Note on a duality of topological branes

We show a duality of branes in the topological B-model by inserting two kinds of the non-compact branes simultaneously. We explicitly derive the integral formula for the matrix model partition function describing this situation, which correspondingly includes both of the characteristic polynomial and the external source. We show that these two descriptions are dual to each other through the Fourier transformation, and the brane partition function satisfies integrable equations in one and two dimensions.


Introduction
D-brane is the most fundamental, and also non-perturbative object in superstring theory.
It gives a non-trivial boundary condition for open strings, and consequently a lot of gauge theories can be constructed as effective theories for stringy modes on them, by considering combinations of branes. When we implement various gauge groups and matter contents, branes often have to be arranged in a complicated way. In such a case, we have to deal with dynamics of brane intersection appropriately, and discuss what kind of stringy excitation can be allowed there. In this sense it is important to study properties of intersection of branes in detail.
In this article we investigate some aspects of the intersecting branes in topological string theory, especially through its matrix model description. It is well known that the Riemann surface plays a crucial role in the topological B-model on Calabi-Yau threefolds, and this Riemann surface can be identified with the spectral curve appearing in the large N limit of the matrix model [1]. A brane is also introduced into the B-model, and its realization in terms of the matrix model was extensively investigated [2]: There are seemingly two kinds of non-compact branes, which correspond to the characteristic polynomial and the external source in the matrix model. While these two descriptions are apparently different, we will see that they are essentially the same. By considering both kinds of the non-compact branes at once, we can discuss intersection of branes in the topological B-model. From the viewpoint of the matrix model, this situation just corresponds to insertion of both of the characteristic polynomial and the external source. We will explicitly show that two descriptions are dual to each other by seeing a newly derived matrix integral formula.
It is known that the correlation function of the characteristic polynomial and the matrix integral with the external source are deeply related to the integrable hierarchy [3]. Thus it is naively expected that the hybrid of them also possesses a similar connection to the integrability. Since in this case we have two kinds of variables corresponding to the characteristic polynomials and the external sources, we obtain the two dimensional integrable equation as a consequence, and identify the partition function as the τ -function of the corresponding integrable hierarchy with the Miwa coordinates.
This article is organized as follows. In Sec. 2 we start with a review on how to consider branes in the topological B-model in terms of the matrix model. We introduce two descriptions of branes using the characteristic polynomial and the external source of the matrix integral with emphasis on their similarity and difference between them. In Sec. 3 we consider intersecting branes in the B-model by introducing both kinds of branes at the same time. We derive a new matrix integral formula corresponding to this situation, and show that the characteristic polynomial and the external source are dual to each other in the sense of Fourier transformation. We also show that the brane intersection partition function satisfies the integrable equation in one and two dimensions, which is well-known as the Toda lattice equation. In Sec. 4 we consider the Gaussian matrix model as an example. In this case we can show the duality using the fermionic variables, which is useful to discuss the effective degrees of freedom on the branes. We close this article in Sec. 5 with some discussions and remarks.

Branes in the topological B-model
We first review the large N matrix model description of the topological B-model, mainly following [2], and how to introduce the brane, which is also called the B-brane or the FZZTbrane. The non-compact Calabi-Yau threefold we discuss in this article is obtained as a Here H(p, x) determines the Riemann surface where W (x) and f (x) are polynomials of degree n + 1 and n, respectively. This smooth Calabi-Yau threefold is given by blowing up the singular one The branes, which are compact, are wrapping n P 1 at critical points W (x) = 0, and the sizes of P 1 's are parametrized by f (x), which describes the quantum correction around them.
The holomorphic (3, 0)-form Ω in the Calabi-Yau threefold (2.1) is chosen to be Ω = du ∧ dp ∧ dx u . (2.5) Then the periods of this (3, 0)-form Ω over three-cycles reduce to the integral of the symplectic In this way we can focus only on the complex one-dimensional subspace Σ by keeping the dependence on u and v fixed.

Characteristic polynomial
The algebraic curve (2.2) can be identified with the spectral curve of the matrix model: It is just given by the loop equation in the large N limit of the matrix integral gs Tr W (X) . (2.8) The polynomial W (x) is the matrix potential. This is the reason why we can discuss the topological B-model using the matrix model. In this description the genus expansion with 1/N corresponds to the quantum correction, and the 't Hooft parameter t = g s N gives the size of P 1 .
The other canonical variable in (2.3) is related to the resolvent of the matrix model The saddle point equation of the matrix model is equivalent to the condition p(x) = 0. Since the one-form on the spectral curve is given by these two canonical variables we can naturally introduce the chiral boson φ(x) on the Riemann surface Σ, which is interpreted as the Kodaira-Spencer field describing deformation of the complex structure at infinity φ(x) = W (x) − 2g s Tr log(x − X) . (2.11) The vertex operator, which creates a non-compact brane at a position x, is constructed by the standard bosonization scheme The prefactor e − 1 2gs W (x) corresponds to the classical part of the operator, while the determinant part, namely the characteristic polynomial, gives the quantum fluctuation as a gravitational back reaction. This brane creation operator gives a pole at x on the Riemann surface, and its residue is given by This is just interpreted as one brane contribution.
The partition function of the branes is represented as a correlation function of the characteristic polynomials, e.g. k-point function is given by This expectation value is taken with respect to the matrix measure with the standard normalization 1 = 1. Including the classical part, the brane partition function is then given by This correlation function can be exactly evaluated using the orthogonal polynomial method.
We will come back to this formula in Sec. 3.2 (see (3.11)).

