Exact solution of matricial $\Phi^3_2$ quantum field theory

We apply a recently developed method to exactly solve the $\Phi^3$ matrix model with covariance of a two-dimensional theory, also known as regularised Kontsevich model. Its correlation functions collectively describe graphs on a multi-punctured 2-sphere. We show how Ward-Takahashi identities and Schwinger-Dyson equations lead in a special large-$\mathcal{N}$ limit to integral equations that we solve exactly for all correlation functions. Remarkably, these functions are analytic in the $\Phi^3$ coupling constant, although bounds on individual graphs justify only Borel summability. The solved model arises from noncommutative field theory in a special limit of strong deformation parameter. The limit defines ordinary 2D Schwinger functions which, however, do not satisfy reflection positivity.


Introduction
Matrix models [1] were intensely studied around 1990. Highlights include the nonperturbative solution of the Hermitean one-matrix model [2,3,4] and the understanding that it gives a rigorous meaning to quantum gravity in two dimensions. As proved by Kontsevich [5], there is an equivalent formulation by a model for Hermitean matrices Φ with action tr(EΦ 2 + i 6 Φ 3 ), where E is a fixed external matrix. Equivalently, the external structure can be moved to the linear term. The resulting partition function (all matrices self-adjoint) was solved by Makeenko and Semenoff [6]. The strategy consists in a diagonalisation of Φ thanks to the Itzykson-Zuber-Harish-Chandra formula, leaving an integral over the eigenvalues x i of the random matrix Φ. Since these x i are dummy integration variables, the partition function is invariant under variations x i → x i +ǫ n x n+1 i . These give rise to Virasoro constraints on Z[j 1 , . . . , j N ], which Makeenko-Semenoff were able to solve.
A renewed interest in matrix models came from field theories on noncommutative spaces of Moyal-Weyl type. We mention the magnetic field model studied in [7], which is also exactly solvable but trivial as a field theory. The field theory of the Φ 3 model on Moyal space with harmonic term (see below) has been studied by one of us (HG) and H. Steinacker in [8,9]. The novel aspect was a renormalisation procedure for the Kontsevich model. Only partial information on correlation functions were obtained; this is the point where the present paper goes much further.
Two of us (HG+RW) worked on the Φ 4 -theory on four-dimensional Moyal-Weyl deformed space and cured the ultraviolet-infrared mixing by adding a harmonic oscillator potential to the action. This leads to a renormalisable model [10], which develops a zero of the β-function of the coupling constant [11] at a special value of the parameter space. At this special point the model becomes a dynamical matrix model. In [12] we (HG+RW) extended the idea of [11] to an alternative solution strategy for matrix models, avoiding the diagonalisation (which is useless for the Φ 4 interaction). We used instead the Ward-Takahashi identities which result from a variation Φ → U * ΦU, with U = exp(iǫB) unitary, to derive a different type of Schwinger-Dyson equations. We proved that one of them consists in a non-linear singular integral equation for the 2-point function alone (first obtained in [13]), which then determines all higher correlation functions. We subsequently reduced the problem to a fixed point equation for a single function on R + and proved that a solution exists [14]. If one could prove that the solution is the Stieltjes transform of a positive measure, which is true for the computer [15], then one could convert the model into a 4-dimensional Euclidean quantum field theory with reflection-positive Schwinger 2-point function [16].
In this paper we apply the strategy of [12] to the Φ 3 2 matrix model 1 . Since a linear term would be generated by loop corrections, we add it from the beginning. We define first the model with cut-offs and give next Ward-Takahashi (WT) identities and Schwinger-Dyson (SD) equations. The 1-point function requires renormalisation, after which the cut-off can be sent to ∞ in the usual way [6]; for noncommutative field theory this corresponds to a limit of large matrices coupled with an infinitely strong deformation parameter -a limit which is called the "Swiss cheese limit". This way one projects onto the genus zero sector, but keeps all possible boundary components. In this limit the infinite hierarchy of SD-equations decouples (as in the Φ 4 -model [12]). We find that a function W (X) related to the 1-point function satisfies a non-linear integral equation which, up to the renormalisation problem, is identical to an equation solved by Makeenko-Semenoff [6] in the framework of the Kontsevich model. This coincidence is by no means surprising! We then proceed by resolving the entire hierarchy of linear equations for all genus-zero 1 In our subsequent paper [17] we extend this work to four and six dimensions. Whereas the renormalisation of Φ 3 4 and Φ 3 6 is much more involved, the solution of the Schwinger-Dyson equations is easily adapted from the Φ 3 2 case. To avoid duplication of material we introduce in some formulae parameters Z, ν which at the end are set to Z = 1 and ν = 0 for Φ 3 2 . matrix correlation functions. Here combinatorial identities on Bell polynomials play a crucial rôle.
In the final section we relate the Φ 3 matrix model to field theory on noncommutative Moyal space. We also perform in position space the limit of large deformation parameter. In this way a Euclidean quantum field theory on standard (undeformed) R 2 is obtained for which we can explicitly describe all connected Schwinger functions. We deduce that already the Schwinger 2-point function does not fulfil reflection positivity for whatever (real or imaginary) non-zero coupling constant. This is in sharp contrast with the φ 4model where numerical and partial analytic evidence was given that the Schwinger 2-point function is reflection positive.
Associating a quantum field theory with a matrix model is somewhat unusual in the traditional setup. We therefore begin in section 2 with a description of this relation, thereby giving a precise definition of correlation functions on the multi-punctured sphere, with N β fields attached to the β th boundary component (= puncture). We also point out that from the graphical perspective the perturbation series cannot be expected to converge; it is at best Borel summable. This highlights our achievement of explicit analytic formulae for any correlation function.

