Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures

We provide strong evidence for the conjecture that the analogue of Kontsevich's matrix Airy function, with the cubic potential $\mathrm{Tr}(\Phi^3)$ replaced by a quartic term $\mathrm{Tr}(\Phi^4)$, obeys the blobbed topological recursion of Borot and Shadrin. We identify in the quartic Kontsevich model three families of correlation functions for which we establish interwoven loop equations. One family consists of symmetric meromorphic differential forms $\omega_{g,n}$ labelled by genus and number of marked points of a complex curve. We reduce the solution of all loop equations to a straightforward but lengthy evaluation of residues. In all evaluated cases, the $\omega_{g,n}$ consist of a part with poles at ramification points which satisfies the universal formula of topological recursion, and of a part holomorphic at ramification points for which we provide an explicit residue formula.


Introduction
This paper achieves decisive progress in the complete solution of a quartic analogue of the Kontsevich model. The Kontsevich model [1] is a N × N Hermitian matrix model obtained by deforming a Gaußian measure dµ 0 (Φ) with covariance Φ(e ij )Φ(e kl ) c = δ il δ jk λ k + λ l (1.1) (where (e kl ) is the standard matrix basis and λ 1 , . . . , λ N are positive real numbers which we rename to E 1 , . . . , E N in this paper) by a cubic term exp( i 6 Tr(Φ 3 )). Under 'quartic analogue' we understand the deformation of the same Gaußian measure (1.1) by a quartic term exp(− λ 4 Tr(Φ 4 )). The Kontsevich model proves a conjecture by Witten [2] that the generating function of intersection numbers of tautological characteristic classes on the moduli space M g,n of stable complex curves is a τ -function for the KdV hierarchy. Thereby it beautifully connects several areas of mathematics and physics such as integrable models, matrix models, 2D quantum gravity, enumerative geometry, complex algebraic geometry and also noncommutative geometry. Some 15 years ago it was understood that the Kontsevich model is also a prime example for a universal structure called topological recursion [3]. It starts with the initial data (Σ, Σ 0 , x, ω 0,1 , B), called the spectral curve. Here x : Σ → Σ 0 is a ramified covering of Riemann surfaces, ω 0,1 is a meromorphic differential 1-form on Σ regular at the ramification points of x, and B the Bergman kernel, a symmetric meromorphic bidifferential form on Σ × Σ with double pole on the diagonal and no residue. From these initial data, topological recursion constructs a hierarchy {ω g,n } with ω 0,2 = B of symmetric meromorphic differential forms on Σ n and understands them as spectral invariants of the curve. Other examples besides the Kontsevich model (which is described e.g. in [4,Sec 6]) are the oneand two-matrix models [5], Mirzakhani's recursions [6] for the volume of moduli spaces of hyperbolic Riemann surfaces, recursions in Hurwitz theory [7] and in Gromov-Witten theory [8].
This paper provides strong evidence that our quartic analogue of the Kontsevich model is a prime example 1 for blobbed topological recursion, an extension of topological recursion developed by Borot and Shadrin [9]. In this setting the differential forms ω g,n (..., z) = P z ω g,n (..., z) + H z ω g,n (..., z) decompose into a part P z ω g,n with poles (in a selected variable z) at ramification points of x : Σ → Σ 0 and a part H z ω g,n with poles somewhere else. The P z ω g,n are recursively given by the universal formula (for simple ramifications) P z ω g,n+1 (I, z) = β i Res q→β i K i (z, q) ω g−1,n+2 (I, q, σ i (q)) (1.2) + g 1 +g 2 =g I 1 I 2 =I (g 1 ,I 1 ) =(0,∅) =(g 2 ,I 2 ) ω g 1 ,|I 1 |+1 (I 1 , q)ω g 2 ,|I 2 |+1 (I 2 , σ i (q)) of topological recursion. Here I = {z 1 , . . . , z n } collects the other variables besides z, the sum is over the ramification points β i of x defined by x (β i ) = 0. The kernel K i (z, q) is defined in the neighbourhood of β i by K i (z, q) = 1 2 q σ i (q) B(z,q ) ω 0,1 (q)−ω 0,1 (σ i (q)) , where σ i = id is the local Galois involution x(q) = x(σ i (q)) near β i . There is no general formula for the other part H z ω g,n . The only requirement is that ω g,n = P z ω g,n + H z ω g,n satisfy abstract loop equations [10]. The ω g ,n on the rhs of (1.2) contain both parts P and H. This paper identifies the ω g,n for the quartic analogue of the Kontsevich model (which is probably the most innovative step) and establishes loop equations for them and for two families of functions interweaved with the ω g,n . These loop equations are very complicated. We succeed in solving them for ω 0,2 , ω 0,3 , ω 1,1 and ω 0,4 . The results are remarkably simple and structured. We prove that, although our loop equations are much more complicated than familiar equations of topological recursion, the solutions satisfy exactly the blobbed topological recursion (1.2) in all four cases. This statement boils down to equality of more than 10 rational numbers. This is unlikely a mere coincidence so that we conjecture that the quartic analogue of the Kontsevich model obeys exactly the structures of blobbed topological recursion. In a subsequent work [11] we confirm the conjecture for genus g = 0 (i.e. for all ω 0,n ). The loop equations established in this paper are shown in [11] to be equivalent to equations which express ω 0,n+1 (z 1 , ..., z n , −z) in terms of ω 0,m+1 (z 1 , ..., z m , +z) with m ≤ n. Their solution is a residue formula which implements blobbed topological recursion.
All these structures make the quartic analogue of the Kontsevich model a member of the family of models associated with the moduli space M g,n of stable complex curves.
We summarise central steps which went into the result. The model under consideration is the result of attempts to understand quantum field theories on noncommutative geometries. These QFT models have a matrix formulation [12,13] which was a main tool in establishing perturbative renormalisability in four dimensions [14] and vanishing of the β-function [15]. Exact solutions of particular choices of these matrix models have been established in [13] for a complex model and most importantly in [16,17] for a quantum field theory limit of the Kontsevich model (completed much later in [18]).
Building on [15], one of us (RW) with H. Grosse proved in [19] that the planar 2-point function of the Quartic Kontsevich Model satisfies a non-linear integral equation Here 0 is the spectral measure of a Laplacian on the noncommutative geometry, λ the coupling constant and µ 2 (Λ), Z(Λ) are renormalisation parameters to achieve existence of lim Λ→∞ G (0) (x, t). For the purpose of this paper it is safe to set µ 2 = 0 = Z − 1. This equation is the first instance of a Dyson-Schwinger equation (or loop equation) in the Quartic Kontsevich Model. In [20] a fixed point formulation of (1.3) was found from which in the following years some qualitative results about the solution were deduced. But in spite of considerable efforts, a solution of (1.3) remained out of reach for 9 years. In 2018, one of us (RW) with E. Panzer found in [21] the solution of (1.3) for 0 (t) = 1, µ 2 = 1 − 2λ log(1 + Λ 2 ) and Z = 1. A year later, two of us (AH, RW) with H. Grosse extended in [22] this solution to any Hölder-continuous 0 with ∞ 0 dt (1+t) 3 0 (t) < ∞. The limit of (1.3) back to a matrix measure 0 (t) = 1 N d k=1 r k δ(t − e k ), already considered in [22], was understood by RW with J. Schürmann in [23] as a problem in complex algebraic geometry. Also the next equation for the planar 2-point function of cycle type (2, 0) was solved in [23].
The present paper is a large-scale extension of [22,23] to higher topological sectors. It was already pointed out in [20,23] that, although all Dyson-Schwinger equations for higher topological sectors are affine equations, no solution theory for them seemed to exist. We succeed in finding one. In Definition 2.3 we identify three families T q 1 ,...,qm pq| , T q 1 ,...,qm p|q| and Ω q 1 ,...,qm of auxiliary functions for which we derive in sec. 2.3 loop equations. These have a graphical interpretation which we provide in Appendix D. Knowing Ω ... and T ... permits a straightforward solution of all cumulants G ... of the quartically deformed measure along the lines of [23]. Section 3 extends the loop equations of sec. 2.3 to functions of several complex variables. The solution for the function Ω (0) 2 (u, z) in Proposition 3.11 makes first contact with the Bergman kernel of topological recursion. We describe in sec. 4 how all equations can be solved by evaluation of residues. Doing this in practice can be a longer endeavour, as demonstrated in Appendix G. The results strongly suggest that our auxiliary functions Ω q 1 ,...qm descent from symmetric meromorphic differential forms ω g,m which satisfy the main equation (1.2) of blobbed topological recursion. Moreover, we provide explicit residue formulae for H z ω g,n (..., z).
The Fourier transform of the measure is conveniently used to organise moments and cumulants As explained in [23], the cumulants (2.4) are only non-zero if (l 1 , . . . , l n ) = (k σ(1) , . . . , k σ(n) ) is a permutation of (k 1 , . . . , k n ), and in this case 4 the cumulant only depends on the cycle type of this permutation σ in the symmetric group S n . A non-vanishing cumulant of b cycles is thus of the form (e k 1 1 ) c and gives, after rescaling by appropriate powers of N , for pairwise different k j i rise to The goal is to compute these 'correlation functions' G ... after (at this point formal) expan- ... , at least in principle. This was achieved in [22,23] for G (0) |k 1 k 2 | and in [23] for G (0) |k 1 |k 2 | . The results of [24] extend this solution to all G (0) |k 1 k 2 | a new structure arises which cannot be treated by the methods developed so far. The present paper gives the solution and shows that precisely these new structures provide the connection to the world of topological recursion [3,4,9].

