textbf{E}(R(z))}\cdot \frac{\prod_{k=1}^d \!(R(\alpha_l)-R(\varepsilon_k))}{(R(\alpha_l)-R(z))\prod_{j=1,j\neq l}^d \!\big(R(\alpha_l)-R\big(\alpha_j\big)\big)}\end{equation*}

and we recall

\begin{equation*}\mathcal{G}^{(0)}(\alpha_l,w)=-\frac{1}{(R(\alpha_l)-R(w))}\frac{\prod_{j=1}^d R(\alpha_l)-R\big({-}\hat{w}^j\big)}{\prod_{j=1}^d \!\big(R(\alpha_l)-R\big(\varepsilon_j\big)\big)}\end{equation*}

from (4.5). Inserting both identities into (6.6) gives after a first partial fraction decomposition

(6.7) \begin{align}\mathcal{G}^{(0)}(z|w)&= \frac{\lambda}{(R(z)-R({-}z))(R(z)-R(w))}\Bigg\{\mathcal{G}^{(0)}(z,w)-\mathcal{G}^{(0)}(w,w)\nonumber\\&\quad -\frac{\textbf{A}(R(z))}{\textbf{E}(R(z))}\mathcal{G}^{(0)}(w,w)\Bigg(\sum_{l=1}^{d}\frac{\prod_{k=1}^d \!(R(\alpha_l)-R(\varepsilon_k))}{(R(\alpha_l)-R(z))\prod_{j=1,j\neq l}^d \!\big(R(\alpha_l)-R\big(\alpha_j\big)\big)}\nonumber\\&\qquad \qquad \qquad -\sum_{l=1}^{d} \frac{\prod_{k=1}^d \!(R(\alpha_l)-R(\varepsilon_k))}{(R(\alpha_l)-R(w))\prod_{j=1,j\neq l}^d \!\big(R(\alpha_l)-R\big(\alpha_j\big)\big)}\Bigg)\nonumber\\&-\frac{\textbf{A}(R(z))}{\textbf{E}(R(z))}\Bigg(\sum_{l=1}^{d}\frac{\prod_{k=1}^d \!\big(R(\alpha_l)-R\big({-}\hat{w}^k\big)\big)}{(R(\alpha_l){-}R(z))(R(\alpha_l){-}R(w))\prod_{j=1,j\neq l}^d \!\big(R(\alpha_l){-}R\big(\alpha_j\big)\big)}\nonumber\\&\qquad \qquad \qquad -\sum_{l=1}^{d}\frac{\prod_{k=1}^d \!\big(R(\alpha_l)-R\big({-}\hat{w}^k\big)\big)}{(R(\alpha_l){-}R(w))^2\prod_{j=1,j\neq l}^d \!\big(R(\alpha_l){-}R\big(\alpha_j\big)\big)}\Bigg)\Bigg\}\,.\end{align}

The second and third line are converted via an identity (4.6) with substitution $\varepsilon_i\mapsto \alpha_i$ and $-\hat{w}^j\mapsto\varepsilon_j$ . One of the surviving terms cancels $\mathcal{G}^{(0)}(w,w)$ in the first line of (6.7). Another partial fraction decomposition in the fourth line of (6.7) and ${1}/({(R(\alpha_l)-R(w))^2})=\lim_{u\to w} {1}/({(R(u)-R(w))})\big({1}/({(R(\alpha_l)-R(u))})- {1}/({(R(\alpha_l)-R(w))}))$ in the fifth line of (6.7) also give rise to expressions (4.6) with substitution $\varepsilon_i\mapsto \alpha_i$ . We thus find

\begin{align*}\mathcal{G}^{(0)}(z|w)& = \frac{\lambda}{(R(z){-}R({-}z))(R(z){-}R(w))} \Bigg\{\mathcal{G}^{(0)}\:\!\!(z,w){-}\frac{\textbf{A}(R(z))\textbf{E}(R(w))}{\textbf{E}(R(z))\textbf{A}(R(w))}\mathcal{G}^{(0)}\:\!(w,w)\nonumber\\& \quad +\frac{\displaystyle\prod_{k=1}^{d}\frac{R(z)-R\big({-}\hat{w}^k\big)}{R(z)-R(\varepsilon_k)}-\frac{\textbf{A}(R(z))\textbf{E}(R(w))}{\textbf{E}(R(z))\textbf{A}(R(w))}\prod_{k=1}^{d}\frac{R(w)-R\big({-}\hat{w}^k\big)}{R(w)-R(\varepsilon_k)}}{R(z)-R(w)}\nonumber\\& \quad -\frac{\textbf{A}(R(z))}{\textbf{E}(R(z))}\lim_{u\to w}\frac{\displaystyle\prod_{k=1}^{d}\frac{R(u)-R\big({-}\hat{w}^k\big)}{R(u)-R(\alpha_k)}-\prod_{k=1}^{d}\frac{R(w)-R\big({-}\hat{w}^k\big)}{R(w)-R(\alpha_k)}}{R(u)-R(w)}\Bigg\}\;.\end{align*}

