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This paper completes the reverse engineering of a special quantum field theory on noncommutative geometries. The final step could be of interest in other areas of mathematics:

Theorem 1.1


$$\begin{aligned} F_g(t_0,t_2,t_3,\dots ,t_{3g-2}) :=\sum _{(k)} \frac{\langle \tau _2^{k_2}\tau _3^{k_3}\dots \tau _{3g-2}^{k_{3g-2}} \rangle }{(1-t_0)^{2g-2+\sum _i k_i}} \prod _{i=2}^{3g-2} \frac{t_i^{k_i}}{k_i!},\quad \sum _{i\ge 2} (i-1) k_i=3g-3, \end{aligned}$$

be the generating function of intersection numbers of \(\psi \)-classesFootnote 1 on the moduli spaces \(\overline{{{\mathcal {M}}}}_{g,n}\) of stable complex curves of genus g [1, 2]. The stable partition function satisfies, as a formal power series in \(N^{-2}\),

$$\begin{aligned} \boxed {\displaystyle \exp \Big (\sum _{g=2}^\infty N^{2-2g} F_g(t)\Big ) =\exp \Big (-\frac{1}{N^2} \Delta _t +\frac{F_2(t)}{N^2}\Big ) 1} \end{aligned}$$

where \(\displaystyle F_2(t)= \frac{7}{240} \cdot \frac{t_2^3}{3! T_0^5} + \frac{29}{5760} \frac{t_2 t_3}{T_0^4} + \frac{1}{1152} \frac{t_4}{T_0^3}\) generates the intersection numbers of genus 2, \(T_0:=(1-t_0)\) and \(\Delta _t=-\sum _{i,j}{\hat{g}}^{ij}\partial _i\partial _j- \sum _{i}{\hat{\Gamma }}^{i}\partial _i\) is a Laplacian on the formal parameters \(t_0,t_2,t_3,\dots \) given by

$$\begin{aligned} \Delta _t&:=-\Big (\frac{2t_2^3}{45 T_0^3} + \frac{37 t_2t_3}{1050 T_0^2} + \frac{t_4}{210 T_0}\Big )\frac{\partial ^2}{\partial t_0^2} -\Big ( \frac{2t_2^3}{27 T_0^4} + \frac{1097 t_2t_3}{12600 T_0^3} +\frac{41 t_4}{2520 T_0^2}\Big ) \frac{\partial }{\partial t_0}\\&\quad -\sum _{k=2}^\infty \Big ( \Big (\frac{2t_2^2}{45 T_0^3} + \frac{2 t_3}{105 T_0^2}\Big ) t_{k+1} +\frac{t_2 {{\mathcal {R}}}_{k+1}(t) }{2T_0} +\frac{3 {{\mathcal {R}}}_{k+2}(t)}{2(3+2k)} \Big ) \frac{\partial ^2}{\partial t_k \partial t_0} \\&\quad -\sum _{k,l=2}^\infty \Big ( \frac{ t_2 t_{k+1}t_{l+1}}{90 T_0^2} + \frac{t_{k+1} {{\mathcal {R}}}_{l+1}(t)}{4T_0} + \frac{t_{l+1} {{\mathcal {R}}}_{k+1}(t)}{4T_0} \\&\quad +\frac{(1{+}2k)!!(1{+}2l)!!{{\mathcal {R}}}_{k+l+1}(t)}{4(1{+}2k{+}2l)!!} \Big ) \frac{\partial ^2}{\partial t_k\partial t_l} \\&\quad -\sum _{k=2}^\infty \Big ( \Big (\frac{19 t_2^2}{540 T_0^4} + \frac{5 t_3}{252 T_0^3}\Big ) t_{k+1} +\frac{t_2 {{\mathcal {R}}}_{k+1}(t)}{48T_0^2} +\frac{{{\mathcal {R}}}_{k+2}(t)}{16(3+2k) T_0} + \frac{t_2t_{k+2}}{90T_0^3} \\&\quad +\frac{{{\mathcal {R}}}_{k+2}(t)}{2T_0} \Big )\frac{\partial }{\partial t_{k}} \end{aligned}$$

with \({{\mathcal {R}}}_m(t):= \displaystyle \frac{2}{3} \sum _{k=1}^m \frac{(2m{-}1)!! \,kt_{k+1}}{(2k{+}3)!! T_0} \sum _{l=0}^{m-k} \frac{l!}{(m{-}k)!} B_{m-k,l} \Big (\Big \{\frac{j!t_{j+1}}{(2j{+}1)!!T_0}\Big \}_{j=1}^{m-l+1}\Big )\).

The \(F_g(t)\) are recursively extracted from \({{\mathcal {Z}}}_g(t):=\frac{1}{(g-1)!} (-\Delta _t+F_2(t))^{g-1} 1\) via

$$\begin{aligned} F_g(t)&=\textstyle {{\mathcal {Z}}}_g(t) - \frac{1}{(g-1)!}\sum _{k=2}^{g-1} B_{g-1,k}\big ( \big \{h! F_{h+1}(t)\}_{h=1}^{g-k}\big ) \\&= \textstyle {{\mathcal {Z}}}_g(t)- \frac{1}{(g-1)!} \sum _{k=2}^{g-1} (-1)^{k-1}(k-1)!B_{g-1,k}\big ( \big \{h! {{\mathcal {Z}}}_{h+1}(t)\}_{h=1}^{g-k}\big ). \end{aligned}$$

Here and in Theorem 1.1, \(B_{m,k}(\{x\})\) are the Bell polynomials (see Definition 4.10). These equations are easily implemented in any computer algebra system.

Theorem 1.1 seems to be closely related with \(\exp (\sum _{g\ge 0} F_g)= \exp ({\hat{W}})1\) proved by Alexandrov [3],Footnote 2 where \({\hat{W}}:=\frac{2}{3}\sum _{k=1}^\infty (k+\frac{1}{2}) t_k {\hat{L}}_{k-1}\) involves the generators \({\hat{L}}_n\) of the Virasoro algebra. Including N and moving \(\exp (N^2 F_0+F_1)\) to the other side, our \(\Delta _t\) is in principle obtained via Baker-Campbell-Hausdorff formula from Alexandrov’s equation. Of course, evaluating the necessary commutators is not viable.

Theorem 1.1 suggests several fascinating questions:

  • Is \({\hat{\Gamma }}^i\) a Levi-Civita connection for \({\hat{g}}^{ij}\), i.e. \({\hat{\Gamma }}^i= \sum _j {\hat{g}}^{ij}\sqrt{\det {\hat{g}}^{-1}} \partial _j (\frac{1}{\sqrt{\det {\hat{g}}^{-1}}})\)? Here \(\det {\hat{g}}^{-1}\) would be the determinant of \(( {\hat{g}}^{ij})\), whatever this means.

  • Is there a reasonable definition of a volume \(\int dt \frac{1}{\sqrt{\det {\hat{g}}^{-1}(t)}}\)?

  • Is it possible to prove that \(\sum _{g=2}^\infty N^{2-2g} F_g(t)\) is Borel summable for \(t_l<0\)?

Theorem 1.1 is a by-product of our effort to construct the non-planar sector of the renormalised \(\Phi ^3_D\)-matricial quantum field theory in any dimension \(D\in \{0,2,4,6\}\). These models are closely related to the Kontsevich model [2] so that a link to intersection numbers is not surprising.

2 Introduction

Matrix models have a huge scope of applications [4, 5], ranging from combinatorics over 2D quantum gravity [6, 7] up to quantum field theory on noncommutative spaces [8,9,10,11,12,13,14,15]. Of particular interest is the Kontsevich model [2], which was designed to prove Witten’s conjecture [1] that the generating function of intersection numbers of \(\psi \)-classes on the moduli space \(\overline{{{\mathcal {M}}}}_{g,n}\) of stable complex curves satisfies the KdV equations. See also [16,17,18].

More recent investigations of matrix models led to the discovery of a universal structure called topological recursion [19, 20]. Topological recursion was subsequently identified in many different areas of mathematics and theoretical physics [18, 21]. The Kontsevich model is, next to the Hermitian 1-matrix model (to which it is related via Miwa transformation; see e.g. [22]), the most basic example for topological recursion.

On the other hand, renormalisation of quantum field theories on noncommutative geometries generically leads to matrix models similar to the Kontsevich model. The crucial difference is that convergence of the (usually) formal series in the coupling constant is addressed, and achieved by renormalisation [11,12,13]. Renormalisation is sensitive to the dimension encoded in the covariance of the matrix model. For historical reasons, namely the perturbative renormalisation [10] of the \(\Phi ^4\)-model on Moyal space and its vanishing \(\beta \)-function [14], also the quartic analogue of the Kontsevich model was intensely studied. In [15] the simplest topological sector was reduced to a closed equation for the 2-point function. This equation was recently solved in [23] after understanding the pattern behind the solution for the 2D-Moyal space [24]. On the 4D-Moyal space, the solution for the 2-point function was derived in [25], which has resolved the triviality problem for this specific noncommutative \(\Phi ^4_4\)-model. All correlation functions with simplest topology (genus \(g=0\) and one boundary) can be explicitly described by a nested Catalan problem [26].

In [27, 28] these methods developed for the \(\Phi ^4\)-model were reapplied to the cubic (Kontsevich-type) model. The new tools, together with the Makeenko-Semenoff solution [29] of a non-linear integral equation, permitted an exact solution of all planar (i.e. genus-0) renormalised correlation functions in dimension \(D\in \{2,4,6\}\). In particular, exact (and surprisingly compact) formulae for planar correlation functions with \(B\ge 2\) boundary components were obtained. The simplicity of the formulae [27] for \(B\ge 2\) suggests an underlying pattern. It is traced back to the universality phenomena captured by topological recursion.Footnote 3 We refer to the book [18].

In this article we give the complete description of the non-planar sector of the renormalised \(\Phi _D^3\)-model. The notation defined in [28] will be used and recalled in Sect. 3. We borrow from topological recursion the notational simplification to complex variables z for the previous \(\sqrt{X+c}\) and the vision that the correlation functions are holomorphic in \(z \in {{\mathbb {C}}}\setminus \{0\}\) (see Sect. 4.1). Knowing this, we proceed however in a different way. In Sect. 4.2 we introduce our main tool, a boundary creation operator \(\hat{{\textrm{A}}}^{\dag g}_{z_1,\dots ,z_B}\) which, when applied to a genus-g correlation function \({{\mathcal {G}}}_g(z_1|\dots |z_{B-1})\) with \(B-1\) boundaries labelled \(z_1,\dots ,z_{B-1}\), creates a \(B^{\text {th}}\) boundary labelled \(z_B\). The existence of such an operator is suggested by the ‘loop insertion operator’ in topological recursion [18].

We rely on the sequence \(\{\varrho _l\}_{l\in {{\mathbb {N}}}}\) of moments of a measure arising from the renormalised planar 1-point function [27, 28]. This sequence is uniquely defined by the renormalised covariance of the model, the renormalised coupling constant and the dimension \(D\in \{0,2,4,6\}\). The boundary creation operator acts on Laurent polynomials in the \(z_i\) with coefficients in rational functions of the \(\varrho _l\). The heart of this paper is a combinatorial proof, independent of topological recursion, that the boundary creation operator does what it should (Theorem 4.6, portioned into Lemmas proved in an appendix). It is then (to our taste) considerably easier compared with topological recursion to derive in Sect. 4.3 structural results about the \({{\mathcal {G}}}_g(z_1|\dots |z_{B})\) such as the degree of the Laurent polynomials, the maximal number of occurring \(\{\varrho _l\}\) and the weight of the rational function.

The main part of Sect. 4.3 is devoted to the solution of \({{\mathcal {G}}}_g(z)\) for \(g\ge 1\). We find \({{\mathcal {G}}}_1(z)\) by a simple ansatz and \({{\mathcal {G}}}_g(z)\) for \(g \ge 2\) from a Dyson-Schwinger type equation (4.3),

$$\begin{aligned} {\hat{K}}_z{{\mathcal {G}}}_g(z)+ \lambda \sum _{h=1}^{g-1} {{\mathcal {G}}}_{h}(z){{\mathcal {G}}}_{g-h}(z) +8\lambda ^4 \hat{{\textrm{A}}}^{\dag \,g{-}1}_{z,z} {{\mathcal {G}}}_{g-1}(z)=0, \end{aligned}$$

where \({\hat{K}}_z\) an integral operator. Thus, all \({{\mathcal {G}}}_g(z)\) can be recursively evaluated if \({\hat{K}}_z\) can be inverted. Topological recursion tells us that the inverse is a residue combined with a special kernel operator. We give a direct combinatorial proof that the same method works in our case.

We show in Sect. 5.1 that the \({{\mathcal {G}}}_g(z)\) arise for \(g\ge 1\) by application of the boundary creation operator to a uniquely defined ‘free energy’ \(F_g(\varrho )\). These \(F_g(\varrho )\) are characterised by ’only’ \(p(3g-3)\) rational numbers, where p(n) is the number of partitions of n. We show in Sect. 5.2 that (2.1) can be written as a second-order differential operator acting on \(\exp (\sum _{g\ge 1} N^{2-2g} F_g)\) in which it is convenient to eliminate \(F_1\). The result is Theorem 1.1 expressed in terms of \(\varrho _0=1-t_0\) and \(\varrho _l =-\frac{t_l+1}{(2l+1)!!}\). In other words, to construct the non-planar sector of the \(\Phi ^3_D\)-matricial QFT model one has to replace the formal parameters \(t_l\) in the generating function \(F_g(t)\) of intersection numbers by precisely determined moments \(\{\varrho _l\}\) resulting from the renormalisation of the planar sector of the model. We finish with a discussion of the (deformed) Virasoro algebra in Sect. 5.3 and a short summary (Sect. 6).

In the meantime an analogue of the boundary creation operator \(\hat{{\textrm{A}}}^{\dag g}_{z_J,z}\) was found for the matricial \(\Phi ^4\)-model [30]. In this setting, two of us with J. Branahl introduced generalised correlation functions which satisfy an interwoven system of Dyson-Schwinger equations [31]. The solution for low \(|\chi |\) provided strong evidence for the conjecture that the matricial \(\Phi ^4\)-model (in that paper called quartic analogue of the Kontsevich model) satisfies blobbed topological recursion [32], a generalisation of topological recursion. The proof that the genus \(g=0\) sector satisfies blobbed topologcal recusion was achieved by two of us in [33], based on a functional relation related to the x-y-symmetry in topological recursion [34, 35]. Some geometrical aspects of these generalised correlation functions were analysed in [36]. Furthermore, the free energies \(F_0\) and \(F_1\) are computed in [37]. All these results are reviewed in [38]. It is also worth to mention that the complex \(\Phi ^4\)-model with complex instead of Hermitian matrix (also known as the LSZ-model [9] from noncommutative geometry perspective) was proved by one of us with J. Branahl to be governed by topological recusion [39].

3 Summary of Previous Results

3.1 Setup

We follow [28] to introduce tuples \(\underline{n}=(n_1,n_2,\ldots ,n_{\frac{D}{2}})\) of non-negative integers. The number of tuples \(\underline{n}\) of given \(|\underline{n}|=n_1+n_2+\cdots +n_{\frac{D}{2}}\) is \(\left( {\begin{array}{c}|\underline{n}|+\frac{D}{2}-1\\ \frac{D}{2}-1\end{array}}\right) \). Let \({{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}:=\{ \underline{m}\in {{\mathbb {N}}}^{D/2}: |\underline{m}|\le {{\mathcal {N}}}\}\). The action of the \(\Phi ^3_D\)-model is then defined by

$$\begin{aligned} S[\Phi ]&=V\bigg (\sum _{\underline{n},\underline{m} \in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \!\!\!\!\!\! Z\frac{H_{\underline{n}\underline{m}}}{2} \Phi _{\underline{n}\underline{m}}\Phi _{\underline{m}\underline{n}} + \!\!\!\! \sum _{\underline{n}\in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \!\!\!\! (\kappa {+}\nu E_{\underline{n}}{+}\zeta E_{\underline{n}}^2) \Phi _{\underline{n}\underline{n}} \nonumber \\&\quad +\frac{\lambda _{bare}Z^{3/2}}{3} \!\!\!\!\!\!\! \sum _{\underline{n},\underline{m},\underline{k} \in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \!\!\!\!\!\!\!\! \Phi _{\underline{n}\underline{m}} \Phi _{\underline{m}\underline{k}} \Phi _{\underline{k}\underline{n}}\bigg ), \end{aligned}$$

where \(H_{\underline{n}\underline{m}}:= E_{\underline{n}}+E_{\underline{m}}\). The constant V is first of all a formal parameter; for a noncommutative quantum field theory model, \(V=(\frac{\theta }{4})^{D/2}\) will be related to the deformation parameter of the Moyal plane. The parameters \(\lambda _{bare},\kappa ,\nu ,\zeta ,Z\) and soon \(\mu _{bare}\) are \({{\mathcal {N}}}\)-dependent renormalisation parameters. They are determined by normalisation conditions parametrised by physical mass \(\mu \) and coupling constant \(\lambda \). The matrices \((\Phi _{\underline{n}\underline{m}})\) are multi-indexed Hermitian matrices, \(\Phi _{\underline{n}\underline{m}} =\overline{\Phi _{\underline{m}\underline{n}}}\). The external matrix \(E=(E_{\underline{m}}\delta _{\underline{n},\underline{m}})\) can be assumed to be diagonal and has the eigenvalues \(E_{\underline{n}}=\frac{\mu _{bare}^2}{2} +e\big (\frac{|\underline{n}|}{\mu ^2 V^{2/D}}\big )\), where e(x) is a monotonously increasing differentiable function with \(e(0)=0\) (on the noncommutative Moyal plane, \(e(x)=x\)).