Dyson-Schwinger Equations
The Fourier transform (2.3) satisfies the equations of motion [23, Lemma 1+2] The first one can be converted into [23, eq. (50)] ∂M pk ∂M kp has to be taken from (2.7). Example 2.1 For p = q one has with Z(0) = 1 The second line results when differentiating (2.6) with respect to M qp . We split the l-sum into l = p where (2.7) is used and l = p where (2.8) for q → l is inserted: , the differentiation yields the Dyson-Schwinger equation (DSE) for the 2-point function The second line results when differentiating (2.6) taken at q → p with respect to M qq . We split the l-sum into l = p where (2.7) is used and l = p where (2.8) for q → l is inserted: , the differentiation yields the Dyson-Schwinger equation for the (1+1)-point function The arising repeated differentiations with respect to the E q suggest to introduce: Moreover, assume that for Then functions Ω q1,...,qm , symmetric in their indices, are defined by For the aim of the paper it will be sufficient to focus on the following special cases, the generalised 2-point and 1 + 1-point functions: These functions will be considered as formal power series in 1 where g plays the rôle of a genus. It is well-known that the 2-point function G have an expansion into ribbon graphs, see [25] for the definition of those ribbon graphs associated to the quartic Kontsevich model. The generalised correlation functions of Def. 2.3 have a topological meaning as well, since the derivative ∂ ∂Eq i acts on an internal edge and fixes it to E q i . This operation is extensively discussed in [25] and depends topologically on the labelling of the two faces adjacent to the edge. Let us briefly sum up the major ideas on which the detailed graphical discussion in App. D will build: We consider genus-g Riemann surfaces X with b boundaries and m marked points. We let χ = 2 − 2g − m − b be the Euler characteristic of X.

The Ω
(g) q 1 ,...,qm correspond to no boundary (b = 0) and m marked points; 2. The T (g) q 1 ,...,qm pq| correspond to b = 1 boundary and m marked points; 3. The T (g) q 1 ,...,qm p|q| correspond to b = 2 boundaries and m marked points. The ribbon graphs, consisting of vertices, edges and faces, are embedded into these Riemann surfaces. The faces carry a label from 1 to N , and we distinguish three types: 1. internal faces whose labellings have to be summed over; 2. external faces in which at least one edge is part of the boundary (with fixed label); 3. marked faces (with fixed label, too), where no edge is part of a boundary.
For the vertices, also three types show up: 1. 4-valent vertices in the interior, carrying the weight (−λ); 2. 1-valent vertices on the boundaries belonging to one (in G |p|q| ) or two (in G |pq| ) external face(s); 3. one square-vertex within every marked face (see Fig. 1).
|pq| (shown on the left, with b = 1 boundary and no marked point) turns an inner face into a marked face with a square-vertex. The result of m such operations is depicted on the right; it corresponds to a genus g-Riemann surface with one boundary (carrying two 1-valent vertices) and m marked points. The operation is described in [25] in details.
Let us return to the actual aim of the paper: We will show in this paper that a complexification of the Ω (g) ... gives rise to meromorphic differentials ω g,n conjectured to obey blobbed topological recursion [9]. This conjecture has been proved for g = 0 in [11]. In a supplement [25] we express the functions Ω q 1 ,...,qm as distinguished polynomials in the correlation functions G |...| , which themselves are quickly growing series in Feynman ribbon graphs. Both results together imply that intersection numbers [9] are accessible by a particular combination of graphs. We let I1 I2=I be the sum over all disjoint partitions of I into two possibly empty subsets I 1 , I 2 .