After evaluation of the limit we reconstruct in the last two lines $\mathcal{G}^{(0)}(z,w)$ and $\mathcal{G}^{(0)}(w,w)$ via (4.5):

\begin{align*}\mathcal{G}^{(0)}(z|w)&= \frac{\lambda}{(R(z)-R(w))^2} \Bigg(\mathcal{G}^{(0)}(z,w)\nonumber\\& \quad -\frac{\textbf{A}(R(z))\textbf{E}(R(w))}{\textbf{E}(R(z))\textbf{A}(R(w))}\frac{R(w)-R({-}w)}{R(z)-R({-}z)}\mathcal{G}^{(0)}(w,w)\Bigg\{1 \nonumber\\& \quad +\frac{R(z)-R(w)}{R(w)-R({-}w)}+\sum_{k=1}^d \frac{R(z)-R(w)}{R(w)-R\big({-}\hat{w}^k\big)}- \sum_{k=1}^d\frac{R(z)-R(w)}{R(w)-R(\alpha_k)}\Bigg\}\Bigg)\;.\end{align*}

With Lemma 5.2 the terms in $\{ \ \}$ can be reduced to $\{ \ \}= ({R(z)+R(w)-2R(0)})/$ $({2(R(w)-R(0))})$ . Inserting (5.2) for $\mathcal{G}^{(0)}(w,w)$ gives the final result (6.5).

7. Outlook

We have developed a new algebraic solution strategy for the two initial cumulants of a quartic analogue of the Kontsevich model. Our results have been extended in [Reference Branahl, Hock and Wulkenhaar3] to an algorithm which allows to recursively compute all other cumulants. The key discovery of [Reference Branahl, Hock and Wulkenhaar3] was to understand that one first has to focus on three families $\Omega^{(g)}_{m}(u_1,...,u_m)$ , $\mathcal{T}^{(g)}(u_1,...,u_m\|z,w|)$ and $\mathcal{T}^{(g)}(u_1,...,u_m\|z|w|)$ of auxiliary functions. Their simplest cases are the functions $\mathcal{T}^{(0)}(\emptyset \|z,w|)\,:\!=\,\mathcal{G}^{(0)}(z,w)$ and $\mathcal{T}^{(0)}(\emptyset \|z|w|)\,:\!=\,\mathcal{G}^{(0)}(z|w)$ analysed in this paper. The auxiliary functions are special polynomials [Reference Branahl, Hock and Wulkenhaar2] in the original cumulants. One first solves a coupĺed system of equations for $(\Omega^{(g)}_{m},\mathcal{T}^{(g)})$ and then uses the result to turn the Dyson–Schwinger equations for the cumulants into a problem which can easily be solved by inversion of Cauchy matrices.

Of particular interest are the functions $\Omega^{(g)}_{n}$ which give rise to a family of meromorphic differentials

(7.1) \begin{align}\omega_{g,n}(z_1,...,z_n)=\Omega^{(g)}_{n}(z_1,...,z_n)dR(z_1)\ldots dR(z_n)\end{align}

which starts with $\omega_{0,2}(z_1,z_2)= {dz_1\,dz_2}/{(z_1-z_2)^2} + {dz_1\,dz_2}/{(z_1+z_2)^2}$ . Also the next forms $\omega_{0,3}$ , $\omega_{0,4}$ and $\omega_{1,1}$ have been found in [Reference Branahl, Hock and Wulkenhaar3], where $\omega_{1,1}$ needs Propositions 5.1 and 6.1 of this paper. Remarkably, all forms computed so far satisfy abstract loop equations [Reference Borot, Eynard and Orantin1] if one sets $\omega_{0,1}(z)=y(z)dx(z)$ with $x(z)=R(z)$ and $y(z)=-R({-}z)$ . It was shown in [Reference Borot and Shadrin7] that the solution of abstract loop equations is blobbed topological recursion, a systematic extension of topological recursion [Reference Eynard and Orantin10, Reference Eynard11] by additional terms which are holomorphic at ramification points of x. The natural conjecture is that all $\omega_{g,n}$ of the quartic analogue of the Kontsevich model obey blobbed topological recursion. The conjecture was proved for genus $g=0$ in [Reference Hock and Wulkenhaar18] by relating it to an equation which expresses $\omega_{g,n+1}(z_1,...,z_n,-z)$ in terms of $\omega_{g,m+1}(z_1,...,z_m,+z)$ with $m\leqslant n$ .

In an early version of this paper we had speculated that the exact solution of the non-linear equation (3.2) might be caused by a hidden integrable structure. The discovery in [Reference Branahl, Hock and Wulkenhaar3, Reference Hock and Wulkenhaar18] that the quartic analogue of the Konsevich model obeys blobbed topological recursion questions this interpretation: integrability is not known in blobbed topological recursion. The relation to intersection theory on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable complex curves extends, however, to blobbed topological recursion [Reference Borot and Shadrin7]. The discovery in [Reference Hock and Wulkenhaar18] that (at least the planar sector of) the quartic analogue of the Kontsevich model is completely governed by the behaviour of the $\omega_{g,n}$ under a global (and canonical) involution makes us confident that the intersection numbers generated by this model will have a geometric significance. It will be an exciting programme to make this precise.