Bulk-boundary correlators in the hermitian matrix model and minimal Liouville gravity

We construct the one matrix model (MM) correlators corresponding to the general bulk-boundary correlation numbers of the minimal Liouville gravity (LG) on the disc. To find agreement between both discrete and continuous approach, we investigate the resonance transformation mixing boundary and bulk couplings. It leads to consider two sectors, depending on whether the matter part of the LG correlator is vanishing due to the fusion rules. In the vanishing case, we determine the explicit transformation of the boundary couplings at the first order in bulk couplings. In the non-vanishing case, no bulk-boundary resonance is involved and only the first order of pure boundary resonances have to be considered. Those are encoded in the matrix polynomials determined in our previous paper. We checked the agreement for the bulk-boundary correlators of MM and LG in several non-trivial cases. In this process, we developed an alternative method to derive the boundary resonance encoding polynomials.


Introduction
The correspondence between the two descriptions of 2D quantum gravity given by the matrix model (MM) and the Liouville gravity (LG) has been widely investigated for more than thirty years [1,2]. One of the key steps was the matching of the MM critical exponents with the gravitational dimension of the LG correlators [3][4][5]. Technically more elaborate, the matching between correlators was first performed in [6] for the sphere one and two points functions. This correspondence involves the so-called resonance transformation, a finite renormalization of the couplings arising due to the ambiguity lying in the presence of contact terms. Since these early steps, various matrix models have been introduced (one and two hermitian matrix models, ADE, matrix chains, O(n) loop gas model, q-Potts,...) to describe a discrete quantum gravity with different kind of matter fields. We concentrate here on the simplest one, the one hermitian matrix model which continuum description is provided by a minimal Liouville gravity, i.e. a quantum gravity which matter sector is given by a minimal model. In this context, the full resonance transformation on the sphere has been conjectured in [7], and checked up to the fifth order [7,8] (see also [9] for the case of the gravitational scaling Lee-Yang model). Agreement between the two descriptions has also been verified on the disc with the trivial boundary conditions (BC) in [10].
Recently, the correspondence between the one matrix model and minimal LG has been extended to worldsheets with arbitrary boundaries. More precisely, matrix correlators describing a disc with any boundary conditions have been constructed by two of the authors in [11]. It was later shown [12] that this construction relies on a linear relation of decomposition satisfied by the FZZT branes [13,14] of LG found in [15]. In [16], we constructed the matrix boundary operators introduced between two arbitrary boundary conditions. This construction involves the first order of a pure boundary resonance transformation that takes the form of a polynomial in the matrix. 1 The purpose of this paper is to extend the previous constructions by considering matrix correlators describing a disc with an insertion of one bulk and one boundary operator. In the search for agreement between LG and MM correlators, we are led to distinguish two different cases. When the boundary operator Kac index does not belongs to the fusion rules of two copies of the bulk operator index, the matter part of the LG correlator has to vanish. This case involves a mixed bulk-boundary resonance transformation that will be determined at the first order by imposing the cancellation of MM correlators combinations. On the other hand, if the LG correlator is non-vanishing, no mixed bulk-boundary resonance arises for dimensional reasons. Thus, we only have to consider the pure boundary resonance already determined in [16]. We explicitly checked the agreement between MM and LG in a few non-trivial cases, confirming our general approach to the resonance transformation. In this process, we derived an alternative expression for the polynomials encoding the resonance transformation. This expression proved to be convenient when the Kac index of the boundary operators are small.
In the first section, we concentrate on the LG side and specialize the expression of the Liouville bulk-boundary correlator [17] to the case of a coupling with a degenerate matter operator. In section two, we explain how to construct the bulk-boundary matrix correlators and derive their expression in the continuum limit. The section three is the core of the paper, starting with general considerations on the resonance transformation and deriving the relation between MM and LG correlators. Then we show the agreement between the two approaches for both vanishing and non-vanishing LG correlators. Most of the technical details are assigned to the appendix.

On the Liouville gravity side 2.1 Bulk-boundary correlator in Liouville gravity
The Liouville gravity is the description of 2D quantum gravity with critical matter in the conformal gauge [18]. The total action obeys the conformal symmetry and consists of three components, the Liouville, matter and ghost parts The bulk Liouville action S L is given as where µ is the bulk cosmological constant and R the Ricci scalar associated to the fixed worldsheet metric g ab . The background charge Q is related to the Liouville parameter b by Q = b + 1/b, which provides the central charge c L = 1 + 6Q 2 . The matter part is described by a conformal action with a central charge c M = 1 − 6q 2 where q = b − 1/b. The ghost part is the usual (b, c) system of the bosonic string theory, with central charge c gh = −26.