Dyson-Schwinger Equations for Generalised Correlation Functions
Equations for T ... result by application of D I := (−N ) m ∂ m ∂Eq 1 ···∂Eq m to (2.9) and (2.10) when taking Definition 2.3 into account. We will give three important types of Dyson-Schwinger equations that will determine the recursive structure of our model: 1) The DSE determining the generalised 2-point function: 2) The DSE determining the generalised 1+1-point function: 3) The DSE for Ω I,q in terms of T : where T ∅ qq j | := G |qq j | and T ∅ q j |q| := G |q j |q| .
Proof We apply the operator D I := (−N ) m ∂ m ∂Eq 1 ···∂Eq m to (2.9) and take Definition 2.3 into account. Here we have and, separating the cases l ∈ I and l / ∈ I, (**) The last term in (**) cancels the contribution with δ |I1|,1 in (*). All other derivatives are clear and arrange into (2.12). The same considerations give (2.13) when applying D I to (2.10). Equation (2.14) is clear.

Remark 2.6
In a similar manner one can derive Dyson-Schwinger equations for G |k 1 | . In fact [20] they can be reduced to polynomials in correlation functions with n s ∈ {1, 2}, in λ and in . We refer to [26] for the precise form of the recursion. Applying D I then gives rise to Dyson-Schwinger equations for generalised correlation functions T q1,...,qm k 1 We have provided them in prior versions of this paper (e.g. https://arxiv.org/abs/2008.12201v3). For the sake of readability we decided to suppress them here.
The Dyson-Schwinger equations of Proposition 2.5 are in one-to-one correspondence with a graphical representation via a perturbative expansion into ribbon graphs. We derive them once again in App. D for illustrative purposes.

Complexification
The equations in Proposition 2.5 are not sufficient to determine the functions G, T, Ω because there is no equation for derivatives with respect to matrix indices (e.g. in ∂T I pq| ∂Ep ) or functions with coincident matrix indices (e.g. G |qq| , G |q|q| or T p pq| ), which however are needed. Our strategy is therefore to meromorphically extend these equations, where the extension is not necessarily unique, but unique at the points E p .
(b) Write the equations in Proposition 2.5 in terms of G, T,Ω and postulate that they extend to pairwise different points such that the ∂ ∂Eq Eq →η -derivative acts in the sense of Definition 2.3 with extension to E q → η, and a difference quotient which tends for η → E q to the derivative on the argument of f . with the difference quotient term of (c). Consider the equations for ζ, η, ζ s i , η j ∈ V \ {E 1 , . . . , E N }. (e) Define the values of G, T,Ω at ζ = E p , η = E q , ζ s i = E p s i , η j = E q j and at coinciding points by a limit procedure. Remark 3.2 The complexification of the derivative defined in (c) distinguishes between Definition 2.3 and a derivative acting on the argument of the function. The derivative on the argument is split into a difference quotient to generate all missing terms in the l-summation, e.g. for l = q by where the analyticity property at E q holds by (b). Consequently, the summation over l gets unrestricted in the extension to V and coincides with Proposition 2.5 on the points E p . Notice that the extension of Definition 3.1 is a meaningful extension but cannot be unique.
The complexification procedure allows to relax the condition that all E 1 , . . . , E N are pairwise different. From now on we agree that (E 1 , . . . , E N ) is made of pairwise different e 1 , . . . , e d which arise with multiplicities r 1 , . . . , r d in the tuple, with r 1 + ... + r d = N .
We search for a solution of the equations after expansion (2.11) of all arising functions as formal power series in 1 N 2 . It will become clear later on that g plays the rôle of the genus of a Riemann surface so that we call (2.11) the genus expansion. When splitting the equations into homogeneous powers of N −2 we agree that 1 N d k=1 r k G (g) (e k , . . . ) contributes to order g. Similarly for T (g) andΩ (g) .
For instance, we have Note that the sum over g restricts to g ≥ 1 because the case g = 0 is explicitly included in the lhs.
For g = 0 this becomes a non-linear equation for the function G (0) (ζ, η) of ζ, η ∈ V. It has been recently solved: Theorem 3.4 ([23], building heavily on [22]) Let λ, e k > 0. Assume that there is a rational function R :Ĉ →Ĉ with 1. R has degree d + 1, is normalised to R(∞) = ∞ and biholomorphically maps a domain U ⊂ C to a neighbourhood (which can be assumed to be V) in C of a real interval that contains e 1 , . . . , e d . 2. In terms of G (0) (R(z), R(w)) =: G (0) (z, w) and e k =: R(ε k ) for z, w, ε k ∈ U one has

4)
Then R and G (0) are uniquely determined by the case g = 0 of (3.3) to The ansatz (3.4) is identically fulfilled by (3.5). In these equations, the solutions of R(v) = R(z) are denoted by v ∈ {z,ẑ 1 , . . . ,ẑ d } with z ∈ U when considering R :Ĉ →Ĉ. The function G (0) (z, w) is symmetric. Its poles are located at z + w = 0 and z, w ∈ { ε k j } for k, j ∈ {1, ..., d}. Figure 2 sketches the map R. The rational function R introduces another change of Fig. 2 Illustration of the complexification procedure: The biholomorphic map R : U → V with R(ε k ) = e k will later be enlarged to a ramified cover R :Ĉ →Ĉ. Functions on U will meromorphically continue to the Riemann sphereĈ = C∪ {∞}.

variables.
Definition 3.5 Let G (g) , T (g) ,Ω (g) be the functions in several complex variables obtained by the complexification of Definition 3.1, genus expansion and by admitting multiplicities of the e k . Then functions G (g) , T (g) , Ω (g) m of several complex variables are introduced by , Originally defined as holomorphic functions on Cartesian products of U, we assume (and will show) that G (g) , T (g) , Ω (g) m extend to meromorphic functions onĈ = C ∪ {∞}.

Complexified Dyson-Schwinger Equations
We now combine the complexification according to Definition 3.1 with the change of variables of Definition 3.5: Equation (3.6) reduces for (g, m) = (0, 0) to Remark 3.8 The DSE of Corollary 3.6 has a very intricate form. Actually, this structure of DSEs is well-known from the 2-matrix model. We refer to [27, eq. (2-19)] with the correspondence Ω → W , T → H and k T (I ε k ,w|) R(ε k )−R(z) →Ũ . However, the last term of (3.6) does not have a corresponding counterpart. Thus, the quartic Kontsevich model is a priori similar to the 2matrix model, but in some sense richer in its structure. This difference could explain why it is (conjecturally) governed by the extension to Blobbed Topological Recursion [9]. The fundamental building blocks in the 2-matrix model are the W (g) m,0 which were proved [5] to satisfy topological recursion. For this reason, the main interest lies on computation and structure of Ω (g) m .
3.3 The DSE for Ω (g) m (u 1 , ..., u m ) To solve the system of equations (3.6) and (3.8) we have to establish another DSE for Ω (g) m . The same steps as in Corollaries 3.6 and 3.7 turn (2.14) into Ω (g) We will prove another relation: Proof Take (3.6), multiply it by rn N (R(w)−R(−z)) , set w = ε n and sum over n. The lhs has the term from the case (I 2 , g 2 ) = (∅, 0) in the second line of (3.6). Moving it to the lhs, we reconstruct a common prefactor R (z)G 0 (z) via (3.7).
The two terms violating the exact coincidence are the last term and third last term of (3.10).