Prelude: A QFT toy model
We consider planar graphs Γ on the 2-sphere with two sorts of vertices: any number of black (internal) vertices of valence 3, and B ≥ 1 white vertices {v β } B β=1 (external vertices, or punctures, or boundary components) of any valence N β ≥ 1. Every face is required to have at most one white vertex (separation of punctures). Faces with a white vertex are called external; they are labelled by positive real numbers x 1 1 , . . . , (the upper index labels the unique white vertex of the face). Faces without white vertex are called internal; they are labelled by positive real numbers y 1 , . . . , y L . Such graphs are dual to triangulations of the B-punctured sphere. We associate a weight (−λ) to each black vertex, weight 1 to each white vertex, and weight 1 z 1 +z 2 +1 to an edge separating faces labelled by z 1 and z 2 . These can be internal or external, also z 1 = z 2 can occur. Multiply the weights of all edges and vertices of the graph and integrate over all internal face variables y 1 , . . . , y L from 0 to a cut-off Λ 2 , thus giving rise to a functionG Λ This setting defines a toy model of quantum field theory, sharing all typical features. It has the power-counting behaviour of the Φ 3 2 model, in particular has a single divergence: The limit lim Λ→∞G Λ Γ 1 (x 1 1 ) does not exist. The problem is cured by renormalisation. We assume the reader is familiar with the notion of one-particle irreducible (1PI) subgraphs. The renormalisation of the toy quantum field theory consists in recursively replacing all 1PI one-point subfunctions f (z) by its Taylor subtraction f (z)−f (0). This does more than necessary, but permits the global (i.e. non-perturbative) normalisation ruleG Γ (0) = 0 for any graph Γ with a single white vertex of valence 1. Omitting the superscript Λ onG means recursive renormalisation plus limit Λ → ∞. We notẽ Consider the following challenge: Fix B white vertices of valences N 1 , . . . , N B , take an arbitrary number (there is a lower bound) of black vertices, and connect them in all possible ways to planar graphs. Assign the weights, perform the renormalisation, evaluate the face integrals (for Λ → ∞) and sum everything up. What does this give?
One meets here a main difficulty of quantum field theory: there are too many graphs. The number of connected planar graphs with n black vertices can be estimated by the number n n−2 of ordered trees with n vertices. With the typical tools of quantum field theory, see e.g. [18], one can prove uniform bounds of the type |G Γ | ≤ C 1 · |λ| n C n 2 . This allows to give a meaning toG(x 1 1 , . . . , ) as a Borel resummation, where λ belongs to a sufficiently small disk tangent to the imaginary axis. Absolute convergence is impossible for anyλ = 0. We should remark that more complicated QFT models have an additional renormalon problem which excludes even Borel summability. In such case one has to employ the constructive renormalisation machinery [18] with its infinitely many (but mutually related) effective coupling constants.
We hope that the reader, with these remarks in mind, will appreciate that we will provide exact formulae for anyG(x 1 1 , . . . , Remarkably, these functions are analytic inλ 2 ! For convenience we refer to the simplest cases:G(x 1 1 ) will be given in (4.18),G(x 1 1 , x 1 2 ) implicitly in (4.21) andG(x 1 1 |x 2 1 ) implicitly in (5.9). One has to insert X β i = (2x β i + 1) 2 and the formulae for W (X) and c(λ) given in Proposition 4.1. The order-n Taylor term reproduces the sum of all graphs with n black vertices and B white vertices of valences N 1 , . . . , N B . The reader is invited to convince herself/himself that these formulae (restricted to the relevant order inλ) and the graphical rules agree on the the following examples: In fact we solve a more general case with weight functions 1 e(z 1 )+e(z 2 )+1 for the edges, where e : R + → R + is a differentiable function of positive derivative. Equivalently, one can keep the old face variables y i but assign a weightρ(y i ) = 1 e ′ (e −1 (y i )) to the faces. The asymptotic behaviour ofρ(y) ∼ y

