A Laplacian to compute intersection numbers on $\bar{\mathcal{M}}_{g,n}$ and correlation functions in NCQFT

Let $F_g(t)$ be the generating function of intersection numbers on the moduli spaces $\bar{\mathcal{M}}_{g,n}$ of complex curves of genus $g$. As by-product of a complete solution of all non-planar correlation functions of the renormalised $\Phi^3$-matrical QFT model, we explicitly construct a Laplacian $\Delta_t$ on the space of formal parameters $t_i$ satisfying $\exp(\sum_{g\geq 2} N^{2-2g}F_g(t))=\exp((-\Delta_t+F_2(t))/N^2)1$ for any $N>0$. The result is achieved via Dyson-Schwinger equations from noncommutative quantum field theory combined with residue techniques from topological recursion. The genus-$g$ correlation functions of the $\Phi^3$-matricial QFT model are obtained by repeated application of another differential operator to $F_g(t)$ and taking for $t_i$ the renormalised moments of a measure constructed from the covariance of the model.


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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: be the generating function of intersection numbers of ψ-classes 1 on the moduli spaces 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 , where F 2 (t) = 7 240 · t 3 2 3!T 5 0 + 29 5760 generates the intersection numbers of genus 2, T 0 := (1 − t 0 ) and ∆ t = − i,jĝ ij ∂ i ∂ j − iΓ i ∂ i is a Laplacian on the formal parameters t 0 , t 2 , t 3 , . . . given by The F g (t) are recursively extracted from Z g (t) := 1 (g−1)! (−∆ t + F 2 (t)) g−1 1 via 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( g≥0 F g ) = exp(Ŵ )1 proved by Alexandrov [3] 2 , whereŴ := 2 3 ∞ k=1 (k + 1 2 )t kLk−1 involves the generatorsL n of the Virasoro algebra. Including N and moving exp(N 2 F 0 + F 1 ) to the other side, our ∆ 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 which we haven't studied yet: • IsΓ i a Levi-Civita connection forĝ ij , i.e.Γ i = jĝ ij detĝ −1 ∂ j ( 1 √ detĝ −1 )? Here detĝ −1 would be the determinant of (ĝ ij ), whatever this means.
• Is there a reasonable definition of a volume dt 1 √ detĝ −1 (t) ?
• Is it possible to prove that ∞ g=2 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 Φ 3 D -matricial quantum field theory in any dimension D ∈ {0, 2, 4, 6}. These models are closely related to the Kontsevich model [2] so that a link to intersection numbers is not surprising.
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 Φ 4 -model on Moyal space and its vanishing β-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 Φ 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 Φ 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 ∈ {2, 4, 6}. In particular, exact (and surprisingly compact) formulae for planar correlation functions with B ≥ 2 boundary components were obtained. The simplicity of the formulae [27] for B ≥ 2 suggests an underlying pattern. It is traced back to the universality phenomena captured by topological recursion 3 . We refer to the book [18].
In this article we give the complete description of the non-planar sector of the renormalised Φ 3 D -model. The notation defined in [28] will be used and recalled in section 3. We borrow from topological recursion the notational simplification to complex variables z for the previous √ X + c and the vision that the correlation functions are holomorphic in z ∈ C \ {0} (see section 4.1). Knowing this, we proceed however in a different way. In section 4.2 we introduce our main tool, a boundary creation operatorÂ †g z 1 ,...,z B which, when applied to a genus-g correlation function G g (z 1 | . . . |z B−1 ) with B − 1 boundaries labelled z 1 , . . . , z B−1 , creates a B 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 { l } l∈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 ∈ {0, 2, 4, 6}. The boundary creation operator acts on Laurent polynomials in the z i with coefficients in rational functions of the 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 section 4.3 structural results about the G g (z 1 | . . . |z B ) such as the degree of the Laurent polynomials, the maximal number of occurring { l } and the weight of the rational function.
The main part of section 4.3 is devoted to the solution of G g (z) for g ≥ 1. Starting point a Dyson-Schwinger type equation (4.3), whereK z an integral operator. Thus, all G g (z) can be recursively evaluated ifK 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 section 5.1 that the G g (z) arise for g ≥ 1 by application of the boundary creation operator to a uniquely defined 'free energy' F g ( ). These F g ( ) are characterised by 'only' p(3g − 3) rational numbers, where p(n) is the number of partitions of n. We show in section 5.2 that (2.1) can be written as a secondorder differential operator acting on exp( g≥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 0 = 1 − t 0 and l = − t l +1 (2l+1)!! . In other words, to construct the non-planar sector of the Φ 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 { l } resulting from the renormalisation of the planar sector of the model. We finish with a discussion of the (deformed) Virasoro algebra in section 5.3 and a short summary (section 6).
In the meantime an analogue of the boundary creation operatorÂ †g z J ,z was found for the matricial Φ 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 |χ| provided strong evidence for the conjecture that the matricial Φ 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 Φ 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].