Solution for Ω
The last line follows from (3.7). In [22] the following representation was proved: . On one hand this shows on the other hand we have the partial fraction decomposition (3.14) Both together imply: Then Proof Inserting (3.12) and (3.14) into (3.11) gives Since G 0 (z) has poles at every z = ± ε n j , the rhs of the above equation has poles at most at the zeros of G 0 . By assumption, the lhs is regular there. Thus, both sides must be constant by Liouville's theorem and then, when considering z → ∞, identically zero. Remark 3.12 Proposition 3.11 indicates that we are on the right track. The solution Ω (0) 2 (u, z) is symmetric, and its part 1 (u−z) 2 is closely related to the Bergman kernel B(u, z) = dudz (u−z) 2 of topological recursion. Looking back into Remark 3.2 we can be confident that the complexification procedure of Definition 3.1 is reasonable.
Comparing again with the 2-matrix model, our DSE (3.11) of Ω (0) 2 differs slightly. The last term G (0) (u, z) in (3.11), which generates the pole on the antidiagonal, is not present in the 2matrix model. We refer for instance to [27, eq. (2-20)] with the same identifications as in Remark 3.10 and with g = 0, q = p, p K = ∅ and |q L | = 1. The last term H in [27, eq. (2-20)] corresponds to our G (0) (u, z), which is not present for p K = ∅. The distinction between the two different sets p K and q L is significant for the 2-matrix model.

Recursive Solution
In previous sections we have introduced and studied certain families of functions Ω (g) m (u 1 , .., u m ), T (g) (u 1 , ..., u m z, w|), T (g) (u 1 , ..., u m z|w|) and G (g) (z, w), G (g) (z|w). As already outlined after Def. 2.3, the integers (g, m, b) are interpreted as topological data of a Riemann surface X (see Fig. 3): • g is the genus of X, • m is the number of marked points on X, • b is the number of boundary components of X; more precisely b = 1 for T (g) (u 1 , ..., u m z, w|) and b = 2 for T (g) (u 1 , ..., u m z|w|).
We let χ = 2 − 2g − m − b be the Euler characteristic of X.
... um Fig. 3 The correlation functions relate to topological data of Riemann surfaces X. The Riemann surface corresponding to T (g) (u 1 , ..., um z, w|) has g handles, m marked points labelled u 1 , .., um and one boundary component with two defects (the 1-valent vertices) labelled z and w. The Riemann surface corresponding to T (g) (u 1 , ..., um z|w|) has g handles, m marked points labelled u 1 , ..., um and two boundary components each with one defect labelled z or w, respectively. The Riemann surface corresponding to Ω (g) (u 1 , ..., um) has g handles, m marked points labelled u 1 , ..., um and no boundary.
The Dyson-Schwinger equations for the generalised correlation functions Ω (g) m (u 1 , .., u m ), T (g) (u 1 , ..., u m z, w|) and T (g) (u 1 , ..., u m z|w|) follow a recursion in the Euler characteristic: To compute a generalised correlation function of negative Euler characteristic −χ = 2g + (m + b) − 2 we need generalised correlation functions of negative Euler characteristic −χ < −χ. In case of equality one builds Figure 4 shows the recursive structure in solving the correlation function for small −χ. This structure extends in obvious manner to higher topologies ...
We will prove that the solution of T (g) (u 1 , ..., u m z, w|) and T (g) (u 1 , ..., u m z|w|) are obtained by a simple evaluation of residues. For that the following analyticity property is necessary: (R(u1)−R(u2)) 2 are analytic at any two coinciding variables in the domain U which includes all ε k but excludes 0.
Proof The analyticity property is proved by induction in the Euler characteristic. It is true for G (0) (z, w) = T (0) (∅ z, w|) by the explicit form (3.5) and G (0) (z|w) = T (0) (∅ z|w|) by the explicit form given in [23]. From Proposition 3.11, we have the limit lim u→z Ω 4 . Then by induction all terms T (g) and Ω (g1) |I1|+1 with 2g 1 + |I 1 | ≥ 2 are analytic at coinciding arguments. The only critical terms for z → u i arise in combination which is analytic for z → u i . Regularity for w → u i is obvious, and regularity for z → w holds by induction. Thus, T (g) (I z, w|) is regular for any z, w → u i and z → w. Similarly for (3.8).
The same argument in the rhs of Proposition 3.9 shows analyticity of Ω 4.1 Recursive Solution of T (g) (u 1 , ..., u m z, w|) and T (g) (u 1 , ..., u m z|w|) The main observation when solving the DSEs (3.6) (or (3.8)) is the rationality of the second term of the lhs in R(z). After multiplication with d k=1 (R(ε k )−R(z)), the resulting second term becomes a polynomial in R(z) of degree d − 1. This observation suggests an application of the interpolation formula (see Lemma E.1), where the d distinct numbers are chosen as x j = R(−ŵ j ) (or x j = R(α j )) in order to let the first term of the lhs vanish at z = −ŵ j (or at z = α j ). The analyticity is easily shown by induction and similar to Lemma 4.1. For the sake of readability we have outsourced Propositions E.2 and E.4 and their proofs to the Appendix E. Here we only give their corollaries: The generalised 2-point function is given by Instead of providing the technical proof, we have decided to give a graphical interpretation of Corollary 4.2 by cutting the Riemann surface corresponding to the generalised together with a residue operation of t at z, −ŵ j , u i and w. Now, the generalised 2-point function can be cut in three topologically different ways: 1. The cut starts from t and ends at t by encircling the set I 1 ⊂ I of marked points and g 1 handles. This separates T (g) (I z, w|) into Ω (g 1 ) |I 1 |+1 (I 1 , t) and T (g 2 ) (I 2 t, w|). Take the sum over all possibilities with Euler characteristic χ ≤ 0. 2. The cut starts at t, paces through a handle and ends again at t. This removes the handle (reduces the genus by 1) at expense of an additional marked point labelled t. 3. The cut starts at t, paces through a handle and ends at the boundary next to w (not at t). This reduces the genus by one and generates the factor 1 R(w)−R(t) and two separated boundaries with one defect on each.
It seems that another possible case would be the variant of 3. where the cut does not pace through a handle but ends next to w. This would generate two separated correlation functions of the form T (g ) (I t|), but these do no exist since the quartic Kontsevich model has no 1-point function (and therefore no generalised 1-point function).