The vanishing of the total conformal anomaly leads to a constraint on the central charges that fixes the Liouville parameter b. In the following, we will concentrate on the p-critical models, realized as a LG with a matter sector given by a minimal model of the Lee-Yang series (2, 2p + 1) for which b −2 = p + 1/2. For the critical gravity, the fields describing the conformal matter, metric and ghosts are formally decoupled. But in the correlation functions, any matter field needs to be dressed by the appropriate Liouville vertex operator to form a composite field of dimension (1, 1), Φ 1,j e 2β 1,j φ(z) . The dressing charge β 1,j is related to the bare operator scaling dimension through where the indices (1, j) denotes the position of Φ 1,j in the Kac table, and j = 1, · · · , p. In order to respect the invariance under diffeomorphisms, the LG composite fields should be integrated over the worldsheet. However, the presence of conformal Killing vectors allow to fix the position of some of these operators, provided we multiply them with a suitable ghost factor. To fix the notation, the integrated (or fixed) dressed operator with Kac label (1, p − J) will be denoted V J with J running from zero to p − 1.
On the disc, the Liouville action acquires a boundary term [13], where K is the intrinsic curvature of the boundary. 2 At the quantum level, the boundary states are labeled by the parameter s, which is related to the boundary cosmological constant µ B in the semi-classical limit by an hyperbolic cosine parameterization [14]. The Liouville gravity boundary states are tensor products of these states |s > with a Cardy state |(1, l) > describing the matter boundary conditions. Such states obey a linear decomposition property [15], where the sum runs from −(l − 1) to l − 1 with steps of two. This decomposition has been shown to be consistent with the matrix model boundary construction on the disc [12]. However, its validity for topologies of higher genus may still need to be checked more carefully [19,20]. Likewise in the bulk case, one may introduce dressed conformal matter boundary operators Φ B 1,j e β 1,j ϕ(z) with a dressing charge given by (2.2). The integrated dressed operators with Kac label (1, 2(l − k) − 1) where k is running from zero to l − 1 will be denoted as B k . For convenience, one may consider the dressed operators as a perturbation of the LG action by introducing external sources, (2.5) The symmetry under translation of the Liouville field leads to associate a gravitational dimension to the coupling constants. Assuming a dimension two for the bulk cosmological constant µ, the other coupling dimensions are found to be [τ J ] = p + 1 − J and [τ (l) k ] = l − k. It is noted that by definition, the bulk and boundary cosmological constants couple to the dressed identity operators, τ p−1 = µ and τ (l) l−1 = µ B , [µ B ] = 1. The sphere and disc partition functions respectively scale as 2Q/b and Q/b.
The bulk-boundary correlation number we are considering can be written as Because of the presence of conformal Killing vectors, one may simply fix the position of both boundary and bulk operators, avoiding this integration over coordinates. The coefficient N Jk is independent of the Liouville parameter s and µ, it takes into account the matter and ghost factors, as well as some additional multiplicative constant in the expression of the Liouville correlator. 3 This coefficient vanishes when the boundary operator Kac index does not belong to the fusion algebra of the modules associated to two copies of the bulk operator, i.e. N (l) (2.7) In the section 4.4 below, we will compare only the µand sdependent part of MM and LG correlators. In this manner, we do not need the explicit expression for the coefficients N (l) Jk and the property (2.7) will be sufficient for our purpose.
The s-dependent factor in (2.6) has been obtained in Liouville field theory and is given in [17] as a Fourier transform where δ ± = σ − 1 2 β ∓ P and S(x) denotes the double sine function [13] which satisfies the shift properties S(x + b) = 2 sin(πbx)S(x), S(x + 1/b) = 2 sin(πx/b)S(x), S(x)S(Q − x) = 1. (2.9) In the context of Liouville theory, the momenta P related to the charge of vertex operator by β = Q/2 − P are purely imaginary. In this case, the Fourier transform in [17] is well defined and the contour C in the integral (2.8) is along the imaginary axis. However, in the context of Liouville gravity, these momenta are real and the convergence of the Fourier transform is no longer guaranteed. As a consequence, the contour C should be appropriately deformed (see details in the appendix A.1). One may notice that R(P, β, s) is an even function of P and s; R(P, β, s) = R(−P, β, s) , R(P, β, s) = R(P, β, −s) . (2.10) For the degenerate values β = kb, k ∈ Z + ,R is reduced to a ratio of sines due to the shift properties satisfied by S, The contour integration can be performed, leading to an infinite sum over residues, (2.12) 3 The exact expression of the factor N (l) J k depends on the proper normalization of the operators. This difference of normalization between MM and LG correlators is sometimes referred as "leg factors", they corresponds to a multiplicative degree of freedom for the coupling constants. Here, it will be fixed by requiring monic polynomials in the resonance transformations of the section three.