The setup
Consider the following action functional for Hermitean matrix-valued 'fields' Φ = Φ * ∈ M N (C): 1) or explicitly (in symmetrised form) Here V is a constant discussed later, λ is the coupling constant (real or complex), and κ will be needed for renormalising the 1-point function. The self-adjoint positive matrix E = (E m δ mn ) plays a crucial rôle. We assume that the eigenvalues E m are a discretisation of a monotonously increasing differentiable function e with e(0) = 0, thus identifying 2E 0 = µ 2 with a squared mass. The resulting covariance functions )+1) are nothing else than the (discretised) edge weights considered in section 2. In particular, the discussion on the dimensionality encoded in e (i.e. in the spectrum of E) applies.
Comparison with (1.1) suggests that V is proportional to the size N of the matrices. This is precisely what we will do. The only reason to keep them distinct is the fact that, as recalled in section 6, the action (3.1) naturally arises in noncommutative field theory. There, V is related to the deformation parameter, so that the limit N ∼ V → ∞ defines the strong-deformation regime.
The partition function with an external field J, which is also a self-adjoint matrix, is formally defined by gives exactly the graphical setup described in section 2 -up to discretisation and temporary admission of non-planar graphs. The matrix indices correspond to face variables, edges between faces m, n have weight 1 Hmn , and the Φ 3 vertices are the black ones with weight (−λ). Identifying the white vertices is a little tricky. It turns out that the source matrices J partition into cycles attaches to the source matrices, which graphically means that the white vertex is the common corner of the N β external faces labelled by p 1 , . . . , p N β .
With this identification we can represent log Z as a sum over the number and the valences of the white vertices, i.e. the cycles of source matrices: where the symmetry factor S (N 1 ,...,N B ) is chosen as follows: If we regroup identical valence numbers N β as (N 1 , . . . , The expansion coefficients G |p 1 In principle they further expand into graphs Γ with all possible numbers of black vertices and their connections. As pointed out in section 2, the resummation is a problematic issue. We therefore keep the (N 1 + . . . +N B )-point functions intact and never expand into graphs. We will prove in this paper (similarly to [12]) that these functions have a welldefined large-(N , V ) limit precisely for the given a scaling factor V 2−B in log Z[J]. For later purpose we note the first terms of the resulting expansion of the partition function itself: All sums run from 0 to a cut-off N . We repeat the remark pointed out in [12] that these correlation functions have common source factors on the diagonal, e.g. (V 1 G |aa| + G |a|a| )J aa J aa . The functions G |aa| and G |a|a| are clearly distinguished by their topology (number and valence of white vertices) and most conveniently identified by continuation of G |ab| and G |a|b| to the diagonal.
Finally, we introduce our main tool: the Ward-Takahashi identities. As proved in [11,12], the invariance of the partition function (3.4) under inner automorphisms Φ → U * ΦU boils down to the WT-identities where the precise form of W a (which we shall not need) is given in [12,Thm 2.3]. These identities are exactly the counterpart of the Virasoro constraints in the traditional approach to matrix models [6].

