Solution of all quartic matrix models

TL;DR

This paper provides an exact solution to the quartic matrix model, extending previous results to finite and large N, and connects it to noncommutative quantum field theory without triviality issues.

Contribution

It identifies the exact two-point function solution for the quartic matrix model using complex analysis and extends it to unbounded operators, linking to topological recursion.

Findings

Exact rational function form of the two-point function for finite N

Solution for large N involving spectral regularization of R

Proof that the noncommutative -model is non-trivial

Abstract

We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-{N}\,\mathrm{Tr}(E\Phi^2+(\lambda/4)\Phi^4)) d\Phi$ on Hermitian ${N}\times{N}$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. It was previously established that the large-$N$ limit of the second moment (the planar two-point function) satisfies a non-linear integral equation. By employing tools from complex analysis, in particular the Lagrange-B\"urmann inversion formula, we identify the exact solution of this non-linear problem, both for finite $N$ and for a large-${N}$ limit to unbounded operators $E$ of spectral dimension $\leq 4$. For finite $N$, the two-point function is a rational function evaluated at the preimages of another rational function $R$ constructed from the spectrum of $E$. Subsequent work has constructed from this formula a family $\omega_{g,n}$ of meromorphic differentials which obey blobbed topological recursion. For unbounded operators $E$, the renormalised two-point function is given by an integral formula involving a regularisation of $R$. This allowed a proof, in subsequent work, that the $\lambda\Phi^4_4$-model on noncommutative Moyal space does not have a triviality problem.

Equations

$$\displaystyle\mathcal{Z}_{E}^{3}=\frac{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\,\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}+\tfrac{\lambda}{3}\Phi^{3}\big{)}\!\Big{)}}{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\,\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}\big{)}\Big{)}}\quad\text{and}\quad\mathcal{Z}_{E}^{4}=\frac{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\,\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}+\tfrac{\lambda}{4}\Phi^{4}\big{)}\!\Big{)}}{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\,\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}\big{)}\Big{)}}\,.$$

$$\displaystyle\langle a_{1}b_{1},\dots,a_{N}b_{N}\rangle_{3}:=\frac{1}{\mathcal{N}}\frac{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\;\Phi_{a_{1}b_{1}}\cdots\Phi_{a_{N}b_{N}}\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}+\tfrac{\lambda}{3}\Phi^{3}\big{)}\Big{)}}{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\,\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}+\tfrac{\lambda}{3}\Phi^{3}\big{)}\Big{)}}\;,$$

$$\displaystyle\langle a_{1}b_{1},\dots,a_{N}b_{N}\rangle_{4}:=\frac{1}{\mathcal{N}}\frac{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\;\Phi_{a_{1}b_{1}}\cdots\Phi_{a_{N}b_{N}}\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}+\tfrac{\lambda}{4}\Phi^{4}\big{)}\Big{)}}{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\,\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}+\tfrac{\lambda}{4}\Phi^{4}\big{)}\Big{)}}$$

$$J(z):=z-\frac{\lambda }{\mathcal{N}}\sum _{k=1}^d\frac{{\varrho }_k}{{\epsilon }_k+z},$$

$$E_l$$

$$G_{ab}^{(0)}=\bigg{[}\frac{1}{\mathcal{N}}\frac{\int d\Phi\;\Phi_{ab}\Phi_{ba}\,e^{-\mathcal{N}\,\mathrm{Tr}(E\Phi^{2}+(\lambda/4)\Phi^{4})}}{\int d\Phi\;e^{-\mathcal{N}\,\mathrm{Tr}(E\Phi^{2}+(\lambda/4)\Phi^{4})}}\bigg{]}_{\text{up to }\mathcal{O}(\mathcal{N}^{-1})\text{-terms}}$$

$${\mathcal{G}}^{(0)}(z,w)$$

$$\displaystyle ZG_{ab}:=\frac{1}{\mathcal{N}}\frac{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\;\Phi_{ab}\Phi_{ba}\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}+\tfrac{\lambda}{4}\Phi^{4}\big{)}\Big{)}}{\displaystyle\int_{M_{\mathcal{N}}^{*}}\!\!\!\!\!d\Phi\;\exp\Big{(}{-}\mathcal{N}\mathrm{Tr}\big{(}E\Phi^{2}+\tfrac{\lambda}{4}\Phi^{4}\big{)}\Big{)}}\,,$$

$$\displaystyle ZG_{ab}=\frac{1}{E_{a}+E_{b}}-\frac{\lambda}{\mathcal{N}(E_{a}+E_{b})}\sum_{n=1}^{\mathcal{N}}\Big{(}ZG_{ab}\;ZG_{an}-\frac{ZG_{nb}-ZG_{ab}}{E_{n}-E_{a}}\Big{)}+\mathcal{O}(\mathcal{N}^{-1})\;.$$

$$\displaystyle G_{ab}^{(0)}:=G(x,y)\Big{|}_{x=E_{a}-\mu^{2}_{bare}/2,\;y=E_{b}-\mu^{2}_{bare}/2}\;,$$

$$\displaystyle(\mu_{bare}^{2}{+}x{+}y)ZG(x,y)=1-\lambda\int_{0}^{\Lambda^{2}}\!\!dt\;\rho_{0}(t)\Big{(}ZG(x,y)\;ZG(x,t)-\frac{ZG(t,y)-ZG(x,y)}{t-x}\Big{)}\;.$$

$$\displaystyle\rho_{0}(t):=\frac{1}{\mathcal{N}}\sum_{n=1}^{\mathcal{N}}\delta\Big{(}t-\Big{(}E_{n}{-}\frac{\mu^{2}_{bare}}{2}\Big{)}\Big{)}\;,\quad\frac{\mu_{bare}^{2}}{2}=\min_{n}(E_{n})\;,~{}~{}\Lambda^{2}=\max_{n}(E_{n})-\frac{\mu^{2}_{bare}}{2}\;.$$

$$({\mu }_{bare}^2+x+y)ZG(x,y)$$

$$=1+\lambda {\int }_0^{{\Lambda }^2}dt{\rho }_0(t)\frac{ZG(t,y)-ZG(x,y)}{t-x}+\lambda {\int }_0^{{\Lambda }^2}ds{\rho }_0(s)\frac{ZG(x,s)-ZG(x,y)}{s-y}$$

$$-{\lambda }^2{\int }_0^{{\Lambda }^2}dt{\rho }_0(t){\int }_0^{{\Lambda }^2}ds{\rho }_0(s)\frac{ZG(x,y)ZG(t,s)-ZG(x,s)ZG(t,y)}{(t-x)(s-y)}.$$

$$(W(x){)}^2+{\lambda }^2{\int }_0^{{\Lambda }^2}dt{\rho }_0(t)\frac{W(t)-W(x)}{t-x}=x,{\rho }_0(t)=\frac{8}{\mathcal{N}}\sum _{n=1}^{\mathcal{N}}\delta (t-(2E_n{)}^2)$$

$$\displaystyle(-c)\Big{(}\frac{2}{1+\sqrt{1+c}}\Big{)}^{\delta_{D,2}+\delta_{D,4}}$$

$${\rho }_c(y)$$

$$ZG(a,b)=\frac{e^{{\mathcal{H}}_a^{\Lambda }[{\tau }_b(\bullet )]}\sin {\tau }_b(a)}{\lambda \pi {\rho }_0(a)}=\frac{e^{{\mathcal{H}}_b^{\Lambda }[{\tau }_a(\bullet )]}\sin {\tau }_a(b)}{\lambda \pi {\rho }_0(b)},$$

$$\displaystyle\mathcal{H}^{\Lambda}_{a}[f(\bullet)]:=\frac{1}{\pi}\lim_{\epsilon\to 0}\int_{[0,\Lambda^{2}]\setminus[a-\epsilon,a+\epsilon]}\frac{dt\;f(t)}{t-a}=\lim_{\epsilon\to 0}\mathrm{Re}\Big{(}\frac{1}{\pi}\int_{0}^{\Lambda^{2}}\frac{dt\;f(t)}{t-(a+\mathrm{i}\epsilon)}\Big{)}$$

$$\displaystyle\Big{(}\mu_{bare}^{2}+a+b+\lambda\pi\mathcal{H}^{\Lambda}_{a}[\rho_{0}(\bullet)]+\frac{1}{\pi}\int_{0}^{\Lambda^{2}}dt\;e^{\mathcal{H}_{t}^{\Lambda}[\tau_{a}(\bullet)]}\sin\tau_{a}(t)\Big{)}ZG(a,b)$$

$$\displaystyle=1+\mathcal{H}^{\Lambda}_{a}\big{[}e^{\mathcal{H}_{\bullet}^{\Lambda}[\tau_{b}]}\sin\tau_{b}(\bullet)\big{]}\;.$$

$$\displaystyle\mathcal{H}^{\Lambda}_{a}\big{[}e^{\mathcal{H}_{\bullet}^{\Lambda}[\tau]}\sin\tau(\bullet)\big{]}$$

$$\displaystyle\tau_{a}(p)=\arctan\bigg{(}\frac{\lambda\pi\rho_{0}(p)}{\mu_{bare}^{2}+a+p+\lambda\pi\mathcal{H}^{\Lambda}_{p}[\rho_{0}(\bullet)]+\frac{1}{\pi}\int_{0}^{\Lambda^{2}}dt\;\tau_{p}(t)}\bigg{)}\;,$$

$$g(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n!}\frac{d^{n-1}}{dw^{n-1}}\Big{|}_{w=0}(\phi(w))^{n}\;.$$