In the simple case k = 1, the summation can be easily done and we get the simple expression Unfortunately, the formula (2.12) is not convenient for comparison with the matrix model results. This is why in the next section we shall exploit the recursion relations over s satisfied by R in order to propose an alternative expression.

Shift identities
The bulk-boundary correlation R(P J , kb, s) obeys several shift relations that are supposed to be related to the bulk [21,22] and boundary [23,24] ground ring structures. They provide a nice recursion relation which bypass the complicated summed expression and will turned to be useful below to relate MM and LG correlators. The readers who are not interested in the technical details might skip this subsection. The simplest relation is obtained from the expression (2.12), when s is shifted by ±i/b; (2.14) Another relation can be derived using the properties of the double sine function to show that Plugging this into the Fourier transform, the first cosine just gives a sum over shifts of s, and we end up with where a similar relation was also derived for negative shifts. The most interesting relation is obtained if by shifting s and k but not P J . The shift relation is properly derived in the appendix (A.2) and it reads cosh πbs R(P J , (k + 1)b, s) = 2 cos(2πbP J ) R(P J , kb, s) − c k [R(P J , kb, s + ib) + R(P J , kb, s − ib)] sinh πbs R(P J , (k + 1)b, s) = is k [R(P J , kb, s + ib) − R(P J , kb, s − ib)] , (2.17) or equivalently where we used the shortcut notation c k = cos πb 2 k, s k = sin πb 2 k .
Together with the knowledge of the expression (2.13) for the operator B 1 , this last relation fully determines R at any β k . Indeed, it is easy to show that the following expression satisfies recursively the second shift relation in (2.17) with the initial condition given by (2.13) (see appendix A.3), with s α = s + iαb and x α = x(s α ) = u cosh πbs α . The expression for k = 2 can be simplified into

On the matrix side
The one matrix model partition function is given by an integral 4 over an N × N hermitian matrix M , where V (M ) is a polynomial potential chosen to achieve the (p + 1)-th Kazakov multi-critical point [25] (for a review of this model, see [1,2]). The partition function and the correlators can be expanded in powers of N −2 , each term being associated to a different topology. We focus here on the first term of the series which describe the planar topologies (sphere or disc). These first order terms depend on the t'Hooft parameter κ 2 = 1/g which weight the number of vertices of the planar Feynman diagrams. Since the parameter κ controls the area of the discrete surfaces dual to the Feynman diagrams, it is sometimes referred as the bare cosmological constant. In the continuum limit, κ is sent to a critical value κ * where the mean area diverge, and we define the renormalized coupling In order to compare the (p + 1)-th multi-critical MM with the (2, 2p + 1) minimal Liouville gravity, we first need to introduce the KdV deformations [26][27][28]. These linear deformations of the potential V (M ) by the other multi-critical potentials lead to the string equation that determines the string susceptibility -or two-punctured sphere partition function -u as a function of the deformations. Under a suitable normalization, the string equation reads where t p−1 corresponds to the LG cosmological constant µ. 5 In particular, u| * = √ µ and the resolvent defined below is identified with the LG identity boundary 1pt function on the disc. As mentioned in the introduction, the bulk resonance transformation has already been treated in [7]. At the level of insertion of a single bulk operator, this unnecessary complication can be avoided by a linear redefinition of the polynomials that perturb the matrix potential. These redefined potentials, denoted here V J , absorb the first order (but all orders in µ) of the resonance transformation. After this redefinition, the new potentials couple to bare parameters 4 This integral should be understood here as a formal series obtained by expanding the exponential, keeping only the quadratic term, the coefficient of which must be negative. 5 Contrary to the unitary models (e.g. the O(n) matrix model [29]), here the area is no longer measured by the dressed identity operators, but by the operator of highest dimension V0. In this sense, the MM renormalized cosmological constant t t J that can be directly identified with the parameters τ J perturbing the minimal LG in the continuum limit. In this setting, the bulk 1-pt functions are obtained from the matrix model correlators as, where the bulk couplings are turned off at * , except for t p−1 = µ. With this choice, all the MM bulk 1-pt functions vanishes apart from the one with J = p − 1.