1-and 2-point functions
We now derive a formula for the connected 1-point function G |a| by inserting (3.4), (3.5) into the corresponding term of (3.6): The last line follows from a two-fold differentiation of (3.7). Of course the sum N m=0 G |am| includes m = a! The connected 2-point function G |ab| is computed for a = b as follows: In the step from the 2nd to 3rd line we have used the Ward-Takahashi identity (3.8). The equation extends by continuity to Ea−E b . The limit is well-defined in perturbation theory where G |a| is, before performing the loop sum, a rational function of the E n so that a factor E a − E b can be taken out of G |a| − G |b| . We shall later see that our large-(N , V ) limit automatically gives a meaning also to lim b→a .
The naïve limit N → ∞ in (4.1) will diverge unless κ = κ(N ) is carefully adjusted. We chose a renormalisation condition where a well-defined limit G |00| is assumed. Substituting (4.2) and (4.3) into (4.1), the Schwinger-Dyson equations are obtained as This equation suggests to introduce (4.7)

Large-(N , V ) limit and integral equations
Let us take the limit N , V → ∞ subject to fixed ratio N V = µ 2 Λ 2 , in which the sum converges to a Riemann integral Expressing discrete matrix elements as a =: V µ 2 x, the eigenvalues of E take the form E a = µ 2 (e(x) + 1 2 ), see (3.3). We introduce the dimensionless 2 coupling constantλ := λ µ 2 and define . Now the limit of (4.6) becomes We assume here G |V µ 2 x|V µ 2 x| = O(V 0 ) so that this term does not contribute to the limit; this will be checked later. It can be seen graphically that this term generates higher genus contributions, which are scaled away in the large-N limit. A final transformation and similarly for other capital letters Y (y), T (y) and functions G(X, Y ) =G(x(X), y(Y )) etc., simplifies (4.10) to , Equation (4.12) closely resembles a problem solved in the appendix of Makeenko-Semenoff [6]. We take their solution (obtained by solving a Riemann-Hilbert problem) as an ansatz 3 2 From the partition function (3.4) and its expansion (3.6) one reads off the following mass dimensions: 3 Our ansatz is more general than necessary in 2 dimensions. We need with Z, ν in 4 and 6 dimensions [17] and treat already here the general case in order to avoid duplication in [17].
with constants Z, ν, c determined by normalisation and consistency conditions (thus becoming functions of λ, Ξ). Straightforward computation using In the last line we can symmetrise 1 Y +c so that the double integral factors. Converting the second line by rational fraction expansion, we arrive at This equation takes the form of (4.12) if we choose ν = 0, Z = 1 and adjust 4 c by For ρ(T ) ∼ T −α and α > 0, realised in our case, the formula (4.13) and the resulting condition on c have a limit Ξ → ∞.
Inserting ρ(T ) from (4.12) into (4.15) we have an explicit expression ofλ 2 in terms of c, either with c > −1 real or c ∈ C \ ]−∞, −1]. Obviously, c = 0 corresponds tõ λ = 0. The implicit function theorem then provides a unique diffeomorphismλ 2 → c(λ) on a neighbourhood of 0 ∈ R or 0 ∈ C. Since we will be able to express all correlation functions in terms of elementary functions of c(λ, e) and ρ(λ, e), this proves analyticity of all correlation functions in these neighbourhoods.

Linearly spaced eigenvalues of E
The noncommutative field theory model of section 6 translates to linearly spaced eigenvalues with e(x) = x. This yields X = (2x + 1) 2 and ρ(Y ) = 2λ 2 √ Y . The integral can be evaluated for Ξ → ∞: 4 In [6], c is determined by c + We thus get for the renormalised 1-point functioñ again with c being the inverse solution of (4.17).
A numerical investigation shows that (4.17) has a solution 5 for −λ c ≤λ ≤λ c and λ c = 0.490686 . . . attained at c c = −0.873759 . . . . By choosing c > 0 it is possible to simulate purely imaginaryλ. A perturbative solution of (4.17) gives as first terms This leads to the following series expansion of the renormalised 1-point function: .
We refrain from spelling out the insertion of (4.18). There is no problem going to the diagonal:G(x, x) = 2W ′ (X).

N -points functions
According to (3.6) the connected (N>2)-point functions are (4.22) 5 In general, the critical value corresponds to ρ 0 : For pairwise different indices we compute, similarly to (4.2), .
Proof. The formula is proved by induction, starting with N = 2 which is formula (4.7) when inserting P a 1 a 2 = −P a 2 a 1 . Assume it holds for N. Then using (4.23) and P a 1 a 2 = −P a 2 a 1 we have Now the definition on P a k a l implies P a 1 a 2 (P a k a 1 − P a k a 2 ) = P a k a 1 P a k a 2 , so that (4.24) follows for N → N + 1.