$$H(g(z))=\sum_{n=1}^{\infty}\frac{z^{n}}{n!}\frac{d^{n-1}}{dw^{n-1}}\Big{|}_{w=0}\Big{(}H^{\prime}(w)\big{(}\phi(w)\big{)}^{n}\Big{)}\;.$$

$${\tau }_a(p)$$

$$h_{0D}(z)$$

$$h_2(z)$$

$$h_4(z)$$

$$J_D(z):=z-\lambda h_D(-{\mu }^2-z)\equiv z-\lambda (-z{)}^{\frac{D}{2}}{\int }_{{\nu }_D}^{{\Lambda }_D^2}\frac{dt{\rho }_c(t)}{(t+{\mu }^2{)}^{\frac{D}{2}}(t+{\mu }^2+z)}.$$

$$D\in \{0,2\}\Rightarrow J_D^{\prime }(z)$$

$$J_4^{\prime }(z)$$

Full text

Solution of all quartic matrix models

Harald Grosse1,3, Alexander Hock2, Raimar Wulkenhaar2

1 Fakultät für Physik, Universität Wien, Boltzmanngasse 5, A-1090 Vienna, Austria

*2 Mathematisches Institut der Westfälischen Wilhelms-Universität,

Einsteinstraße 62, D-48149 Münster, Germany*

*3 Erwin Schrödinger International Institute for Mathematics and Physics,

University of Vienna, Boltzmanngasse 9, A-1090 Vienna, Austria*

E-mails: harald.grosse@univie.ac.at, a_hock03@wwu.de, raimar@math.uni-muenster.de

Abstract

We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-\mathcal{N}\,\mathrm{Tr}(E\Phi^{2}+(\lambda/4)\Phi^{4}))d\Phi$ on Hermitean $\mathcal{N}\times\mathcal{N}$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. We prove that the two-point function is a rational function evaluated at roots of another rational function $J$ constructed from the spectrum of $E$. This rationality is strong support for the conjecture that the quartic analogue of the Kontsevich model is integrable. We also solve the large-$\mathcal{N}$ limit to unbounded operators $E$. The renormalised two-point function is given by an integral formula involving a regularisation of $J$.

MSC 2010: 30C15, 14H70, 37F10, 81Q80

Keywords:

matrix models, solvable non-linear integral equations, integrability, complex curves

1 Introduction

This paper establishes a structural relationship between

[TABLE]

Each integral is over self-adjoint $\mathcal{N}\times\mathcal{N}$-matrices, $\lambda$ is a scalar and $E$ a positive111In moments of the measure the denominator drops out. Then in the quartic case and for $\mathrm{Re}(\lambda)>0$, $E$ is chosen self-adjoint but positivity is not necessary. $\mathcal{N}\times\mathcal{N}$-matrix. The left ‘cubic’ case $\mathcal{Z}_{E}^{3}$ is the partition function of the Kontsevich model [Kon92]; it is of paramount importance in several areas of mathematics and mathematical physics. We recall the main result, the proof of a conjecture due to Witten [Wit91]:

Theorem 1.1** ([Kon92]).**

The logarithm $\log\mathcal{Z}^{3}_{E}$ of the partition function is a formal power series in $\{t_{n}\}$, where $t_{n}:=-(2n-1)!!\frac{1}{\mathcal{N}}\mathrm{Tr}(E^{-2n-1})$. The coefficients in this series are intersection numbers of characteristic classes on the moduli spaces $\overline{\mathcal{M}}_{g,s}$ of stable complex curves.

On the other hand, the Kontsevich integral can be extended to define moments

[TABLE]

which can be interpreted as correlation functions in a matricial QFT model. In this setting, the large-$\mathcal{N}$ limit of $E$ is allowed to be an unbounded operator of dimension $\leq 6$ (see Definition 2.1); the correlation functions then need renormalisation. By solving Dyson-Schwinger equations, the genus-$g$ contributions to all renormalised correlation functions (1.2) have been determined in [GSW17, GSW18] for $g=0$ and in [GHW19a] for $g\geq 1$. The main tools are a Ward-Takahashi identity similar to [DGMR07], the solution [MS91] of a non-linear integral equation as well as differential operators in the $t_{n}$ and residue techniques inspired by topological recursion [EO07, Eyn16].

From a quantum field theoretical point of view, the cubic interaction $\mathrm{Tr}(\Phi^{3})$ in (1.2) is not the first choice. One would rather be interested in the quartic analogue

[TABLE]

of the Kontsevich model. Such correlation functions arose in studies of quantum field theories on noncommutative geometries [GW05].

This paper proves that both cases in (1.1) share identical structures, at least in their simplest topological sector. We prove that also the quartic model is exactly solvable in terms of implicitly defined functions of the same type as in the cubic Kontsevich model:

Theorem 1.2**.**

Consider the quartic matrix model with measure $e^{-\mathcal{N}\,\mathrm{Tr}(E\Phi^{2}+(\lambda/4)\Phi^{4})}d\Phi$, in which the self-adjoint $\mathcal{N}\times\mathcal{N}$-matrix $E$ has eigenvalues $0<E_{1}<E_{2}<\dots<E_{d}$ of multiplicities $r_{1},r_{2},\dots,r_{d}$. These data encode a meromorphic function

[TABLE]

where $\{\varepsilon_{k},\varrho_{k}\}_{k=1,\dots,d}$ are the unique solutions in an open neighbourhood of $\lambda=0$ of

[TABLE]

with $\lim_{\lambda\to 0}\varepsilon_{k}=E_{k}$ and $\lim_{\lambda\to 0}\varrho_{k}=r_{k}$. For any $u\neq-\varepsilon_{k}$, let $z\in\{u,\hat{u}^{1},\dots,\hat{u}^{d}\}$ be the list of roots of $J(z)=J(u)$. Then the planar two-point function

[TABLE]

satisfies an equation that extends into the complex plane where, in an open neighbourhood of $\lambda=0$, it is solved by the rational function

[TABLE]

with $G_{ab}^{(0)}=\mathcal{G}^{(0)}(\varepsilon_{a},\varepsilon_{b})$. This function $\mathcal{G}^{(0)}(z,w)$ is symmetric in $z,w$ and defined outside poles located at $z+w=0$, at $z=\widehat{\varepsilon_{k}}^{m}$ and at $w=\widehat{\varepsilon_{l}}^{n}$, for $k,l,m,n=1,\dots,d$.

Theorem 1.2, combined with previous work [GW14a, dJHW19], shows that all moments (1.3) of the simplest topology $(B=1,g=0)$ can be exactly solved, as convergent functions of $\lambda$, for any operator $E$ (of dimension $\leq 4$ to be introduced). It is plausible that all other topological sectors become accessible, and that they will all be rational functions evaluated at implicitly defined points. Working out the details is a long-term programme.

Theorem 1.2 confirms a conjecture which crystallised during a decade of work of two of us (HG, RW). Building on a Ward-Takahashi identity found in [DGMR07], we derived 10 years ago in [GW09] a closed non-linear integral equation for $G_{ab}^{(0)}$ in the large-$\mathcal{N}$ limit. Over the years we found so many surprising facts about this equation that the quartic matrix model being solvable is the only reasonable explanation. A key step was the reduction to an equation for an angle functions of essentially only one variable [GW14a]. Moreover, a recursive formula to determine all planar $N$-point functions $G^{(0)}_{b_{0}\dotsb_{N-1}}$ from the planar two-point function $G_{ab}^{(0)}$ was found in [GW14a]. This recursion was recently solved in terms of a combinatorial structure named ‘Catalan table’ [dJHW19]. Last year, one of us with E. Panzer obtained in [PW18] the exact solution of $G_{ab}^{(0)}$ (at large $\mathcal{N}$) in the case $E=\mathrm{diag}(1,2,3,4,\dots)$. The solution is expressed in terms of the Lambert function defined by the implicit equation $W(z)\exp(W(z))=z$.

The final step achieved in this paper is the understanding that the structure discovered in [PW18] generalises to any $E$ of dimension $\leq 4$. The main obstacle was to identify a family $\rho_{\lambda}$ of deformed measures defined by an implicit equation in terms of $E,\lambda$. In the case of finite matrices, the deformed measure defines a rational function $J$ with $d+1$ inverses, where $d$ is the number of distinct eigenvalues of $E$.

This paper is organised as follows. In sec. 2 we recall from [GW14a] the non-linear equation for the planar two-point function and the solution ansatz in terms of an angle function. We also introduce a dimension $D$ encoded in $E$. The resulting non-linear integral equation for the angle function is solved in sec. 3. We construct a sectionally holomorphic function $J_{D}$ out of the data $(E,\lambda)$ of the model and its dimension $D$. We guess a formula for the angle function in terms of $J_{D}$ and prove that this formula satisfies the non-linear equation. Then in sec. 4 we compute the two-point function $G_{ab}^{(0)}$ from the angle function. In dimension $D\in\{2,4\}$ we obtain an explicit integral representation of $G_{ab}^{(0)}$. In the case of finite matrices (necessarily $D=0$) this integral can also be evaluated and gives the result stated in Theorem 1.2. Here $J$ and $J_{0}$ are related by a shift. In sec. 5 we outline structures which could be relevant for the solution of higher topological sectors. These structures share some aspects with topological recursion [EO07] which organises the solution of the Kontsevich model, but there are important differences. A few examples are given in sec. 6. We finish by a summary (sec. 7).

