The null identities for boundary operators in the ( 2 , 2 p + 1 ) minimal gravity

By using the matrix-model representation, we show that correlation numbers of boundary changing operators (BCO) in (2, 2p+1) minimal Liouville gravity satisfy some identities, which we call the null identities. These identities enable us to express the correlation numbers of BCO in terms of those of boundary preserving operators. We also discuss a physical implication of the null identities as the manifestation of the boundary interaction.


Introduction
The 2-dimensional gravity coupled with a minimal model of CFT has been studied as a good example of well-defined quantum gravitational theories [1], which also allows a non-perturbative discrete formulation given by matrix models [2,3,4,5].
In this paper, we follow the one-matrix model description [6,7] of the (2, 2p + 1) minimal gravity on Riemann surfaces but focus on the description in the presence of boundaries [8,9]. The boundary conditions of the minimal gravity, also referred to as FZZT branes [10], are specified by the value of the boundary cosmological constant µ B and the Kac label (1, m) of the matter Cardy state. In [9], it was shown that such boundary conditions are realized in the matrix model by introducing a generalization of the resolvent operators. The disk partition function for the (1, m) Cardy state is given by where f m (M ) is a monic polynomial of the Hermitian matrix M with degree m and · · · stands for the expectation value of the one-matrix model. After some renormalizations, the coefficients of f m (M ) are related to the sources of boundary operators, which preserves the (1, m) boundary condition.
One can introduce some impurities on the boundary, which interpolate two different boundary conditions. These are called the boundary changing operators (BCO). Between two boundary segments of (1, m 1 ) and (1, m 2 ) with different boundary cosmological constants, one can put a (1, k) primary operator dressed by the Liouville factor e β k φ , where k = |m 1 − m 2 | + 1, |m 1 − m 2 | + 3, · · · , m 1 + m 2 − 1, β k = (k+1)b 2 and b 2 = 2/(2p + 1). It was shown in [9] that these operators are described in the one-matrix model as follows. One extends the disk partition functions to the 2 × 2 block matrix of the form . ( Here, g m 1 m 2 (M ) is a polynomial of M with degree less than min(m 1 , m 2 ). The coefficients of g m 1 m 2 (M ) provides sources of BCOs between the (1, m 1 ) and (1, m 2 ) boundaries. Correlation numbers with more different boundary conditions can also be treated in the similar way by introducing more block structures. It was shown that this formulation correctly reproduces the correlation numbers of BCOs, computed in the Liouville theory approach.
In this paper, we demonstrate that the correlation numbers of BCOs satisfy some nontrivial identities, which we call null identities. The idea for deriving the null identities is the following. The perturbed partition function (2) can be diagonalized to the form, where f ′ m 1 and f ′ m 2 are new polynomials of M with degree m 1 and m 2 , respectively. This shows that the sources of BCOs, which were originally encoded in the coefficients of g m 1 m 2 (M ), are actually redundant and can be absorbed into the redefinitions of the sources of the boundary preserving operators in f m 1 and f m 2 . Thus, after the redefinitions, the partition function becomes independent of the sources of BCOs. In terms of the original parametrization, this implies that there exist differentials ∇ n (n = 1, 2, · · · , min(m 1 , m 2 )) such that they are given by linear combinations of the derivatives of the sources and satisfy ∇ n (F m 1 m 2 ) = 0. This is the simplest example of what we call the null identities. These identities enable us to write the correlation numbers of BCOs in terms of those of boundary preserving operators. We will present a general derivation of the null identities and show that ∇ n can be constructed in such a way that the curvature is vanishing (i.e. [∇ n , ∇ l ] = 0). Then, we will discuss physical implications of the identities. This paper is organized as follows. In Section 2, we derive the null identities. In section 3 we show some examples of the differentials ∇ n and the null identities, and discuss physical implications of them. Section 4 is devoted to conclusion and discussion on a possible extension to the cases where more than two boundary conditions are allowed. We present the case with three boundary parameters in some detail.