Let us turn to the boundary effect. The disc with one marked point and trivial boundary conditions (i.e. leading to (1, 1) matter BC in the continuum limit) is given by the resolvent, defined as This quantity is the Stielges transform of the eigenvalue density for the matrix M . The bare boundary cosmological constantx controls the length of the boundary of the discretized surfaces. In the continuum limit, it is also sent to a critical valuex * where the mean boundary length diverges. This critical value can be taken to be zero by a shift of the matrix, and we define the renormalized boundary cosmological constant as ǫx =x, [x] = 1. In the process, we have to throw away the non-universal contributions which are polynomials inx such as the term V ′ (x) appearing in the expression (3.4) of the resolvent. Since in the following we focus on the expression of the correlators in this continuum limit, we automatically throw away all the non-critical terms and write directly [30]: The uniformizing parameterization of x has been introduced in order to resolve the branch cut over ] − ∞, −u] of the resolvent. The right bound of the branch cut is identified with the "string susceptibility" u.
On the disc topology, the boundary describing a matter with (1, l) BC have been constructed in [11] by allowing the matrix to additionally interact with Gaussian vectors with flavors. For instance, the one and two-point functions of minimal LG with matter BC (1, l) and cosmological constants x(s) (two identical boundaries for the two-point function) have been identified in [16] as LG = tr where the leg factors have been included in the definition of the LG correlators and creates the corresponding MM boundary. The polynomials P (l),k (x, M ), determined in [16], encode the pure boundary resonance transformation and will be defined more precisely in the section 4.1 below. The MM correlators with identical boundaries F l (x, M ) can be derived from the following disc partition function in the presence of boundary sources t To investigate the bulk-boundary matrix correlator, we need to consider the disc partition function of Eq. (3.8) in the presence of the bulk sources in the CFT frame, as in (3.3). The matrix correlator we are interested in is defined as where 'c' denotes the connected part and * all the non-trivial couplings turned off. This quantity is easily evaluated from the expansion of the product F l (see identity (A.1) of [12]), (3.10) We recursively make use of the property in order to rewrite the matrix correlators as This decomposition involves the derivative of the known bulk 1-pt functions with respect to the boundary cosmological constant [10], where the normalization has been fixed such that (3.14) We end up with the following expression for the bulk-boundary matrix correlators, It should be noted that the s-dependent part of O Finally, let us stress that the expression (3.15) satisfies shift relations similar to those obtained for the LG correlator in section 1.2.
The third identity is derived from a relation that can be linked to the insertion of a boundary ground ring operator, as explained in the appendix of [16].

The resonance transformation
In order to compare the matrix model with the LG results, we have to take into account a possible redefinition of the coupling constants known as the resonance transformation. This transformation is a consequence of the ambiguity in the definition of the so-called contact terms, i.e. the values of the correlators that contain operators taken at coinciding points. Such contact terms can be reabsorbed into a finite renormalization of the couplings. Coinciding bulk operators lead to pure bulk coupling resonance. This phenomenon was first introduced in the matrix model context in [6] and then later investigated for the perturbed sphere partition function in [7]. When dealing with a worldsheet having a boundary, we have to take into account boundary operators at coinciding points which leads to pure boundary couplings resonance, i.e. redefinition of the boundary couplings involving only other boundary couplings. In addition, we also have to take into account a bulk operator coinciding with the boundary (or equivalently with its mirror image) and bulk operators coinciding with boundary ones. These two phenomena lead to a bulk-boundary resonance which translates into the presence of bulk couplings in the resonance transformation of the boundary ones. It is stressed that no boundary couplings can be involved in the resonance of the bulk ones. The resonance transformation can be seen as a finite renormalization and this should be related to the study made in [31], where it was noticed that no boundary couplings arise in the RG equations of bulk couplings. On the contrary bulk couplings in the RG equations of boundary ones generate induced boundary flows.