2 The setup

It was proved in [GW14a] that the second moment $ZG_{ab}$ of the quartic model,

[TABLE]

satisfies in the large-$\mathcal{N}$ limit the closed equation

[TABLE]

Here, $\{E_{a}\}$ are the eigenvalues of $E$, and the $\Phi_{ab}$ in (2.1) are the matrix elements of $\Phi$ in the eigenbasis of $E$. Equivalently, dropping $\mathcal{O}(\mathcal{N}^{-1})$ amounts to saying that $G_{ab}\mapsto G_{ab}^{(0)}$ is the planar 2-point function. The key observation is that, writing

[TABLE]

then $G(x,y)$ originally defined only on the (shifted) spectrum of $E$ extends to a sectionally holomorphic function which satisfies the integral equation222Strictly speaking, we should write $\int_{-\epsilon}^{\Lambda^{2}+\epsilon}$ for the integrals in (2.4) and (2.6).

[TABLE]

Here and from now on, the $\mathcal{O}(\mathcal{N}^{-1})$-correction is ignored, and we have defined

[TABLE]

Following [PW18] one can also derive a symmetric equation equivalent to (2.4):

[TABLE]

These equations are the analogue of the equation

[TABLE]

in the Kontsevich model (in dimension $D=0$; generalised in [GSW17, GSW18] to $D\in\{2,4,6\}$). Its solution found by Makeenko and Semenoff [MS91] was later understood as key ingredients of the spectral curve of topological recursion [EO07, Eyn16]. The solution is universal in terms of an implicitly defined parameter $c$, which depends on $E,\lambda$ and the dimension $D\in\{0,2,4,6\}$:

[TABLE]

This parameter $c$ effectively deforms the initial matrix $E$ to $(E^{2}+c/4)^{\frac{1}{2}}$ and thereby the measure $\rho_{0}$ into an implicitly defined deformed measure $\rho_{c}$.

We will see that exactly the same is true for the quartic model. Employing the same complex analysis techniques as in [MS91], we prove that equations (2.4) or (2.6) have a universal solution in terms of a deformation $\rho_{\lambda}$ of the measure (2.5).

In fact we solve the problem in a larger quantum field theoretical perspective. This refers to a limit $\mathcal{N}\to\infty$ in which the matrix $E$ becomes an unbounded operator on Hilbert space. For the Kontsevich model, the same quantum field theoretical extension was solved in [GSW17, GSW18, GHW19a]. Of course one can study a large-$\mathcal{N}$ limit in which $E-\frac{\mu^{2}_{bare}}{2}$ is resized to keep a finite support $[0,\Lambda^{2}]$ of the measure. We call this the dimension-0 case. It is only little more effort to solve the problem for two classes of unbounded operators $E$. Our strategy follows closely the usual renormalisation procedure in quantum field theory. This means that $\mu_{bare}^{2}$ and possibly $Z$ are carefully chosen functions of $\Lambda^{2}$, i.e. of the largest eigenvalue of $E$ (see (2.5)). As $\mathcal{N}\to\infty$, this largest eigenvalue also tends to $\infty$ with a certain rate that encodes a dimension according to Weyl’s law for the Laplacian:

Definition 2.1**.**

The spectral dimension of a spectral measure function $\rho$ is defined by $D_{spec}(\rho):=\inf\{p\;:~{}\int_{0}^{\infty}\frac{dt\;\rho(t)}{(t+1)^{p/2}}\text{ converges}\}$. The renormalisation procedure is classified by the number $D=2[\frac{1}{2}D_{spec}(\rho)]\in\{0,2,4,>4\}$ as follows:

$D=0:$

One can set $Z,\mu_{bare}^{2}$ to any finite value, e.g. $Z=1=\mu^{2}_{bare}$.

$D=2:$

One can set $Z$ to any finite value (e.g. $Z=1$), but $\mu^{2}_{bare}(\Lambda)$ diverges with $\Lambda^{2}$. The asymptotic behaviour determines itself, but there remains a freedom parametrised by a finite parameter $\mu^{2}>0$.

$D=4:$

The asymptotic behaviour of the divergent $\mu^{2}_{bare}(\Lambda)$ and $Z(\Lambda)$ for $\Lambda\to\infty$ determines itself. There remains a freedom parametrised by a finite parameter $\mu^{2}>0$ and a global finite factor multiplying $G(x,y)$. This irrelevant factor can be chosen to adjust $G(0,0)=1$.

$D>4:$

This case cannot be renormalised anymore.

We will determine a precise dependence $\mu^{2}(\Lambda)$ in $D\in\{2,4\}$ and $Z(\Lambda)$ in $D=4$ so that $G(x,y)$ has a well-defined limit $\mathcal{N}\to\infty$ (or $\Lambda^{2}\to\infty$).

We temporarily assume that the distributional measure $\rho_{0}(t)$ can be approximated by a Hölder-continuous function. The final result will make perfect sense for $\rho_{0}$ being a linear combination of Dirac measures; it is only that intermediate steps become more transparent if $\rho_{0}\in C^{0,\alpha}([0,\Lambda^{2}])$ is assumed. Using techniques for boundary values of sectionally holomorphic functions, explained in detail in [GW14a, GW14b, PW18], one finds that a solution for $G(a,b)$ at $0<a,b<\Lambda^{2}$ should be searched in the form

[TABLE]

where the angle function $\tau_{a}:(0,\Lambda^{2})\to[0,\pi]$ for $\lambda>0$ and $\tau_{a}:(0,\Lambda^{2})\to[-\pi,0]$ for $\lambda<0$ remains to be determined. Here,

[TABLE]

denotes the finite Hilbert transform. We go with the ansatz (2.9) into (2.4) at $x=a+\mathrm{i}\epsilon$ and $y=b$:

[TABLE]

A Hölder-continuous function $\tau:(0,\Lambda^{2})\to[0,\pi]$ or $\tau:(0,\Lambda^{2})\to[-\pi,0]$ satisfies

[TABLE]

The first identity appeared in [Tri57], the second one was proved in [PW18]. Inserting both identities into (2.11) gives with (2.9) a consistency relation for the angle function:

[TABLE]

where the $\arctan$-branch in $[0,\pi]$ is selected for $\lambda>0$ and the branch in $[-\pi,0]$ for $\lambda<0$.

Equation (2.13) was solved in [PW18] for $\rho_{0}(t)\equiv 1$. The hard part was to guess a possible solution. This guess was achieved by an ansatz for $\tau_{a}(p)$ as formal power series in $\lambda$ and evaluation of the first coefficients by the HyperInt package [Pan15]. These first coefficients had a striking pattern so that the whole series could be conjectured and resummed via Lagrange inversion theorem [Lag70] and Bürmann formula [Bür99]. Since these are the main tools also in this paper, we recall:

Theorem 2.2**.**

Let $\phi(w)$ be analytic at $w=0$ with $\phi(0)\neq 0$ and $f(w):=\frac{w}{\phi(w)}$. Then the inverse $g(z)$ of $f(w)$ with $z=f(g(z))$ is analytic at $z=0$ and given by

[TABLE]

More generally, if $H(z)$ is an arbitrary analytic function with $H(0)=0$, then

[TABLE]

The solution of (2.13) for $\rho_{0}(x)\equiv 1$ and in the limit $\Lambda\to\infty$ guessed in [PW18] is

[TABLE]

where $W_{0}$ is the principal branch of the Lambert function [CGH*+*96]. It was not so difficult to verify that (2.16) satisfies (2.13) for $\mu_{bare}^{2}=1-2\lambda\log(1+\Lambda^{2})$.

3 Solution of the angle function

In this paper we succeed in solving (2.13) for any Hölder-continuous $\rho_{0}$ of dimension $\leq 4$. Again the difficulty was to guess the solution; verifying it is a straightforward exercise in complex analysis. The main step is to deform the measure function. We first introduce structures for a fictitious measure $\rho_{c}$; later $\rho_{c}$ will be particularly chosen.

Definition 3.1**.**

Let $\rho_{c}$ be a Hölder-continuous function on some interval $[\nu_{D},\Lambda_{D}^{2}]$. For $\mu^{2}=\mu_{bare}^{2}$ in $D=0$, and $\mu^{2}>\min(0,-\nu_{D})$ a free parameter in $D\in\{2,4\}$, define functions $h_{00}\equiv h_{0},h_{02},h_{04},h_{2},h_{4}$ on $\mathbb{C}\setminus[\nu_{D},\Lambda_{D}^{2}]$ by

[TABLE]

Definition 3.2**.**

For $\lambda\in\mathbb{C}$ and $h_{D}$ as given in Definition 3.1, we introduce functions $J_{D}$ on $\mathbb{C}\setminus[-\mu^{2}-\Lambda_{D}^{2},-\mu^{2}-\nu_{D}]$ by

[TABLE]

The limits $\lim_{\Lambda_{D}^{2}\to\infty}h_{D}(z)$ and $\lim_{\Lambda_{D}^{2}\to\infty}J_{D}(z)$ exist for $\rho_{c}$ of dimension $D$ according to Definition 2.1. We have

[TABLE]

which is uniformly positive on $\mathbb{R}_{+}$ for real $\lambda>-(h_{0D}^{\prime}(-\mu^{2}))^{-1}$ in $D\in\{0,2\}$. In contrast,

[TABLE]

which is uniformly positive in the opposite region of real $\lambda<(h_{04}^{\prime}(-\mu^{2}))^{-1}$.