The next step is to define (and rearrange) the partition function

$$\begin{aligned} {{\mathcal {Z}}}[J]&=\int {{\mathcal {D}}} \Phi \exp \left( -S[\Phi ]+V{\textrm{Tr}}(J\Phi ) \right) \nonumber \\&= \int {{\mathcal {D}}}\Phi \exp \Big (-VZ \sum _{\underline{n},\underline{m}\in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \frac{H_{\underline{n}\underline{m}}}{2} \Phi _{\underline{n}\underline{m}} \Phi _{\underline{m}\underline{n}}\Big ) \nonumber \\&\quad \times \exp \Big (-\frac{\lambda _{bare}Z^{3/2} }{3V^2} \sum _{\underline{n},\underline{m},\underline{k} \in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \frac{\partial ^3}{\partial J_{\underline{n}\underline{m}} \partial J_{\underline{m}\underline{k}} \partial J_{\underline{k}\underline{n}}} \Big ) \nonumber \\&\quad \bigg [\exp \bigg (V \sum _{\underline{n},\underline{m}\in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \frac{(J_{\underline{n}\underline{m}} {-}(\kappa {+}\nu E_{\underline{n}}{+}\zeta E_{\underline{n}}^2) \delta _{\underline{m},\underline{n}}) (J_{\underline{m}\underline{n}}{-}(\kappa {+}\nu E_{\underline{n}} {+}\zeta E_{\underline{n}}^2) \delta _{\underline{m},\underline{n}})}{ 2ZH_{\underline{n}\underline{m}}}\bigg )\bigg ], \end{aligned}$$

where source \((J_{\underline{n}\underline{m}})\) is a multi-indexed Hermitian matrix of rapidly decaying entries.

The correlation functions are defined as moments of the partition function. It turns out by earlier work [15] that the correlation functions expand into multi-cyclic contributions. It is therefore convenient to work with \({{\mathbb {J}}}_{\underline{p}_1\ldots \underline{p}_{N_\beta }} :=\prod _{j=1}^{N_\beta } J_{\underline{p}_j\underline{p}_{j+1}}\) with \(\underline{p}_{N_\beta +1}\equiv \underline{p}_{1}\). Taking into account that genus-g correlation functions scale with \(V^{-2g}\) [4, 15], the following expansion of the partition function is obtained:

$$\begin{aligned}&\log \frac{{{\mathcal {Z}}}[J]}{{{\mathcal {Z}}}[0]} \nonumber \\&\quad =:\sum _{B=1}^\infty \sum _{1\le N_1 \le \ldots \le N_B}^\infty \sum _{\underline{p}_i^\beta \in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \sum _{g=0}^{\infty }V^{2-B-2g}\frac{G^{(g)}_{|\underline{p}_1^1\ldots \underline{p}^1_{N_1}|\ldots |\underline{p}_1^B\ldots \underline{p}^B_{N_B}|}}{ \prod _{i=1}^{N_B} \nu _i!} \prod _{\beta =1}^B\frac{{{\mathbb {J}}}_{\underline{p}^\beta _1\ldots \underline{p}^\beta _{N_\beta }}}{N_\beta }, \end{aligned}$$

where \(\nu _i=\# \{\beta \in \{1,\ldots ,B\}\;:~N_\beta =i\}\). We call the cumulant \(G^{(g)}_{|\underline{p}_1^1\ldots \underline{p}^1_{N_1}|\ldots |\underline{p}_1^B\ldots \underline{p}^B_{N_B}|}\) an \((N_1+\cdots +N_B)\)-point function of genus g; when the \(N_\beta \) do not matter, a correlation function of genus g with B boundary components. hese \((N_1+\cdots +N_B)\)-point functions have a perturbative expansion into ribbon graphs drawn on genus-g Riemann surfaces with B boundary components. These ribbon graphs are dual to maps and as such can also be studied from a point of view of enumerative geometry.

Finally, we recall from [28] the Ward-Takahashi identity for \(|\underline{q}|\ne |\underline{p}|\)

$$\begin{aligned} \sum _{\underline{m}\in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \frac{\partial ^2}{\partial J_{\underline{q}\underline{m}} \partial J_{\underline{m}\underline{p}}}{{\mathcal {Z}}}[J]&= \sum _{\underline{m}\in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \frac{V}{(E_{\underline{q}}-E_{\underline{p}})Z} \Big (J_{\underline{m}\underline{q}} \frac{\partial }{\partial J_{\underline{m}\underline{p}}} -J_{\underline{p}\underline{m}} \frac{\partial }{\partial J_{\underline{q}\underline{m}}}\Big ) {{\mathcal {Z}}}[J] \nonumber \\&\quad -\frac{V}{Z}(\nu +\zeta H_{\underline{p}\underline{q}}) \frac{\partial {{\mathcal {Z}}}[J]}{\partial J_{\underline{q}\underline{p}}}. \end{aligned}$$

It arises from invariance of the partition function under unitary transformation \(\Phi \mapsto U^\dagger \Phi U\) of the integration variable [14], or directly from the structure of \({{\mathcal {Z}}}[J]\) [40].

3.2 Dyson–Schwinger equation for \(B=1\)

The Dyson-Schwinger equations are determined in [27, 28] for \(g=0\) and solved for all planar correlation functions. Inserting there the formal genus expansion \( G_{|\underline{p}_1^1\ldots \underline{p}^1_{N_1}|\ldots |\underline{p}_1^B\ldots \underline{p}^B_{N_B}|} :=\sum _{g=0}^\infty V^{-2g} G^{(g)}_{|\underline{p}_1^1\ldots \underline{p}^1_{N_1}|\ldots | \underline{p}_1^B\ldots \underline{p}^B_{N_B}|}\) it is straightforward to extract from [27, 28] Dyson-Schwinger equations for the cumulants \(G^{(g)}_{|\underline{p}_1^1\ldots \underline{p}^1_{N_1}|\ldots | \underline{p}_1^B\ldots \underline{p}^B_{N_B}|}\). It turns out that the planar 1-point function plays a special rôle; we shift it to

$$\begin{aligned} \frac{W^{(0)}_{|\underline{p}|}}{2\lambda }&:=G^{(0)}_{|\underline{p}|} +\frac{F_{\underline{p}}}{\lambda },&F_{\underline{p}}&:=E_{\underline{p}}-\frac{\lambda \nu }{2} =\frac{\mu ^2}{2}+e\Big (\frac{|\underline{p}|}{\mu ^2 V^{2/D}}\Big ). \end{aligned}$$

Three relations between renormalisation parameters are immediate [28]: \(\lambda =Z^{1/2}\lambda _{bare}\), \(\frac{\lambda \zeta }{Z}=1-\frac{1}{Z}\) and \(\mu _{bare}^2=\mu ^2 + \lambda \nu \). Now we can use all the Dyson-Schwinger equations evaluated in [28]. The planar 1-point function satisfies [28, eq. (3.12)]

$$\begin{aligned} (W^{(0)}_{|\underline{p}|})^2+2\lambda \nu W^{(0)}_{|\underline{p}|} +\frac{2\lambda ^2}{V}\sum _{\underline{n}\in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \frac{W^{(0)}_{|\underline{p}|}-W^{(0)}_{|\underline{n}|}}{ F^2_{\underline{p}}-F^2_{\underline{n}}}&=\frac{4F_{\underline{p}}^2}{Z}+C, \end{aligned}$$

where \(C:=-\frac{\lambda ^2\nu ^2(1+Z)+4 \kappa \lambda }{Z}\), whereas for \(g\ge 1\) we get

$$\begin{aligned} W^{(0)}_{|\underline{p}|} G^{(g)}_{|\underline{p}|} +\lambda \nu G^{(g)}_{|\underline{p}|} +\lambda \sum _{h=1}^{g-1} G^{(h)}_{|\underline{p}|} G^{(g-h)}_{|\underline{p}|} +\frac{\lambda ^2}{V} \sum _{\underline{n}\in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \frac{G^{(g)}_{|\underline{p}|}-G^{(g)}_{|\underline{n}|}}{ F^2_{\underline{p}}-F^2_{\underline{n}}} +\lambda G^{(g-1)}_{|\underline{p}|\underline{p}|}=0. \end{aligned}$$

3.3 Integral equations

Introducing the measure

$$\begin{aligned} \varrho (X):= \frac{2(2\lambda )^2}{V} \sum _{\underline{n}\in {{\mathbb {N}}}_{{\mathcal {N}}}^{D/2}} \delta (X-4F_{\underline{n}}^2), \end{aligned}$$

we can rewrite (3.7) as an integral equation. The measure has support in \([4 F_{\underline{0}}^2,\Lambda _{{{\mathcal {N}}}}^2]\) where \(\Lambda _{{{\mathcal {N}}}}^2=\max (4F_{\underline{n}}^2\;:~ |\underline{n}|={{\mathcal {N}}})\). For quantum field theory it is necessary to take a large-\({{\mathcal {N}}}\) limit. In general this produces divergences which need renormalisation. Optionally the large-\({{\mathcal {N}}}\) limit can be entangled with a limit \(V\rightarrow \infty \) which, supposing the \(F_{\underline{n}}\) scale down with V (as e.g. in (3.5)), can be designed to let \(\varrho (X)\) converge to a continuous function. We also pass to mass-dimensionless quantities via multiplication by specified powers of \(\mu \) [28]. This amounts to choose the mass scale as \(\mu =1\).

To keep maximal flexibility we consider a measure \(\varrho \) with support in \([1,\Lambda ^2]\) of which a limit \(\Lambda \rightarrow \infty \) has to be taken for quantum field theory. As already observed in [29], equation (3.6) extends to the closed equation

$$\begin{aligned} (W_0(X))^2 +2\lambda \nu W_0(X) +\int _1^{\Lambda ^2} dY\,\varrho (Y)\frac{W_0(X)-W_0(Y)}{X-Y}&=\frac{X}{Z}+C \end{aligned}$$

for a sectionally holomorphic function \(W_0(X)\) from which one extracts \(W^{(0)}_{|\underline{p}|}=W_0(4F_{\underline{p}}^2)\). The corresponding relation

$$\begin{aligned} G^{(g)}_{|\underline{p}_1^1\ldots \underline{p}^1_{N_1}|\ldots | \underline{p}_1^B\ldots \underline{p}^B_{N_B}|}&= G_g\big (4F_{\underline{p}_1^1}^2,\ldots ,4F_{\underline{p}^1_{N_1}}^2|\ldots | 4F_{\underline{p}_1^B}^2,\ldots ,4F_{\underline{p}^B_{N_B}}^2\big ), \end{aligned}$$

for \(2g+B>1\) extends (3.7) to

$$\begin{aligned} 0&=W_0(X) G_g(X) +\lambda \nu G_g(X) +\frac{1}{2} \int _1^{\Lambda ^2} dY\,\varrho (Y)\frac{G_g(X)-G_g(Y)}{X-Y} \nonumber \\&\quad +\lambda \sum _{h=1}^{g-1} G_h(X) G_{g-h}(X) +\lambda G_{g-1}(X|X). \end{aligned}$$

Using techniques for boundary values of sectionally holomorphic functions [29], easily adapted to include \(Z-1,\nu ,C\ne 0\) [28], one obtains the following solution of (3.9):

$$\begin{aligned} W_0(X)=&\frac{\sqrt{X+c}}{\sqrt{Z}}-\lambda \nu +\frac{1}{2}\int _1^{\Lambda ^2} dY\frac{\varrho (Y)}{ (\sqrt{X+c}+\sqrt{Y+c})\sqrt{Y+c}}. \end{aligned}$$

Here, the finite parameter c and the (for \(\Lambda ^2\rightarrow \infty \)) possibly divergent \(Z,\nu \) are determined by renormalisation conditions depending on the dimension:

$$\begin{aligned} \underbrace{ W_0(1)=1}_{D\ge 2},\qquad \underbrace{W_0'(1)=\frac{1}{2}}_{D\ge 4},\qquad \underbrace{W_0''(1)=-\frac{1}{4}}_{D=6}, \end{aligned}$$

together with the convention \(Z=1\) for \(D\in \{2,4\}\) and \(\nu =0\) for \(D=2\). For given coupling constant \(\lambda \) as the only remaining parameter, these equations can be solved for c:

$$\begin{aligned} (1-\sqrt{1{+}c}) \Big (\frac{1+\sqrt{1{+}c}}{2}\Big )^{\!\!\delta _{D,0}{+}\delta _{D,6}} =\frac{1}{2}\int _{1}^{\Lambda ^2}dY\frac{\varrho (Y)}{ (\sqrt{1{+}c}+\sqrt{Y{+}c})^{D/2}\sqrt{Y{+}c}}. \end{aligned}$$

By the implicit function theorem, (3.13) has a smooth solution in an inverval \(-\lambda _c<\lambda <\lambda _c\), in any dimension \(D\in \{0,2,4,6\}\). The Lagrange inversion theorem gives the expansion of c in \(\lambda ^2\):

$$\begin{aligned} c=\sum _{n=1}^\infty \frac{1}{n!}\frac{d^{n-1}}{dw^{n-1}} \Bigg \vert _{w=0}\bigg (\frac{\frac{w}{2}\int _{1}^{\Lambda ^2}dY \frac{\varrho (Y)}{(\sqrt{1+w}+\sqrt{Y+w})^{D/2}\sqrt{Y+w}}}{ (1-\sqrt{1+w})\left( \frac{1+\sqrt{1+w}}{2}\right) ^{\delta _{D,0}+\delta _{D,6}}}\Bigg )^n. \end{aligned}$$

After that renormalisation procedure the limit \(\Lambda ^2\rightarrow \infty \) is safe in all correlation function and any dimension \(D\in \{2,4,6\}\).

3.4 Dyson–Schwinger equation for \(B>1\)

An \((N_1+\cdots +N_B)\)-point function of genus g can be obtained from the \((1+1+\cdots +1)\)-point function of genus g with B boundary components through the explicit formula [28, Prop. 4.1]

$$\begin{aligned} G_g&(X_1^1,\ldots ,X_{N_1}^1|\ldots |X_1^B,\ldots ,X_{N_B}^B) \nonumber \\&=\lambda ^{N_1+\cdots +N_B-B}\sum _{k_1=1}^{N_1}\ldots \sum _{k_B=1}^{N_B} G_g(X_{k_1}^1|\ldots |X_{k_B}^{N_B}) \prod _{\beta =1}^B\prod _{\begin{array}{c} l_\beta =1\\ l_\beta \ne k_\beta \end{array}}^{N_\beta } \frac{4}{X^\beta _{k_\beta }-X^\beta _{l_\beta }}. \end{aligned}$$

For \(g=0\) one has to write \(\frac{W_0(X_{k_1}^1)}{2\lambda }\) instead of \(G_0(X_{k_1}^1)\). Furthermore, a \((1+1+\dots +1)\)-point function \(G_{g}(X_1|X\triangleleft _J)\) with \(B>1\) boundary components and genus g fulfils the linear integral equation [28, eq. (4.5)]

$$\begin{aligned} 0&=\big (W_0(X_1)+\lambda \nu \big ) G_{g}(X_1|X\triangleleft _J) +\frac{1}{2}\int _{1}^{\Lambda ^2}dY\varrho (Y)\frac{G_g(X_1|X\triangleleft _{J})- G_g(Y|X\triangleleft _{J})}{X_1-Y} \nonumber \\&\quad +\lambda \sum _{\begin{array}{c} h+h'=g \\ I\uplus I'= J\\ (h,I),(h',I) \ne (0,\emptyset ) \end{array}} G_h(X_1|X\triangleleft _{I})G_{h'} (X_1|X\triangleleft _{I'}) +\lambda \sum _{\beta \in J}G_g(X_1,X_\beta ,X_\beta |X \triangleleft _{J\backslash \{\beta \}}) \nonumber \\&\quad + \lambda G_{g-1}(X_1|X_1|X\triangleleft _J), \end{aligned}$$

where \(J=\{2,3,..,B\}\). Here and throughout the paper (for z instead of X) we abbreviate \(G_g(X_0|X\triangleleft _I):=G_g(X_0|X_{i_1}|X_{i_2}|\ldots |X_{i_p})\) and \(G_g(X\triangleleft _I):=G_g(X_{i_1}|X_{i_2}|\ldots |X_{i_p})\) if \(I=\{i_1,\ldots ,i_p\}\). We let \(G_g(X_0|X\triangleleft _\emptyset )=G_g(X_0)\). In the sum, \(I\uplus I'=J\) indicates summation over all possibly empty subsets \(I\subset J\), with \(I':=J\setminus I\). The difference to the planar sector \((g=0)\) is the last term indexed \(g-1\) which only contributes if \(g\ge 1\). Furthermore, the entire sector of genus \(h<g\) contributes to the genus-g sector.

The equations (3.15) for \(g=0\) have been solved in [27]:

$$\begin{aligned}&G_0(X|Y)=\frac{4\lambda ^2}{\sqrt{X+c}\sqrt{Y+c} (\sqrt{X+c}+\sqrt{Y+c})^2},\nonumber \\&G_0(X^1|\dots |X^B)= \frac{d^{B-3}}{dt^{B-3}} \Big (\frac{(-2\lambda )^{3B-4}}{(R(t))^{B-2} \sqrt{X^1{+}c{-}2t}^3 \cdots \sqrt{X^B{+}c {-}2t}^3}\Bigg )\Bigg |_{t=0} \end{aligned}$$

for \(B\ge 3\), where

$$\begin{aligned} R(t) := \lim _{\Lambda ^2\rightarrow \infty }\Big (\frac{1}{\sqrt{Z}}- \int _1^{\Lambda ^2} \frac{dT \varrho (T)}{\sqrt{T{+}c}} \frac{1}{(\sqrt{T{+}c}+\sqrt{T{+}c{-}2t}) \sqrt{T{+}c{-}2t}}\Big ). \end{aligned}$$

Note that multiple t-derivatives of R(t) at \(t=0\) produce renormalised moments of the measure (3.8):

$$\begin{aligned} \varrho _l := \lim _{\Lambda ^2\rightarrow \infty } \Big ( \frac{\delta _{l,0}}{\sqrt{Z}}- \frac{1}{2} \int _1^{\Lambda ^2} \frac{dT\, \varrho (T)}{(\sqrt{T{+}c})^{3+2l}}\Big ). \end{aligned}$$

In fact the proof of (3.16) consists in a resummation of an ansatz which involves Bell polynomials (see Definition 4.10) in the \(\{\varrho _l\}\).

The next goal is to find solutions for (3.11) and (3.15) at any genus by employing techniques of complex analysis. The moments (3.17) will be of paramount importance for that. We will find that all solutions are universal in terms of \(\{\varrho _l\}\). The concrete model characterised by the sequence \(E_{\underline{n}}\), coupling constant \(\lambda \) and the dimension D only affects the values of \(\{\varrho _l\}\) via the measure (3.8) and the D-dependent solution c of (3.13).