In the problem we are considering, we have already taken into account the bulk resonance transformation in a suitable definition of the matrix potential deformations V J , and the general boundary resonance transformation writes where the c ρ,ν are just numerical constants. The resonance condition on the coupling constants dimension that restricts the form of the RHS gave its name to the transformation. We recall the dimensions of the couplings: l−1 is due to the fact that the monomial of degree l − 1 in F l (x, M ) is proportional to − s l s 1 x, see [11,12]). The boundary resonance transformation relevant for the study of boundary one and two points functions, as well as bulk-boundary correlators, writes The function P ··· a and Q ··· a are polynomials in µ and µ B and the summations are restricted because of the resonance condition on the gravitational dimensions. Multiplication of the boundary couplings by a constant will only change the normalization of the boundary operator, leading to a different leg factor. In order to fix this degree of freedom, we assume here that at the first order t since the matrix correlators with powers of M a , a < l − 1 are vanishing due to (3.11). In a similar way for the identity operator we get This is in indeed what we observe [12] since P (l) l−1 (µ, µ B ) = −(s l /s 1 )µ B . We now turn to the boundary 2pt functions and start again with the non-trivial operators k = l − 1 = m, As in the case of the boundary 1pt functions, the first sum vanishes so that the quadratic term in the resonance transformation (4.1) plays no role here. The second term defines the two polynomials that were determined in [16], Since we imposed P (l),k k = 1, P (l),k is a monic polynomial in M . When one of the operator is trivial (m = l − 1), the computation is similar to (4.5) but we have to define the matrix polynomial separatly as where the normalization has been fixed in order to have again a monic polynomial in M . In the following we will not specify anymore whether or not the operators are different from the identity, but just keep in mind this subtlety in the definition of P (l),l−1 . Finally, let us investigate the case of interest for this paper, the bulk-boundary correlators As in the case of the boundary 1pt and 2pt functions, the second term vanishes because either the correlator is zero (a < l − 1) or a = l − 1 and the coupling t Introducing the previous polynomials P (l),k and the bulk-boundary encoding polynomials 6 we finally get LG = tr The gravitational dimension of the polynomials can be deduced from this relation, The formula (4.11) implies that we have to consider two different cases. When the LG correlator vanishes, both terms in (4.11) contribute. It appears that imposing the vanishing of the combination of the two MM correlators provides a sufficient number of constraints to fully determine the bulk-boundary resonance term Q (l),J . It will be done in the section 4.3.
On the other hand, when the LG correlator is non-zero, i.e. l−k ≤ p−J, we observe that the term which involves Q (l),J is not present in (4.11). Indeed, in this case the boundary operator B k does not belong to the fusion outcome of the two copies of the bulk operators involved in the resonance of t (l) b<l+J−p−1 . Because of the orthogonality property of the boundary 2pt functions, there can be no contribution from a contact term arising when a bulk operator meets the boundary. We will explicitly check this result by relating MM and LG correlators for several non-trivial cases (k = 0, 1, 2, l − 2, l − 1) in the section 4.4 below.

Preliminary checks
As the first verification of the formula derived for both MM and LG correlator, we check the agreement of both expressions when no resonance is involved. This is the case for the boundary operator B 0 since the boundary coupling τ (4.13) As the second verification of the previous formalism, we may investigate the shift relations involving s → s±i/b. More precisely, we should check that the sign flips arising due to such shifts (see (2.14) and (3.16)) are compatible with the formula (4.11) and the definition of the polynomials P (l),k . In particular, the coefficients P (l),k a (µ, x) are polynomials of degree x k−a in x. Since µ has a gravitational dimension two and x one, their monomials are of the form x k−a−2n where n is an integer. This implies P (l),k a (µ, −x) = (−1) k−a P (l),k a (µ, x). Focusing again on the non-vanishing LG correlators, we have  as required.