Lemma 3.3**.**

Let |\lambda|<\big{(}\int_{\nu_{D}}^{\Lambda_{D}^{2}}dt\frac{\rho_{c}(t)}{(t+\mu^{2}/2)^{2}}+\delta_{D,4}\int_{\nu_{D}}^{\Lambda_{D}^{2}}dt\frac{\rho_{c}(t)}{(t+\mu^{2})^{2}}\big{)}^{-1}. Then:

$J_{D}$* is a biholomorphic map from a right half plane $R_{\mu}:=\{z\in\mathbb{C}\;:~{}\mathrm{Re}(z)>-\frac{\mu^{2}}{2}\}$ onto a domain $U_{D}\subset\mathbb{C}$. For $\lambda$ real, $U_{D}$ contains $[J_{D}(\nu_{D}),\infty)$.* 2. 2.

For $\lambda$ real, $\mathrm{Im}(J_{D}(z))$ and $\mathrm{Im}(z)$ have the same sign for every $z\in R_{\mu}$.

Proof. 1. We show that $J_{D}$ is injective on $R_{\mu}$. Any two points $z_{0}\neq z_{1}\in R_{\mu}$ can be connected by a straight line $[0,1]\ni s\mapsto c(s)=z_{0}+(z_{1}-z_{0})s\in R_{\mu}$. Then for $D\in\{0,2\}$

[TABLE]

For $D=4$ we have

[TABLE]

which under the adapted condition leads to the same conclusion \big{|}J_{4}(z_{1})-J_{4}(z_{0})\big{|}>0.

It follows from basic properties of holomorphic functions that $J_{D}$ is, as holomorphic and injective function, even a biholomorphic map $J_{D}:R_{\mu}\to U_{D}:=J_{D}(R_{\mu})$.

2. For $\lambda$ real we have

[TABLE]

The term in $\{~{}\}$ is strictly positive by the same reasoning as above. $\square$

We can now define the ‘$\lambda$-deformed’ measure:

Definition 3.4**.**

Given $\lambda\in\mathbb{R}$, $\mu^{2}>0$ and a Hölder-continuous function $\rho_{0}:[0,\Lambda^{2}]\to\mathbb{R}_{+}$ of dimension $D$ according to Definition 2.1. Then a function $\rho_{\lambda}$ on $[\nu_{D},\Lambda_{D}^{2}]$ is implicitly defined by the equations

[TABLE]

where $h_{D}$ in $J_{D}$ is defined via (3.2) and (3.1) by the same function $\rho_{c}\mapsto\rho_{\lambda}$.

Remark 3.5*.*

The deformation from $\rho_{0}$ to $\rho_{\lambda}$ is the analogue of the deformation from $E$ to $E_{c}:=\sqrt{E^{2}+\frac{1}{4}c(\lambda)}$ in the Kontsevich model. There the deformation parameter $c(\lambda)$ is implicitly defined via (2.8). Neither that equation nor (3.8) in the quartic model can in general be solved in terms of ‘known’ functions.

The converse problems are easy: In the Kontsevich model, choose $E$ together with some parameter $c$ and take $\lambda(c)$ according to (2.8). In the quartic model, start with $\rho_{c}(x)$, read off $\rho(J_{D}(x))$ and reconstruct the deformed matrix $E_{c}$ in the partition function via (2.5).

Definition 3.6**.**

Given $\lambda\in\mathbb{R}$, $\mu^{2}>0$ and a Hölder-continuous function $\rho_{0}:[0,\Lambda^{2}]\to\mathbb{R}_{+}$ of dimension $D\in\{0,2,4\}$ according to Definition 2.1. Let $\rho_{\lambda}$ be its associated deformed measure according to Definition 3.4, and let $\lambda$ satisfy the requirements of Lemma 3.3 so that $J_{D}:R_{\mu}\to U_{D}$ is biholomorphic. Then a holomorphic function $I_{D}:U_{D}\setminus[0,\Lambda^{2}]\ni w\mapsto I_{D}(w)\in\mathbb{C}$ is defined by

[TABLE]

where $\mu$ in (3.9) and in Definition 3.2 are the same and $J_{D},h_{D}$ are defined with the deformed measure $\rho_{c}\mapsto\rho_{\lambda}$.

Theorem 3.7**.**

Let $\rho_{0}:[0,\Lambda^{2}]\to\mathbb{R}_{+}$ be a Hölder-continuous measure of dimension $D\in\{0,2,4\}$ and $\rho_{\lambda}$ its deformation according to Definition 3.4 for a real coupling constant $\lambda$ with |\lambda|<\big{(}\int_{\nu_{D}}^{\Lambda_{D}^{2}}dt\frac{\rho_{\lambda}(t)}{(t+\mu^{2}/2)^{2}}+\delta_{D,4}\int_{\nu_{D}}^{\Lambda_{D}^{2}}dt\frac{\rho_{\lambda}(t)}{(t+\mu^{2})^{2}}\big{)}^{-1}. Then the consistency equation (2.13) for the angle function is solved by

[TABLE]

with $I_{D}$ given by Definition 3.6, provided that the following relations between $\mu_{bare}$ and $\mu$ are arranged: $\mu_{bare}^{2}=\mu^{2}$ for $D=0$ and

[TABLE]

Proof. Assume (3.10). Then for the given range of $\lambda$ we have for $0<p<\Lambda^{2}$

[TABLE]

where 2. of Lemma 3.3, the definition of $h_{D}$ and the defining relation (3.8) between $\rho_{0}$ and $\rho_{\lambda}$ have been used. The $\arctan$ ranges in $[0,\pi]$ for $\lambda>0$ and in $[-\pi,0]$ for $\lambda<0$. Comparison with (2.13) shows (after renaming variables) that we have to prove

[TABLE]

We evaluate the integral over $\tau_{a}$. For $p>\Lambda^{2}$ we have $J_{D}^{-1}(p)>\Lambda_{D}^{2}$ and consequently $\mathrm{Im}(h_{D}(J_{D}^{-1}(p+\mathrm{i}\epsilon)))=0$. This implies

[TABLE]

In the second line, the contour $\gamma_{\epsilon}$ encircles $[\nu_{D},\infty)$ clockwise at distance $\epsilon$, i.e. it goes straight from $+\infty-\mathrm{i}\epsilon$ to $\nu_{D}-\mathrm{i}\epsilon$, in a left half circle to $\nu_{D}+\mathrm{i}\epsilon$ and straight again to $+\infty+\mathrm{i}\epsilon$. The denominator included in $T^{\epsilon}_{D}(a)$ is holomorphic in $U_{D}$ and does not contribute for $\epsilon\to 0$. The constants $c_{D}$ and $\kappa_{D}$ are chosen as $c_{2}=c_{0}=1$, $\kappa_{0}=0$ and

[TABLE]

We insert (3.9) and transform to $w=J(z)$:

[TABLE]

The function $h_{D}$ in (3.16) is defined with the $\lambda$-deformed measure $\rho_{\lambda}$. We will now

• •

rename $\rho_{\lambda}$ to $\rho_{c}$ and the given coupling constant $\lambda$ to $\lambda_{c}\in\mathbb{R}$,

• •

consider a general complex $\lambda\in\mathbb{C}$ (i.e. $h_{0D}$ will be taken as in (3.1) without any relation between $\rho_{c}$ and $\lambda$),

• •

take $\epsilon$ a fixed positive number.

In this setting, $z$ in (3.16) keeps distance $\epsilon$ from $[\nu_{D},\infty)$ so that (3.16) becomes a holomorphic function of $\lambda$ in a sufficiently small open ball around the origin. We choose its radius so small that the logarithm admits a uniformly convergent power series expansion on $\gamma_{\epsilon}$. Hence, integral and series commute:

[TABLE]

Since $h_{0D}(z)\propto z^{-1}$ for $|z|\to\infty$, we can close $\gamma_{\epsilon}$ by a large circle to a closed contour $\bar{\gamma}_{\epsilon}$ which avoids $[\nu_{D},\infty)$.

We first evaluate the part without $h_{0D}^{\prime}$ (and the global factor $c_{D}^{-1}$) by the residue theorem. Since $h_{0D}(z)$ is holomorphic in $\mathbb{C}\setminus[\nu_{D},\infty)$, only the pole of order $n$ at $z=-c_{D}(a+\kappa_{D})-\mu^{2}$ contributes:

[TABLE]

Setting $\phi(w)=h_{0D}(w-c_{D}(a+\kappa_{D})-\mu^{2})$, the Lagrange inversion formula (2.14) shows that $w=-K_{D}^{\epsilon}(a)$ is the inverse solution of the equation $-\lambda c_{D}=f(-K_{D}^{\epsilon}(a))$, where $f(w)=\frac{w}{\phi(w)}$. This means

[TABLE]

Introducing $z(a):=K_{D}^{\epsilon}(a)+c_{D}(a+\kappa_{D})$, equation (3.19) becomes

[TABLE]

Comparing with Definition 3.2, equation (3.20) boils down for any $D\in\{0,2,4\}$ to $a=J_{D}(z(a))$. But $a\in[0,\Lambda^{2}]\subset U_{D}$ so that we can invert to $z(a)=J_{D}^{-1}(a)$. In summary, we have proved a useful perturbative formula for $J_{D}^{-1}$:

Lemma 3.8**.**

For any $a\in[0,\Lambda^{2}]$ and $\lambda$ satisfying the assumptions of Lemma 3.3, the inverse function of $J_{D}$ defined in (3.2) admits a convergent representation

[TABLE]

We continue with (3.17). We insert (3.1) for $h_{0D}^{\prime}$ and change the integration order:

[TABLE]

We first look at generic points $t\neq c_{D}(a+\kappa_{D})$. This is no restriction because for Hölder-continuous $\rho_{c}$, ordinary and improper integral (the point $t=c_{D}(a+\kappa_{D})$ removed) agree. The residue theorem picks up the simple pole at $z=-\mu^{2}-t$, for which we resum the series to the logarithm, and the pole of order $n$ at $z=-c_{D}(a+\kappa_{D})-\mu^{2}$:

[TABLE]

where $H_{t,a}(w):=\log\frac{w+t-c_{D}(a+\kappa_{D})}{t-c_{D}(a+\kappa_{D})}$. The original dependence on $\epsilon$ dropped out. The Bürmann formula (2.15) identifies the term in $\{~{}\}$ of the last line of (3.23) as $H_{t,a}(-K^{\epsilon}_{D}(a))$:

[TABLE]

We have used $K^{\epsilon}_{D}(a)+c_{D}(a+\kappa_{D})=J_{D}^{-1}(a)$ and rearranged the denominator with (3.2) to $J_{D}(t)-a$.