External source
Let us then consider another kind of the non-compact brane in the B-model, which is described by the external source in the matrix model. We consider the matrix action written in a form of where the potential is regarded as an integral of the one-form along an open path to a certain point p on the Riemann surface We can assume the matrix A is diagonal A = diag(a 1 , · · · , a N ) without loss of generality.
The action (2.17) corresponds to the matrix model with the external source which is analogous to the Kontsevich model [4]. The external source implies positions of N branes, at least at the classical level, because the extremum of the action W (P ) − A = 0 gives the classical solution, X = A, since we have W (P ) = X according to (2.18).
The one-form used in (2.18) is apparently different from (2.10), but they are equivalent due to the symplectic invariance for a pair of the canonical variables (p, x). This symmetry is manifest by construction of the topological B-model as seen in (2.5) and (2.6). Therefore two descriptions of the non-compact branes based on the characteristic polynomial and the external source in the matrix model are dual to each other in this sense. We will show in Sec. 3 that they are converted through the Fourier transformation by deriving the explicit matrix integral representation.

Topological intersecting branes
As seen in the previous section, there are two kinds of non-compact branes in the topological B-model, which are related through the symplectic transformation. We then study the situation such that both kinds of branes are applied at once. This is realized by inserting the characteristic polynomial to the matrix model in the presence of the external source. Let us consider the corresponding partition function denoted by We now have to be careful of the meaning of the matrix potential and the external source.
As discussed in Sec. 2.2, if the matrix potential is given by the integral of the one-form in the form of (2.18), the external source gives classical positions of branes in the x-coordinate.
In this case, however, the roles of x and p are exchanged. Instead of (2.18), the potential should be written as With this choice, the corresponding external source determines positions of branes in the pcoordinate. We note that, in this case, the extremum of the action does not simply imply the classical positions of branes as W (X) = A, because of potential shift due to the characteristic polynomial. Although there exists this kind of back reaction, the interpretation of the external source as the p-coordinate should still hold.
In Fig. 1 we depict the situation such that both kinds of branes are inserted to the Riemann surface Σ. Since their positions are labeled by the x and p coordinates, respectively, these branes are extended along perpendicular directions. We call this the brane intersection in the topological B-model. We will see in the following that these coordinates can be exchanged through the Fourier transformation.

Matrix integral formula
We then provide an explicit formula for the partition function (3.1). In order to derive the formula, we apply a method to compute the matrix model partition function with the external source [5,6], which is also applicable to this situation.
First of all, we move to "eigenvalue representation" from the N × N Hermitian matrix integral (3.1) by integrating out the angular part of X. This can be done by using Harish-Chandra-Itzykson-Zuber formula [7,8] where the integral is taken over U(N ) group, and ∆(x) is the Vandermonde determinant.
Thus we have the eigenvalue representation of (3.1) We here apply the formula We recall that j, k = 1, · · · , N and α, β = 1, · · · , M . Since each matrix element can be replaced with any monic polynomials, P k (x) = x k + · · · , this determinant (3.6) is written in a more generic form .

(3.7)
Thus the integral is given by To perform this integral, we then introduce an auxiliary function Using this function, we arrive at the final expression of the partition function (3.1) This expression is manifestly symmetric under the exchange of (a 1 , · · · , a N ) and (λ 1 , · · · , λ M ) with the transformation: P k (λ) ↔ Q k (a). As seen in (3.9), this is nothing but a Fourier string duality [9,10,11]. We also note that this kind of symplectic invariance appears quite generally in the topological expansion of the spectral curve [12,13,14].