Vanishing correlators and their constraints
We consider the case l − k > p − J for which the LHS of (4.11) is equal to zero. It is easy to see by recursion over k that all the monomial terms have to vanish independently, resulting in l + J − p constraints Ja has been computed in the section 3. It is convenient to introduce the Chebyshev polynomials of the first and second kind, T k (x) = u k cosh kπbs, U k (x) = u k sinh(k + 1)πbs sinh πbs , (4.16) and to rewrite the expression (3.15) as The general boundary 2pt functions have been studied in [16], but a simplification occurs when the two boundaries are identical. It can be seen by looking at the following correlator, When y belongs to the set of {x α } α , all the terms vanish apart from x α = y. For this remaining term, both numerator and denominator cancels and we end up with the resolvent derivative at x α , where q (l),J (x α ) is a polynomial of degree p − J − 1 in x α whose coefficients depend on s. This term arises because of the freedom in the identification to add x c α sinh πbs α with c < l − 1 − a to T p−j (x α ) within the sum over α. Thus, the expression of Q (l),J (x, x α ) contains p − J free parameters that will be determined a posteriori. Knowing l values of Q (l),J , we can use the Lagrange interpolation formula to obtain Q (l),J (x, M ) as a polynomial of degree l − 1, To solve these constraints, we first note that we can replace x n α in the previous equations by a Chebyshev polynomial of the first kind with the same degree. Then we decompose q (l),J over a basis of Chebyshev polynomials of the second kind, We end up with a linear system overq that can be solved by inverting the (p − J) × (p − J) matrix U (s). 7 This technique provides a unique expression for the polynomials Q (l),J (x, M ) that solves the constraints (4.15). From this expression, it is not obvious that the coefficients of Q (l),J (x, M ) are polynomials in x and µ. This can be shown, provided that the determinant of U cancels with the numerators of each coefficients. This is indeed what happened in the few cases we checked. 7 For a matter of clarity, here the vectors and matrix indices start from zero instead of the usual convention one.
As a crosscheck for the expression (4.23) of Q (l),J , we consider the insertion of a bulk identity operator, J = p − 1. In this simple case, there is only one constraint that can be easily solved, But since the cosmological constant µ = t p−1 is not turned off at the critical point * and is not resonant, Q (l),p−1 is just the derivative of the boundary matrix operator F l (x, M ) with respect to µ at fixed x, This result is in agreement with the expression (4.23) and (4.29).
As another example, let us investigate the case J = p − 2 where the matrix U is simply 2 × 2. The inversion givesq Even in the case of l = 4, the expression for Q (4),p−2 is rather complicated and we will not give it here. However, we have been able to check that the factor x 2 − u 2 c 2 1 appearing in the denominator of the two previous coefficients cancels so that each monomial of Q (4),p−2 (M ) is indeed a polynomial in x and µ. We believe that the cancelation of det U is a general feature such that the polynomial Q (l),p−J (M ) determined by this method always has the required form.

Non-vanishing correlators
We investigate here the relation (4.11) in the case of non-vanishing LG correlators. As already mentioned above, there is no bulk-boundary resonance involved. The non-resonant case k = 0 has already been treated as a preliminary check. In this section, we restrict ourselves to the cases k = 1, 2, l − 2 and l − 1. The aim is not to derive a general proof but only to provide convincing arguments for the consistency of the approach presented in section 4.1 and the pure boundary resonance expression found in [16]. Accordingly, we will concentrate on the µand sdependent part of the LG and MM correlators which are already non-trivial. In particular, we neglect the coefficient N (l) Jk that contains the matter and ghost sectors contributions. With a more careful analysis, one could also a priori derive their expression from the MM side but this is beyond the scope of this paper.
The pure boundary resonance transformation, encoded in the polynomials P (l),k (x, M ), has been determined in [16] by solving the orthogonality constraints on the boundary 2pt functions. Unfortunately, their explicit calculation is tedious when k is large since it involves the inversion of a matrix of size (k + 1) × (k + 1). This is why in appendix B we developed an alternative derivation which is useful for k close to l.
Case k = 1: The resonance polynomial is of degree one and reads [16], To compute the MM correlator involving P (l),1 (x, M ), we need the expression of O  The sum of the two shifted MM correlators can identified with LG correlators through (4.13), and simplified using (2.17), where we denoted r = 2P J /b. We deduce Case k = 2: The expression of P (l),2 is known explicitly (see the formula (4.13) of [16]), To avoid too complicated expressions, we simply show the proportionality of the MM correlators with R(P J , (l − 2)b, s) but the coefficient N J2 could also be determined. We notice that the insertion of the polynomial within the MM correlator trivially gives an expression proportional to R(P J , (l − 2)b, s). Consequently, we only need to show that the difference of the two polynomials, now of degree one in M , is also proportional to this LG correlator,P It can be rewritten by introducing the polynomial P (l),1 of (4.32), and inserted into to the MM correlator to give a sum of two LG correlators through (4.13) and (4.35): (4.40) The expression inside the parenthesis can be simplified using the recursion relation (2.22), it is indeed proportional to R(P J , (l − 2)b, s). This shows the validity of (4.11) with k = 2.