We stress that (3.24) is proved for complex $\lambda$ in a ball about the origin of small radius determined by $\epsilon$. The identity theorem for holomorphic functions allows us to enlarge the domain of $\lambda$ on both sides back to the original domain of the Theorem. This includes the original real value $\lambda=\lambda_{c}$ we started with, where $\rho_{c}=\rho_{\lambda}$ on the rhs and $\lim_{\epsilon\to 0}T^{\epsilon}_{D}(a)=\frac{1}{\pi}\int_{0}^{\Lambda^{2}}dp\;\tau_{a}(p)$ on the lhs. Therefore, for the original real $\lambda$,

[TABLE]

where also $J_{D}$ is built from $\rho_{\lambda}$.

The $t$-integral in (3.25) does not need any exception point. But for the next step it is useful to remove an $\epsilon$-interval about $t=J_{D}^{-1}(a)$ to take the logarithms apart. These principal value integrals can equivalently be written as limit of the real part when shifting $a$ to $a+\mathrm{i}\epsilon$:

[TABLE]

Here we have completed the first $t$-integral $h_{0D}(J_{D}^{-1}(a+\mathrm{i}\epsilon))$ with (3.1) to $h_{D}(J_{D}^{-1}(a+\mathrm{i}\epsilon))$ and transformed in the second integral to $x=J_{D}(t)$. Taking the relation (3.8) to the original measure into account and recalling the definition (3.9) of $I_{D}(a)$, we precisely confirm our aim (3.13) provided that

[TABLE]

This finishes the proof. $\square$

4 Solution of the 2-point function

4.1 Hilbert transform of the angle function

With $\tau_{a}(p)$ determined, it remains to evaluate the Hilbert transform in the equation (2.9) for the two-point function $G(a,b)$. We first establish a general integral representation. In the next subsection this integral will be evaluated for the case of finite matrices.

Proposition 4.1**.**

The renormalised two-point function of the $D$-dimensional quartic matrix model is given by

[TABLE]

where

[TABLE]

and $J_{D}$ is built via (3.2) and (3.1) with the deformed measure $\rho_{\lambda}$ defined in (3.8). In $D=4$ dimensions, $G(a,b)$ is only determined up to a multiplicative constant which here is normalised to $G(0,0)=1$ independently of $\mu$. For $D\in\{0,2\}$ there is an alternative representation

[TABLE]

Proof. We rely on structures developed during the proof of Theorem 3.7. The Hilbert transform of $\tau_{a}$ given by (3.10) can be written as

[TABLE]

In the second line, $\epsilon^{\prime}$ must be chosen much larger than $\epsilon$ so that $J_{D}(\gamma_{\epsilon})$ separates $b+\mathrm{i}\epsilon^{\prime}$ from $\mathbb{R}_{+}$. As before, we are allowed to include a holomorphic denominator $\mu^{2}+a+J_{D}^{-1}(w)$. In contrast to the procedure in Theorem 3.7 we choose it such that it has individually a limit for $\Lambda\to\infty$. This leads to the large-$w$ behaviour

[TABLE]

Thus, in dimension $D\in\{0,2\}$ where $c_{D}=1$, the integrand decays sufficiently fast to deform $\gamma_{\epsilon}$ near $\infty$. For $D=4$, however, $c_{D}\neq 1$ prevents the deformation. This forces us to subtract the Hilbert transform \mathcal{H}^{\Lambda}_{r}\big{[}\tau_{r}(\bullet)] at some reference point $a=b=r>0$. We first move $J_{D}(\gamma_{\epsilon})$ past the pole $w=b+\mathrm{i}\epsilon^{\prime}$ at expense of its residue. In the remaining integral (which is automatically real) we transform to $w=J_{D}(z)$:

[TABLE]

The line (4.5a) evaluates to

[TABLE]

where real and imaginary part of $a+I_{D}(b+\mathrm{i}\epsilon^{\prime})$ are rearranged to $\tau_{a}(b)$ as in (3.12).

In the last line (4.5b) we write

[TABLE]

Inserted back into (4.5b) we deform in the parts with products of logarithms the contour $\gamma_{\epsilon}$ into the straight line $-\frac{\mu^{2}}{2}+\mathrm{i}\mathbb{R}$. No poles or branch cuts are hit during this deformation because $J_{D}(z)$ and $J_{D}(-\mu^{2}-z)$ are holomorphic on the slit half plane $\{\mathrm{Re}(z)>-\frac{3}{5}\mu^{2}\}\setminus[\nu_{D},\infty)$. In this way we produce integrals which are manifestly symmetric in both variables:

[TABLE]

The counterterm for $D=4$ is indispensable for convergence. Now the line (4.5b) becomes

[TABLE]

For any $D\in\{0,2,4\}$ we can add the convergent integral \frac{1}{2\pi\mathrm{i}}\int_{\gamma_{\epsilon}}dz\;\big{(}\frac{\log c_{D}}{z-b}-\delta_{D,4}\frac{\log c_{D}}{z-r}\big{)}=0 (in $D\in\{0,2\}$ we have $c_{D}=1$ whereas for $D=4$ we close $\gamma_{\epsilon}$ and use the residue theorem).

We follow the same strategy as in Theorem 3.7: $\rho_{\lambda}$ is renamed to $\rho_{c}$ and held fixed, $h_{D}$ and $J_{D}$ are built with $\rho_{c}$ and an independent complex $\lambda$ in a sufficiently small ball about the origin. Its radius is determined by $\epsilon$ which is also kept fixed. Also $\Lambda_{D}^{2}$ is still finite, and $a-ac_{D}^{-1}$ and $\kappa_{D}$ have according to (3.15) a factor $\lambda$ in front of them. After all, the logarithm in (4.8) admits a uniformly convergent power series expansion for any $z$ on $\gamma_{\epsilon}$. Every term of the expansion decays sufficiently fast for $z\to\infty$ to admit a closure of $\gamma_{\epsilon}$ to the contour $\bar{\gamma}_{\epsilon}$ that avoids $[\nu_{D},\infty)$. We proceed by the residue theorem. This is simpler than in Theorem 3.7 because $\frac{1}{z-b},\frac{1}{z-r}$ and $h_{0D}(z)$ are holomorphic in the interior of $\bar{\gamma}_{\epsilon}$ and on $\bar{\gamma}_{\epsilon}$ itself:

[TABLE]

where H_{a,b}(w)=\log\big{(}\frac{w-\mu^{2}-a-b}{-\mu^{2}-a-b}\big{)}. We apply the Bürmann formula (2.15). For that we need the auxiliary series

[TABLE]

In the same way as in the proof of (3.19), the Lagrange inversion formula (2.14) yields

[TABLE]

which by (3.1) and (3.2) rearranges into $a=J_{D}(a+L_{D}(a))$ in any dimension $D\in\{0,2,4\}$. We invert it to $a+L_{D}(a)=J_{D}^{-1}(a)$, but question this step for $D=4$ in Remark 4.2. The Bürmann formula (2.15) now gives

[TABLE]

By the identity theorem for holomorphic functions, this equation holds in the larger common $\lambda$-holomorphicity domain of both sides. It contains the original real coupling constant so that $J_{D}$ in (4.10) extends to the situation formulated in the Proposition.

It remains to collect the pieces: We want to evaluate (2.9). We set $Z=1$ in $D\in\{0,2\}$ and $Z=C_{r}e^{\mathcal{H}^{\Lambda}_{r}[\tau_{r}(\bullet)]}$ in $D=4$, where $C_{r}$ is a finite number. We thus need the exponential of (4.5), which is the exponential of (4.6) times the exponential of (4.10). This is to be multiplied by $\frac{\sin\tau_{b}(a)}{\lambda\pi\rho_{0}(a)}$ which cancels with the corresponding term in (4.6):

[TABLE]

For $D\in\{0,2\}$ this already gives (4.3) after reconstructing $J_{D}$ from $h_{D}$.

As discussed in Remark 4.2 after the proof, this equation is not appropriate for all cases of $D=4$. We can already in (4.5b) deform the contour $\gamma_{\epsilon}$ to the straight line $-\frac{\mu^{2}}{2}+\mathrm{i}\mathbb{R}$. After trading $h_{D}$ in (4.7) for $J_{D}$ via (3.2), equation (4.10) can be written as

[TABLE]

Inserting this and its flip $a\leftrightarrow b$ back into (4.11) gives rise to a representation where $J_{D}^{-1}$ is avoided completely:

[TABLE]

We can absorb the $r$-dependent factors arising for $D=4$ by an appropriate choice of $C_{r}$ and then adjust $C_{r}$ further to have $G(0,0)=1$. This amounts to replace $\tilde{\tilde{N}}_{4}(a,b)$ by $N_{4}(a,b):=\tilde{\tilde{N}}_{4}(a,b)-\tilde{\tilde{N}}_{4}(0,0)$. $\square$

Remark 4.2*.*

The representation (4.3), renormalised to $\frac{G(a,b)}{G(0,0)}$, might fail for $D=4$. For finite $\Lambda$, as seen in the proof above, the representations (4.3) and (4.1)+(4.2) are equivalent for $|\lambda|$ small enough. But in the limit $\Lambda\to\infty$ it can happen that $J_{4}(\mathbb{R}_{+})$ defined by (3.2) develops an upper bound for any $\lambda>0$, independently of whether $\rho_{c}$ is discrete or continuous. In such a case $J_{4}^{-1}(a)$ does not exist for all $a\in\mathbb{R}$ and (4.3) becomes meaningless for $\Lambda\to\infty$, whereas (4.1)+(4.2) do not show any problem.