4 Solution of the Non-planar Sector

4.1 Change of variables

As already mentioned, the equation for \(W_0\) and its solution holomorphically extend to (certain parts of) the complex plane. The corresponding techniques have been brought to perfection by Eynard. We draw a lot of inspiration from the exposition given in [18]. Starting point is another change of variables:

$$\begin{aligned}&z:=\sqrt{X+c},\qquad {{\mathcal {W}}}_0(z(X)):=W_0(X)\qquad \text {and} \nonumber \\&{{\mathcal {G}}}_g(z^1_1(X^1_1),\ldots ,z_{N_1}^1(X^1_{N_1})|\ldots | z_1^B(X_1^B),\ldots ,z_{N_B}^B(X^B_{N_B})) \nonumber \\&\quad :=G_g(X^1_1,\ldots ,X^1_{N_1}|\ldots |X^B_1,\ldots ,X^B_{N_B}) \end{aligned}$$

for \(2g+B>1\). In the beginning, z is defined to be positive; nevertheless all correlation functions have an analytic continuation. We define them by the complexification of the equations (3.11) and (3.15), where we assume that the complex variables fulfil the equations if they lie on the interval \([\sqrt{1+c},\sqrt{1+\Lambda ^2}]\). By recursion hypothesis each correlation function is analytic for non-vanishing imaginary part of the complex variables \(z_i\), possibly with the exception of diagonals \(z_i=\pm z_j\).

We rephrase some of the earlier results in this setup. The solutions (3.11), (3.16) and the formula for the (\(1+1+1\))-point function given in [27] are easily translated into

$$\begin{aligned}&{{\mathcal {W}}}_0(z)=\frac{z}{\sqrt{Z}}-\lambda \nu +\frac{1}{2}\int _{\sqrt{1+c}}^{\sqrt{\Lambda ^2+c}} dy \frac{\tilde{\varrho }(y)}{(z+y)y},\qquad \tilde{\varrho }(y):=2y\varrho (\sqrt{y^2-c}), \nonumber \\&{{\mathcal {G}}}_0(z_1|z_2)= \frac{4\lambda ^2}{z_1z_2(z_1+z_2)^2}, \qquad \qquad {{\mathcal {G}}}_0(z_1|z_2|z_3)=-\frac{32 \lambda ^5}{\varrho _0z_1^3z_2^3z_3^3}. \end{aligned}$$

Note that \(\tilde{\varrho }(y)\) has support in \([\sqrt{1+c},\sqrt{\Lambda ^2+c}]\subset {{\mathbb {R}}}_+\) because of \(c>-1\) [28]. Furthermore, \({{\mathcal {W}}}_0(z)\) extends to a sectionally holomorphic function with branch cut along \([-\sqrt{1+\Lambda ^2}, -\sqrt{1+c}]\), the \((1+1)\)-point function of genus zero is holomorphic outside \(z_i=0\) and the diagonals \(z_1=-z_2\), whereas the \((1+1+1)\)-point function (and all higher-B functions) at genus 0 are meromorphic with only pole at \(z_i=0\).

Definition 4.1

Let \({\hat{K}}_z\) be the integral operator of the linear integral equation,

$$\begin{aligned} {\hat{K}}_z f(z):={{\mathcal {W}}}_0(z) f(z) +\lambda \nu f(z) +\frac{1}{2}\int _{\sqrt{1+c}}^{\sqrt{\Lambda ^2+c}} dy\, \tilde{\varrho }(y)\frac{f(z)-f(y)}{z^2-y^2}, \end{aligned}$$

where \({{\mathcal {W}}}_0(z)\) is given by (4.2).

In this notation, (3.11) takes the form

$$\begin{aligned} 0&={\hat{K}}_z {{\mathcal {G}}}_g(z) +\lambda \sum _{h=1}^{g-1}{{\mathcal {G}}}_h(z) {{\mathcal {G}}}_{g-h}(z) +\lambda {{\mathcal {G}}}_{g-1}(z|z). \end{aligned}$$

We will heavily rely on:

Lemma 4.2

The operator \({\hat{K}}_z\) defined in Definition 4.1 satisfies

$$\begin{aligned} {\hat{K}}_z\Big (\frac{1}{z}\Big )&=\frac{1}{\sqrt{Z}} ,&{\hat{K}}_z\Big (\frac{1}{z^{3+2n}}\Big )&= \sum _{k=0}^{n}\frac{\varrho _k}{z^{2n+2-2k}}. \end{aligned}$$


This is a reformulation of [27, Lemma 5.5]. \(\quad \square \)

The first step beyond [27] is to determine the 1-point function at genus 1:

Proposition 4.3

The solution of (4.3) for \(g=1\) is

$$\begin{aligned} {{\mathcal {G}}}_1(z)=\frac{\lambda ^3\varrho _1}{\varrho _0^2z^3} -\frac{\lambda ^3}{\varrho _0z^5}, \end{aligned}$$

where the \(\varrho _l\) are given in (3.17).


From (4.2) we have \(\lambda {{\mathcal {G}}}_0(z|z)=\frac{\lambda ^3}{z^4}\). Lemma 4.2 suggests the ansatz \({{\mathcal {G}}}_1(z)=\frac{\beta }{z^3}+\frac{\gamma }{z^5}\) with \({\hat{K}}_z {{\mathcal {G}}}_1(z)=\frac{\beta \varrho _0 }{z^2} +\frac{\gamma \varrho _0}{z^4}+\frac{\gamma \varrho _1}{z^2}\). Comparison of coefficients yields the assertion.

\(\square \)

From (3.14) we get the N-point function of genus 1 which in complex variables reads

$$\begin{aligned} {{\mathcal {G}}}_1(z_1,\ldots ,z_N)=\sum _{k=1}^N {{\mathcal {G}}}_1(z_k) \prod _{l=1,l\ne k}^N\frac{4\lambda }{z_k^2-z_l^2}. \end{aligned}$$

Next we express equation (3.15) in the new variables. To find more convenient results we use (3.14) to write with \(J=\{2,..,B\}\)

$$\begin{aligned}&{{\mathcal {G}}}_g(z_1,z_\beta ,z_\beta |z \triangleleft _{J\backslash \{\beta \}})=\lim _{z_\alpha \rightarrow z_\beta }{{\mathcal {G}}}_g(z_1,z_\alpha ,z_\beta |z \triangleleft _{J\backslash \{\beta \}}) \nonumber \\&\quad =16\lambda ^2\left[ \frac{{{\mathcal {G}}}_g(z_1|z \triangleleft _{J\backslash \{\beta \}})}{(z_1^2-z_\beta ^2)^2}-\lim _{z_\alpha \rightarrow z_\beta } \frac{ \frac{{{\mathcal {G}}}_g(z_\alpha |z \triangleleft _{J\backslash \{\beta \}})}{z_1^2-z_\alpha ^2}-\frac{{{\mathcal {G}}}_g(z_\beta |z \triangleleft _{J\backslash \{\beta \}})}{z_1^2-z_\beta ^2}}{(z_\alpha +z_\beta ) (z_\alpha -z_\beta )}\right] \nonumber \\&\quad =16\lambda ^2\frac{\partial }{2z_\beta \partial z_\beta }\frac{{{\mathcal {G}}}_g(z_1|z \triangleleft _{J\backslash \{\beta \}})-{{\mathcal {G}}}_g(z_\beta |z \triangleleft _{J\backslash \{\beta \}})}{z_1^2-z_\beta ^2} \end{aligned}$$

and \({{\mathcal {G}}}_0(z_1,z_2,z_2)=8\lambda \frac{\partial }{2z_2\partial z_2 }\frac{{{\mathcal {W}}}_0(z_1)-{{\mathcal {W}}}_0(z_2)}{z_1^2-z_2^2}\).

Inserting (4.4) into (3.15) gives with Definition 4.1 the following formula for \({{\mathcal {G}}}_g(z_1|z\triangleleft _J)\), for \(2g+|J| \ge 2\):

$$\begin{aligned} 0&={\hat{K}}_{z_1} {{\mathcal {G}}}_g(z_1|z\triangleleft _J) + \lambda {{\mathcal {G}}}_{g-1}(z_1|z_1|z\triangleleft _{J}) +\lambda \sum _{\begin{array}{c} h+h'=g \\ I\uplus I'= J\\ (h,I),(h',I')\ne (0,\emptyset ) \end{array}} {{\mathcal {G}}}_h(z_1|z\triangleleft _{I}){{\mathcal {G}}}_{g-h} (z_1|z\triangleleft _{J\backslash I})\nonumber \\&\quad +(2\lambda )^3 \sum _{\beta \in J} \frac{\partial }{z_\beta \partial z_\beta }\frac{{{\mathcal {G}}}_g(z_1|z \triangleleft _{J\backslash \{\beta \}})-{{\mathcal {G}}}_g(z_\beta |z \triangleleft _{J\backslash \{\beta \}})}{z_1^2-z_\beta ^2}. \end{aligned}$$

Note that this covers also (4.3) as \(J=\emptyset \).

4.2 Boundary creation operator

We are going to construct an operator which plays the rôle of the formal \(T_{\underline{n}}:= \frac{1}{E_{\underline{n}}} \frac{\partial }{\partial E_{\underline{n}}}\) applied to the logarithm of the partition function \({{\mathcal {Z}}}[0]\) given in (3.2). In dimension \(D=0\) where \(Z-1=\kappa =\nu =\zeta =0\) and \(\mu _{bare}=\mu \), \(\lambda _{bare}=\lambda \) we formally have

$$\begin{aligned}&T_{\underline{n}} \log \Big (\int d\Phi \; e^{-{\textrm{tr}}(E\Phi ^2+\frac{\lambda }{3}\Phi ^3)}\Big ) \nonumber \\&\quad =-\frac{1}{{{\mathcal {Z}}}[0]} \int d\Phi \; \sum _{\underline{m}} \frac{\Phi _{\underline{m}\underline{n}} \Phi _{\underline{n}\underline{m}}}{E_{\underline{n}}} e^{-{\textrm{tr}}(E\Phi ^2+\frac{\lambda }{3}\Phi ^3)} \nonumber \\&\quad =\frac{1}{\lambda {{\mathcal {Z}}}[0]} \int d\Phi \; \frac{1}{E_{\underline{n}}} \Big (\frac{\partial }{\partial \Phi _{\underline{n}\underline{n}}} + 2 E_{\underline{n}} \Phi _{\underline{n}\underline{n}} \Big ) e^{-{\textrm{tr}}(E\Phi ^2+\frac{\lambda }{3}\Phi ^3)} \nonumber \\&\quad =\frac{2}{\lambda {{\mathcal {Z}}}[0]} \int d\Phi \; \Phi _{\underline{n}\underline{n}} e^{-{\textrm{tr}}(E\Phi ^2+\frac{\lambda }{3}\Phi ^3)} =\frac{2}{\lambda } G_{|\underline{n}|}. \end{aligned}$$

By repeated application of \(T_{\underline{n}_i}\) we formally produce an \((1+\dots +1)\)-point function. Of course, these operations are not legitimate: in dimensions \(D\in \{2,4,6\}\) we have to include for renormalisation the \(\Phi \)-linear terms in (3.1), and the partition function has no chance to exist for real \(\lambda \).

Nevertheless, we are able to show that \(T_{\underline{n}_i}\) admits a rigorous replacement which we call the boundary creation operator. It will be our main device:

Definition 4.4

For \(J=\{1,\dots ,p\}\) let \(|J|:=p\) and \(z_J:=(z_1,\dots ,z_{p})\). Then

$$\begin{aligned} \hat{{\textrm{A}}}^{\dag g}_{z_J,z} := \sum _{l= 0}^{3g-3+|J|} \Big (-\frac{(3+2l) \varrho _{l+1}}{ \varrho _0 z^3}+\frac{3+2l}{z^{5+2l}}\Big ) \frac{\partial }{\partial \varrho _l} + \sum _{i\in J}\frac{1}{\varrho _0 z^3 z_i} \frac{\partial }{\partial z_i}. \end{aligned}$$

Note that the last variable z in \(\hat{{\textrm{A}}}^{\dag g}_{z_J,z}\) plays a very different rôle than the \(z_J\)!

Lemma 4.5

The differential operators \(\hat{{\textrm{A}}}^{\dag g}_{z_J,z}\) commute,

$$\begin{aligned} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_p,z_q} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_p} =\hat{{\textrm{A}}}^{\dag g}_{z_J,z_q,z_p} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_q}. \end{aligned}$$


Being a derivative, it is enough to verify \(\hat{{\textrm{A}}}^{\dag g}_{z_J,z_p,z_q} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_p}(\varrho _k) =\hat{{\textrm{A}}}^{\dag g}_{z_J,z_q,z_p} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_q}(\varrho _k)\) for any k and \(\hat{{\textrm{A}}}^{\dag g}_{z_J,z_p,z_q} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_p}(z_i) =\hat{{\textrm{A}}}^{\dag g}_{z_J,z_q,z_p} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_q}(z_i)\) for any \(i\in J\). This is guaranteed by

$$\begin{aligned} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_p,z_q} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_p}(\varrho _k)&= \frac{(3+2k)(5+2k) \varrho _{k+2}}{ \varrho _0^2 z_q^3z_p^3} -\frac{(3+2k)(5+2k)}{ \varrho _0 z_q^{7+2k} z_p^3} - \frac{3(3+2k)\varrho _{k+1}\varrho _1}{\varrho _0^3 z_q^3z_p^3} \\&\quad + \frac{3(3+2k)\varrho _{k+1}}{\varrho _0^2 z_q^5z_p^3} + \frac{3(3+2k) \varrho _{k+1}}{\varrho _0^2 z_q^3z_p^5} -\frac{(3+2k)(5+2k)}{\varrho _0 z_q^3z_p^{7+2k}}, \\ \hat{{\textrm{A}}}^{\dag g}_{z_J,z_p,z_q} \hat{{\textrm{A}}}^{\dag g}_{z_J,z_p}(z_i)&= \frac{3\varrho _1}{\varrho _0^3 z_q^3 z_p^3 z_i} -\frac{3}{\varrho _0^2 z_q^5 z_p^3 z_i} -\frac{3}{\varrho _0^2 z_q^3z_p^5 z_i} - \frac{1}{\varrho _0^2 z_q^3 z_p^3 z_i^3}. \end{aligned}$$

\(\quad \square \)

This shows that boundary components labelled by \(z_i\) behave like bosonic particles at position \(z_i\). The creation operator \((2\lambda )^3 \hat{{\textrm{A}}}^{\dag g}_{z_J,z}\) adds to a |J|-particle state another particle at position z. The |J|-particle state is precisely given by \({{\mathcal {G}}}_g(z\triangleleft _J)\):

Theorem 4.6

Assume that \({{\mathcal {G}}}_g(z)\) is, for \(g\ge 1\), an odd function of \(z\ne 0\) and a rational function of \(\varrho _0,\dots ,\varrho _{3g-2}\) (true for \(g=1\)). Then the \((1+1+\cdots +1)\)-point function of genus \(g\ge 1\) and B boundary components of the renormalised \(\Phi ^3_D\)-matricial QFT model in dimension \(D\in \{0,2,4,6\}\) has the solution

$$\begin{aligned} {{\mathcal {G}}}_g(z_1|\ldots |z_B)=(2\lambda )^{3B-3}\hat{{\textrm{A}}}^{\dag g}_{z_1,\dots ,z_B} \big (\hat{{\textrm{A}}}^{\dag g}_{z_1,\dots ,z_{B-1}} \big (\cdots \hat{{\textrm{A}}}^{\dag g}_{z_1,z_2} {{\mathcal {G}}}_g(z_1)\ldots \big )\big ),\qquad z_i\ne 0, \end{aligned}$$

where \({{\mathcal {G}}}_g(z_1)\) is the 1-point function of genus \(g\ge 1\) and the boundary creation operator \(\hat{{\textrm{A}}}^{\dag g}_{z_J}\) is defined in Definition 4.4. For \(g=0\) the boundary creation operators act on the \((1+1)\)-point function

$$\begin{aligned} {{\mathcal {G}}}_0(z_1|\ldots |z_B)=(2\lambda )^{3B-6} \hat{{\textrm{A}}}^{\dag 0}_{z_1,\dots ,z_B} \big ( \hat{{\textrm{A}}}^{\dag 0}_{z_1,\dots ,z_{B-1}} \big (\cdots \hat{{\textrm{A}}}^{\dag 0}_{z_1,z_2,z_3} {{\mathcal {G}}}_0(z_1|z_2)\ldots \big )\big ). \end{aligned}$$


We rely on several Lemmas proved in Appendix A. Regarding (4.8) as a definition, we prove in Lemma A.6 an equivalent formula for the linear integral equation (4.5). This expression is satisfied because Lemmas A.3 and A.5 add up to 0. Consequently, the family of functions (4.8) satisfies (4.5). This solution is unique because of uniqueness of the perturbative expansion. \(\quad \square \)

Remark 4.7

The assumption that \({{\mathcal {G}}}_g(z)\) is an odd function of z and rational in \(\varrho _0,\ldots ,\varrho _{3g-2}\) will inductively follow from Theorem 4.12 and Proposition 4.14 for genus \(h\le g\). On the other hand, to prove Theorem 4.12 and Proposition 4.14 for genus h, we need to apply Theorem 4.6 for genus \(h'<h\). In this way, Theorems 4.6,  4.12 and Proposition 4.14 are all inductively proved step by step when increasing the genus in each Theorem/Proposition.

Corollary 4.8

Let \(J=\{2,\ldots ,B\}\). Assume that \(z\mapsto {{\mathcal {G}}}_g(z)\) is holomorphic in \({{\mathbb {C}}}\setminus \{0\}\) with \({{\mathcal {G}}}_g(z)=-{{\mathcal {G}}}_g(-z)\) for all \(z\in {{\mathbb {C}}}\setminus \{0\}\) and \(g\ge 1\). Then all \({{\mathcal {G}}}_g(z_1|z\triangleleft _J)\) with \(2-2g-B<0\)

  1. 1.

    are holomorphic in every \(z_i \in {{\mathbb {C}}}\setminus \{0\}\)

  2. 2.

    are odd functions in every \(z_i\), i.e. \( {{\mathcal {G}}}_g(-z_1|z\triangleleft _J)=-{{\mathcal {G}}}_g(z_1|z\triangleleft _J)\) for all \(z_1,z_i\in {{\mathbb {C}}}\setminus \{0\}\).