The null identities
Under the double scaling limit of the one-matrix model, insertions of the matrix M in the path integral can be replaced with insertions of a quadratic differential operator Q, which acts on the space of eigenvalues of M [11,12]. Additive and multiplicative constants appear in this replacement: M → ǫQ + c. For insertions of polynomials of M , these constants can be absorbed into renormalizations of the coefficients of the polynomials and the overall factors of the operators. After the renormalization, the perturbed partition function takes the form, where C m i (Q) and c(Q) are polynomials obtained by renormalizing f m i and g m 1 m 2 , respectively. They are written as where d = −1 + min (m 1 , m 2 ) and c k real. The coefficients of C m i (Q) and c(Q) correspond to the sources of boundary preserving and changing operators, respectively. We will discuss this correspondence later in more detail after we derive the null identities in the following. By the formula tr log R(Q) = log detR(Q), the perturbed partition function (4) can be written as the expectation value of the logarithm of det(R(Q)). As a polynomial of Q, the degree of det(R(Q)) is m = m 1 + m 2 and it has m independent coefficients. However, the matrix R(Q) has m + d + 1 parameters in (4). Hence, d + 1 parameters are redundant and those extra coefficients can be absorbed into redefinitions of the coefficients. This implies that there exist d + 1 constraints on the partition function: which we refer to as null identities. Here n = 1, 2, · · · , d + 1 and ∇ n are linear differential operators given by combinations of The differential operators ∇ n are specified by the condition where x is a formal parameter representing Q. We express tr R m 1 m 2 (x) as The operators ∇ n specified by (7) are equivalently defined by requiring the following conditions: For ∀k ∈ {1, 2, . . . , m} ∇ n ζ k = 0.
A general solution to (9) can be constructed as follows. First, note that for each c n there should exists an independent differential operator satisfying (9). Then, we put an ansatz, where∇ n is a linear differential operator consisting of Specifically,∇ is written as∇ where η Then, if ζ k − ξ k (∀k ∈ {1, 2, . . . , m}) has no explicit a i -dependence (which is always the case for 2 × 2 block matrix dealing with two boundary parameters), the differential operator is given by The conditions for the differential operators (9) allow an ambiguity in the overall normalizations. This ambiguity is fixed in (10) by setting the coefficients of the c n -derivatives to be unity. This choice is very useful, since with this choice, the operators mutually commute: [∇ n , ∇ l ] = 0. This can be seen as follows. In general, [∇ n , ∇ l ] is a linear differential operator. Since both of ∇ n and ∇ l satisfies (9), their commutator [∇ n , ∇ l ] should also satisfy (9). Then [∇ n , ∇ l ] should be again given by a linear combination of {∇ n }: With the choice of (10), the left-hand side of (14) does not contain c n -derivatives, while the right-hand side does. This means that α nlk = 0 and thus [∇ n , ∇ l ] = 0.

Physical implications for correlation numbers
The null identities (6) provide important facts that any correlation numbers of boundary changing operators can be rewritten in terms of the correlation of boundary preserving operators. The possibility is due to the fusion rule between BCO operators.
Let us consider the simplest case F 11 = − log det(R 11 (Q)) , where R 11 (Q)) is a 2 × 2 matrix where a i 's are cosmological constants of (1,1) boundaries and assumed to take different values a 1 = a 2 . The off-diagonal component c couples to the boundary changing operator B 11 intertwining two different (1,1) boundaries and produces one null operator with ξ 1 = a 1 + a 2 and ξ 2 = a 1 a 2 . We have the null identity ∇ N F 11 = 0 where N is a positive integer. For N = 1, the identity shows which is consistent with the fact that the one-point correlation of BCO is not allowed, since the boundary conditions contradict with each other. For N = 2, the two-point correlation of BCO is given in terms of one-point boundary preserving correlation numbers: where we define Using ∂/∂ξ 2 = −(1/a 12 )(∂/∂a 1 − ∂/∂a 2 ) with a 12 = a 1 − a 2 , the result can be rewritten as where The one-point correlation O i becomes u 1/b 2 cosh πs i b if one evaluates it at value a i = u cosh (πbs i ), where u is a scale factor and s i a boundary parameter.
It is noted that the free energy is given as where F