In [GHW19b] we prove that for the measure function $\rho_{0}(t)=t$, of spectral dimension exactly $4$, there is no such problem. But other cases with $4<D_{spec}(\rho_{0})<6$ are very likely affected. It is the identification $a+L_{4}(a)=J_{4}^{-1}(a)$ made before (4.10) which might fail for $\Lambda\to\infty$. For the same reasons, also $\tau_{a}(p)$ given in (3.10) with (3.9) does not have a limit $\Lambda\to\infty$ for $D=4$ and $\lambda>0$. Such problems have been noticed in [GW14b]. They concern only auxiliary functions; the final result (4.1)+(4.2) is consistent for all $\lambda>0$.

4.2 The deformed measure for finite matrices

For the original problem of (finite) $\mathcal{N}\times\mathcal{N}$-matrices, the construction of the deformed measure is particularly transparent. It gives rise to a rational function $J$ for which the remaining integral of Proposition 4.1 can be evaluated.

In dimension $D=0$ the special treatment of the lowest eigenvalue $E_{1}=\frac{\mu_{bare}^{2}}{2}=\frac{\mu^{2}}{2}$ is no longer necessary. The notation simplifies considerably when redefining $J(z):=\frac{\mu^{2}}{2}+J_{0}(z-\frac{\mu^{2}}{2})$. Let $0<E_{1}<E_{2}<\dots<E_{d}$ be the eigenvalues of $E$ and $r_{1},\dots,r_{d}$ their multiplicities, with $\sum_{k=1}^{d}r_{k}=\mathcal{N}$. We shift the measure (2.5) to $\rho(t):=\rho_{0}(t-\frac{\mu^{2}}{2})$:

[TABLE]

The deformed measure is according to (3.8) given by

[TABLE]

where $J_{0}$, and thus $J$, arises via (3.2) and (3.1) from the same measure $\rho_{\lambda}$:

[TABLE]

This equation and its derivative evaluated at $z_{l}=J^{-1}(E_{l})=\varepsilon_{l}$ for $l=1,\dots,d$ provide a system of $2d$ equations for the $2d$ parameters $\{\varepsilon_{k},\varrho_{k}\}$:

[TABLE]

The implicit function theorem guarantees a solution in an open $\lambda$-interval, and one explicitly constructs a sequence converging to the solution $\{\varepsilon_{k},\varrho_{k}\}$. Alternatively, (4.17) can be interpreted as a system of $2d$ polynomial equations ($d$ of them of degree $d+1$, the other $d$ of degree $2d+1$). Such system have many solutions, and they will indeed be needed in intermediate steps. The right solution is the one which for $\lambda\to 0$ converges to $\{E_{k},r_{k}\}$.

4.3 Proof of Theorem 1.2

Recall from (2.3) that $G^{(0)}_{ab}=G(x,y)$ where $x+\frac{\mu^{2}}{2}=J(\varepsilon_{a})$ and $y+\frac{\mu^{2}}{2}=J(\varepsilon_{b})$. The ansatz (2.9) for $G(x,y)$ is turned with (4.5) and (4.6) into the representation

[TABLE]

Here the integration variable $z$ in (4.5b) is shifted into $z+\frac{\mu^{2}}{2}\mapsto z$, and $\gamma_{\epsilon}^{\prime}$ is the shifted contour which encircles $[\nu_{0}+\frac{\mu^{2}}{2},\infty)$. We have $\lambda h_{0}(z-\mu^{2}/2)=-z-J(-z)$ from (3.2).

For any $v,z\notin\{-\varepsilon_{1},\dots,-\varepsilon_{d}\}$ we can expand the rational function $J(z)-J(v)$ according to (4.16) into

[TABLE]

Here $\hat{v}^{1},\dots,\hat{v}^{d}$ are the other roots of the numerator polynomial; they are functions of $v$ and the initial data $E,\lambda$. For real $v$ it follows from the intermediate value theorem that these roots are interlaced between the poles $\{-\varepsilon_{k}\}$ of $J$. In particular, for $v\geq 0$ and $\lambda>0$ all $\hat{v}^{k}$ are real and located in $-\varepsilon_{k+1}<\hat{v}^{k}<-\varepsilon_{k}$ for $k=1,\dots d-1$ and $\hat{v}^{d}<-\varepsilon_{d}$.

Inserting (4.19) into (4.18) gives

[TABLE]

Lemma 4.3**.**

For $u,v>0$, a posteriori extented to a neighbourhood of $[\nu_{0}+\frac{\mu^{2}}{2},\infty)$, one has

[TABLE]

Proof. As before, for finite $\epsilon$ and for $\lambda$ in a small open ball, the logarithm in (4.20) can be expanded. After closing the integration contour, the residue theorem picks up the simple poles at $z=\hat{v}^{k}$ and $z=-\varepsilon_{k}$ and the poles of $n$-th order at $z=-J(v)$. The other candidates $z=v$ and $z=\varepsilon_{k}$ are outside the contour for real $u,v$. The poles of $n$-th order combine (up to a global sign) to a Bürmann formula (2.15) for H_{u,v}(w):=\log\big{(}\frac{J(w-J(u))-J(v)}{J(-J(u))-J(v)}\big{)}, where $w\mapsto-L(u)$ solves the auxiliary integral

[TABLE]

The Lagrange inversion formula (2.14) gives $-\lambda\frac{1}{\mathcal{N}}\sum_{l=1}^{d}\frac{\varrho_{l}}{\varepsilon_{l}+J(u)+L(u)}=-L(u)$, which is solved by $J(u)+L(u)=u$. Putting everything together, the integral (4.20) evaluates to

[TABLE]

The identity (4.19) applied for $z=-J(u)$ simplifies this to (4.21). $\square$

The representation (4.21) is rational in the first variable. There are two ways to proceed. First, we can expand (4.21) via (4.19) to

[TABLE]

This formula is manifestly symmetric in $u,v$ — a crucial property below. But it needs all roots of $J$, which exist only in a neighbourhoud of $[\nu_{0}+\frac{\mu^{2}}{2},\infty)$, not globally.

The limit $u\to\varepsilon_{a}$ of (4.21) gives with $r_{a}=\varrho_{a}J^{\prime}(\varepsilon_{a})$:

Corollary 4.4**.**

For any $a=1,\dots,d$ and $v$ in a neighbourhood of $\mathbb{R}_{+}$ one has

[TABLE]

In particular, for any $a,b=1,\dots,d$ one has

[TABLE]

Next we recall the basic lemma

[TABLE]

valid for pairwise different $x_{0},\dots,x_{d}$ and any $c_{1},\dots,c_{d}$. (The rational function of $x_{0}$ has potential simple poles at $x_{0}=x_{k}$, $k=1,\dots,d$, but all residues cancel. Hence, it is an entire function of $x_{0}$, by symmetry in all $x_{k}$. The behaviour for $x_{0}\to\infty$ gives the assertion.) We use (4.26) for $x_{0}=J(u)$, $x_{k}=J(\varepsilon_{k})$ and $c_{k}=J(-\hat{v}^{k})$ to rewrite (4.21) as

[TABLE]

The second line results from (4.24). Using the symmetry $\mathcal{G}^{(0)}(\varepsilon_{k},v)=\mathcal{G}^{(0)}(v,\varepsilon_{k})$, the previous formulae give rise to a representation of $\mathcal{G}^{(0)}(u,v)$ which is rational in both variables. The assertion (1.5) in Theorem 1.2 follows from symmetry $\mathcal{G}(\varepsilon_{k},w)=\mathcal{G}(w,\varepsilon_{k})$ and insertion of $\mathcal{G}(w,\varepsilon_{k})$ given by (4.21) into (4.27). We could also insert the symmetrised version of (4.27),

[TABLE]

back into (4.27). The remaining assertion of Theorem 1.2 about the poles of $\mathcal{G}^{(0)}(z,w)$ will be established in Proposition 4.6 below. $\square$

Remark 4.5*.*

For any $z\neq\varepsilon_{k}$ one has

[TABLE]

We already know this identity. The original equation (2.2) for $G^{(0)}_{ab}=\mathcal{G}^{(0)}(\varepsilon_{a},\varepsilon_{b})$ with $\mathcal{O}(\mathcal{N}^{-1})$-contributions dropped (in accordance with planarity) extends to complex variables $\varepsilon_{a}\mapsto z$ and $\varepsilon_{b}\mapsto w$, with $w+z\neq 0$ and $w,z\notin\{\varepsilon_{k}\}_{k=1,\dots d}\cup\{\widehat{\varepsilon_{k}}^{l}\}_{k,l=1,\dots d}$:

[TABLE]

Now (4.29) follows by comparison with (4.27).