Boundary identity operator (k = l − 1): All the bulk-boundary LG correlators containing the identity boundary operator B l−1 are non-vanishing, regardless of the bulk operator. The simplest way to determine the polynomial P (l),l−1 is to use the fact that the boundary cosmological constant is not turned off at the * -critical point. Since this coupling is not resonant, P (l),l−1 is simply the derivative of F l with respect to t Pluging this result into the expression (3.15) for the MM correlator and using (4.27) to perform the α-summation, we obtain Case k = l − 2: The polynomial P (l),l−2 can be computed using the method given in appendix B. The matrix U to be inverted is only of size 2 × 2, and we find with the determinant Using these results, we can derive the MM correlator associated to the polynomial P (l),l−2 (x, M ), .

(4.46)
This coefficient is in agreement with the formula (4.13) for l = 2 and (4.35) for l = 3.

Summary and discussion
In this paper we investigated the bulk-boundary disc correlators of the hermitian matrix model and the ( 2 , 2 p + 1 ) minimal Liouville gravity. In our study of LG, we specialized to the case of a coupling to degenerate matter operators for which the Liouville dressing charges takes only a finite set of values. We derived two different expression, (2.12) and (2.20), for the cosmological constants dependent part of the correlator. We also provided various useful relations involving either a shift of the boundary parameter ((2.14),(2.16) and (2.17)) or a recursion relation on the boundary operator momentum (2.22). Given the large range of applications for the Liouville theory, such relations could also be useful in other contexts.
In the second section, we constructed the MM bulk-boundary correlator following the method of [11,12]. We obtained its expression in the continuum limit (3.15) and successfully identified it to the LG correlator in the nonresonant case. But our main interest was in the study of the resonance transformation at the bulk-boundary level. For this purpose, we considered this transformation in a very general setting in section three. It led us to distinguish between two different cases. In the first case, the bulk-boundary LG correlator is vanishing due to the fusion rules obeyed by its matter part. Imposing this vanishing condition on the MM correlators, we were able to find the first order of a bulk-boundary resonance (4.22). This transformation can be encoded in a matrix polynomial whose coefficients satisfy a the linear system of equations (4.28).
In the second case, the LG correlator is non-vanishing. There, no bulk-boundary resonance arises -except for the identity operator -and the MM and LG correlators were seen to agree in a number of specific cases, provided we take into account the boundary resonance investigated in [16]. In this process, we derived an alternative method to compute the matrix polynomials encoding the transformation which is particularly efficient for the boundary operators having a small Kac index.
The explicit agreement between LG and MM correlators in the non-vanishing case has been shown only for some particular cases. Even though these cases are enough to draw the general pattern, a complete proof is still lacking. Such a proof would provide the expression of the factors N (l) Jk that can be identified, once a proper normalization is fixed, to the matter contribution of the LG correlator, namely to the bulk-boundary disc correlation function of the (2, 2p + 1) minimal model.
Our ultimate purpose is to conjecture the expression of the resonance transformation including the boundary effects at all order. Such a proposal was made for the bulk case in [7] after the study of the 3-pt functions. The boundary disc 3-pt function can be investigated in a similar way. In particular, the LG correlator is known and obeys the property of factorization into ghost, matter and Liouville sectors. The expression of the Liouville correlator found in [32] is rather complicated but should simplify in the case of a coupling to degenerate matter operators.
We defined the MM boundary perturbations to be simple power of the matrix, in contrast with the bulk case where these perturbations are critical potentials related to the underlying KdV hierarchy. There may exist a better definition of the perturbations in the boundary case, possibly related to an underlying integrable hierarchy. This change of basis for the MM perturbation may allow to rewrite the resonance transformation in a simpler form.
The relations we found in section 1.2 for the LG correlators are believed to be consequences of the presence of the ground ring structures and the connection still need to be specified, as we did for the boundary 2-pt function in [16]. The exact realization of such structures within matrix models is still an open problem. For instance, in the O(n) model the boundary ground ring identities have been related to the continuum limit of MM loop equations in [33]. At the moment, no general picture including these two examples has emerged.