The boundary creation operator \(\hat{{\textrm{A}}}^{\dag g}_{z_J,z}\) of Definition 4.4 preserves holomorphicity in \({{\mathbb {C}}}\setminus \{0\}\) and maps odd functions into odd functions. Thus only the initial conditions need to be checked. They are fulfilled for \({{\mathcal {G}}}_0(z_1|z_2|z_3)\) and \({{\mathcal {G}}}_1(z_1)\) according to (4.2); for \(g\ge 2\) by assumption. \(\quad \square \)

The assumption will be verified later in Proposition 4.14.

Corollary 4.9

The boundary creation operator \(\hat{{\textrm{A}}}^{\dag g}_{z_J,z_1}\) acting on an \((N_1+\cdots +N_B)\)-point function of genus g gives the following \((1+N_1+\cdots +N_B)\)-point function of genus g

$$\begin{aligned}&{{\mathcal {G}}}_g(z_1|z^1_1,..,z^1_{N_1}|..|z^B_1,..,z^B_{N_B})\\&\quad =(2\lambda )^3 \hat{{\textrm{A}}}^{\dag g}_{z^1_1,..,z^1_{N_1},\dots ,z^B_1,..,z^B_{N_B},z_1} ({{\mathcal {G}}}_g(z^1_1,..,z^1_{N_1}|..|z^B_1,..,z^B_{N_B})). \end{aligned}$$


This follows from the change to complex variables in equation (3.14) and \(\hat{{\textrm{A}}}^{\dag g}_{z_J,z_1} (\frac{1}{z_i^2-z_j^2})=0\) for \(1\ne i\ne j\ne 1\). \(\quad \square \)

4.3 Solution of the 1-point function for \(g\ge 1\)

It remains to check that the 1-point function \({{\mathcal {G}}}_g(z)\) at genus \(g\ge 1\) satisfies the assumptions of Theorem 4.6 and Corollary 4.8, namely:

  1. 1.

    \({{\mathcal {G}}}_g(z)\) depends only on the moments \(\varrho _0,\dots ,\varrho _{3g-2}\) of the measure,

  2. 2.

    \(z\mapsto {{\mathcal {G}}}_g(z)\) is holomorphic on \({{\mathbb {C}}}\setminus \{0\}\) and an odd function of z.

We establish these properties by solving (4.3) via a formula for the inverse of \({\hat{K}}_z\). This formula is inspired by topological recursion, see e.g. [18]. We give a few details in Sect. 4.4.

Definition 4.10

The Bell polynomials are defined by

$$\begin{aligned} B_{n,k}(x_1,\ldots ,x_{n-k+1})=\sum \frac{n!}{j_1!j_2!\ldots j_{n-k+1}!} \left( \frac{x_1}{1!}\right) ^{j_1}\left( \frac{x_2}{2!}\right) ^{j_2}\ldots \left( \frac{x_{n-k+1}}{(n-k+1)!}\right) ^{j_{n-k+1}} \end{aligned}$$

for \(n\ge 1\), where the sum is over non-negative integers \(j_1,\ldots ,j_{n-k+1}\) with \(j_1+\cdots +j_{n-k+1}=k\) and \(1j_1+2j_2+\cdots +(n-k+1)j_{n-k+1}=n\). Moreover, one defines \(B_{0,0}=1\) and \(B_{n,0}=B_{0,k}=0\) for \(n,k>0\).

An important application is Faà di Bruno’s formula, the n-th order chain rule:

$$\begin{aligned} \frac{d^n}{dx^n}f(g(x))=\sum _{k=1}^{n}f^{(k)}(g(x)) \, B_{n,k}(g'(x),g''(x),\ldots ,g^{(n-k+1)}(x)). \end{aligned}$$

Proposition 4.11

Let \(f(z)=\sum _{k=0}^\infty \frac{a_{2k}}{z^{2k}}\) be an even Laurent series about \(z=0\) bounded at \(\infty \). Then the inverse of the integral operator \({\hat{K}}_z\) of Definition 4.1This suggests to introd is given by the residue formula

$$\begin{aligned} \Big [z^2{\hat{K}}_z\frac{1}{z}\Big ]^{-1}f(z)&=- \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0}\left[ K(z,z')\, f(z')dz'\right] ,\\ K(z,z')&:=\frac{2}{({{\mathcal {W}}}_0(z')-{{\mathcal {W}}}_0(-z'))(z'^2-z^2)}. \end{aligned}$$


The formulae (4.2) give rise to the series expansion

$$\begin{aligned} \frac{1}{2}({{\mathcal {W}}}_0(z')-{{\mathcal {W}}}_0(-z'))= \sum _{l=0}^\infty \varrho _l \cdot (z')^{2l+1}, \end{aligned}$$

where the \(\varrho _l\) are given in (3.17) (either take \(\lim _{\Lambda \rightarrow \infty }{{\mathcal {W}}}_0(z)\) in (4.2) or leave out the limit in (3.17)). The series of its reciprocal is found using (4.9):

$$\begin{aligned}&\frac{2}{({{\mathcal {W}}}_0(z')-{{\mathcal {W}}}_0(-z'))} = \frac{1}{z' \varrho _0} \sum _{m=0}^\infty \frac{(z')^{2m}}{m!} S_m, \nonumber \\&S_m:= \frac{d^m}{d \tau ^m}\Big |_{\tau =0} \Big (\sum _{l=0}^\infty \frac{\varrho _l}{\varrho _0} \tau ^l\Big )^{-1} = \sum _{i=0}^m\frac{(-1)^i i!}{\varrho _0^{i}} B_{m,i}(1!\varrho _1,2! \varrho _2,\ldots ,(m{-}i{+}1)!\varrho _{m-i+1}). \end{aligned}$$

Multiplication by the geometric series gives

$$\begin{aligned} K(z,z')=-\frac{1}{z^2 z' \varrho _0} \sum _{n,m=0}^\infty \frac{(z')^{2m+2n}}{m! z^{2n}} S_m. \end{aligned}$$

The residue of a monomial in \(f(z')=\sum _{k=0}^\infty \frac{a_{2k}}{(z')^{2k}}\) is then

$$\begin{aligned} \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0}\left[ K(z,z')\frac{dz'}{(z')^{2k}}\right] =-\frac{1}{\varrho _0}\sum _{j=0}^{k} \frac{S_j}{j!z^{2k-2j+2}}. \end{aligned}$$

In the next step we apply the operator \(z^2{\hat{K}}\frac{1}{z}\) to (4.13), where Lemma 4.2 is used:

$$\begin{aligned}&z^2 {\hat{K}}_z\Big (\frac{1}{z} \frac{(-1)}{\varrho _0} \sum _{j=0}^{k} \frac{S_j}{j!z^{2k-2j+2}}\Big ) =-\frac{z^2}{\varrho _0} \sum _{j=0}^{k}\sum _{i=0}^{k-j}\frac{S_j\varrho _i}{j! z^{2k-2j-2i+2}} \nonumber \\&\quad =-\sum _{j=0}^{k}\frac{S_{k-j}}{(k-j)! z^{2j}} -\frac{1}{\varrho _0} \sum _{i=0}^{k-1} \sum _{j=i+1}^{k} \frac{S_{k-j}\varrho _{j-i}}{(k-j)! z^{2i}}. \end{aligned}$$

The last sum over j is treated as follows, where the Bell polynomials are inserted for \(S_m\):

$$\begin{aligned}&\sum _{j=i+1}^{k}\frac{S_{k-j} \varrho _{j-i}}{(k-j)!} =\sum _{j=1}^{k-i}\frac{S_{k-j-i}\varrho _{j}}{(k-j-i)! }\\&\quad =\sum _{j=1}^{k-i}\sum _{l=0}^{k-j-i}\frac{(-1)^{l}l!}{(k-j-i)! \varrho _0^{l}}\,\varrho _jB_{k-j-i,l}(1!\varrho _1,\ldots ,(k-j-i-l+1)! \varrho _{k-j-i-l+1}) \\&\quad =\sum _{l=0}^{k-i-1}\frac{(-1)^{l}l!}{\varrho _0^{l}(k-i)!} \sum _{j=1}^{k-i-l}\left( {\begin{array}{c}k{-}i\\ j\end{array}}\right) j!\,\varrho _j B_{k-j-i,l}(1!\varrho _1,\ldots ,(k{-}j{-}i{-}l{+}1)!\varrho _{k-j-i-l+1}) \\&\quad =\sum _{l=0}^{k-i}\frac{(-1)^{l}(l+1)!}{\varrho _0^{l}(k-i)!} B_{k-i,l+1}(1!\varrho _1,\ldots ,(k-i-l)!\varrho _{k-i-l}) \\&\quad =-\varrho _0\frac{S_{k-i}}{(k-i)!}. \end{aligned}$$

We have used \(B_{n,0}=0\) and \(B_{0,n}=0\) for \(n>0\) to eliminate some terms, changed the order of sums, and used the following identity for the Bell polynomials [27, Lemma 5.9]

$$\begin{aligned} \sum _{j=1}^{n-k}\left( {\begin{array}{c}n\\ j\end{array}}\right) x_jB_{n-j,k}(x_1,\ldots ,x_{n-j-k+1}) =(k+1)B_{n,k+1}(x_1,\ldots ,x_{n-k}). \end{aligned}$$

Inserted back we find that (4.14) reduces to the (\(j=k\))-term of the first sum in the last line of (4.14), i.e.

$$\begin{aligned}&z^2 {\hat{K}}_z \Big ( \frac{1}{z} \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0}\Big [ K(z,z')\frac{dz'}{(z')^{2k}}\Big ]\Big )= -\frac{1}{z^{2k}}. \end{aligned}$$

This finishes the proof. \(\quad \square \)

Theorem 4.12

For any \(g\ge 1\) and \(z\in {{\mathbb {C}}}\setminus \{0\}\) one has

$$\begin{aligned} {{\mathcal {G}}}_g(z) = \frac{\lambda }{z} \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0} \left[ K(z,z') \Big \{ \sum _{h=1}^{g-1}{{\mathcal {G}}}_h(z') {{\mathcal {G}}}_{g-h}(z') + {{\mathcal {G}}}_{g-1}(z'|z')\Big \} (z')^2 dz'\right] . \end{aligned}$$


The formula arises when applying Proposition 4.11 to (4.3) and holds if the function in \(\{~\}\) is an even Laurent polynomial in \(z'\) bounded in \(\infty \). This is the case for \(g=1\) where only \({{\mathcal {G}}}_0(z'|z') =\frac{\lambda ^2}{(z')^4}\) contributes. Evaluation of the residue reconfirms Proposition 4.3. We proceed by induction in \(g\ge 2\), assuming that all \({{\mathcal {G}}}_h(z')\) with \(1\le h<g\) on the rhs of (4.16) are odd Laurent polynomials bounded in \(\infty \); their product is even. The induction hypothesis also verifies the assumption of Theorem 4.6 so that \({{\mathcal {G}}}_{g-1}({-}z'|{-}z')=-{{\mathcal {G}}}_{g-1}(z'|{-}z')={{\mathcal {G}}}_{g-1}(z'|z')\) is even and, because of \({{\mathcal {G}}}_{g-1}(z'|z'')=(2\lambda )^3 \hat{{\textrm{A}}}^{\dag g-1}_{z'',z'} {{\mathcal {G}}}_{g-1}(z'')\), inductively a Laurent polynomial bounded in \(\infty \). Thus, equation (4.16) holds for genus \(g\ge 2\) and, as consequence of (4.13), \({{\mathcal {G}}}_g(z)\) is again an odd Laurent polynomial bounded in \(\infty \). Equation (4.16) is thus proved for all \(g\ge 1\), and the assumption of Theorem 4.6 is verified. \(\quad \square \)

A more precise characterisation can be given. It relies on

Definition 4.13

A polynomial \(P(x_1,x_2,\dots )\) is called n -weighted in \((x_1,x_2,\ldots )\) if \(\sum _{k=1}^\infty k x_k \frac{\partial }{\partial x_k} P(x_1,x_2,\dots ) =nP(x_1,x_2,\dots )\).

The Bell polynomials \(B_{n,k}(x_1,\dots ,x_{n-k+1})\) are n-weighted. The number of monomials in an n-weighted polynomial is p(n), the number of partitions of n. The product of an n-weighted by an m-weighted polynomial is \((m+n)\)-weighted.

In the following we let \(P^{decoration}_j(\varrho )\) be some j-weighted polynomial in \((\frac{\varrho _1}{\varrho _0},\dots ,\frac{\varrho _j}{\varrho _0})\) with rational coefficients. A decoration (empty or primes) distinguishes several such polynomials, but is of no relevance.

Proposition 4.14

For \(g\ge 1\) one has

$$\begin{aligned} {{\mathcal {G}}}_g(z)= (2 \lambda )^{4g-1} \sum _{k=0}^{3g-2} \frac{P_{3g-2-k}(\varrho )}{\varrho _0^{2g-1} z^{2k+3}}, \end{aligned}$$

where \(P_0\in {{\mathbb {Q}}}\) and the \(P_j(\varrho )\) for \(j\ge 1\) are some j-weighted polynomials in \((\frac{\varrho _1}{\varrho _0},\dots ,\frac{\varrho _j}{\varrho _0})\) with rational coefficients.


The case \(g=1\) is directly checked. We proceed by induction in g for both terms in \(\{~\}\) in (4.16). The hypothesis gives \({{\mathcal {G}}}_h(z'){{\mathcal {G}}}_{g-h}(z')= (2\lambda )^{4g-2} \sum _{k=0}^{3g-4} \frac{P'_{3g-4-k}(\varrho )}{\varrho _0^{2g-2} (z')^{2k+6}}\). In the second term in \(\{~\}\), \({{\mathcal {G}}}_{g-1}(z'|z')=(2\lambda )^3 \hat{{\textrm{A}}}^{\dag \,g{-}1}_{z',z'} {{\mathcal {G}}}_{g-1}(z')\), the three types of contributions in the boundary creation operator act as follows:

$$\begin{aligned}&(2\lambda )^3 \sum _{k=0}^{3g-5}(2\lambda )^{4g-5} \frac{P_{3g-5-k}(\varrho )}{(z')^{2k+3}} \Big (-\frac{3\varrho _1}{\varrho _0 (z')^3}+\frac{3}{(z')^5}\Big ) \frac{\partial }{\partial \varrho _0} \frac{1}{\varrho _0^{2g-3}}\\&\quad =\frac{(2\lambda )^{4g-2}3(3-2g)}{\varrho _0^{2g-2}} \Big (\sum _{k=0}^{3g-5}\frac{\frac{\varrho _1}{\varrho _0} P_{3g-5-k}(\varrho )}{(z')^{2k+6}} +\sum _{k=0}^{3g-5}\frac{ P_{3g-4-(k+1)}(\varrho )}{(z')^{2(k+1)+6}} \Big ),\\&(2\lambda )^3 \sum _{k=0}^{3g-5}(2\lambda )^{4g-5} \frac{P_{3g-5-k}(\varrho )}{\varrho ^{2g-3}} \frac{1}{\varrho _0(z')^4} \frac{\partial }{\partial z'} \frac{1}{(z')^{2k+3}} \\&\quad =-\frac{(2\lambda )^{4g-2}}{\varrho _0^{2g-2}} \sum _{k=0}^{3g-5}\frac{(2k+3)P_{3g-4-(k+1)}(\varrho )}{(z')^{2(k+1)+6}} \\&(2\lambda )^3 \sum _{k=0}^{3g-5}\frac{(2\lambda )^{4g-5} }{\varrho ^{2g-3}(z')^{2k+3}} {\hat{A}}^{\dag {g-1}}_{z',z'}P_{3g-5-k}(\varrho ) \\&\quad = \sum _{k=0}^{3g-5}\frac{(2\lambda )^{4g-2}}{\varrho _0^{2g-3}(z')^{2k+3}} \Big \{ \Big (-\frac{3\varrho _1}{\varrho _0 (z')^3}+\frac{3}{(z')^5}\Big ) \sum _{l=1}^{3g-5-k} \Big ({-}\frac{\varrho _l}{\varrho _0^2}\Big ) \frac{\partial }{\partial \frac{\varrho _l}{\varrho _0}} P_{3g-5-k}(\varrho ) \\&\qquad + \sum _{l=1}^{3g-5-k}\Big (-\frac{(3+2l)\frac{\varrho _{l+1}}{\varrho _0}}{(z')^3} +\frac{3+2l}{(z')^{5+2l}} \Big )\frac{1}{\varrho _0} \frac{\partial }{\partial \frac{\varrho _l}{\varrho _0}} P_{3g-5-k}(\varrho ) \Big \} \end{aligned}$$

which are all of the same structure \((2\lambda )^{4g-2} \sum _{k=0}^{3g-4} \frac{P''_{3g-4-k}(\varrho )}{\varrho _0^{2g-2} (z')^{2k+6}}\)

$$\begin{aligned}&\frac{\lambda }{z} \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0} \Big [K(z,z') dz' (z')^2 (2\lambda )^{4g-2} \sum _{k=0}^{3g-4} \frac{P_{3g-4-k}(\varrho )}{\varrho _0^{2g-2} (z')^{2k+6}}\Big ]\\&\quad =(2\lambda )^{4g-1}\sum _{k=0}^{3g-4} \sum _{j=0}^{k+2} \frac{P_{3g-4-k}(\varrho )S_j(\varrho )}{2j!\varrho _0^{2g-1} z^{2k+7-2j}} =(2\lambda )^{4g-1}\sum _{k=0}^{3g-2} \frac{P'_{3g-2-k}(\varrho )}{\varrho _0^{2g-1} z^{2k+3}}, \end{aligned}$$

because \(S_j(\varrho )\) is also a j-weighted polynomial by (4.11). \(\quad \square \)

In particular, this proves the assumption of Theorem 4.6, namely that \({{\mathcal {G}}}_g(z)\) depends only on \(\{\varrho _0,\dots , \varrho _{3g-2}\}\). To be precise, we reciprocally increase the genus in Theorem 4.6 and Proposition 4.14 as described in Remark 4.7.