Equation (4.29) has also been established for Hölder-continuous measure in Theorem 3.7. Namely, when expressed in terms of the angle function and variables $a+\frac{\mu^{2}}{2}=J(\varepsilon_{a})$ and $b+\frac{\mu^{2}}{2}=J(\varepsilon_{b})$, the terms $\{~{}\}$ in the first line of (4.30) become $\lambda\pi\rho_{0}(a)\cot\tau_{b}(a)$. In (3.12) we had found $\lambda\pi\rho_{0}(a)\cot\tau_{b}(a)=\lim_{\epsilon\to 0}\mathrm{Re}(b+I_{0}(a+\mathrm{i}\epsilon))$, which translates into $\lambda\pi\rho_{0}(a)\cot\tau_{b}(a)=J(w)-J(-z)$ for $z\neq\varepsilon_{k}$. From that starting point we had derived (4.21) so that finding (4.29) from (4.27) is no surprise.

But there is another line of arguments. We could have started with (4.21) as an ansatz, from which alone we arrive at (4.27). If we could also prove (4.29) from (4.21) alone, then (4.30) is a consequence of the ansatz (4.21), and we have proved that (4.27) solves (4.30). To directly verify (4.29) as identity for rational functions, note that both sides approach $z$ for $z\to\infty$. The rhs has poles only at $z=\varepsilon_{k}$ with residue $\frac{\lambda}{\mathcal{N}}\varrho_{k}$. The same poles with the same residues also arise on the lhs, taking $r_{k}/J^{\prime}(\varepsilon_{k})=\varrho_{k}$ into account. But the lhs also has potential poles at $z=-\varepsilon_{j}$ and at all $z=\widehat{\varepsilon_{j}}^{n}$. We have $\operatorname{Res}_{z\to-\varepsilon_{j}}J(z)=-\frac{\lambda}{\mathcal{N}}\varrho_{j}$. Taking (4.21) for $\mathcal{G}^{(0)}(z,\varepsilon_{l})$ in which we have $\lim_{z\to-\varepsilon_{j}}\frac{J(z)-J(-\widehat{\varepsilon_{l}}^{k})}{J(z)-J(\varepsilon_{k})}=1$ for any $k,l$, one easily finds that $\mathcal{G}^{(0)}(-\varepsilon_{j},\varepsilon_{l})$ is regular for $j\neq l$ and that $\operatorname{Res}_{z\to-\varepsilon_{j}}\frac{\lambda}{\mathcal{N}}r_{j}\mathcal{G}^{(0)}(z,\varepsilon_{j})=\frac{\lambda}{\mathcal{N}}\frac{r_{j}}{J^{\prime}(\varepsilon_{j})}$, which thus cancels $\operatorname{Res}_{z\to-\varepsilon_{j}}J(z)=-\frac{\lambda}{\mathcal{N}}\varrho_{j}$.

Finally, from (4.27) we conclude

[TABLE]

where (4.25) together with $\varrho_{k}J^{\prime}(\varepsilon_{k})=r_{k}$ has been used in the second equality. The basic lemma (4.26) in $d$ variables $x_{k}=J(\varepsilon_{k})$ gives $\mathrm{Res}_{z\to\widehat{\varepsilon_{j}}^{n}}\sum_{k=1}^{d}r_{k}\mathcal{G}^{(0)}(z,\varepsilon_{k})=\frac{r_{j}}{J^{\prime}(\widehat{\varepsilon_{j}}^{n})}$, which precisely cancels $\operatorname{Res}_{z\to\widehat{\varepsilon_{j}}^{n}}\sum_{k=1}^{d}\frac{r_{k}}{J(\varepsilon_{k})-J(z)}=-\frac{r_{j}}{J^{\prime}(\widehat{\varepsilon_{j}}^{n})}$. In summary, (4.29) is a corollary of (4.21). $\square$

Proposition 4.6**.**

The planar two-point function has the (manifestly symmetric) rational fraction expansion

[TABLE]

Proof. Expanding the first denominator in (4.27) via (4.19), $\mathcal{G}^{(0)}(u,v)$ has potential poles at $u=-\hat{v}^{n}$ for every $n=1,\dots,d$. However, for $u=-\hat{v}^{n}$ the sum in the first line of (4.27) becomes $\sum_{k=1}^{d}\frac{1}{(J(-\hat{v}^{n})-J(\varepsilon_{k}))}\frac{\prod_{l=1}^{d}(J(\varepsilon_{k})-J(-\hat{v}^{l}))}{\prod_{k\neq j=1}^{d}(J(\varepsilon_{k})-J(\varepsilon_{j}))}=-1$ when using the basic lemma (4.26). Consequently, $\mathcal{G}^{(0)}(z,w)$ is regular at $z=-\hat{w}^{n}$ and by symmetry at $w=-\hat{z}^{n}$.

This leaves the diagonal $z+w=0$ and the complex lines ($z=\widehat{\varepsilon_{k}}^{m}$, any $w$) and ($w=\widehat{\varepsilon_{l}}^{n}$, any $z$) as the only possible poles of $\mathcal{G}^{(0)}(z,w)$. The function $(z+w)\mathcal{G}^{(0)}(z,w)$ approaches $1$ for $z,w\to\infty$. Its residues at $z=\widehat{\varepsilon_{k}}^{m},w=\widehat{\varepsilon_{l}}^{n}$ are obtained from (4.27):

[TABLE]

The second line follows from (4.28). $\square$

5 Remarks on higher topological sectors

and topological recursion

The solution of a large class of matrix models is governed by a universal structure called topological recursion [EO07]. Its initial data is given by a spectral curve $(\Sigma,\Sigma_{0},x,\omega_{0,1},\omega_{0,2})$ in which

• •

$x:\Sigma\to\Sigma_{0}$ is a covering (with branch points) of Riemann surfaces,

• •

$\omega_{0,1}$ is a meromorphic 1-form on $\Sigma$ which is regular at the branch points of $x$, and

• •

$\omega_{0,2}$ is a symmetric meromorphic $2$-form on $\Sigma\times\Sigma$ with a double pole on the diagonal.

From these data a family $\omega_{g,B}$ of differential forms on $\Sigma^{\times B}$ is constructed which in examples of matrix models are directly related to distinguished correlation functions.

A main example for this construction is the Kontsevich model [Kon92]. Its spectral curve is constructed from $x,y:\bar{\mathbb{C}}\to\bar{\mathbb{C}}$ given by333Set $\lambda^{2}\mapsto\frac{1}{8}$ to match conventions of [Eyn16, §6.4].

[TABLE]

where $\varepsilon_{k}$ are the eigenvalues of the deformed matrix $\sqrt{4E^{2}+c}$. Here the shift $c$ obeys the implicit equation $c=-\frac{8}{\mathcal{N}}\sum_{k=1}^{d}\frac{\lambda^{2}r_{k}}{\epsilon_{k}}=-\frac{8}{\mathcal{N}}\sum_{k=1}^{d}\frac{\lambda^{2}r_{k}}{\sqrt{4E_{k}^{2}+c}}$, where $E_{k}$ are the eigenvalues of the original matrix $E$ and $r_{k}$ the multiplicity of $E_{k}$. From $x,y$ in (5.1) one builds $\omega_{0,1}(z)=y(-z)x^{\prime}(z)dz$ and $\omega_{0,2}(z,z^{\prime})=\frac{dz\,dz^{\prime}}{(z-z^{\prime})^{2}}$. We refer to [EO07, Eyn16] for details.

Identifying similar structures in the quartic matrix model could be the key to its complete solution. A first indication comes from comparing the equations for the planar cylinder $(g=0,B=2)$ amplitude $\mathcal{G}^{(0)}(z|w)$ in the Kontsevich model and the quartic model. In [Eyn16, §6.4] the cylinder amplitude is denoted $\mathcal{G}^{(0)}(z|w)=16\lambda^{2}\Omega_{0,2}(z,w)$; it satisfies${}^{\ref{fn}}$ the equation

[TABLE]

In the quartic model, the planar cylinder amplitude satisfies the equation (see [GW14a], here with multiplicities admitted)

[TABLE]

Proceeding as before, i.e. setting $G_{|a|c|}^{(0)}=\mathcal{G}^{(0)}(\varepsilon_{a}|\varepsilon_{c})$ and inserting $E_{a}=J(\varepsilon_{a})$, (5.3) extends into the complex plane to an equation for $\mathcal{G}^{(0)}(z|w)$:

[TABLE]

By (4.29), the function in $\{~{}\}$ on the lhs of (5.4) simplifies to $J(z)-J(-z)$. Therefore, introducing two coverings $x,y:\overline{\mathbb{C}}\to\overline{\mathbb{C}}$ of Riemann surfaces by

[TABLE]

equation (5.4) becomes

[TABLE]

Up to the irrelevant prefactor $(8\lambda^{2})$ versus $(-\lambda)$, (5.2) and (5.6) have identical lhs. But incidentally, the choice (5.5) gives a vanishing rhs in (5.2), and no recursion takes place. Thus, despite remarkable similarities, the quartic model does not seem to fit into the standard scenario of topological recursion. It remains to be seen whether at least some quantities of the quartic matrix model follow a topopogical recursion [EO07] for the classical spectral curve $\mathcal{E}(x,y)$ determined by (5.5). This curve would be the resultant

[TABLE]

which is a polynomial of total degree $2d+1$ in both $x,y$ and of degree $d$ in every single variable $x,y$.

Developing from scratch the recursion of topological sectors in the quartic model is a longer programme. We can give a few comments on the first step, the solution of (5.6).