Finally, we could also apply this boundary construction to other matrix models, such as the O(n) [34], ADE [35] or dimers [36] models. These models have an interpretation as statistical models defined on a fluctuating lattice. We hope to develop a statistical interpretation of the matrix boundaries in this context.
and the Fourier transformR(P J , kb, σ) is analytic in this strip. We now examine the poles of the expression (2.11) forR. Since some of the zeros in the denominator cancel with zeros in the numerator, we are left with two infinite series of poles centered on ±P J , poles being separated by a distance b. In addition, the two intervals of length kb, ] ± P J − kb/2, ±P J + kb/2[ are free of poles, When σ 0 < 0, the two intervals overlap and the strip Re σ ∈] − σ 0 , σ 0 [ around the origin is indeed free of poles. Closing the contour on the right, the sum of residues leads to the formula (2.12) for the LG correlator. When σ 0 > 0, the two intervals free of poles does not overlap anymore. However, we can still make sense ofR as a Fourier transform by smoothly deforming the contour from the case σ 0 > 0 in the inversion formula. This contour, denoted C, is such that only the poles of (A.3) with n ≥ 0 are picked up. The residue formula again lead us to the expression (2.12) for the LG correlator. We can check that this expression has the correct asymptotic (A.1) and satisfies the two reflection properties,

A.2 Derivation of the shift equation
To derive the two shift relations (2.17), we first notice that δ ± is invariant under k → k + 1, σ → σ + b/2 and since the shift of k increment the number of terms in the numerator product of (2.11), we get R(P J , (k + 1)b, σ) = 4 sin πb(σ + (k − 1)b/2 + P J ) sin πb(σ + (k − 1)b/2 − P J )R(P J , kb, σ − b/2) (A.5) This relation can be inserted within the Fourier transform, and the shift over σ can be reabsorbed under a change of variable σ → σ − b/2. This change of variable shifts the contour of b/2 to the left, so that we could a priori pick up an extra pole. However, since the poles ofR in σ are separated by a distance b and J, k are integer, the contour C can always be chosen such that no additional poles are picked up. This choice will be confirmed below by the reflection properties P J → −P J of the recursion relation, and we can safely write R(P J , (k + 1)b, s) = −ie −πbs C dσe −2πsσ 4 sin πb(σ + kb/2 + P J ) sin πb(σ + kb/2 − P J )R(P J , kb, σ). (A.6) Then, we use a trigonometric identity to transform the product of sines into a difference of cosines, R(P J , (k + 1)b, s) = ie −πbs C dσe −2πsσ e 2iπbσ e iπb 2 k + e −2iπbσ e −iπb 2 k − 2 cos 2πbP J R (P J , kb, σ). (A.7) Inside the brackets, the first two terms can be absorbed by a shift of the variable s, the third one does not depend on the integration variable, so we end up with e πbs R(P J , (k + 1)b, s) = 2 cos 2πbP J R(P J , kb, s) − e iπb 2 k R(P J , kb, s − ib) − e −iπb 2 k R(P J , kb, s + ib) (A.8) Note that this relation is invariant under the bulk reflection P J → −P J which confirms the fact that no extra poles were picked up since individual residues does not obey this symmetry. Finally, using the property R(P, β, −s) = R(P, β, s) we easily derive the two equations (2.17).

A.3 Solution of the second shift equation (2.17)
To show that the expression (2.20) satisfies the recursion relation given by the second equation of (2.17), we just plug it in the RHS of this equation and then shift the indices α to get R(P J , (k + 1)b, s) = u k−1 Γ k sinh iπb 2 k sinh πbs where we denoted (A.10) We first examine the term α = k appearing in the first sum and see that in order to form a complete product over β running form −k to k with steps of two (excepting α), we need an extra factor (x k − x −k ). Similarly the same extra factor is needed for the term α = −k. Then, all the other terms have contributions of both sums which differences leads to the same factor:

B Alternative solution for the pure boundary resonance encoding polynomial
The solution found in [16] for P (l),k is manageable when k is small since it involves the computation of determinants of size (k + 1) × (k + 1). The alternative solution we demonstrate here is convenient when l − k is small, relying on the inversion of a matrix of size (l − k) × (l − k). However this solution is specific to the case of two identical boundaries. The orthogonality conditions arising from the boundary 2pt function can be written as [16], This matrix U is the same that appeared in (4.28) for the determination of the bulk-boundary resonance encoding polynomials Q (l),J . As a crosscheck, we can consider the case of the insertion of the boundary identity operator k = l − 1. The relation (B.5) supplies only one constraint that can be solved, leading tõ p (l),l−1 = s 1 s l 1 sinh πbs (B.9) which indeed gives a polynomial P (l),l−1 that corresponds to the expression (4.41) obtained as a derivative of F l with respect to x. As another crosscheck, the explicit expression for the polynomial with l = 4 and k = 2 has also been derived, it matches the formula (4.13) given in [16].