(D) 11
is the c-independent part. This shows that the cubic correlation of the BCO is absent and four-point correlation In a similar manner, from null identities obtained by successive applications of ∇, one can find identities relating higher-point correlation numbers of BCOs with lower-point correlation numbers of boundary preserving operators.
One may look into a little complicated case: BCO between (1, 1) boundary and (1,2) boundary. This can be investigated using F 12 = − log det(R 12 (Q)) , where In this case also there is one off-diagonal parameter which couples to BCO B 12 . The null operator is given as with ξ 3 = µ 1 µ 2 µ 3 and provides a similar null identity as in between (1, 1) boundaries: ∇ N F 12 = 0. It is obvious that one has an alternative expression of the free energy as in (22) In this case as well, correlation numbers with insertions of odd number of BCO B 12 are prohibited. Twopoint correlation is similarly given as in (18) Here I 3 is given in terms of one-point correlations of the boundary preserving operator O i : with a ij := a i − a j . It is noted that (1, 2) boundary condition is realized when a 2 = µ + and a 3 = µ − with µ ± = u cosh (πb(s 2 ± ib)) and s 2 real. In this case one has O 2 | a 2 =µ + = O 3 | a 3 =µ − = −u 1/b 2 cosh(πb/s 2 ) and the two-point correlation of BCO becomes 1 with s p = s 1 + s 2 and s m = s 1 − s 2 [9]. Suppose we consider BCO between two different (1, 2) boundaries: The off-diagonal terms has two real parameters c 1 and c 2 and thus there are two independent commuting null operators: where ξ i is defined by (12), implying null identities ∇ N 1 1 ∇ N 2 2 F 22 = 0. The free energy can be written in the form and therefore, no correlations with odd number of BCOs B where B 22 /∂ξ i . To find BCO correlations between (1, 2) boundaries we need to put correct parameterization of a i 's: a 1,2 = u cosh (πb(s 1 ± ib)), a 3,4 = u cosh (πb(s 2 ± ib)). It is notable that B as given in [13]. One can extend the discussion to BCOs between (1, m 1 ) and (1, m 2 ) boundaries without any difficulties using the null operator (13). It is noted that (∂ζ k /∂c n ) has no ξ i -dependence. As a result, the free energy has no correlations with odd number insertions of BCOs.

Conclusion and discussion
In this paper, we considered correlation numbers of boundary changing and preserving operators in the (2, 2p + 1) minimal Liouville gravity on disk. In terms of the one-matrix model, we showed that those correlation numbers satisfy some identities, called the null identities in this paper. These identities enable us to express correlation numbers including boundary changing operators in terms of correlation numbers with only boundary preserving operators. This means that the correlation numbers of the boundary changing operators can be determined from those of boundary preserving operators. In addition, the null operator shows that the free energy has no cubic correlation of BCOs.
One may extend the matrix into n × n blocks to describe correlation numbers with n different boundary parameters: with C i (Q) and c ij (Q) respectively being where d ij = −1 + min (m i , m j ). The coefficients of C i (Q) and c ij (Q) are identified with the sources of boundary preserving and changing operators. Under this setup, as opposed to 2 × 2 block diagonal case, there seems in general no explicit formula for mutually commuting differential operators ∇ n that satisfies ∇ n (det R m 1 m 2 ···mn (x)) = 0. However, still it is possible to find them under making ansatz (10) by requiring the conditions (9), where the parameters {ζ i } and {ξ i } are understood as straightforward extensions of (8) and (12), respectively. For example, with 3 × 3 block matrix: There are three commuting differential operators that provides the null identities: where ξ 2 = a 1 a 2 + a 2 a 3 + a 3 a 1 , ξ 3 = a 1 a 2 a 3 . The coefficients e i (i = 1, 2, 3) are explicitly given by which depends on ξ i 's unlike in the 2 × 2 case. As a result, the free energy, satisfying the null identity ∇ N 1 1 ∇ N 2 2 ∇ N 3 3 F 111 = 0, has non-vanishing cubic correlation of BCOs, for example, B a 2 a 3 11 B a 3 a 1 11 B a 1 a 2 11 = ∂ 3 F 111 ∂c 1 ∂c 2 ∂c 3 c=0 = −2 a 12 O 3 + a 23 O 1 + a 31 O 2 a 12 a 23 a 31 .
As one considers bigger size matrix with its block components of higher order polynomials, their expression becomes more and more complicated, still one can expect to find out the differential operators case-by-case.