Remark 4.3
This graphical description was already invented for the 2-matrix model to understand graphically any correlation function as a recursion depending on correlation functions of lower topology [28]. However, in the 2-matrix model two different sets of marked points exists, whereas the quartic Kontsevich model has a mixing of those sectors. In general, a proof of graphical rules is achieved by direct derivation via DSEs.  The graphical intepretation of Corollary 4.4 differs slightly from Corollary 4.2 since the initial topological data of T (g) (I z|w|) differs from T (g) (I z, w|). The cutting operation itself generates the factor where residues of t are taken at z, α j , u i , w. This is due to the fact that the starting boundary has only one defect. The first two cases are essentially the same as in Corollary 4.2. However, the third case differs since two boundaries each with one defect are present.
The third case has a cut starting at t and merging into the second boundary next to w. Both boundaries are merged to a single boundary with two defects. Furthermore, the cut not ending at its starting point generates again a factor 1 R(w)−R(t) , similar to the third case of Corollary 4.2.
Remark 4. 5 We have focused the computation in this paper to the generalised 2-point and 1 + 1point function. In a previous version of this paper (e.g. https://arxiv.org/abs/2008.12201v3) we have also defined more general correlation functions. These can be solved exactly with the same graphical rules, but with more possibilities in cutting the Riemann surfaces into different topologies.

Recursive Solution for Ω (g)
m under Assumptions on its Poles The solution of the DSE for Ω We prove this conjecture in Appendix F for the planar sector g = 0. Conjecture 4.6 and Lemma 4.1 imply that Ω We evaluate in Appendix G the residues of Corollary 4.7 for Ω  m are closely related to structures in blobbed topological recursion (BTR) [9]. We review in Appendix B central aspects of the BTR. To make contact with BTR we reformulate the solution formulae of our loop equations in terms of meromorphic differential forms: Corollary 4.7 takes the following form: where d u j is the exterior differential in u j and t g,m (u 1 , ..., u m z, w|) := λ 2g−m T (g) (z 1 , ..., z m z, w|) m j=1 R (z j )dz j as well as t g,m (u 1 , ..., u m z|w|) . . . Res . . .
which is holomorphic in z at the ramification points. We will discuss these projectors in the context of blobbed topological recursion in Appendix B, too.

Solution of ω g,m to Low Degree
This subsection lists the results for ω 0,3 , ω 0,4 and ω 1,1 obtained by evaluating the residues in the system (4.2), (E.2) and (E.4). Appendix G gives details about the procedure and provides a few intermediate results.
We let σ i be the local Galois involution near the ramification point β i , i.e. R(z) = R(σ i (z)), lim z→β i σ i (z) = β i and σ i = id. We let B(z, w) = dz dw (z−w) 2 be the Bergman kernel and define x(z) = R(z) and y(z) = −R(−z). Moreover, we introduce two kernel forms .
We remark that ω 0,m given by Conjecture 4.9 automatically satisfy the linear and quadratic loop equations given later in Definition B.1 or inside Conjecture 5.1.
The residues at q = β i can be evaluated with the formulae given in Appendix C; the residues at q = u k are straightforward. In terms of and with Q(u; z) : and P z ω 0,4 (u 1 , u 2 , u 3 , z) (4.9) where in Q (u; z), Q (u; z) the derivative is with respect to the second argument. We have simplified (4.10) using the reflection (G.14).
We also have a result for g = 1: Proposition 4.10 The solution of (4.2) for m = 0 and g = 1 is The differential form ω 1,1 satisfies for z near β i the loop equations given in Definition B.1.

Main Conjecture
We established with the proof of Conjecture 4.9 for m = 2 and m = 3 as well as with Proposition 4.10 the unexpected result that all ω g,m evaluated so far satisfy the linear and quadratic loop equations. This is very unlikely a mere coincidence, which suggests: Conjecture 5.1 Let R :Ĉ →Ĉ be the ramified covering defined in (3.5). Let β 1 , ..., β 2d be its ramification points and σ i the corresponding local Galois involution in the vicinity of β i . For all g ≥ 0 and m ≥ 1, the meromorphic differentials ω g,m given by ω 0, If the conjecture is true 6 , it is a general fact established in blobbed topological recursion [9] (and recalled in Appendix B) that the projection to the polar part is given by the universal formula of topological recursion: where B(u, z) = du dz (u−z) 2 is the Bergman kernel.