4.4 Remarks on topological recursion

We return to equation (4.5) for \(g>0\) or \(|J|\ge 3\) in case of \(g=0\). In the first term, \({{\mathcal {G}}}_g(z_1|z \triangleleft _J)\) is given by Theorem 4.6 so that \(z_1{{\mathcal {G}}}_g(z_1|z \triangleleft _J)\) is by Proposition 4.14 and Definition 4.4 of \(\hat{{\textrm{A}}}^{\dag g}_{z_J,z_1}\) an even Laurent polynomial about \(z_1=0\) bounded at \(\infty \). Therefore, Proposition 4.11 applies and gives the representation

$$\begin{aligned}&z_1 {{\mathcal {G}}}_g(z_1|z\triangleleft _J) \nonumber \\&\quad =\mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0} K(z_1,z')(z')^2 dz' \Big [ \lambda \sum _{\begin{array}{c} h+h'=g\\ I\uplus I'= J\\ (h,I),(h',I')\ne (0,\emptyset ) \end{array}} {{\mathcal {G}}}_h(z'|z\triangleleft _I) {{\mathcal {G}}}_{h'}(z'|z\triangleleft _{I'}) \nonumber \\&\qquad +\lambda {{\mathcal {G}}}_{g-1}(z'|z'|z\triangleleft _J) +(2\lambda )^3 \sum _{\beta \in J} \frac{\partial }{z_\beta \partial z_\beta } \frac{{{\mathcal {G}}}_g(z'|z\triangleleft _{J\setminus \beta }) -{{\mathcal {G}}}_g(z_\beta |z\triangleleft _{J\setminus \beta })}{(z')^2-z_2^2}\Big ]. \end{aligned}$$

The last term \({{\mathcal {G}}}_g(z_\beta |z\triangleleft _{J\setminus \beta })\) does not contribute to the residue. The other term \({{\mathcal {G}}}_g(z'|z\triangleleft _{J\setminus \beta })\) also arises for \((h,I)=(g,\{\beta \})\) and \((h',I')=(0,\{\beta \})\) in the sum, where it has the prefactor \({{\mathcal {G}}}_0(z'|z_\beta )\) given in the second line of (4.2). The total contribution of this term inside [  ] is

$$\begin{aligned}&\frac{\lambda {{\mathcal {G}}}_g(z'|z\triangleleft _{J\setminus \beta })}{z'z_\beta } \Big (\frac{4\lambda ^2}{(z'-z_\beta )^2} +\frac{4\lambda ^2}{(z'+z_\beta )^2}\Big )\;. \end{aligned}$$

This suggests to introduce

$$\begin{aligned} \omega _{g,B}(z_1,\ldots ,z_B)&:=\Big (\prod _{i=1}^B z_i\Big ) {{\mathcal {G}}}_g(z_1|\ldots |z_B) \quad \text {for } 2g+B>2, \nonumber \\ \omega _{0,2}(z_1,z_2)&:=\frac{4\lambda ^2}{(z_1-z_2)^2}. \end{aligned}$$

By Theorem 4.6 and Proposition 4.14 together with Definition 4.4, each \(\omega _{g,B}(z_1,\ldots ,z_B)\) with \(2g+B>2\) is an even Laurent-polynomial in every \(z_i\) so that we can replace \(z'\mapsto -z'\) in one of the arguments. The structure then matches the combination \(\omega _{0,2}(z',z_\beta )+\omega _{0,2}(z',-z_\beta )\) needed as prefactor of \(\omega _{g,B-1}(z',z_2,{\mathop {\check{\dots }}\limits ^{\beta }},z_B)\). Multiplying (4.17) by \(\prod _{i\in J}z_i\) and taking the previous discussion into account, we have proved (after a shift \(B\mapsto B+1\)):

Theorem 4.15

(cf. [18, Thm. 6.4.4]). The functions \(\omega _{g,B+1}(z_0,\ldots ,z_B)\) given via (4.18) in terms of the complexification (3.10) and (4.1) of the cumulants \({{\mathcal {G}}}_g(z_0|z\triangleleft _{\{1,\ldots ,B\}})\) in (3.3) satisfy for \(2g+B\ge 2\) the recursive equation

$$\begin{aligned}&\omega _{g,B+1}(z_0,\ldots ,z_B) \nonumber \\&\quad =\lambda \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0} \bigg [K(z_0,z')\,dz'\Big (\omega _{g-1,B+2}(z',{-}z', z\triangleleft _{\{1,\ldots ,B\}})\nonumber \\&\qquad + \sum _{\begin{array}{c} h+h'=g\\ I\uplus I' =\{1,\ldots ,B\} \\ (h,I),(h',I')\ne (0,\emptyset ) \end{array}} \omega _{h,|I|+1}(z',z\triangleleft _I) \omega _{h',|I'|+1}({-}z',z\triangleleft _{I'})\Big )\bigg ], \end{aligned}$$

where \(\omega _{g,|I|+1}(z_0,z\triangleleft _I)= \omega _{g,n+1}(z_0,z_{i_1},\ldots .,z_{i_n})\) if \(I=\{{i_1},\ldots .,{i_n}\}\) and the the kernel K is given in Proposition 4.11.

The case (\(g=0\), \(B=3\)) was excluded in the above discussion, but can be checked directly.

After rescaling \(\omega _{g,B}(z_1,..,z_B)= (2\lambda )^{4g+3B-4}\omega ^{TR}_{g,B}(z_1,..,z_B)\) we have reproved the known result [18, Thm. 6.4.4] of topological recursion when taking a factor 2 out of K and writing

$$\begin{aligned} \frac{1}{({{\mathcal {W}}}_0(z')-{{\mathcal {W}}}_0(-z'))(z'^2-z^2)}&=\frac{\frac{1}{2}(\frac{1}{z-z'}-\frac{1}{z+z'})}{ x'(z') (y(z')-y(-z'))} \;,\\ \text {where } \quad x(z)&:=\frac{z^2}{2},~y(z):=-{{\mathcal {W}}}_0(z). \end{aligned}$$

In fact, [18, Thm. 6.4.4] motivated our ansatz for an inverse of \({\hat{K}}_z\) as the residue involving \(K(z,z')\). Our proof of Theorem 4.15 is of comparable difficulty and length as [18, Thm. 6.4.4].

For the \(\omega _{g,|J|}(z\triangleleft _J)\) with \(J\ge 2\) we have the alternative representation of Proposition 4.6. There is, however, one property where (4.19) is really needed: for the proof of the symmetry \({{\mathcal {G}}}_g(z_1|z_2)={{\mathcal {G}}}_g(z_2|z_1)\) for \(g\ge 1\). The complete symmetry of \(\omega _{g,B}(z_1,\ldots ,z_B)\) or equivalently \({{\mathcal {G}}}_g(z_1|\ldots |z_B)\) in all arguments then follows from Proposition 4.6. For \(g=0\) the symmetry of the starting point \({{\mathcal {G}}}_0(z_1|z_2|z_3)\) is manifest in (4.2). The proof of \({{\mathcal {G}}}_g(z_1|z_2)={{\mathcal {G}}}_g(z_2|z_1)\) is the same as [20, Thm. 4.6] (which uses a slightly different notation). There is no need to repeat it in this paper.

5 A Laplacian to Compute Intersection Numbers

5.1 Free energy and boundary annihilation operator

Definition 5.1

We introduce the operators

$$\begin{aligned}&\hat{{\textrm{A}}}^{\dag }_z := \sum _{l= 0}^{\infty } \Big (-\frac{(3+2l) \varrho _{l+1}}{ \varrho _0 z^3}+\frac{3+2l}{z^{5+2l}}\Big ) \frac{\partial }{\partial \varrho _l},&\hat{{\textrm{N}}}&= - \sum _{l=0}^\infty \varrho _l\frac{\partial }{\partial \varrho _l} , \nonumber \\&\hat{{\textrm{A}}}_{{\check{z}}} f(\bullet ) :=- \sum _{l=0}^\infty \mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow 0} \Big [\frac{z^{4+2l} \varrho _l}{3+2l} f(z) dz\Big ] \end{aligned}$$

and the free energies

$$\begin{aligned} F_1=-\frac{1}{24} \log \varrho _0,\qquad F_g&= \frac{1}{(2g-2)(2\lambda )^3} \hat{{\textrm{A}}}_{{\check{z}}} {{\mathcal {G}}}_g(\bullet ) \text { for } g> 1. \end{aligned}$$

We call \(\hat{{\textrm{A}}}_{{\check{z}}}\) a boundary annihilation operator acting on Laurent polynomials f.

Proposition 5.2

The \(F_g\) have for \(g>1\) a presentation as

$$\begin{aligned} F_g= (2 \lambda )^{4g-4} \frac{P_{3g-3}(\varrho ) }{\varrho _0^{2g-2}}, \end{aligned}$$

where \(P_{3g-3}(\varrho )\) is some \((3g-3)\)-weighted polynomial in \((\frac{\varrho _1}{\varrho _0},\dots , \frac{\varrho _{3g-3}}{\varrho _0})\). They satisfy

$$\begin{aligned} {{\mathcal {G}}}_g(z) = (2\lambda )^3 \hat{{\textrm{A}}}^{\dag }_z F_g \qquad \text {for } g\ge 1. \end{aligned}$$


From Proposition 4.14 we conclude for \(g\ge 2\)

$$\begin{aligned} F_g&:=\frac{1}{(2\lambda )^3(2g-2)} \hat{{\textrm{A}}}_{{\check{z}}} {{\mathcal {G}}}_g(\bullet )\\&=- \frac{(2 \lambda )^{4g-4}}{2g-2} \mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow 0}\Big [ \sum _{l=0}^\infty \frac{\varrho _lz^{4+2l}}{(3+2l)} \sum _{k=0}^{3g-2} \frac{P_{3g-2-k}(\varrho )}{\varrho _0^{2g-1} z^{2k+3}} dz\Big ]\\&= \frac{(2 \lambda )^{4g-4}}{2-2g} \sum _{k=1}^{3g-2} \frac{ \frac{\varrho _{k-1}}{(2k+1)\varrho _0} \cdot P_{3g-2-k}(\varrho )}{\varrho _0^{2g-2}} = (2 \lambda )^{4g-4} \frac{P'_{3g-3}(\varrho ) }{\varrho _0^{2g-2}}, \end{aligned}$$

which confirms the structure (5.2).

Equation (5.3) is straightforward to check for \(g=1\). Equation (5.3) for \(g\ge 2\) is equivalent to \((2g-2){{\mathcal {G}}}_g(z) = A^\dag _z A_{{\check{w}}} {{\mathcal {G}}}_g(\bullet )\). Inserting the definitions we have

$$\begin{aligned} \hat{{\textrm{A}}}^{\dag }_z \hat{{\textrm{A}}}_{{\check{w}}} {{\mathcal {G}}}_g(\bullet )&=-\hat{{\textrm{A}}}^{\dag }_z \mathop {{{\,\textrm{Res}\,}}}\limits _{w\rightarrow 0} \Big [\sum _{l=0}^\infty \frac{\varrho _l w^{4+2l}dw}{3+2l} {{\mathcal {G}}}_g(w)\Big ]\\&=\mathop {{{\,\textrm{Res}\,}}}\limits _{w\rightarrow 0} \Big [\sum _{l=0}^\infty w^{4+2l}dw \Big (-\frac{1}{z^{5+2l}} +\frac{\varrho _{l+1}}{\varrho _0 z^3} \Big ) {{\mathcal {G}}}_g(w)\Big ]\\&\quad -\mathop {{{\,\textrm{Res}\,}}}\limits _{w\rightarrow 0} \Big [\sum _{l=0}^\infty \frac{\varrho _l w^{4+2l}dw}{3+2l} \hat{{\textrm{A}}}^{\dag }_z {{\mathcal {G}}}_g(w)\Big ]\\&=\mathop {{{\,\textrm{Res}\,}}}\limits _{w\rightarrow 0} \Big [ dw\Big (-\frac{w^{2}}{z(z^2-w^2)} +\sum _{l=0}^\infty w^{2+2l}\frac{\varrho _{l}}{\varrho _0 z^3} \Big ) {{\mathcal {G}}}_g(w)\Big ]\\&\quad -\mathop {{{\,\textrm{Res}\,}}}\limits _{w\rightarrow 0} \Big [\sum _{l=0}^\infty \frac{\varrho _l w^{4+2l}dw}{3+2l} \Big ( \hat{{\textrm{A}}}^{\dag g}_{w,z} {{\mathcal {G}}}_g(w) -\frac{1}{\varrho _0z^3 w} \frac{\partial {{\mathcal {G}}}_g(w)}{\partial w} \Big ) \Big ]\\&=-\mathop {{{\,\textrm{Res}\,}}}\limits _{w\rightarrow 0} \Big [ \frac{w^{2}dw}{z(z^2-w^2)} {{\mathcal {G}}}_g(w)\Big ] +\hat{{\textrm{A}}}_{{\check{w}}}\hat{{\textrm{A}}}^{\dag g}_{\bullet ,z} {{\mathcal {G}}}_g(\bullet ), \end{aligned}$$

where we have completed \(\hat{{\textrm{A}}}^{\dag }_{z}\) to \(\hat{{\textrm{A}}}^{\dag g}_{w,z}\) and used that the residue of a total differential vanishes. With Proposition 4.14 it is easy to see that \(\mathop {{{\,\textrm{Res}\,}}}\limits _{w\rightarrow 0} \Big [ \frac{w^{2}dw}{z(z^2-w^2)} {{\mathcal {G}}}_g(w)\Big ]={{\mathcal {G}}}_g(z)\). Since \({{\mathcal {G}}}_g(z|w)={{\mathcal {G}}}_g(w|z)\) is symmetric for \(g\ge 2\) according to the discussion at the end of Sect. 4.4, we have \((2\lambda )^{-3}{{\mathcal {G}}}_g(z|w) =\hat{{\textrm{A}}}^{\dag g}_{w,z} {{\mathcal {G}}}_g(w) =\hat{{\textrm{A}}}^{\dag g}_{z,w} {{\mathcal {G}}}_g(z)\) by Theorem 4.6. We have \(\hat{{\textrm{A}}}_{{\check{w}}} \hat{{\textrm{A}}}^{\dag g}_{z,\bullet } =\hat{{\textrm{N}}}\), and since \(\hat{{\textrm{N}}} P_j(\varrho )\equiv 0\) for any j-weighted polynomial \(P_j(\varrho )\) in \((\frac{\varrho _1}{\varrho _0},\ldots \frac{\varrho _j}{\varrho _0})\), we have \(\hat{{\textrm{N}}} {{\mathcal {G}}}_g(z)=(2g-1){{\mathcal {G}}}_g(z)\) by Proposition 4.14. Altogether we have proved \(\hat{{\textrm{A}}}^{\dag }_z \hat{{\textrm{A}}}_{{\check{w}}} {{\mathcal {G}}}_g(\bullet ) =-{{\mathcal {G}}}_g(z) + \hat{{\textrm{N}}} {{\mathcal {G}}}_g(z) =(2g-2){{\mathcal {G}}}_g(z)\). \(\quad \square \)

Remark 5.3

Proposition 5.2 shows that the \(F_g\) provide the most condensed way to describe the non-planar sector of the \(\Phi ^3\)-matricial QFT model. All information about the genus-g sector is encoded in the \(p(3g-3)\) rational numbers which form the coefficients in the \((3g-3)\)-weighted polynomial in \((\frac{\varrho _1}{\varrho _0}, \frac{\varrho _2}{\varrho _0},\dots )\). From these polynomials we obtain the \((1+\dots +1)\)-point function with B boundary components via \({{\mathcal {G}}}_g(z)=(2\lambda )^3 \hat{{\textrm{A}}}^{\dag }_z F_g\) followed by Theorem 4.6.

Lemma 5.4

Whenever \((2g+B-2)>0\), the operator \(\hat{{\textrm{N}}}\) measures the Euler characteristics,

$$\begin{aligned} \hat{{\textrm{N}}} {{\mathcal {G}}}_g(z_1|\dots |z_B) = (2g+B-2){{\mathcal {G}}}_g(z_1|\dots |z_B) . \end{aligned}$$


Both cases with \((2g+B-2)=1\) are directly checked. The general case follows by induction from \([\hat{{\textrm{N}}},\hat{{\textrm{A}}}^{\dag g}_{z_J,z}]=\hat{{\textrm{A}}}^{\dag g}_{z_J,z}\) in combination with Theorem 4.6 and \(\hat{{\textrm{N}}} F_g=(2g-2)F_g\) for \(g\ge 2\). \(\quad \square \)

Corollary 5.5

$$\begin{aligned} \hat{{\textrm{A}}}_{{\check{z}}} {{\mathcal {G}}}_g(\bullet |z_2|\dots |z_B) = (2\lambda )^{3} (2g+B-3){{\mathcal {G}}}_g(z_2|\dots |z_B) \end{aligned}$$

whenever \((2g+B-3)>0\).

Hence, up to a rescaling, \(\hat{{\textrm{A}}}_{{\check{z}}}\) indeed removes the boundary component previously located at z. We also have \(\hat{{\textrm{A}}}_{{\check{z}}} F_g=0\) for all \(g\ge 1\) so that the \(F_g\) play the rôle of a vacuum. Note that \({{\mathcal {W}}}_0(z)\) cannot be produced by whatever \(F_0\).