SD
with notations introduced in the proposition. We multiply by Haa λ and bring −2G |a 1 | G |a 1 |a 2 |...|a B | to the lhs, thus reconstructing the function W |a 1 | defined in (4.5): With these notations, and including ν (here = 0) from (4.13) for later use in [17], we can express the limit of (5.4) in terms of X i := (2e(x i ) + 1) 2 as follows: The measure ρ(T ) was defined in (4.12). In presence of ν = 0 we need a finite cut-off Ξ; the limit Ξ → ∞ is only possible for the solutions. The inhomogeneity only involves known functions with < B boundary components.

Solution for the (1+1+1)-point function
We specify the problem (5.7) to the (1+1+1)-point function and consequently Because of the factorisation the only reasonable ansatz is This gives as prefactor of 3 in (5.12) (with exchanged lhs and rhs and use of (4.13)): For linearly spaced eigenvalues e(x) = x and Z = 1, i.e. ρ(T ) = 2λ 2 √ T , this amounts to

Solution for the
with ρ l defined in (5.18). The step from the first to second line relies on with compensation of l = 0 with the integral in W (X 1 ) according to (4.13). After a reflection l → j − l we arrive at (5.17).
A shift in k yields the result.
Proposition 5.10. The solution of (5.27) for l = −2 and l = −1, We exchange the summation order s . Therefore (5.31) solves (5.27) for l ≥ 0 iff the following is true: Combined with the ansatz (5.16) and with Z = 1 in 2 dimensions we have proved (provided that Conjecture 5.11 is true): Theorem 5.12. The (1 + · · · + 1)-point function with B ≥ 3 boundary components of the Φ 3 2 matricial QFT-model has the solution . This section parallels the treatment of the φ ⋆4 4 case in [12]. We refer to that paper for more details. The φ ⋆3 2 -model on Moyal-deformed 2D Euclidean space with harmonic propagation is defined by the action The tadpole contribution proportional to κ ∈ R is required for renormalisation. By ⋆ we denote the 2D-Moyal product parametrised by θ ∈ R, The Moyal space possesses a convenient matrix basis where the L α m (t) are associated Laguerre polynomials of degree m in t and (ξ 1 , ξ 2 ) k := (ξ 1 + iξ 2 ) k . The matrix basis satisfies (f kl ⋆ f mn )(ξ) = δ ml f kn (ξ) and R 2 dξ f mn (ξ) = (2πθ)δ mn . A convenient regularisation consists in restricting the fields φ to those with finite expansion φ(ξ) = N m,n=0 Φ mn f mn (ξ). Using formulae for Laguerre polynomials, the action (6.1) takes precisely the form (3.1) of a matrix model for φ = φ * ∈ M N (C), with the following identification: This explains our interest in linearly spaced eigenvalues e(x) = x.
It was also pointed out in [16] and [15] that the Schwinger 2-point function is reflection positive iff the function p 2 →Ŝ 2 (p) is a Stieltjes function. This is not the case, neither for real nor purely imaginary non-vanishingλ! For c > 0 and thusλ ∈ iR, the integrand has a pole (or end point of a branch cut) in the complex plane at p 2 = µ 2 (−1 ± i √ c), contradicting holomorphicity in C \ R − . For −1 < c < 0 and thusλ ∈ R one finds that the imaginary part ofŜ 2 (p) at p 2 = (−3 − i |c| 10 )µ 2 is negative 8 . This contradicts the anti-Herglotz property of Stieltjes functions. A rigorous proof that the 2-point function of Φ 3 2 is not reflection positive will be given in [17].

Summary
We have given an alternative solution strategy for the large-N limit of the Φ 3 2 matrix model (= renormalsed Kontsevich model). This limit suppresses non-planar graphs. In principle, punctures (or boundary components) are also suppressed, but special limits of noncommutative field theory amplify them to the same level as the disk topology. We have established exact formulae, analytic in the (squared) coupling constant, for all these correlation functions. Correlation functions of disk topology (single puncture) can certainly be derived from previous results on the Kontsevich model. The complete treatment of the multi-punctured cases is new (to the best of our knowledge).
In our subsequent paper [17] we extend this work to the Φ 3 4 and Φ 3 6 models. There the renormalisation is much more involved, whereas the solution of Schwinger-Dyson equations is easily adapted from Φ 3 2 . We will discuss the issue of overlapping divergences and renormalons in Φ 3 6 . The main result will be the proof that Φ 3 4 and Φ 3 6 , but not Φ 3 2 , have reflection positive 2-point functions.
Reflection posivity of higher correlation functions is work in progress. Another interesting question concerns the identification of the KdV hierarchy in the solution we found.
We also hope that these investigations provide new ideas for attacking the more difficult equations of the Φ 4 4 model.