Conclusion and Outlook
This paper makes the Quartic Kontsevich Model a member of a rich family of models affiliated with the moduli space M g,n of stable complex curves. Common to all these models is the possibility to construct all functions of interest (cumulants of a measure, correlation functions, generating functions of something) recursively in decreasing Euler characteristic χ = 2−2g−n. The quartic analogue of the Kontsevich model originates from attempts to put the λϕ 4 -quantum field theory model on a noncommutative geometry. It is a Hermitian matrix model in which a Gaußian measure with non-trivial covariance (2.1) is deformed by a quartic potential, see (2.2). This paper shows that the loop equation for the planar 2-point function of the Quartic Kontsevich model, found in [19] and eventually solved in [22], is indeed the initial datum for a novel structure affiliated with M g,n .
We find that the primary structure of the Quartic Kontsevich Model is not the entirety of cumulants of the quartically deformed measure (as thought before) but a family of auxiliary functions Ω This distinction is nothing new for matrix models. It already appeared for the Hermitian 2-matrix model (2MM) [29] which has mixed-coloured and non-mixed coloured boundaries. The underlying structure of monochromatic boundary correlation functions of the 2MM was proved to follow a topological recursion [5]. However, to compute nonmixed coloured boundary correlation functions the knowledge of mixed-coloured boundary correlation functions is inevitable [27].
The Quartic Kontsevich Model, discussed here, almost shares its structure with the 2MM (cf. (3.6) with [28, eq. (1-3)]), even though it is by definition a completely different model. We have shown that the resulting Dyson-Schwinger equations are structurally almost of the same form. We have found an algorithm consisting of three steps (see Figure  4) to compute a given correlation function of Euler characteristic χ − 1 from correlation functions of Euler characteristic ≥ χ. We showed that this calculation reduces to an evaluation of residues.
A look upon the explicitly given results for small (−χ) suggests that the quartic analogue of the Kontsevich model is governed by blobbed topological recursion [9]. This is an extension of topological recursion by an infinite family of initial data φ g,m . For convenience we provide in Appendix B some background information about the BTR.
The final proof of our Main Conjecture 5.1 is on the way. The proof for g = 0 is accomplished in [11]; there remains little doubt that the result holds in general. The geometric structure is apparent: The spectral curve (of genus zero) is identified and parametrised by The numbers ε n are related by e p = R(ε p ) to the distinct values e p occurring with multiplicity r p in the parameters E 1 , ..., E N of the Gaußian measure (2.1). Our blobbed topological recursion is defined by: 1. the covering x = R :Ĉ →Ĉ of the Riemann sphere ramified at {β 1 , . . . , β 2d }; 2. two meromorphic differentials both regular at the ramification points, where B(z, u) = dz du (z−u) 2 is the usual Bergman kernel and φ 0,2 (z, u) = dz du (z+u) 2 a symmetric 2-form blob with a double pole on the antidiagonal;
The presence of a blob φ 0,2 (z, u) is an important difference to the standard approach [9]. Moreover, we recall that for the proof of BTR [9] it was sufficient to assume ω g,m to be defined on disjoint unions ∪ i U i about the ramification points. In contrast, our differential forms ω g,m are globally defined meromorphic forms onĈ m .
We noticed an intriguing rôle of the global involution z → −z onĈ. This involution is of central importance in [11] for proving Conjecture 5.1 for genus g = 0. The blobs of higher genus have poles at the fixed point z = 0 of this involution; also the other poles at z i = −z j are related in this way. Since z → −z is a very natural structure, we expect that the corresponding intersection numbers have a topological significance. It seems worthwhile to work out details and to compute these numbers. Moreover, comparing our spectral curve (6.1) to [30], we already realised that a subset of the normalised part generates simple Hurwitz numbers. Our partition function is, however, considerably easier and more natural than that of [30].
One can take the point of view that the linear and quadratic loop equations [10] are the heart of TR. Their general solution is blobbed topological recursion [9]; further conditions are necessary to reduce it to pure TR. This raises the question why the original Kontsecich model [1] and a large class of generalisations [31] satisfy these further conditions, whereas the quartic analogue of the Kontsevich model does not. At the moment we do not have a good intuitive explanation, but on a technical level there are several reasons. In Remarks 3.8, 3.10, 3.12 and 4.3 we have indicated similarities and decisive differences to the Hermitean 2-matrix model. Precisely the additional terms compared with the 2-matrix model are responsible for the poles of ω g,n away from ramification points of x (and the diagonal in case of ω 0,2 ). The Laurent series about these additional poles is completely fixed by our global (onĈ n ) loop equations; there is absolutely no freedom in choosing the blobs. This is a clear difference with the original formulation of blobbed topological recursion [9] in which the abstract loop equations are only considered locally in a neighbourhood of the ramification points (so that the blobs can be chosen freely within the constraints of the loop equations).
Another technical reason for BTR is that the ω g,n in matrix models are typically related to correlations of diagonal matrix elements Φ aa (such as in the (generalised) Kontsevich model [31]) or correlations of resolvents Tr((z − Φ) −1 ) (such as in the 2-matrix model [5]). Because of the invariance of the quartic Kontsevich model under Φ → −Φ, these special correlations only give rise to even Euler characteristics. In particular, the initial Ω (0) 1 cannot be obtained in this way. We have shown in this paper that Ω (0) 1 in the quartic Kontsevich model is built from the two-point function G (0) (z, w), which has a pole at z + w = 0. This pole at opposite diagonals proliferates into the ω g,n for all 2g + n ≥ 2 and induces poles at z i = 0 for g ≥ 1.
Private discussions with B. Eynard and E. Garcia-Failde also suggest that there is hope to formulate the current version of blobbed topological recursion in terms of pure TR by increasing the genus of the spectral curve by 1. The appearance of the same phenomenon in the O(n) model [32] and the remarkable structural analogies of holomorphic and polar parts in the quartic Kontsevich model make this hope a justified research goal for the future, among other stimulating questions arising from this model.

A Notations and Relations
For the sake of readability, and because we partly deviate from conventions in the literature, we list in the table below a few important notations and symbols used in this paper.

B Recap of Blobbed Topological Recursion
The outstanding applicability of topological recursion (TR) to a great bandwidth of mathematical phenomena is clearly undoubted. However, there exist models showing a certain recursive behaviour regarding their solutions of loop equations, but not perfectly fitting into the recursion of ordinary TR, for instance in the Hermitian 1-matrix model extended by multi-trace contributions [33] or in the quartic melonic tensor model [34]. The appearance of poles at z ∈ {0, −u l } in Corollary 4.7 gave a first hint 7 to focus on a framework that even enlarges the mentioned bandwidth. Discovered in 2015, it extends the usual TR by additional topological quantities baptised blobs to blobbed topological recursion [9]. It was observed that the loop equations of several (matrix) models can be reduced to a system of linear and quadratic loop equations: Definition B.1 Let x : Σ → Σ 0 be a ramified covering with simple ramification points β i and σ i be the local Galois involution around β i , i.e. x(z) = x(σ i (z)), lim z→βi σ i (z) = β i and σ i = id. A family of meromorphic differential forms ω g,m on Σ m , with g ≥ 0 and m > 0, fulfils the linear loop equation if is a holomorphic linear form for z → β i with (at least) a simple zero at β i . The family of ω g,m fulfils the quadratic loop equation if is a holomorphic quadratic form with at least a double zero at z → β i .
An important subclass of solutions is given by differentials governed by TR [10]. The entirety of solutions, instead, is provided by BTR. According to Subsection 4.3, the solutions ω 0,2 , ω 0,3 , ω 0,4 and ω 1,1 of the loop equations of the Quartic Kontsevich Model fulfil the linear and quadratic loop equations. We hope to provide in near future the proof of the natural Main Conjecture 5.1 that all ω g,m obey these loop equations.
The suggestive notation in (4.3) was inspired by [9] and shall be explained now. In the framework of BTR, one defines projectors H z and P z acting on ω g,m (..., z) = H z ω g,m (..., z) + P z ω g,m (..., z).
It is shown in [9] that the part P z ω g,m (z 1 ..., z m−1 , z) is produced by the universal formula of topological recursion from ω g ,m with 2g + m − 2 < 2g + m − 2. The mechanism of BTR can be depicted as in Fig. 7. Applying these projections in every . It is recursively generated. The second and third graph on the rhs are copies of ordinary topological recursion; these provide the part of ωg,m with poles in z 1 at ramification points of x = R. The first graph on the rhs, however, depicts the holomorphic part as an additional input of each recursion step. Its poles in z 1 are located outside the ramification points of x = R.
We will slightly deviate from the above conventions by choosing the unstable blobs φ 0,1 , φ 0,2 differently and by adopting a global formulation. First, we set φ 0,1 = 0 and φ 0,2 (z, u) = dzdu (z+u) 2 with ω 0,1 (z) = y(z)dx(z) as usual and ω 0,2 (z, u) = B(z, u) + φ 0,2 (z, u), see (6.1). In the original formulation [9], the Riemann surface C is a disjoint union ∪ i U i of sufficiently small neighbourhoods of the ramification points β i . Then H z ω g,m is indeed holomorphic in every z ∈ C. In contrast, our Quartic Kontsevich Model is defined globally onĈ so that the term holomorphic part should be treated with more caution. It is rather a relic of previous namings and means holomorphic in ramification points, but with poles somewhere else onĈ.
The global formulation also suggests a more natural definition of the projection P z , namely for a 1-form ω in a selected variable (in case there are 2d ramification points). Here B(z, z ) is the Bergman kernel; whereas [9] defines P z with the given bidifferential ω 0,2 (z, z ). The global formulation allows us to start the contour integral at the special point ∞ ∈Ĉ instead of β i chosen in [9]. In particular, our projector (B.3) sees the residue and thus gives the whole principal part of the Laurent series about β i . A main achievement in [9] is a simple proof (which adapts arguments of [10,Prop. 2.7]) that meromorphic m-forms ω g,m which satisfy the abstract loop equations of Definition B.1 have a polar part given by the universal TR-formula. The essence of the proof remains unchanged when defining the polar part via (B.3). We find it convenient to sketch the arguments. For z near β i and I = {z 1 , ..., z m−1 } define S i z ω g,m (I, z) = ω g,m (I, z) + ω g,m (I, σ i (z)) , ∆ i z ω g,m (I, z) = ω g,m (I, z) − ω g,m (I, σ i (z)) .