Let $z\in\{0,\pm\alpha_{1},\dots,\pm\alpha_{d}\}$ be the solutions of $J(z)-J(-z)=0$, with all $\alpha_{k}>0$. When treating (5.6) as a Carleman-type singular integral equation as in [GW14a], it is clear that $\mathcal{G}^{(0)}(\alpha_{k},w)$ is regular for all $w>0$. Therefore, setting $z\mapsto\alpha_{k}$ gives a system of $k$ affine equations

[TABLE]

They are easily solved for $\frac{r_{l}}{\mathcal{N}}\mathcal{G}^{(0)}(\varepsilon_{l}|w)$, $l=1,\dots,d$, in terms of the planar two-point function $\mathcal{G}^{(0)}(z,w)$ already known. Moreover, since $\mathcal{G}^{(0)}(z,w)$ depends on $z$ only via $J(\pm z)$, setting instead $z\mapsto-\alpha_{k}$ in (5.6) gives the same $\mathcal{G}^{(0)}(\varepsilon_{l}|w)$. With these $\mathcal{G}^{(0)}(\varepsilon_{l}|w)$ determined, (5.6) gives the explicit formula for $\mathcal{G}^{(0)}(z|w)$:

[TABLE]

The global denominator $J(z)-J(-z)$ introduces a pole only at $z=0$, but not at $z=\pm\alpha_{k}$.

In principle one can continue this strategy for solving the higher topological sectors. Formulae get extremely cumbersome, and explicit cancellations of the factors $(z\mp\alpha_{k})$ seem desirable. However, leaving $J(z)$ intact seems the only way to keep track of the behaviour at $z=\pm\alpha$. For instance, we immediately conclude $\mathcal{G}^{(0)}(\alpha_{k}|w)=-\mathcal{G}^{(0)}(-\alpha_{k}|w)$ for $w>0$ from (5.9), which would be obscure after any ‘simplification’ of (5.9).

Finally, let us point out that in the quartic model one also has to control correlation functions with an even number of variables per boundary component, such as $\mathcal{G}(z_{1},z_{2}|w_{1},w_{2})$ and $\mathcal{G}(z_{1},z_{2}|w|w^{\prime})$. These satisfy more complicated equations [GW14a] which we have not studied yet.

6 Examples

6.1 A Hermitean one-matrix model

The extreme case of a single $r_{1}=\mathcal{N}$-fold degenerate eigenvalue $E=\frac{\mu^{2}}{2}\cdot\mathrm{id}$ corresponds to a standard Hermitean one-matrix model with measure $\exp(-\mathcal{N}\,\mathrm{Tr}(\frac{\mu^{2}}{2}\Phi^{2}+\frac{\lambda}{4}\Phi^{4}))d\Phi$. This purely quartic case was studied in [BIPZ78]. Transforming $M\mapsto\sqrt{\mathcal{N}}\mu\Phi$ and $g=\frac{\lambda}{4\mu^{4}}$ brings [BIPZ78, eq. (3)] into our conventions. The equations (1.4) reduce for $E_{1}=\frac{\mu^{2}}{2}$ and $d=1$ to

[TABLE]

with principal solution (i.e. $\lim_{\lambda\to 0}\varepsilon_{1}=\frac{\mu^{2}}{2}$)

[TABLE]

The other root $\widehat{\varepsilon_{1}}^{1}$ with $J(\widehat{\varepsilon_{1}}^{1})=\widehat{\varepsilon_{1}}^{1}-\frac{\lambda\varrho_{1}}{\mathcal{N}(\varepsilon_{1}+\widehat{\varepsilon_{1}}^{1})}=J(\varepsilon_{1})=\frac{\mu^{2}}{2}$ is found to be

[TABLE]

The planar two-point function $G_{11}^{(0)}$ can be evaluated via (4.25) or (4.23) to

[TABLE]

The result can be put into $G_{11}^{(0)}=\frac{1}{3\mu^{2}}a^{2}(4-a^{2})$ for $a^{2}=\frac{2\mu^{2}}{\mu^{2}+\sqrt{\mu^{4}+12\lambda}}$ and thus agrees with the literature: This value for $a^{2}$, which corresponds to $\frac{a^{2}\lambda}{\mu^{2}}=\varepsilon_{1}-\frac{\mu^{2}}{2}$, solves [BIPZ78, eq. (17a)] for $g:=\frac{\lambda}{4\mu^{4}}$ so that (6.4) reproduces444thanks to a lucky coincidence: In [BIPZ78] expectation values of traces $\langle\mathrm{Tr}(M^{2p})\rangle$ are studied, whereas we consider $\langle M_{11}M_{11}\rangle$. For constant $E$ all moments of individual matrix elements are equal and agree up to global rescaling by $\mathcal{N}^{\delta}$ with expectation values of traces. [BIPZ78, eq. (27)] for $p=1$ (and the convention $G_{11}^{(0)}=\frac{1}{\mu^{2}}$ for $\lambda=0$).

The meromorphic extension $\mathcal{G}^{(0)}(z,w)$ is most conveniently derived from Proposition 4.6 after cancelling the two representations (6.4) for $G_{11}^{(0)}=\mathcal{G}^{(0)}(\varepsilon_{1},\varepsilon_{1})$:

[TABLE]

where $a^{2}=\frac{2\mu^{2}}{\mu^{2}+\sqrt{\mu^{4}+12\lambda}}$. We have used $J^{\prime}(\widehat{\varepsilon_{1}}^{1})=\frac{\widehat{\varepsilon_{1}}^{1}-\varepsilon_{1}}{\widehat{\varepsilon_{1}}^{1}+\varepsilon_{1}}$.

6.2 A special case in $D=2$: constant

density

The case $\rho_{0}(x)\equiv 1$ was solved in [PW18]. For $\rho_{0}(x)\equiv 1$ (or any other constant), the deformed measure (3.8) is necessarily the same, $\rho_{\lambda}=1$. Only their support at finite $\Lambda$ is different, but the relative error vanishes for $\Lambda^{2}\to\infty$. The dimensional classification of Definition 2.1 gives $D=2$, and from (3.2) we find in the limit $\Lambda\to\infty$ and for $\mu\equiv 1$

[TABLE]

The inverses are provided by the branches of Lambert-W [CGH*+*96], in particular

[TABLE]

Formulae (3.10) for $\tau_{a}(p)$ and (4.3) for $G(a,b)$ specify to their counterparts in [PW18].

To approach the remaining integral $N_{2}(a,b)$ one could try to approximate $J_{2}$ by a rational function. As a Stieltjes function, $\frac{\log(1+z)}{z}$ has uniformly convergent Padé approximants obtained by terminating the continued fraction

[TABLE]

after $2d-1$ or $2d$ fractions. This gives rise to a representation $G(a,b)=\lim_{d\to\infty}\mathcal{G}^{(0)}_{d}(\varepsilon_{a},\varepsilon_{b})$ with every $\mathcal{G}^{(0)}_{d}(z,w)$ of the form (1.5). The $\varepsilon_{a},\varepsilon_{b}$ also depend on $d$.

6.3 A particular case in $D=4$: linear density

$\rho_{0}(x)=x$

The case $\rho_{0}(x)=x$ corresponds to the self-dual $\Phi^{4}$-model on four-dimensional Moyal space [GW05, GW14a] and is therefore of particular interest. Coincidently, the equation for the deformed measure $\rho_{\lambda}$ is of exceptional type, namely a standard Fredholm integral equation of second kind:

[TABLE]

In [GHW19b] we prove that this equation is solved by a hypergeometric function:

[TABLE]

Remarkably, the spectral dimension $D_{spec}$ introduced in Definition 2.1 gets modified by the interaction from $D_{spec}(\rho_{0})=4$ to $D_{spec}(\rho_{\lambda})=4-2\frac{\arcsin(\lambda\pi)}{\pi}$. For $\lambda>0$, this dimension drop makes $J_{4}^{-1}$ globally defined on $\mathbb{R}_{+}$ and thus avoids the triviality problem of the matricial $\lambda\Phi^{\star 4}$-model.

7 Summary

The solution of the equation for the planar 2-point function $G_{ab}^{(0)}$, now achieved, is by far the hardest problem in the complete solution of the quartic matrix model. All other correlation functions satisfy a hierarchy of linear integral equations [GW14a]. Abstract solution formulae could be written down for all of them. We expect, however, that the solutions have a much simpler algebraic structure which is not visible in the abstract formulae. For example, all planar $N$-point functions with a single boundary component are linear combinations of products of two-point functions [GW14a]. The arising terms are in bijection with Catalan tables [dJHW19]. We remark that the same factorisation of planar $N$-point functions into weighted products of two-point functions arises in the Hermitean two-matrix model [EO05].

The next step in this programme is the solution of the linear integral equations for planar correlation functions with several boundary components (i.e. multi-traces), followed by the non-planar sector. Some indications were given in sec. 5. It seems that the recursion of the quartic model does not follow the standard procedure of topological recursion [EO07]. It is nevertheless clear that the tools developed in topological recursion will be highly relevant for the solution.

The solution of an interacting quantum field theory by rational functions is exceptional. We view this rationality as strong support for the conjecture that the quartic analogue of the Kontsevich model is integrable. Working out details and consequences is a fascinating endeavour for the next years.

Acknowledgements

RW would like to thank Erik Panzer for the joint solution of a special case which is indispensable prerequisite of the present paper. AH thanks Akifumi Sako for hospitality during a visit of the Tokyo University of Science where first thoughts to generalise the special case were developed. This work was supported by the Erwin Schrödinger Institute (Vienna) through a “Research in Team” grant and by the Deutsche Forschungsgemeinschaft via the Cluster of Excellence555“Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) im Rahmen der Exzellenzstrategie des Bundes und der Länder EXC 2044 –390685587, Mathematik Münster: Dynamik–Geometrie–Struktur” “Mathematics Münster” and the RTG 2149.