5.2 The Laplacian

Let \({{\mathcal {Z}}}_V^{np} := \exp \big (\sum _{g=1}^\infty V^{2-2g} F_g\big )\) be the non-planar part of the partition function, understood as formal power series in \(\frac{1}{V^2}\). Consider the following operation with \({{\mathcal {Z}}}_V^{np}\):

$$\begin{aligned} O_V:=&\frac{1}{{{\mathcal {Z}}}_V^{np}} \Big [\frac{V^2}{8\lambda ^4} {\hat{K}}_z \hat{{\textrm{A}}}^\dag _z + \Big (\hat{{\textrm{A}}}^\dag _z +\frac{1}{\varrho _0 z^4} \frac{\partial }{\partial z} \Big ) \hat{{\textrm{A}}}^\dag _z +\frac{V^2}{64 \lambda ^4 z^4} \Big ] {{\mathcal {Z}}}_V^{np}. \end{aligned}$$

Its coefficients are

$$\begin{aligned}{}[V^2] O_V =\frac{1}{8\lambda ^4} {\hat{K}}_z \hat{{\textrm{A}}}^\dag _z F_1 + \frac{1}{64 \lambda ^4 z^4} =0 \end{aligned}$$

because of \(\hat{{\textrm{A}}}^\dag _z F_1=\frac{{{\mathcal {G}}}_1(z)}{(2\lambda )^3}\) and \({\hat{K}}_z{{\mathcal {G}}}_1(z)+\frac{\lambda ^3}{z^4}=0\) by (4.3) and (4.2) and

$$\begin{aligned}{}[V^{4-2g}] O_V&= \frac{1}{8\lambda ^4} {\hat{K}}_z \hat{{\textrm{A}}}^{\dag }_z F_g + \sum _{h=1}^{g-1} \big (\hat{{\textrm{A}}}^{\dag }_z F_h\big ) \big (\hat{{\textrm{A}}}^{\dag }_z F_ {g-h}\big ) + \Big (\hat{{\textrm{A}}}^{\dag }_z +\frac{1}{\varrho _0 z^4} \frac{\partial }{\partial z}\Big ) \big (\hat{{\textrm{A}}}^{\dag }_z F_ {g-1}\big ) \end{aligned}$$

for \(g\ge 2\). Inserting \(\hat{{\textrm{A}}}^{\dag }_z F_h =\frac{1}{(2\lambda )^3} {{\mathcal {G}}}_h\) and \(\big (\hat{{\textrm{A}}}^{\dag }_z +\frac{1}{\varrho _0 z^4} \frac{\partial }{\partial z}\big ) {{\mathcal {G}}}_{g-1}(z)=\frac{1}{(2\lambda )^3} {{\mathcal {G}}}_{g-1}(z|z)\) we see that \([V^{4-2g}] O_V\) equals \(\frac{1}{\lambda (2\lambda )^6}\) times the rhs of (4.3), hence vanishes. This means that \(O_V\equiv 0={{\mathcal {Z}}}^{np}_V O_V\).

We invert \({\hat{K}}_z\) via Proposition 4.11, divide by z and apply \(\hat{{\textrm{A}}}_{{\check{z}}}\) given by the residue in Definition 5.1:

$$\begin{aligned} \frac{V^2}{8\lambda ^4} \hat{{\textrm{N}}} {{\mathcal {Z}}}_V^{np}&= {-} \!\sum _{\ell =0}^\infty \mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow 0}\!\Big [dz \frac{z^{3+2\ell } \varrho _\ell }{(3+2\ell )} \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0}\! \Big [ dz'(z')^2 K(z,z')\Big (\!\Big ( \hat{{\textrm{A}}}^\dag _{z'} {+}\frac{1}{\varrho _0 (z')^4} \frac{\partial }{\partial z'} \Big ) \hat{{\textrm{A}}}^\dag _{z'} \\&\quad {+}\frac{V^2}{64 \lambda ^4 (z')^4} \Big ) \Big ]\Big ]{{\mathcal {Z}}}_V^{np}. \end{aligned}$$

We insert \(K(z,z')=\frac{2}{({{\mathcal {W}}}_0(z') -{{\mathcal {W}}}_0(-z')) ({z'}^2-z^2)}\) from Proposition 4.11 and commute the integration contours from \(|z'|<|z|\) to \(|z|<|z'|\). This procedure picks up the poles of \(K(z,z')\) at \(z=z'\) and \(z=-z'\), i.e.

$$\begin{aligned} \mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow 0}\mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0} =\mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0}\Big ( \mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow 0} +\mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow z'} +\mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow -z'} \Big )\;. \end{aligned}$$

There is no pole at \(z=0\); the other two poles give the same contribution. Abbreviating \(f(z',\varrho )= \big (\big (\hat{{\textrm{A}}}^\dag _{z'} {+}\frac{1}{\varrho _0 (z')^4} \frac{\partial }{\partial z'} \big ) \hat{{\textrm{A}}}^\dag _{z'} +\frac{V^2}{64 \lambda ^4 (z')^4} \big ) {{\mathcal {Z}}}_V^{np}\), we thus have

$$\begin{aligned}&{-} \sum _{\ell =0}^\infty \mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow 0}\Big [dz \frac{z^{3+2\ell } \varrho _\ell }{(3+2\ell )} \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0} \Big [ dz'(z')^2 \frac{2}{({{\mathcal {W}}}_0(z') -{{\mathcal {W}}}_0(-z')) ({z'}^2-z^2)} f(z',\varrho )\Big ]\Big ]\\&\quad ={-} \sum _{\ell =0}^\infty \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0}\Big [ dz'(z')^2 \Big (\mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow z'}+\mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow -z'}\Big ) \Big [ \frac{z^{3+2\ell } \varrho _\ell }{(3+2\ell )} \frac{2f(z',\varrho ) dz }{({{\mathcal {W}}}_0(z')-{{\mathcal {W}}}_0(-z')) ({z'}^2{-}z^2)} \Big ]\Big ]\\&\quad = \mathop {{{\,\textrm{Res}\,}}}\limits _{z'\rightarrow 0}\Big [ \sum _{\ell =0}^\infty \frac{dz' (z')^{4+2\ell } \varrho _\ell }{(3+2\ell )} \frac{2}{({{\mathcal {W}}}_0(z') -{{\mathcal {W}}}_0(-z'))} f(z',\varrho )\Big ]\;. \end{aligned}$$

Inserting (4.10) and renaming \(z'\rightarrow z\) we get

$$\begin{aligned} \frac{V^2}{8\lambda ^4} \hat{{\textrm{N}}} {{\mathcal {Z}}}_V^{np}&= \mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow 0} \Big [ dz \,z^3 \tilde{{{\mathcal {R}}}}(z) \Big (\Big ( \hat{{\textrm{A}}}^\dag _{z} +\frac{1}{\varrho _0 (z)^4} \frac{\partial }{\partial z} \Big ) \hat{{\textrm{A}}}^\dag _{z} +\frac{V^2}{64 \lambda ^4 z^4} \Big ) \Big ]{{\mathcal {Z}}}_V^{np}, \end{aligned}$$


$$\begin{aligned} \tilde{{{\mathcal {R}}}}(z)=\frac{\sum _{\ell =0}^\infty \frac{ \varrho _\ell z^{2\ell }}{ (3+2\ell )}}{\sum _{j=0}^\infty \varrho _j z^{2j}} =\sum _{m=0}^\infty \tilde{{{\mathcal {R}}}}_m(\varrho )\, z^{2m}. \end{aligned}$$

The inverse of the denominator is given by (4.11), without the \(\frac{1}{z'}\) prefactor and in variables \(z'\mapsto z\). Its product with the numerator is

$$\begin{aligned} \tilde{{{\mathcal {R}}}}_m(\varrho )&= \frac{S_m(\varrho )}{3\cdot m!} - \sum _{k=1}^m \frac{\varrho _k}{(3+2k)\varrho _0} \frac{S_{m-k}(\varrho )}{(m-k)!} =-\frac{2}{3} \sum _{k=1}^m \frac{k \varrho _k}{(3+2k)\varrho _0} \frac{S_{m-k}(\varrho )}{(m-k)!}, \end{aligned}$$

where we have used (4.15) for the first \(S_m(\varrho )\) to achieve better control of signs.

The residue of \(\frac{V^2}{64 \lambda ^4 z^4}\) is immediate and can be moved to the lhs:

$$\begin{aligned}&\frac{V^2}{8\lambda ^4} \Big (\hat{{\textrm{N}}}-\frac{1}{24}\Big ) {{\mathcal {Z}}}_V^{np}\\&\quad = \sum _{m=0}^\infty \tilde{{{\mathcal {R}}}}_m(\varrho ) \mathop {{{\,\textrm{Res}\,}}}\limits _{z\rightarrow 0} \Big [ z^{3+2m}\,dz \Big [\Big ( \hat{{\textrm{A}}}^\dag _{z} +\frac{1}{\varrho _0 z^4} \frac{\partial }{\partial z} \Big ) \hat{{\textrm{A}}}^\dag _{z} \Big ]\Big ]{{\mathcal {Z}}}_V^{np} \\&\quad =\Big [ \sum _{k=0}^\infty \Big (-\frac{3(3+2k)\varrho _1 \varrho _{k+1}\tilde{{{\mathcal {R}}}}_1(\varrho )}{\varrho _0^3} +\frac{3(3+2k)\varrho _{k+1} \tilde{{{\mathcal {R}}}}_2(\varrho )}{\varrho _0^2} \Big )\frac{\partial }{\partial \varrho _k}\\&\qquad + \sum _{k,l=0}^\infty \frac{(3+2k)(3+2l)\tilde{{{\mathcal {R}}}}_1(\varrho )}{\varrho _0^2} \varrho _{l+1}\frac{\partial }{\partial \varrho _l} \varrho _{k+1}\frac{\partial }{\partial \varrho _k}\\&\qquad -\sum _{k,l=0}^\infty \frac{(3+2k)(3+2l)\tilde{{{\mathcal {R}}}}_{l+2}(\varrho )}{\varrho _0} \Big ( \varrho _{k+1}\frac{\partial }{\partial \varrho _k} \frac{\partial }{\partial \varrho _l} +\frac{\partial }{\partial \varrho _l} \varrho _{k+1}\frac{\partial }{\partial \varrho _k} \Big )\\&\qquad +\sum _{k,l=0}^\infty (3+2k)(3+2l)\tilde{{{\mathcal {R}}}}_{k+l+3}(\varrho ) \frac{\partial }{\partial \varrho _k} \frac{\partial }{\partial \varrho _l}\\&\qquad +\sum _{k=0}^\infty \frac{3(3{+}2k)\varrho _{k+1}\tilde{{{\mathcal {R}}}}_2(\varrho )}{\varrho _0^2} \frac{\partial }{\partial \varrho _k} -\sum _{k=0}^\infty \frac{(3{+}2k)(5{+}2k)\tilde{{{\mathcal {R}}}}_{k+3}(\varrho )}{\varrho _0} \frac{\partial }{\partial \varrho _k} \Big ]{{\mathcal {Z}}}_V^{np}. \end{aligned}$$

Next we separate the \(\varrho _0\)-derivatives:

$$\begin{aligned}&\frac{V^2}{8\lambda ^4} \Big (\hat{{\textrm{N}}}-\frac{1}{24}\Big ) {{\mathcal {Z}}}_V^{np} \\&\quad = \Big [ \Big (\frac{9 \tilde{{{\mathcal {R}}}}_1(\varrho ) \varrho _1^2}{\varrho _0^2} - \frac{18 \tilde{{{\mathcal {R}}}}_{2}(\varrho )\varrho _1}{\varrho _0} +9\tilde{{{\mathcal {R}}}}_{3}(\varrho )\Big ) \frac{\partial ^2}{\partial \varrho _0^2}\\&\qquad +\Big ( -\frac{9\varrho _1^2 \tilde{{{\mathcal {R}}}}_1(\varrho )}{\varrho _0^3} +\frac{18\varrho _{1}\tilde{{{\mathcal {R}}}}_2(\varrho )}{\varrho _0^2} +\frac{15 \tilde{{{\mathcal {R}}}}_1(\varrho ) \varrho _2}{\varrho _0^2} -\frac{30 \tilde{{{\mathcal {R}}}}_{3}(\varrho )}{\varrho _0}\Big ) \frac{\partial }{\partial \varrho _0}\\&\qquad +\sum _{k=1}^\infty 6(3{+}2k)\Big ( \tilde{{{\mathcal {R}}}}_{k+3}(\varrho ) -\frac{\tilde{{{\mathcal {R}}}}_{2}(\varrho ) \varrho _{k+1}}{\varrho _0} -\frac{\tilde{{{\mathcal {R}}}}_{k+2}(\varrho )\varrho _{1}}{\varrho _0} +\frac{\tilde{{{\mathcal {R}}}}_1(\varrho )\varrho _{k+1}\varrho _{1}}{\varrho _0^2}\Big ) \frac{\partial }{\partial \varrho _k} \frac{\partial }{\partial \varrho _0}\\&\qquad + \sum _{k,l=1}^\infty (3{+}2k)(3{+}2l)\Big ( \frac{ \varrho _{l+1}\varrho _{k+1} \tilde{{{\mathcal {R}}}}_1(\varrho )}{\varrho _0^2} + \tilde{{{\mathcal {R}}}}_{k+l+3}(\varrho ) - \frac{2\varrho _{k+1}\tilde{{{\mathcal {R}}}}_{l+2}(\varrho )}{\varrho _0} \Big ) \frac{\partial }{\partial \varrho _k} \frac{\partial }{\partial \varrho _l}\\&\qquad +\sum _{k=1}^\infty (3k+2) \Big (-\frac{3\varrho _1 \varrho _{k+1}\tilde{{{\mathcal {R}}}}_1(\varrho )}{\varrho _0^3} +\frac{6\varrho _{k+1} \tilde{{{\mathcal {R}}}}_2(\varrho )}{\varrho _0^2}\\&\qquad + \frac{(5+2k)\varrho _{k+2} \tilde{{{\mathcal {R}}}}_1(\varrho )}{\varrho _0^2} -\frac{2(5+2k)\tilde{{{\mathcal {R}}}}_{k+3}(\varrho )}{\varrho _0} \Big )\frac{\partial }{\partial \varrho _k} \Big ]{{\mathcal {Z}}}_V^{np}. \end{aligned}$$

We isolate \(F_1\), i.e. \({{\mathcal {Z}}}_V^{np}=\varrho _0^{-\frac{1}{24}} {{\mathcal {Z}}}_V^{stable}\), where \({{\mathcal {Z}}}_V^{stable} := \exp \big (\sum _{g=2}^\infty V^{2-2g} F_g\big )\) is the stable partition function. We commute the factor \(\varrho _0^{-\frac{1}{24}}\) in front of [  ] and move it to the other side:

$$\begin{aligned}&\frac{V^2}{8\lambda ^4} \hat{{\textrm{N}}} {{\mathcal {Z}}}_V^{stable}\\&\quad = \Big [ \Big (\frac{49 \varrho _1^2 \tilde{{{\mathcal {R}}}}_1(\varrho )}{64\varrho _0^4} - \frac{49 \varrho _1 \tilde{{{\mathcal {R}}}}_{2}(\varrho )}{32\varrho _0^3} -\frac{5 \tilde{{{\mathcal {R}}}}_1(\varrho ) \varrho _2}{8\varrho _0^3} +\frac{105 \tilde{{{\mathcal {R}}}}_{3}(\varrho )}{64\varrho _0^2}\Big )\\&\qquad + \Big (\frac{9 \tilde{{{\mathcal {R}}}}_1(\varrho ) \varrho _1^2}{\varrho _0^2} - \frac{18 \tilde{{{\mathcal {R}}}}_{2}(\varrho )\varrho _1}{\varrho _0} +9\tilde{{{\mathcal {R}}}}_{3}(\varrho )\Big ) \frac{\partial ^2}{\partial \varrho _0^2}\\&\qquad +\Big ( -\frac{39\varrho _1^2 \tilde{{{\mathcal {R}}}}_1(\varrho )}{4\varrho _0^3} +\frac{39\varrho _{1}\tilde{{{\mathcal {R}}}}_2(\varrho )}{2\varrho _0^2} +\frac{15 \tilde{{{\mathcal {R}}}}_1(\varrho ) \varrho _2}{\varrho _0^2} -\frac{123 \tilde{{{\mathcal {R}}}}_{3}(\varrho )}{4\varrho _0}\Big ) \frac{\partial }{\partial \varrho _0}\\&\qquad +\sum _{k=1}^\infty 6(3{+}2k)\Big ( \tilde{{{\mathcal {R}}}}_{k+3}(\varrho ) -\frac{\tilde{{{\mathcal {R}}}}_{2}(\varrho ) \varrho _{k+1}}{\varrho _0} -\frac{\tilde{{{\mathcal {R}}}}_{k+2}(\varrho )\varrho _{1}}{\varrho _0} +\frac{\tilde{{{\mathcal {R}}}}_1(\varrho )\varrho _{k+1}\varrho _{1}}{\varrho _0^2}\Big ) \frac{\partial }{\partial \varrho _k} \frac{\partial }{\partial \varrho _0}\\&\qquad + \sum _{k,l=1}^\infty (3{+}2k)(3{+}2l)\Big ( \frac{ \varrho _{l+1}\varrho _{k+1} \tilde{{{\mathcal {R}}}}_1(\varrho )}{\varrho _0^2} + \tilde{{{\mathcal {R}}}}_{k+l+3}(\varrho ) - \frac{2\varrho _{k+1}\tilde{{{\mathcal {R}}}}_{l+2}(\varrho )}{\varrho _0} \Big ) \frac{\partial }{\partial \varrho _k} \frac{\partial }{\partial \varrho _l}\\&\qquad +\sum _{k=1}^\infty (3{+}2k)\Big ( -\frac{\tilde{{{\mathcal {R}}}}_{k+3}(\varrho )}{4 \varrho _0} +\frac{25\varrho _{k+1} \tilde{{{\mathcal {R}}}}_2(\varrho )}{4\varrho _0^2} +\frac{\tilde{{{\mathcal {R}}}}_{k+2}(\varrho )\varrho _{1}}{4\varrho _0^2} -\frac{13\varrho _1 \varrho _{k+1}\tilde{{{\mathcal {R}}}}_1(\varrho )}{4\varrho _0^3}\\&\qquad + \frac{(5{+}2k)\varrho _{k+2} \tilde{{{\mathcal {R}}}}_1(\varrho )}{\varrho _0^2} -\frac{2(5{+}2k)\tilde{{{\mathcal {R}}}}_{k+3}(\varrho )}{\varrho _0} \Big )\frac{\partial }{\partial \varrho _k} \Big ]{{\mathcal {Z}}}_V^{stable}. \end{aligned}$$

Expanding \({{\mathcal {Z}}}_V^{stable} =:1+\sum _{g=2}^\infty V^{2-2g}{{\mathcal {Z}}}_g\), we have

$$\begin{aligned} \hat{{\textrm{N}}}{{\mathcal {Z}}}_V^{stable}&=\sum _{g=2}^\infty (V^{-2})^{g-1}(2g-2) {{\mathcal {Z}}}_g =2V^{-2}\frac{d}{dV^{-2}} \sum _{g=2}^\infty (V^{-2})^{g-1} {{\mathcal {Z}}}_g\\&=2V^{-2}\frac{d}{dV^{-2}} {{\mathcal {Z}}}_V^{stable}. \end{aligned}$$