The quadratic loop equation (B.2) can be written as
Both ω 0,1 (z) and (by the linear loop equation) S i z ω g,m (I, z) have a simple zero at z = β i so that we arrive at .
The second equality follows from the antisymmetry ofQ .
It writes out as in (5.1).
Here B n,k are the Bell polynomials. The first examples are Proof Insert the power series ansatz into the identity 0 = x(σ i (q))−x(q) for q in a neighbourhood of β i . Then all derivatives with respect to q vanish at q = β i so that we have from Faà di Bruno's formula and with x (β i ) = 0 The recursion kernel near a ramification point β i specifies to .
In terms of b n,k,i (x), the terms in the recursion kernel expand into Up to order O((q − β i ) 3 ) we thus get

D Graphical Derivation of the Dyson-Schwinger Equations
Since all the correlation functions have a combinatorial interpretation, also the DSE's of Prop. 2.5 can be described combinatorially in terms of ribbon graphs. The correlation function are generating series of those ribbon graphs; we refer to [25] for more details and precise definitions. Before, we need two further relations between generalised correlation functions achieved by applying (2.8).

D.1 Ward-Takahashi Identity for Generalised Correlation Functions
We will represent higher correlation function through lower ones by summing over one of the indices: I\qi pqinq| + T (g−1) I p|pnq| + T (g−1) I pq|nq| .
Applying the operator D I : to the identity and making use of Definition 2.3 yields the assertion. D.2 Graphical Derivation of (2.14) We will start with the DSE 3) of Prop. 2.5, which is achieved combinatorially by an action of the operation −N ∂ ∂Eq on Ω I with q / ∈ I (this was discussed in [25] in greater detail). The derivative acts on an edge adjacent to an internal face, splitting the edge and fixing the label of the internal face to q. There are three different cases depending on the other adjacent face to this edge: a) If the other adjacent face is also an internal face labelled by l, a generalised 2-point function appears with a sum over this external face l, which is 1 N N l=1 T I ql| . b) If the other adjacent face is labelled by q i ∈ I, a generalised 2-point function appears with external faces q and q i . A derivative of the form −N ∂ ∂Eq i has to be taken only acting on the face labelled by E q i since this face has one additional edge already split, as defined in (3.1). A sum over all possible q i has to be taken. c) If the other adjacent face is the same face, a generalised 1 + 1-point function appears with one genus less and two external faces labelled by q each of length one, that is 1 N 2 T I q|q| . Combining case a) and b) via where we have used (3.1), we obtain the DSE (2.14).