Setup
We follow [28] to introduce tuples n = (n 1 , n 2 , .., n D 2 ) of non-negative integers. The number of tuples n of given |n| = n 1 + n 2 + .. + nD where H nm := E n + E m . The constant V is first of all a formal parameter; for a noncommutative quantum field theory model, V = ( θ 4 ) D/2 will be related to the deformation parameter of the Moyal plane. The parameters λ bare , κ, ν, ζ, Z and soon µ bare are N -dependent renormalisation parameters. They will be determined by normalisation conditions parametrised by physical mass µ and coupling constant λ. The matrices (Φ nm ) are multi-indexed Hermitian matrices, Φ nm = Φ mn . The external matrix E = (E m δ n,m ) can be assumed to be diagonal and has the eigenvalues E n = The next step is to define (and rearrange) the partition function where source (J nm ) 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 J p 1 ...p N β := N β j=1 J p j p j+1 with p N β +1 ≡ 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: | an (N 1 + ... + N B )-point function of genus g; when the N β do not matter, a correlation function of genus g with B boundary components. hese (N 1 + ... + 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 |q| = |p| It arises from invariance of the partition function under unitary transformation Φ → U † ΦU of the integration variable [14], or directly from the structure of Z[J] [40].

Integral equations
Introducing the measure we can rewrite (3.7) as an integral equation. The measure has support in [4F 2 0 , Λ 2 N ] where Λ 2 N = max(4F 2 n : |n| = N ). For quantum field theory it is necessary to take a large-N limit. In general this produces divergences which need renormalisation. Optionally the large-N limit can be entangled with a limit V → ∞ which, supposing the F n scale down with V (as e.g. in (3.5)), can be designed to let (X) converge to a continuous function. We also pass to mass-dimensionless quantities via multiplication by specified powers of µ [28]. This amounts to choose the mass scale as µ = 1.
To keep maximal flexibility we consider a measure with support in [1, Λ 2 ] of which a limit Λ → ∞ has to be taken for quantum field theory. As already observed in [29], equation (3.6) extends to the closed equation for a sectionally holomorphic function W 0 (X) from which one extracts W . The corresponding relation for 2g + B > 1 extends (3.7) to Using techniques for boundary values of sectionally holomorphic functions [29], easily adapted to include Z − 1, ν, C = 0 [28], one obtains the following solution of (3.9): Here, the finite parameter c and the (for Λ 2 → ∞) possibly divergent Z, ν are determined by renormalisation conditions depending on the dimension: together with the convention Z = 1 for D ∈ {2, 4} and ν = 0 for D = 2. For given coupling constant λ as the only remaining parameter, these equations can be solved for c: By the implicit function theorem, (3.13) has a smooth solution in an inverval −λ c < λ < λ c , in any dimension D ∈ {0, 2, 4, 6}. The Lagrange inversion theorem gives the expansion of c in λ 2 : After that renormalisation procedure the limit Λ 2 → ∞ is safe in all correlation function and any dimension D ∈ {2, 4, 6}.

Dyson-Schwinger equation for
For g = 0 one has to write where J = {2, 3, .., B}. Here and throughout the paper (for z instead of X) we abbre- ). In the sum, I I = J means summation over all possibly empty subsets I ⊂ J, with I := J \ I. The difference to the planar sector (g = 0) is the last term indexed g − 1 which only contributes if g ≥ 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]: Note that multiple t-derivatives of R(t) at t = 0 produce renormalised moments of the measure (3.8): In fact the proof of (3.16) consists in a resummation of an ansatz which involves Bell polynomials (see Definition 4.10) in the { 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 { l }. The concrete model characterised by the sequence E n , coupling constant λ and the dimension D only affects the values of { 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: 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 [ . 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 = ±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 Note that˜ (y) has support in [ [28]. Furthermore, W 0 (z) extends to a sectionally holomorphic function with branch cut along [− √ 1 + Λ 2 , − √ 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.
In this notation, (3.11) takes the form We will heavily rely on: Lemma 4.2 The operatorK z defined in Definition 4.1 satisfieŝ Proof This is a reformulation of [27, Lemma 5.5].
The first step beyond [27] is to determine the 1-point function at genus 1: where the l are given in (3.17).
Comparison of coefficients yields the assertion.
From (3.14) we get the N -point function of genus 1 which in complex variables reads 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} and (3.15) gives with Definition 4.1 the following formula for G g (z 1 |z J ), for 2g + |J| ≥ 2: Note that this covers also (4.3) as J = ∅.