Consequently, we obtain a parabolic differential equation in \(V^{-2}\) which is easily solved. Inserting

$$\begin{aligned} \tilde{{{\mathcal {R}}}}_1(\varrho )={-}\frac{2}{15} \frac{\varrho _1}{\varrho _0},~~ \tilde{{{\mathcal {R}}}}_2(\varrho )=\frac{2}{15} \frac{\varrho _1^2}{\varrho _0^2} -\frac{4}{21} \frac{\varrho _2}{\varrho _0},~~ \tilde{{{\mathcal {R}}}}_3(\varrho )= {-}\frac{2}{15} \frac{\varrho _1^3}{\varrho _0^3} +\frac{34}{105} \frac{\varrho _1\varrho _2}{\varrho _0^2} -\frac{2}{9} \frac{\varrho _3}{\varrho _0}, \end{aligned}$$

we have:

Theorem 5.6

When expressed in terms of the moments of the measure \(\varrho \), the stable partition function is given as a formal power series in \(V^{-2}\) by

$$\begin{aligned} {{\mathcal {Z}}}_V^{stable}&:=\exp \Big (\sum _{g=2}^\infty V^{2-2g} F_g(\varrho )\Big ) = \exp \Big ( -\frac{(2\lambda )^4}{V^2} \Delta _{\varrho }+\frac{F_2(\varrho )}{V^2}\Big ) 1,\qquad \end{aligned}$$


$$\begin{aligned} F_2&= (2\lambda )^4 \Big ( -\frac{21 \varrho _1^3 }{160\varrho _0^5} +\frac{29}{128} \frac{\varrho _1\varrho _2}{\varrho _0^4} -\frac{35}{384} \frac{\varrho _3}{\varrho _0^3} \Big ), \end{aligned}$$
$$\begin{aligned} \Delta _\varrho&:= -\Big (-\frac{6\varrho _1^3}{5\varrho _0^3} + \frac{111 \varrho _1\varrho _2}{70 \varrho _0^2} - \frac{\varrho _3}{2\varrho _0}\Big ) \frac{\partial ^2}{\partial \varrho _0^2} -\Big ( \frac{2\varrho _1^3}{\varrho _0^4} - \frac{1097 \varrho _1\varrho _2}{280 \varrho _0^3} +\frac{41 \varrho _3}{24 \varrho _0^2}\Big ) \frac{\partial }{\partial \varrho _0} \nonumber \\&\quad -\sum _{k=1}^\infty (3+2k)\Big ( \Big (-\frac{2\varrho _1^2}{5\varrho _0^3} + \frac{2 \varrho _2}{7\varrho _0^2}\Big )\varrho _{k+1} -\frac{3\tilde{{{\mathcal {R}}}}_{k+2}(\varrho )\varrho _{1}}{2\varrho _0} +\frac{3\tilde{{{\mathcal {R}}}}_{k+3}(\varrho )}{2} \Big ) \frac{\partial ^2}{\partial \varrho _k \partial \varrho _0} \nonumber \\&\quad -\sum _{k,l=1}^\infty (3{+}2k)(3{+}2l)\Big ( {-}\frac{ \varrho _1 \varrho _{l+1}\varrho _{k+1}}{30\varrho _0^2} - \frac{\varrho _{k+1}\tilde{{{\mathcal {R}}}}_{l+2}(\varrho )}{4\varrho _0} -\frac{\varrho _{l+1}\tilde{{{\mathcal {R}}}}_{k+2}(\varrho )}{4\varrho _0} \nonumber \\&\quad +\frac{\tilde{{{\mathcal {R}}}}_{k+l+3}(\varrho )}{4} \Big ) \frac{\partial ^2}{\partial \varrho _k\partial \varrho _l} \nonumber \\&\quad -\sum _{k=1}^\infty (3+2k)\Big ( \Big (\frac{19 \varrho _1^2}{60 \varrho _0^4} - \frac{25 \varrho _2}{84 \varrho _0^3}\Big )\varrho _{k+1} +\frac{\varrho _{1}\tilde{{{\mathcal {R}}}}_{k+2}(\varrho )}{16\varrho _0^2} -\frac{\tilde{{{\mathcal {R}}}}_{k+3}(\varrho )}{16 \varrho _0} \nonumber \\&\quad - \frac{(5+2k)\varrho _1\varrho _{k+2}}{30\varrho _0^3} -\frac{(5+2k)\tilde{{{\mathcal {R}}}}_{k+3}(\varrho )}{2\varrho _0} \Big )\frac{\partial }{\partial \varrho _k} \end{aligned}$$

and \(\tilde{{{\mathcal {R}}}}_m(\varrho )\) given by (5.6).

Because we are essentially treating the Kontsevich model [2], our \(F_g\) are nothing else than the generators of intersection numbers of tautological characteristic classesFootnote 4 on the moduli space of stable complex curves [1, 2, 18, 41]. The free energies \(F_g\) are listed in different conventions in the literature. The translation to e.g. [18, 41] is as follows:

$$\begin{aligned} \text {[41]}:{} & {} (1-I_1)&=\varrho _0,&I_{k+1}&=-(2k+1)!! \varrho _k \quad \text {for } k\ge 1, \\ \text {[18]}:{} & {} (2-t_3)&=\varrho _0,&t_{2k+3}&= -\varrho _k, \quad \text {for } k\ge 1. \end{aligned}$$

It is clear that Theorem 5.6 translates into the same statement for the generating function of intersection numbers. We have given this formulation in the very beginning in Theorem 1.1. There we adopt the conventions in [41] but rename \(I_k\equiv t_k=-(2k-1)!!\varrho _{k-1}\) and \(T_0\equiv (1-I_1)\). We also redefined \({{\mathcal {R}}}_m(t)=(2m-1)!!\tilde{{{\mathcal {R}}}}_m(\varrho )\) as well as \(N=\frac{V}{(2\lambda )^2}\). The formula can easily be implemented in computer algebraFootnote 5 and quickly computes the free energies \(F_g(t)\) to moderately large g. Several other implementations exist, for instance the powerful Sage programme [42] which performs many more natural operations in the tautological ring. An early implementation in Maple (up to \(g=10\)) can be found in [43]. These implementations were an important consistency check for us. Algorithms to compute \(\kappa ,\delta ,\lambda \)-classes from \(\psi \)-classes are given in [44].

Example 5.7

For convenience we list

$$\begin{aligned} F_3&= \frac{1225}{144}{\cdot }\frac{t_2^6}{6!T_0^{10}} + \frac{193}{288} {\cdot } \frac{t_2^4 t_3}{4! T_0^9} + \frac{205}{3456} {\cdot } \frac{t_2^2 t_3^2}{2!2! T_0^8} + \frac{53}{1152} {\cdot } \frac{t_2^3 t_4}{3! T_0^8} + \frac{583}{96768} {\cdot } \frac{t_3^3}{3! T_0^7}\\&\quad + \frac{1121}{241920} {\cdot } \frac{t_2 t_3 t_4}{T_0^7} + \frac{17}{5760} {\cdot } \frac{t_2^2 t_5}{2! T_0^7} + \frac{607}{1451520} {\cdot } \frac{t_4^2}{2! T_0^6} + \frac{503}{1451520} {\cdot } \frac{t_3 t_5}{T_0^6}\\&\quad + \frac{77}{414720} {\cdot } \frac{t_2 t_6}{T_0^6} + \frac{1}{82944} {\cdot } \frac{t_7}{T_0^5} \end{aligned}$$

(already given in [41, eq. (5.30)]) and

$$\begin{aligned} F_4&= \frac{1816871}{48} {\cdot } \frac{t_2^9}{9! T_0^{15}} +\frac{3326267}{1728} {\cdot } \frac{t_2^7 t_3}{7! T_0^{14}} +\frac{728465}{6912} {\cdot } \frac{t_2^5 t_3^2}{5! 2! T_0^{13}} +\frac{43201}{6912} {\cdot } \frac{t_2^3t_3^3}{3! 3! T_0^{12}}\\&\quad +\frac{134233}{331776} {\cdot } \frac{t_2 t_3^4}{4! T_0^{11}} + \frac{70735}{864} {\cdot } \frac{t_2^6 t_4}{6! T_0^{13}} +\frac{83851}{17280} {\cdot }\frac{t_2^4 t_3 t_4}{4! T_0^{12}} +\frac{26017}{82944} {\cdot } \frac{t_2^2t_3^2 t_4}{2!2!T_0^{11}}\\&\quad +\frac{185251}{8294400} {\cdot } \frac{t_3^3 t_4}{3! T_0^{10}} +\frac{5609}{23040} {\cdot } \frac{t_2^3 t_4^2}{3!2!T_0^{11}} + \frac{177}{10240} {\cdot } \frac{t_2 t_3 t_4^2}{2! T_0^{10}} +\frac{175}{165888}{\cdot } \frac{t_4^3}{3! T_0^9}\\&\quad +\frac{21329}{6912} {\cdot } \frac{t_2^5 t_5}{5! T_0^{12}} +\frac{13783}{69120} {\cdot } \frac{t_2^3 t_3 t_5}{3!T_0^{11}} +\frac{1837}{129600} {\cdot } \frac{t_2 t_3^2 t_5}{2! T_0^{10}} +\frac{7597}{691200} {\cdot } \frac{t_2^2 t_4 t_5}{2! T_0^{10}}\\&\quad + \frac{719}{829440} {\cdot } \frac{t_3 t_4 t_5}{T_0^9} +\frac{533}{967680}{\cdot } \frac{t_2 t_5^2}{2! T_0^9} +\frac{2471}{23040} {\cdot } \frac{t_2^4 t_6}{4! T_0^{11}} +\frac{7897}{1036800} {\cdot } \frac{t_2^2 t_3 t_6}{2! T_0^{10}}\\&\quad +\frac{1997}{3317760} {\cdot } \frac{t_3^2 t_6}{2! T_0^9} {+}\frac{1081}{2322432} {\cdot } \frac{t_2 t_4 t_6}{T_0^9} {+}\frac{487}{18579456} {\cdot } \frac{t_5 t_6}{T_0^8} {+}\frac{4907}{1382400} {\cdot } \frac{t_2^3 t_7}{3! T_0^{10}}\\&\quad +\frac{16243}{58060800} {\cdot } \frac{t_2 t_3 t_7}{T_0^9} +\frac{1781}{92897280} {\cdot } \frac{t_4 t_7}{T_0^8} + \frac{53}{460800} {\cdot } \frac{t_2^2 t_8}{2! T_0^9} + \frac{947}{92897280} {\cdot } \frac{t_3 t_8}{T_0^8}\\&\quad + \frac{149}{39813120} {\cdot } \frac{t_2 t_9}{T_0^8} + \frac{1}{7962624} {\cdot } \frac{t_{10}}{T_0^7}. \end{aligned}$$

The first line agrees with [41, Table II]. The notation is such that the intersection numbers are easily identified, e.g. \(\langle \tau _2\tau _3^4\rangle \equiv \int _{\overline{{{\mathcal {M}}}}_{4,5}} \psi _1^2 \psi _2^3 \psi _3^3\psi _4^3\psi _5^3= \frac{134233}{331776}\) or \(\langle \tau _2^2\tau _4\tau _5\rangle \equiv \int _{\overline{{{\mathcal {M}}}}_{4,4}} \psi _1^2 \psi _2^2 \psi _3^4\psi _4^5=\frac{7597}{691200}\). The very last number is \(\langle \tau _{3g-2}\rangle \equiv \int _{\overline{{{\mathcal {M}}}}_{g,1}} \psi _1^{3g-2} =\frac{1}{24^{g}\cdot g!}\) for \(g=4\), in agreement with [41, eq. (5.31)]. The arXiv version v1 of this paper also gave \(F_5\) and \(F_6\) in an appendix, but there is not really a need for them.

Remark 5.8

As explained in [18, sec.], it is more appropriate to view \(F_g\), after (inverse) Schur transform of the \(\varrho _l\),

$$\begin{aligned} (-(2l+1)!!) \frac{\varrho _l}{\varrho _0} =: -[x^l] \exp \Big (-\sum _{i=1}^\infty s_ix^i\Big ) \end{aligned}$$

as generating function of intersection numbers of \(\kappa \)-classes on \(\overline{{{\mathcal {M}}}}_{g,0}\). When passing to \({{\mathcal {G}}}_g(z_1|\dots |z_n)\) by application of the boundary creation operator, equivalently to the \(\omega _{g,n}(z_1,\dots ,z_n)\) of topological recursion, all mixed intersection numbers of \(\psi \)- and \(\kappa \)-classes are generated. A factor \(\frac{-(2l+1)!!}{z_i^{2l+3}}\) translates to \(\psi _i^l\) and differentiation with respect to \(s_l\) gives a factor \(\kappa _l\) under the \(\overline{{{\mathcal {M}}} }_{g,n}\)-integral. In particular, the subfamily of intersection numbers only of \(\psi \)-classes and with power \(\ge 2\) arises by multiple application of \(\prod _{i=1}^n \frac{2+3l_i}{z_i^{5+2l_i}} \frac{\partial }{ \partial \varrho _{l_i}}\Big |_{l_i\ge 1}\) on \(F_g\) and projecting to the part without \(\varrho _{l\ge 1}\). Every term in \(F_g\) with n factors of \(t_{i\ge 2}\) thus gives a unique intersection number of \(\psi \)-classes on \(\overline{{{\mathcal {M}}}}_{g,n}\). This justifies to view \(F_g\) as intersection numbers of \(\psi \)-classes.

Example 5.9

Consider \(g=2\) and \(F_2\) given by (5.7). Inserting \((-7!!) \frac{\varrho _3}{\varrho _0}=s_3-s_2s_1+\frac{s_1^3}{6}\), \((-5!!) \frac{\varrho _2}{\varrho _0}=s_2-\frac{s_1^2}{2}\) and \((-3!!) \frac{\varrho _1}{\varrho _0}=s_1\) from (5.9) we get

$$\begin{aligned} F_2&=\frac{7}{240} \cdot \frac{1}{6!} \Big ((-3!!) \frac{\varrho _1}{\varrho _0}\Big )^3 +\frac{29}{5760} \Big ((-5!!) \frac{\varrho _2}{\varrho _0}\Big ) \Big ((-3!!) \frac{\varrho _1}{\varrho _0}\Big ) +\frac{1}{1152} \Big ((-7!!) \frac{\varrho _3}{\varrho _0}\Big )\\&= \frac{7}{240} \frac{s_1^3}{3!} +\frac{29}{5760} (s_2-\tfrac{s_1^2}{2}) s_1 +\frac{1}{1152}(s_3-s_2s_1+\tfrac{s_1^3}{6})\\&=\frac{43}{2880} \frac{s_1^3}{3!} +\frac{1}{240} s_2s_1 +\frac{1}{1152}s_3 \end{aligned}$$

from which one extracts \(\int _{\overline{{{\mathcal {M}}}}_{2,0}} \kappa _3=\frac{1}{1152}\), \(\int _{\overline{{{\mathcal {M}}}}_{2,0}} \kappa _2\kappa _1=\frac{1}{240}\) and \(\int _{\overline{{{\mathcal {M}}}}_{2,0}} (\kappa _1)^3=\frac{43}{2880}\).

Applying \({\hat{A}}^{\dag }_{z_1}\) directly to (5.7) gives

$$\begin{aligned}&{{\mathcal {G}}}_2(z_1) \nonumber \\&\quad = (2\lambda )^3 {\hat{A}}^\dag _{z_1}F_2\nonumber \\&\quad = \frac{(2\lambda )^7}{\varrho _0^3} \Big \{ \frac{7}{288}\Big (\frac{-3!!\varrho _1 }{\varrho _0}\Big )^4 \frac{(-1!!)}{z_1^3} +\frac{5}{144} \Big ( \frac{-3!!\varrho _1}{\varrho _0} \Big )^2 \Big (\frac{-5!!\varrho _2}{\varrho _0}\Big ) \frac{(-1!!)}{z_1^3} \nonumber \\&\qquad +\frac{29}{5760} \Big (\frac{-5!!\varrho _2}{\varrho _0}\Big )^2 \frac{(-1!!)}{z_1^3} +\frac{11}{1440}\frac{(-7!!\varrho _3)}{\varrho _0} \frac{(-3!!\varrho _1)}{\varrho _0} \frac{(-1!!)}{z_1^3} \nonumber \\&\qquad +\frac{1}{1152} \Big (-9!! \frac{\varrho _4}{\varrho _0}\Big ) \frac{(-1!!)}{z_1^{3}}+ \frac{7}{288}\Big (\frac{(-3!!\varrho _1) }{\varrho _0}\Big )^3 \frac{(-3!!)}{z_1^5} \nonumber \\&\qquad +\frac{29}{1440} \frac{(-3!!\varrho _1)}{\varrho _0} \frac{(-5!!\varrho _2)}{\varrho _0} \frac{(-3!!)}{z_1^5} +\frac{1}{384} \frac{(-7!!\varrho _3)}{\varrho _0} \frac{(-3!!)}{z_1^5}\\&\qquad +\frac{7}{480} \Big ( \frac{-3!!\varrho _1}{\varrho _0} \Big )^2\frac{(-5!!)}{z_1^7} +\frac{29}{5760} \frac{(-5!!\varrho _2)}{\varrho _0} \frac{(-5!!)}{z_1^7} \nonumber \\&\qquad +\frac{29}{5760} \frac{(-3!!\varrho _1)}{\varrho _0} \frac{(-7!!)}{z_1^9} +\frac{1}{1152} \frac{(-9!!)}{z_1^{11}}\Big \}\;. \end{aligned}$$

The very last term encodes the intersection number \(\int _{\overline{{{\mathcal {M}}}}_{2,1}} \psi _1^4=\frac{1}{1152}\). All other involve \(\kappa \)-classes. To extract them we have to pass via (5.9) to the \(s_l\)-variables:

$$\begin{aligned} {{\mathcal {G}}}_2(z_1)&= \frac{(2\lambda )^7}{\varrho _0^3} \Big \{ \Big (\frac{1}{1152}s_4+\frac{13}{1920}s_1s_3 +\frac{53}{5760}\frac{s_2^2}{2} +\frac{259}{5760}\cdot \frac{s_1^2}{2}s_2+\frac{29}{128}\cdot \frac{s_1^4}{24}\Big ) \frac{(-1!!)}{z_1^{3}} \nonumber \\&\quad + \Big (\frac{1}{384}s_3+\frac{101}{5760}s_2s_1 +\frac{169}{1920}\cdot \frac{s_1^3}{6}\Big ) \frac{(-3!!)}{z_1^5} +\Big (\frac{29}{5760}s_2 +\frac{139}{5760}\frac{s_1^2}{2} \Big )\frac{(-5!!)}{z_1^7} \nonumber \\&\quad +\frac{29}{5760}s_1 \frac{(-7!!)}{z_1^9} +\frac{1}{1152}\frac{(-9!!)}{z_1^{11}}\Big \}\;. \end{aligned}$$

We extract e.g. \(\int _{\overline{{{\mathcal {M}}}}_{2,1}} \kappa _2(\kappa _1)^2=\frac{259}{5760}\), \(\int _{\overline{{{\mathcal {M}}}}_{2,1}} \kappa _2\kappa _1 \psi _1=\frac{101}{5760}\), \(\int _{\overline{{{\mathcal {M}}}}_{2,1}} (\kappa _1)^2 (\psi _1)^2=\frac{139}{5760}\) and so on. These intersection numbers agree with the implementation in admcycles [42].