D.3 Graphical Derivation of (2.12)
The DSE 1) of Prop. 2.5 is achieved by a bijection between T (g) I pq| , a generating series of ribbon graphs with a certain structure, and other generating series via an operation we call vertex deletion (of the first vertex). The generic deletion is drawn in Fig. 8. We will However, five further subcases occur, depending on the choice of k: -For k = q: The rhs can be split into two components with the structure T (g 1 ) with g 1 + g 2 = g and I 1 I 2 = I. -For k = q i ∈ I: The rhs has an extra derivative on the face labelled by q i coming from the split edge of the square-vertex, that is − ∂ ext I\q i pq i nq| . -For k = p: A reduction of the genus by one can be generated with two boundary components one of length 1 and the other of length 3, so we obtain T (g−1) I p|pnq| . -For k = n: A reduction of the genus by one can be generated with two boundary components one of length 1 and the other of length 3, so we obtain T (g−1) I n|pnq| . -For k = q: A reduction of the genus by one can be generated with two boundary components of length 2, so we obtain T (g−1) I pq|nq| . These six terms can be combined and surprisingly the identity of Lemma D.1 can be applied exactly. Consequently, including the deleted free propagator and the vertex, we obtain after summation over n I\q i pq i nq| + T (g−1) I p|pnq| + T (g−1) b) For n = p: There is one case where the correlation function is split into two components after deleting the first vertex. Depending on k, the three different subcases are drawn in Fig. 9. -For generic k: The sum over k is taken and the two components have the form T (g 1 ) I 2 pq| with g 1 + g 2 = g and I 1 I 2 = I. -For k = p: A reduction of the genus by one can be generated with two boundaries for one of the correlation functions, we obtain T (g 1 −1) I 2 pq| . -For k = q i ∈ I: One correlation function gets an additional derivative wrt to the external face q i that is (− ∂ ext ∂Eq i T (g 1 ) I 2 pq| . Including the deleted free propagator and applying the DSE (2.14) with g 1 and I 1 , we finally obtain −λ E p + E q g 1 +g 2 =g c) For n = p: There is a second case where the correlation function is not split into two components, see Fig. 10. The genus is reduced by one and we can again distinguish between different k.
-For generic k: There is an internal face labelled by k, we take the sum and have I p|p|pq| . I pqpq| . These cases sum again together to the following I p|p|pq| + T (g−1) I,p pq| . This identity is then applied with p = p and g → g − 1. d) For n = q i ∈ I: There is an overall derivative − ∂ ext ∂Eq i . Combinatorially, all the cases are exactly the same as for generic n, but with I \ q i instead of I, see Fig. 11. So we conclude equivalently to case a) e) For n = q: Fig. 12 shows that for generic k the genus is reduced by one. Here again, we have several subcases: p q q i k q p g q I\i Fig. 11 The generalised 2-point function with n = q i , which is similar to the generic n case.  Fig. 12 The generalised 2-point function with n = q is reduced by one genus, but has at least two boundaries.
-For generic k: The sum over k remains since k is an interal face, that is I q|q|pq| . -For k = p: Also a reduction by one further genus can occur, which has three boundaries of the form T (g−2) I q|p|pq| . Summing all subcases, the identity of Lemma D.2 applies perfectly, and we conclude I\q i pq i q|q| + T (g−1) I pqqq| + g 1 +g 2 =g I 1 I 2 =I T (g 1 ) I\q i pq i qq| . -For k = q: The graph can be split into two after deleting the first vertex. Two correlation functions occur each with boundary of length 2; we obtain T (g 1 ) with g 1 + g 2 = g and I 1 I 2 = I. -For k = q: There is also the case that two separated boundaries remain, but with one genus less, T (g−1) I q|qqp| + T (g−1) I qq|qp| . -For k = p: Two boundaries remain but genus is reduced by one, T (g−1) I p|qqp| . In summary, we find for the case e') together with Lemma D.1 I q|qqp| + T (g−1) Including the first four cases similar to the graphical derivation of the generalised 2-point function, we finally confirm the DSE (2.13) of the generalised 1 + 1-point function T E Proofs for Section 4.1 We will use the following well-known interpolation formula: Lemma E.1 Let f be a polynomial of degree d−1 ≥ 0 and x 1 , ..., x d be pairwise distinct complex numbers. Then, for all x ∈ C, We recall [22] that (3.7) gives rise to the product representation Proposition E.2 Let I = {u 1 , ..., u m }. The DSE (3.6) is solved by where v ∈ {w,ŵ 1 , ...,ŵ d } are the solutions of R(v) = R(w). We employed the short-hand notation Proof The second term of the lhs of the DSE (3.6) is conveniently written as , where f ( . ; w|I) is now a polynomial of degree d − 1. Applying Lemma E.1 with L w (t) := d j=1 (t − R(−ŵ j )), the interpolation formula yields where the analyticity of f (R(z); w|I) at z = −ŵ j was used. Next, insert (3.6) again for z → t near t = −ŵ j at which the first term of the lhs vanishes (here it is important that the integrand has only a simple pole at t = −ŵ j ). Inserting it for f (R(t); w|I) leads to where the product representation (E.1) was inserted. Next, compute for the same integrand the residue at t = z (for arbitrary z) Summing both expressions gives the assertion.

Remark E.3
The residue formula of Proposition E.2 is equivalent to the formula found in [23] via inversion of Cauchy matrices 1 R(εj )−R(−ŵ k ) j,k . This is not surprising as the derivation of the inverse Cauchy matrix in [35] is mainly based on the interpolation formula.

The proof of Corollary 4.2 is:
Proof We rewrite one of the terms in Proposition E.2 as where we substituted t → x = R(t), then moved the integration contour and finally represented the result in form of a residue formula. Proposition 3.11 implies that 1 R (ui)R (t)(t−ui) 2 is partially given in Ω 2 (z, w) is the only correlation function divergent on the diagonal so that the terms in { } extend to I1,I2,g1,g2 Ω (g1) |I1|+1 (I 1 ; t)T (g2) (I 2 t, w|) and finally to I1,I2,g1,g2 Ω (g1) is represented by the w-residue in the Corollary, where again vanishing terms are added after substitution and moving the integration contour.
As argued in the proof of Corollary 4.2, Lemma 4.1 implies that the residue at t = u i contributes only via Ω Proof Similar to the proof of Proposition E.2, but with d distinct points x k = R(α k ) for the interpolation formula of Lemma E.1.
The proof of Corollary 4.4 works in a completely analogous way.
In these variables the DSE (3.6) reads for g = 0 and m ≥ 1 where U (0) (∅ z, w|) = G (0) (z, w). The DSE (3.10) becomes for m ≥ 2 Lemma F.1 For all m ≥ 1, the function W Proof By induction in m, starting with Proposition 3.11 for m = 1. Assume that the assertion concerning W (0) k+1 is true for all k ≤ m − 1. Then (F.4) is recursively obtained when taking the residue in (F.2) and inserting it repeatedly into itself. Next, taking (F.4) into account, the residue of (F.3) at z = u j k collapses to But the rhs is Res z→ uj k W (0) m+1 (u 1 , ..., u m ; z)dz when expressing (3.9) in terms of W and U. With R ( u j k )G 0 ( u j k ) = 1 we finish the proof.
Proof By induction in m, starting from the true statement for G (0) (z, w) and W (0) 2 (u; z). If the statement is true for all W (0) |I|+1 (I; z) with |I| + 1 ≤ m and U (0) (I ..|) with |I| ≤ m − 1, then the rhs of (F.2) has at most simple poles at z = ε k j and no poles at z = − ε k j . The same is true for the second term of the lhs of (F.2) so that the statement extends to U (0) (u 1 , . . . , u m z, w|). This means that the rhs and the second term of the lhs of (F.3) have at most simple poles at z = ± ε k j . Since the prefactor G 0 (z) has simple poles at every z = ± ε k j and R (± ε k j ) is regular, the function W By construction all functions are holomorphic at z = ε n . This leaves the opposite diagonals z = −u k and the ramification points z = β i (from the prefactor R (z) in (F.3)) as the only possible location of poles in W (0) m+1 (u 1 , ..., u m ; z). These are preserved by differentiation to Ω (0) m+1 (u 1 , ..., u m , z), so that the proof of Conjecture 4.6 for g = 0 is complete.
The last two lines only contribute to the residue at q = −u k via the first-order pole of G (0) (u k , q). The residue cancels G 0 and otherwise amounts to replace q → −u k . This produces U (0) (I\u k u k ) according to (G.4).
The explicit formulae for W (0) 3 allow to prove the following identity 8 : (G.14) Inserted back into (G.13) for z → −q j gives the assertion.
We insert (G.11) and (G.10) into (G.6) for m = 3. Let againq j i = σ i (q) be the unique (for simple ramification points) preimage with lim q→β i σ i (q) = β i . Only the first line of (G.11) contributes to the residue at q = β i . More precisely, in the sum in this first line we have contributions from all terms with j = j i , whereas for j = j i we only have poles inǓ j (u q), namely in the single term .