Boundary creation operator
We are going to construct an operator which plays the rôle of the formal T n := 1 En ∂ ∂En applied to the logarithm of the partition function Z[0] given in (3.2). In dimension D = 0 where Z − 1 = κ = ν = ζ = 0 and µ bare = µ, λ bare = λ we formally have By repeated application of T n i we formally produce an (1 + · · · + 1)-point function.
Of course, these operations are not legitimate: in dimensions D ∈ {2, 4, 6} we have to include for renormalisation the Φ-linear terms in (3.1), and the partition function has no chance to exist for real λ. Nevertheless, we are able to show that T n i admits a rigorous replacement which we call the boundary creation operator. It will be our main device: This shows that boundary components labelled by z i behave like bosonic particles at position z i . The creation operator (2λ) 3Â †g z J ,z adds to a |J|-particle state another particle at position z. The |J|-particle state is precisely given by G g (z J ): Theorem 4.6 Assume that G g (z) is, for g ≥ 1, an odd function of z = 0 and a rational function of 0 , . . . , 3g−2 (true for g = 1). Then the (1 + 1 + ... + 1)-point function of genus g ≥ 1 and B boundary components of the renormalised Φ 3 D -matricial QFT model in dimension D ∈ {0, 2, 4, 6} has the solution where G g (z 1 ) is the 1-point function of genus g ≥ 1 and the boundary creation operator A †g z J is defined in Definition 4.4. For g = 0 the boundary creation operators act on the (1 + 1)-point function

Remark 4.7
The assumption that G g (z) is an odd function of z and rational in 0 , ..., 3g−2 will inductively follow from Theorem 4.12 and Proposition 4.14 for genus h ≤ 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, Theorem 4.6, Theorem 4.12 and Proposition 4.14 are all inductively proved step by step when increasing the genus in each Theorem/Proposition.
Proof The boundary creation operatorÂ †g z J ,z of Definition 4.4 preserves holomorphicity in C \ {0} and maps odd functions into odd functions. Thus only the initial conditions need to be checked. They are fulfilled for G 0 (z 1 |z 2 |z 3 ) and G 1 (z 1 ) according to (4.2); for g ≥ 2 by assumption.
The assumption will be verified later in Proposition 4.14.
Proof This follows from the change to complex variables in equation (3.14) and A †g z J ,z1 ( 1

Solution of the 1-point function for g ≥ 1
It remains to check that the 1-point function G g (z) at genus g ≥ 1 satisfies the assumptions of Theorem 4.6 and Corollary 4.8, namely: 1. G g (z) depends only on the moments 0 , . . . , 3g−2 of the measure, 2. z → G g (z) is holomorphic on C \ {0} and an odd function of z.
We establish these properties by solving (4.3) via a formula for the inverse ofK z . This formula is inspired by topological recursion, see e.g. [18]. We give a few details in section 4.4.
An important application is Faà di Bruno's formula, the n-th order chain rule: d n dx n f (g(x)) = n k=1 f (k) (g(x)) B n,k (g (x), g (x), ..., g (n−k+1) (x)). (4.9) z 2k be an even Laurent series about z = 0 bounded at ∞. Then the inverse of the integral operatorK z of Definition 4.1 is given by the residue formula .
Proof The formulae (4.2) give rise to the series expansion where the l are given in (3.17) (either take lim Λ→∞ W 0 (z) in (4.2) or leave out the limit in (3.17)) The series of its reciprocal is found using (4.9): Multiplication by the geometric series gives The residue of a monomial in f (z ) = ∞ k=0 a 2k (z ) 2k is then (4.13) In the next step we apply the operator z 2K 1 z to (4.13), where Lemma 4.2 is used: (4.14) The last sum over j is treated as follows, where the Bell polynomials are inserted for S m : 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] n−k j=1 n j x j B n−j,k (x 1 , ..., x n−j−k+1 ) = (k + 1)B n,k+1 (x 1 , ..., x n−k ). 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.
This finishes the proof.
Theorem 4.12 For any g ≥ 1 and z ∈ C \ {0} one has Proof 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 ∞. This is the case for g = 1 where only G 0 (z |z ) = λ 2 (z ) 4 contributes. Evaluation of the residue reconfirms Proposition 4.3. We proceed by induction in g ≥ 2, assuming that all G h (z ) with 1 ≤ h < g on the rhs of (4.16) are odd Laurent polynomials bounded in ∞; their product is even. The induction hypothesis also verifies the assumption of Theorem 4.6 so that G g−1 (−z |−z ) = −G g−1 (z |−z ) = G g−1 (z |z ) is even and, because of G g−1 (z |z ) = (2λ) 3Â †g−1 z ,z G g−1 (z ), inductively a Laurent polynomial bounded in ∞. Thus, equation (4.16) holds for genus g ≥ 2 and, as consequence of (4.13), G g (z) is again an odd Laurent polynomial bounded in ∞. Equation (4.16) is thus proved for all g ≥ 1, and the assumption of Theorem 4.6 is verified.
A more precise characterisation can be given. It relies on The Bell polynomials B n,k (x 1 , . . . , 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 ( ) be some j-weighted polynomial in { 1 0 , . . . , j 0 } with rational coefficients. A decoration (empty or primes) distinguishes several such polynomials, but is of no relevance. Proposition 4.14 For g ≥ 1 one has where P 0 ∈ Q and the P j ( ) for j ≥ 1 are some j-weighted polynomials in { 1 0 , . . . , j 0 } with rational coefficients.
In particular, this proves the assumption of Theorem 4.6, namely that G g (z) depends only on { 0 , . . . , 3g−2 }. To be precise, we reciprocally increase the genus in Theorem 4.6 and Proposition 4.14 as described in Remark 4.7.