On the other hand we can apply \({\hat{A}}_{z_1,z_2}\) to (5.10) in order to extract intersection numbers on \(\overline{{{\mathcal {M}}}_{2,2}}\). We only give the part without \(\varrho _{l\ge 1}\) which encodes intersection numbers of \(\psi \)-classes:

$$\begin{aligned} {{\mathcal {G}}}_2(z_1|z_2)&= \frac{(2\lambda )^{10}}{\varrho _0^4} \Big \{ \frac{1}{384} \frac{(-9!!)}{z_2^{11}} \frac{(-3!!)}{z_1^5} +\frac{29}{5760} \frac{(-7!!)}{z_2^9} \frac{(-5!!)}{z_1^7} +\frac{29}{5760} \frac{(-5!!)}{z_2^7} \frac{(-7!!)}{z_1^9}\\&\quad +\frac{1}{1152} \frac{(-11!!)}{z_2^{13}} \frac{(-1!!)}{z_1^{3}} +\frac{1}{384} \frac{(-3!!)}{z_2^5} \frac{(-9!!)}{z_1^{11}} +\frac{1}{1152} \frac{(-1!!)(-11!!)}{z_2^3z_1^{13}}\\&\quad +{{\mathcal {O}}}(\varrho _{l\ge 1})\Big \}\;. \end{aligned}$$

This gives the intersection numbers \(\int _{\overline{{{\mathcal {M}}}}_{2,2}} \psi _1^0\psi _2^5=\frac{1}{1152}\), \(\int _{\overline{{{\mathcal {M}}}}_{2,2}} \psi _1\psi _2^4=\frac{1}{384}\) and \(\int _{\overline{{{\mathcal {M}}}}_{2,2}} \psi _1^2\psi _2^3=\frac{29}{5760}\) as well as the symmetric copies \(\psi _1\leftrightarrow \psi _2\). The last case arises by application of \(\prod _{i=1}^2(2\lambda )^3\frac{2+3l_i}{z_i^{5+2l_i}} \frac{\partial }{ \partial \varrho _{l_i}}\Big |_{l_1=1,l_2=2}\) to the term \((2\lambda )^4\frac{29}{128} \frac{\varrho _1\varrho _2}{\varrho _0^4}\) of \(F_2\). In this sense \(F_2\) itself can be viewed as generating function of the subset of intersection numbers of \(\psi \)-classes on \(\overline{{{\mathcal {M}}}}_{2,n}\) where the power of every \(\psi _i\) is \(\ge 2\).

5.3 A deformed Virasoro algebra

We return to (5.4) with \(O_V\equiv 0\), but instead of applying the inverse of \({\hat{K}}_z\) we directly take the residue to define a family of operators on rational functions \(f(\varrho )\) of \(\varrho _l\):

$$\begin{aligned} {\tilde{L}}_n f(\varrho ):= {{\,\textrm{Res}\,}}_{z\rightarrow 0} \Big [z^{3+2n} \Big (\frac{V^2}{8\lambda ^4} {\hat{K}}_z \hat{{\textrm{A}}}^\dag _z + (\hat{{\textrm{A}}}^\dag _z )^2 +\frac{1}{\varrho _0 z^4} \frac{\partial \hat{{\textrm{A}}}^\dag _z }{\partial z} +\frac{V^2}{64\lambda ^4 z^4} \Big ) f(\varrho ) dz\Big ]. \end{aligned}$$

By construction, \({\tilde{L}}_n {{\mathcal {Z}}}^{np}_V=0\). Recall that in the Kontsevich model one has \(L_n{{\mathcal {Z}}}\) for the full partition function and generators \(L_n\) of a Virasoro algebra (or rather a Witt algebra). As explained below, these \({\tilde{L}}_n\) do not satisfy the commutation relations of the Virasoro algebra exactly. An explicit expression is obtained from

$$\begin{aligned} {\hat{K}}_z \hat{{\textrm{A}}}^{\dag }_z&= \sum _{l= 0}^{\infty } \sum _{j=0}^{l} \frac{(3+2l)\varrho _{l-j} }{z^{4+2j}} \frac{\partial }{\partial \varrho _l},\\ \frac{1}{\varrho _0 z^4} \frac{\partial }{\partial z} \hat{{\textrm{A}}}^{\dag }_z&= \sum _{l= 0}^\infty \Big (\frac{3(3+2l) \varrho _{l+1}}{ \varrho _0^2 z^8}-\frac{(3+2l)(5+2l)}{\varrho _0 z^{10+2l}}\Big ) \frac{\partial }{\partial \varrho _l},\\ \hat{{\textrm{A}}}^{\dag }_z\hat{{\textrm{A}}}^{\dag }_z&=\sum _{k= 0}^{\infty } \Big (\frac{(5+2k) \varrho _{k+2}}{ \varrho _0 z^3}-\frac{5+2k}{z^{7+2k}}\Big )\frac{(3+2k)}{ \varrho _0 z^3} \frac{\partial }{\partial \varrho _k}\\&\quad + \sum _{k= 0}^{\infty } \Big ({-}\frac{3\varrho _{1}}{ \varrho _0 z^3}+\frac{3}{z^{5}}\Big ) \Big (\frac{(3{+}2k) \varrho _{k+1}}{ \varrho _0^2 z^3}\Big ) \frac{\partial }{\partial \varrho _k}\\&\quad + \sum _{l,k= 0}^{\infty } \frac{(3{+}2l) (3{+}2k) \varrho _{k+1} \varrho _{l+1}}{ \varrho _0^2 z^6} \frac{\partial ^2}{\partial \varrho _l \partial \varrho _k}\\&\quad - \sum _{l,k= 0}^{\infty } \frac{2(3{+}2l)(3{+}2k) \varrho _{l+1}}{ \varrho _0 z^{8+2k}} \frac{\partial ^2}{\partial \varrho _l \partial \varrho _k} + \sum _{l,k= 0}^{\infty } \frac{(3{+}2l)(3{+}2k)}{z^{10+2l+2k}} \frac{\partial ^2}{\partial \varrho _l \partial \varrho _k}. \end{aligned}$$

Evaluating the residues and defining \(A=\frac{(2\lambda )^4}{4V^2}\) and rescaling \(L_n:=A {\tilde{L}}_n\) gives

$$\begin{aligned} L_0&=\frac{1}{16}+ \frac{1}{2}\sum _{l=0}^\infty (3+2l)\varrho _l \frac{\partial }{\partial \varrho _l},\\ L_1&=\frac{1}{2}\sum _{l=0}^\infty (5{+}2l)\varrho _{l}\frac{\partial }{\partial \varrho _{l+1}} +A\Big (\sum _{k=0}^\infty \frac{(3{+}2k)}{\varrho _0^2} \varrho _{k+1}\frac{\partial }{\partial \varrho _k} {-} \frac{3\varrho _1}{\varrho _0^2}\Big ) \sum _{l=0}^\infty (3{+}2l)\varrho _{l+1} \frac{\partial }{\partial \varrho _l} \\&\quad \text {and for }\,n\ge 2: \\ L_{n}&=\frac{1}{2} \sum _{l=0}^\infty (3{+}2n{+}2l)\varrho _{l}\frac{\partial }{\partial \varrho _{n+l}} +A \delta _{n,2} \sum _{l=0}^\infty \frac{6(3+2l) \varrho _{l+1}}{ \varrho _0^2}\frac{\partial }{\partial \varrho _{l}} \\&\quad -A\frac{2(2n{-}3)(2n{-}1)}{\varrho _0} \frac{\partial }{\partial \varrho _{n-3}} +A \sum _{l= 0}^{n-3}(3+2l)(2n-2l-3) \frac{\partial ^2}{\partial \varrho _l \partial \varrho _{n-3-l}}\\&\quad -A\sum _{l= 0}^{\infty } \frac{2(3+2l)(2n-1) \varrho _{l+1}}{ \varrho _0} \frac{\partial ^2}{\partial \varrho _{n-2} \partial \varrho _l}. \end{aligned}$$

To write it in a more compact way, it is convenient to introduce the differential operator

$$\begin{aligned} {\hat{D}}:= \sum _{l=0}^\infty \frac{(3+2l)\varrho _{l+1}}{\varrho _0} \frac{\partial }{\partial \varrho _l}\;. \end{aligned}$$

Note that \(\frac{\partial }{\partial \varrho _{l}} {\hat{D}}\ne {\hat{D}}\frac{\partial }{\partial \varrho _{l}}\). The result is:

Lemma 5.10

The nonplanar partition \({{\mathcal {Z}}}_V^{np} := \exp \Big (\sum _{g=1}^\infty V^{2-2g} F_g\Big )\) satisfies the constraints \(L_n{{\mathcal {Z}}}^{np}_V=0\) for all \(n\in {{\mathbb {N}}}\), where

$$\begin{aligned} L_0&=\frac{1}{16}+ \frac{1}{2}\sum _{l=0}^\infty (3+2l)\varrho _l \frac{\partial }{\partial \varrho _l},\nonumber \\ L_1&=\frac{1}{2}\sum _{l=0}^\infty (5+2l)\varrho _{l}\frac{\partial }{\partial \varrho _{l+1}} +A{\hat{D}}^2 \nonumber \\&\quad \text {and for}\, n\ge 2{:} \nonumber \\ L_{n}&=\frac{1}{2} \sum _{l=0}^\infty (3{+}2n{+}2l)\varrho _{l}\frac{\partial }{\partial \varrho _{n+l}} +A \sum _{l= 0}^{n-3}(3+2l)(2n-2l-3) \frac{\partial ^2}{\partial \varrho _l \partial \varrho _{n-3-l}} \nonumber \\&\quad -2A(2n-1)\frac{\partial }{\partial \varrho _{n-2}}{\hat{D}}, \end{aligned}$$

where \({\hat{D}}\) is the differential operator defined by (5.13) and \(A=\frac{(2\lambda )^4}{4V^2}\).

With the commutation rules

$$\begin{aligned} \Big [{\hat{D}},\varrho _l\Big ]&= \frac{3+2l}{\varrho _0} \varrho _{l+1}\;,&l\ge 0\;,\\ \Big [{\hat{D}},\frac{\partial }{\partial \varrho _l} \Big ]&= -\frac{1+2l}{\varrho _0} \frac{\partial }{\partial \varrho _{l-1}}&l\ge 1\;,{} & {} \Big [{\hat{D}},\frac{\partial }{\partial \varrho _0} \Big ]&= \frac{1}{\varrho _0} {\hat{D}}\; , \end{aligned}$$

we end up after long but straightforward computation:

Lemma 5.11

The generators \(L_n\) of Lemma 5.10 obey the commutation relation

$$\begin{aligned}{}[L_0,L_n]=-nL_n \end{aligned}$$

and for any \(m,n\ge 1\),

$$\begin{aligned} {[}L_m,L_n]&= (m-n) L_{m+n} -4A (m+1)B_{n-2}L_{m-1} +4A (n+1)B_{m-2}L_{n-1}\\&\quad - 4A \delta _{m,1} \frac{n(n+1)}{\varrho _0^2} L_{n-2} + 4A \delta _{n,1} \frac{m(m+1)}{\varrho _0^2} L_{m-2} \end{aligned}$$


$$\begin{aligned} B_m:= (2m+3)\frac{\partial }{\partial \varrho _{m}} \frac{1}{\varrho _0} \quad \text {for }m\ge 0\;,\qquad B_{-1}:= -\frac{1}{2}\Big \{{\hat{D}}, \frac{1}{\varrho _0} \Big \} \;. \end{aligned}$$

Remark 5.12

The differential operator \({\hat{D}}\) has its origin in the implicit definition of the constant c (3.13) and the dependence of \(\varrho _l\) on c. Since the expression

$$\begin{aligned} \varrho _{-1}:=-\int _1^{\Lambda ^2} dY\;\frac{\varrho (Y)}{\sqrt{Y+c}}, \qquad (\varrho _{-1}=c\;\text {for}\;D=0) \end{aligned}$$

diverges for any \(D>0\) in the limit \(\Lambda \rightarrow \infty \), it is necessary to reconstruct the analogue of the derivative \(\frac{\partial }{\partial c}\) through the differential operator \({\hat{D}}\). Replacing the differential operator by \({\hat{D}}\mapsto \frac{\partial }{\partial \varrho _{-1}}\) and the generators by

$$\begin{aligned} L_n \mapsto L_n+\frac{1}{2}\varrho _{-1}\frac{\partial }{\partial \varrho _{-1}}, \end{aligned}$$

recovers the original undeformed Virasoro algebra. As explained above, \(\varrho _{-1}\) and consequently the standard Virasoro generators do not exist in dimension \(D>0\). The renormalisation necessary for \(D>0\) alters the definition of c and prevents the construction of \(L_{-1}\) and \(F_0\) which in \(D=0\) depend on \(\varrho _{-1}\). Higher topologies (\(\chi \le 0\)) are not affected because any explicit \(\varrho _{-1}\)-dependence drops out.

6 Summary

The construction of the renormalised \(\Phi ^3_D\)-QFT model on noncommutative geometries of dimension \(D\le 6\) is now complete. After the previous solution of the planar sector in [27, 28] we established in this paper an algorithm to compute any correlation function \( G^{(g)}_{|\underline{p}_1^1\ldots \underline{p}^1_{N_1}|\cdots | \underline{p}_1^B\cdots \underline{p}^B_{N_B}|}\) of genus \(g\ge 1\):

  1. 1.

    Compute the free energy \(F_g(t)\) via Theorem 1.1 and the note thereafter. It encodes the \(p(3g-3)\) intersection numbers of \(\psi \)-classes (all with a power \(\ge 2\)) on the moduli spaces \(\overline{{{\mathcal {M}}}}_{g,n}\) of stable complex curves of genus g. Take \(F_1=-\frac{1}{24} \log T_0\) for \(g=1\). Alternatively, start from intersection numbers obtained by other methods (e.g. [42]).

  2. 2.

    Change variables to \(\varrho _0=1-t_0\) and \(\varrho _l =-\frac{t_{l+1}}{(2l+1)!!}\), where \(\varrho _l\) are given by (3.17) for the measure (3.8) and with c implicitly defined by (3.13).

  3. 3.

    Apply to the resulting \(F_g(\varrho )\) according to Proposition 5.2 and Theorem 4.6 the boundary creation operators \( \hat{{\textrm{A}}}^{\dag g}_{z_1,\dots ,z_B}\circ \dots \hat{{\textrm{A}}}^{\dag g}_{z_1,z_2} \circ \hat{{\textrm{A}}}^{\dag g}_{z_1}\) defined in Definition 4.4. Multiply by \((2\lambda )^{4g+3B-4}\) to obtain \({{\mathcal {G}}}_g(z_1|\cdots |z_B)\).

  4. 4.

    Pass to \({{\mathcal {G}}}_g(z_1^1\cdots z^1_{N_1}|\cdots |z_1^B\cdots z^B_{N_B})\) via difference quotients (3.14), where \(X^\beta _{k_\beta }=(z^\beta _{k_\beta })^2-c\).

  5. 5.

    Specify to \(z^\beta _{k_\beta }\mapsto ( 4F_{\underline{p}^\beta _{k_\beta }}^2+c)^{1/2}\) to obtain \(G^{(g)}_{|\underline{p}_1^1\cdots \underline{p}^1_{N_1}|\cdots | \underline{p}_1^B\cdots \underline{p}^B_{N_B}|}\), where \(F_{\underline{p}}\) arises by mass-renormalisation from the \(E_{\underline{p}}\) in the initial action (3.1) of the model.

Our work was essentially a reverse engineering in opposite order. The last step 6. was given to us by the formal partition function of the model. From there we had to climb up to the formula for the intersection numbers.

We remark that, in spite of the relation to the integrable Kontsevich model [2], this \(\Phi ^3_D\)-model provides a fascinating toy model for a quantum field theory which shows many facets of renormalisation. Our exact formulae can be expanded about \(\lambda =0\) via (3.13) and agree with the usual perturbative renormalisation which in \(D=6\) needs Zimmermann’s forest formula [45] (see [28]). Also note that at fixed genus g one expects \({{\mathcal {O}}}(n!)\) graphs with n vertices so that a convergent summation at fixed g cannot be expected a priori. Moreover, in \(D=6\) the \(\beta \)-function of the coupling constant is positive for real \(\lambda \), which in this particular case poses not the slightest problem for summation.

What remains to understand is the resummation in the genus, i.e. \(\sum _{g=1}^\infty V^{2-2g} {{\mathcal {G}}}_g(z)\) or \(\sum _{g=2}^\infty N^{2-2g} F_g(t)\). All intersection numbers are positive for \(t_l>0\), which corresponds to \(\varrho _l<0\) for \(l\ge 1\). Because of the \(\lambda ^2\)-prefactor in front of (3.8) and the definition (3.17) of the \(\varrho _l\), we have \(t_l>0\) for real \(\lambda \). Therefore, the sum over the genus must diverge for \(\lambda \in {{\mathbb {R}}}\), which is not surprising because in this case the action (3.1) is unbounded from below. In contrast, it was observed in [28] that for the planar sector it is better to take \(\lambda \in {{\mathbb {R}}}\). The final challenge of this model is to establish that \(\sum _{g=2}^\infty N^{2-2g} F_g(t)\) is Borel summable for \(t_l<0\), which would achieve convergence of the genus expansion in two disks in the complex \(\lambda \)-plane tangent from above and below the real axis at \(\lambda =0\). Important progress in this direction was recently achieved in [46] with the proof of factorial bounds \(|F_g| \le r^{-g}\Gamma (\beta g) \) for some \(r>0\) and \(\beta \le 5\).