Remarks on topological recursion
We return to equation (4.5) for g > 0 or |J| ≥ 3 in case of g = 0. In the first term, G g (z 1 |z J ) is given by Theorem 4.6 so that z 1 G g (z 1 |z J ) is by Proposition 4.14 and Definition 4.4 ofÂ †g z J ,z 1 an even Laurent polynomial about z 1 = 0 bounded at ∞. Therefore, Proposition 4.11 applies and gives the representation The last term G g (z β |z J\β ) does not contribute to the residue. The other term G g (z |z J\β ) also arises for (h, I) = (g, {β}) and (h , I ) = (0, {β}) in the sum, where it has the prefactor G 0 (z |z β ) given in the second line of (4.2). The total contribution of this term inside [ ] is This suggests to introduce By Theorem 4.6 and Proposition 4.14 together with Definition 4.4, each ω g,B (z 1 , ..., z B ) with 2g + B > 2 is an even Laurent-polynomial in every z i so that we can replace z → −z in one of the arguments. The structure then matches the combination ω 0,2 (z , z β ) + ω 0,2 (z , −z β ) needed as prefactor of ω g,B−1 (z , z 2 , β . . ., z B ). Multiplying (4.17) by i∈J z i and taking the previous discussion into account, we have proved (after a shift B → B + 1): where ω g,|I|+1 (z 0 , z I ) = ω g,n+1 (z 0 , z i1 , ...., z in ) if I = {i 1 , ...., 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.
For the ω g,|J| (z J ) with J ≥ 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 G g (z 1 |z 2 ) = G g (z 2 |z 1 ) for g ≥ 1. The complete symmetry of ω g,B (z 1 , ..., z B ) or equivalently G g (z 1 |...|z B ) in all arguments then follows from Proposition 4.6. For g = 0 the symmetry of the starting point G 0 (z 1 |z 2 |z 3 ) is manifest in (4.2). The proof of G g (z 1 |z 2 ) = 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.

Free energy and boundary annihilation operator
Res z→0 z 4+2l l 3 + 2l f (z)dz (5.1) and the free energies We callÂž a boundary annihilation operator acting on Laurent polynomials f .

Proposition 5.2
The F g have for g > 1 a presentation as Proof From Proposition 4.14 we conclude for g ≥ 2 , which confirms the structure (5.2).
. Inserting the definitions we havê where we have completedÂ † z toÂ †g w,z and used that the residue of a total differential vanishes. With Proposition 4.14 it is easy to see that Res w→0 w 2 dw z(z 2 −w 2 ) G g (w) = G g (z). Since G g (z|w) = G g (w|z) is symmetric for g ≥ 2 according to the discussion at the end of section 4.4, we have (2λ) −3 G g (z|w) =Â †g w,z G g (w) =Â †g z,w G g (z) by Theorem 4.6. We havê AwÂ †g z,• =N, and sinceNP j ( ) ≡ 0 for any j-weighted polynomial P j ( ) in { 1 0 , ... j 0 }, we haveNG g (z) = (2g −1)G g (z) by Proposition 4.14. Altogether we have provedÂ † zÂw G g (•) = −G g (z) +NG g (z) = (2g − 2)G g (z). Remark 5.3 Proposition 5.2 shows that the F g provide the most condensed way to describe the non-planar sector of the Φ 3 -matricial QFT model. All information about the genusg sector is encoded in the p(3g − 3) rational numbers which form the coefficients in the (3g − 3)-weighted polynomial in { 1 0 , 2 0 , . . . }. From these polynomials we obtain the (1 + · · · + 1)-point function with B boundary components via G g (z) = (2λ) 3Â † z F g followed by Theorem 4.6.
Corollary 5.5Âž Hence, up to a rescaling,Âž indeed removes the boundary component previously located at z. We also haveÂžF g = 0 for all g ≥ 1 so that the F g play the rôle of a vacuum. Note that W 0 (z) cannot be produced by whatever F 0 .

The Laplacian
Let Z np V := exp ∞ g=1 V 2−2g F g be the non-planar part of the partition function, understood as formal power series in 1 V 2 . Consider the following operation with Z np V : Its coefficients are 6 times the rhs of (4.3), hence vanishes. This means that O V ≡ 0 = Z np V O V . We invertK z via Proposition 4.11, divide by z and applyÂž given by the residue in Definition 5.1:

.3) and (4.2) and
We insert K(z, z ) = 2 (W 0 (z )−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. There is no pole at z = 0; the other two poles give the same contribution.

Res
Inserting (4.10) and renaming z → z we get The inverse of the denominator is given by (4.11), without the 1 z prefactor and in variables z → z. Its product with the numerator is where we have used (4.15) for the first S m ( ) to achieve better control of signs. The residue of V 2 64λ 4 z 4 is immediate and can be moved to the lhs: Next we separate the 0 -derivatives: Consequently, we obtain a parabolic differential equation in V −2 which is easily solved. Inserting andR m ( ) 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 classes 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: [41] : (1 − I 1 ) = 0 , I k+1 = −(2k + 1)!! k for k ≥ 1, [18] : 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 ≡ t k = −(2k − 1)!! k−1 and T 0 ≡ (1 − I 1 ). We also redefined R m (t) = (2m − 1)!!R m ( ) as well as N = V (2λ) 2 . The formula can easily be implemented in computer algebra 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 κ, δ, λ-classes from ψ-classes are given in [44].
Remark 5.8 As explained in [18, sec. 6.7.3.1], it is more appropriate to view F g , after (inverse) Schur transform of the l , as generating function of intersection numbers of κ-classes on M g,0 . When passing to G g (z 1 | . . . |z n ) by application of the boundary creation operator, equivalently to the ω g,n (z 1 , . . . , z n ) of topological recursion, all mixed intersection numbers of ψ-and κ-classes are generated. A factor −(2l+1)!! z 2l+3 i translates to ψ l i and differentiation with respect to s l gives a factor κ l under the M g,n -integral. In particular, the subfamily of intersection numbers only of ψ-classes and with power ≥ 2 arises by multiple application of on F g and projecting to the part without l≥1 . Every term in F g with n factors of t i≥2 thus gives a unique intersection number of ψ-classes on M g,n . This justifies to view F g as intersection numbers of ψ-classes.

A deformed Virasoro algebra
We return to (5.4) with O V ≡ 0, but instead of applying the inverse ofK z we directly take the residue to define a family of operators on rational functions f ( ) of l : By construction,L n Z np V = 0. Recall that in the Kontsevich model one has L n Z for the full partition function and generators L n of a Virasoro algebra (or rather a Witt algebra). As explained below, theseL n do not satisfy the commutation relations of the Virasoro algebra exactly. An explicit expression is obtained from Evaluating the residues and defining A = (2λ) 4 4V 2 and rescaling L n := AL n gives and for n ≥ 2: To write it in a more compact way, it is convenient to introduce the differential operatorD (5 + 2l) l ∂ ∂ l+1 + AD 2 and for n ≥ 2: whereD is the differential operator defined by (5.12) and A = (2λ) 4 4V 2 .
With the commutation rules , we end up after long but straightforward computation: Lemma 5.11 The generators L n of Lemma 5.10 obey the commutation relation diverges for any D > 0 in the limit Λ → ∞, it is necessary to reconstruct the analogue of the derivative ∂ ∂c through the differential operatorD. Replacing the differential operator byD → ∂ ∂ −1 and the generators by recovers the original undeformed Virasoro algebra. As explained above, −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 −1 . Higher topologies (χ ≤ 0) are not affected because any explicit −1 -dependence drops out.
was recently achieved in [46] with the proof of factorial bounds |F g | ≤ r −g Γ(βg) for some r > 0 and β ≤ 5.