Integrability
The formula (3.10) is a quite natural generalization of the well-known formulae for the expectation value of characteristic polynomial product where P k (x) is k-th monic orthogonal polynomial with respect to the weight function w(x) =  It is convenient to apply the simplest choice of the polynomial P k (x) = x k to this formula (3.12). In this case the function Q k (a) is given by with an Airy-like function See [3] and references therein for details It is known that this kind of determinantal formula generically plays a role as the τfunction [6], and satisfies the Toda lattice equation by taking the equal parameter limit [15].
We show that the formula (3.10) indeed satisfies a similar integrable equation in the following.
Let us parametrize the positions of branes by "center of mass" and deviations from it as a j = a + δa j and λ α = λ + δλ α . We rewrite the numerator in terms of the deviations {δa j } and {δλ α } by considering the Taylor expansion around the center of masses and so on. The first determinant in the RHS is almost canceled by the Vandermonde determinants in the denominator of (3.10), since they are invariant under the constant shift as ∆(a) = ∆(δa) = det(δa k ) j−1 and ∆(λ) = ∆(δλ) = det(δλ β ) α−1 , respectively. Therefore the partition function in the equal position limit becomes This expression is seen as a hybridized version of the Wronskian. In the following we apply P k (x) = x k and (3.13) as the case of (3.12) for simplicity.
In order to derive the integrable equation, we now use the Jacobi identity for determinants, which is given by where D is a determinant, and the minor determinant D (3.20) Here we have used a formula (A.1) discussed in Appendix A. This provides the following relation for the equal position partition function This is just the Toda lattice equation along the a-direction, but with a trivial factor which can be removed by rescaling the function.
We can assign another relation to the partition function by the identity  We then obtain the two-dimensional Toda lattice equation [16] with an extra factor. In order to remove this irrelevant factor, we rescale the partition functioñ This means that the brane intersection partition function (3.24) plays a role of the τ -function for the one-dimensional, and also the two-dimensional Toda lattice equation simultaneously.
We note that this is an exact result for finite N (and also M ). If one takes the large N limit, corresponding to the continuum limit for the Toda lattice equations, it reduces to the KdV/KP equations. We also comment that the τ -function of the two-dimensional Toda lattice hierarchy can be realized as the two-matrix model integral [18,19].
Although, in this section we have focused only on the equal position limit of the partition function (3.10), it can be regarded as the τ -function for the corresponding integrable hierarchy. In this case we can introduce two kinds of the Miwa coordinates t n = 1 n Tr A −n ,t n = 1 n tr Λ −n . (3.26) It is shown that all the time variables, t n andt n , are trivially related to each other in the equal parameter limit. After taking the continuum limit, namely the large N limit of the matrix model, it shall behave as the τ -function for the KdV/KP hierarchies.

Gaussian matrix model
We now study a specific example of the matrix model with the harmonic potential W (x) = 1 2 x 2 , namely the Gaussian matrix model. In this case we can check the duality formula more explicitly [20,21,22]: det(a j +iY ) , This expression is essentially the same as (3.13) in the case of the harmonic potential. This is a specific property for the Gaussian model.

Fermionic formula
For the Gaussian matrix model, there is another interesting derivation of the duality (4.1) using fermionic variables, instead of the method used in Sec. 3.1. Following the approach applied in [20,21] basically, we discuss it from the view point of the topological strings. We will actually show that the effective fermionic action for this partition function gives a quite natural perspective on the topological intersecting brane. Using this formula, the LHS of (4.2) becomes Then the effective action yields Since the four-point interaction is also represented in terms of the M × M matrix as Tr ψ αψα 2 = −tr ψ iψi 2 , this term can be removed by inserting an M × M auxiliary This is just the RHS of the duality formula (4.2).
Let us comment on the meaning of this formula in terms of the topological strings. When we apply m distinct values to A as A = diag(a (1) , · · · , a (1) , a (2) , · · · , a (2) , · · · · · · , a (m) , · · · , a (m) 9) stacked N branes are decoupled into N 1 + · · · + N m as shown in Fig. 2. This means that the U(N ) symmetry of the original matrix model is broken into its subsector We find a similar symmetry breaking in the dual representation. In particular, when we put Λ as Λ = diag(λ (1) , · · · , λ (1) , · · · · · · , λ (l) , · · · , λ (l) the U(M ) symmetry is broken as We can also discuss the symmetry breaking of the fermions by seeing the fermionic effective action in (4.7),

Bosonic formula
We can extend the duality formula (4.2) for the inverse characteristic polynomial [22]  (4.14) In this case a bifundamental bosonic field plays a similar role to the fermionic field, which is used to represent the characteristic polynomial in the numerator. Actually we can derive this duality formula in almost the same manner discussed in Sec. 4.1.
The average shown in the LHS of (4.14) is explicitly written as Since the inverse of a determinant is written as a Gaussian integral with a bosonic variable in the bifundamental representations, Then, inserting an auxiliary M × M Hermitian matrix Y in order to eliminate the four-point interaction, it is written as This is the RHS of the duality formula (4.14).
The In this case, the correlation function is written in terms of the Cauchy transform of the corresponding orthogonal polynomial [28].

Discussion
In this article we have investigated the intersection of branes in the topological B-model using its matrix model description. In particular, since two different descriptions of the non-compact brane correspond to the characteristic polynomial and the external source in the matrix model, we have considered the brane partition function given by inserting both of them simultaneously. We have derived the determinantal formula for this partition function, and shown that two descriptions of the branes are dual to each other in the sense of the Fourier transformation. We have also shown that the brane partition function plays a role of the τ -function, and satisfies the Toda lattice equations in one and two dimensions. We have investigated the Gaussian matrix model as an example, and discussed the effective action of the intersecting branes in terms of the bifundamental fermion/boson.
Although we have focused on the U(N ) symmetric matrix model all through this article, we can apply essentially the same argument to O(N ) and Sp(2N ) symmetric matrix models.
In such a case the Hermitian matrix is replaced with real symmetric and self-dual quaternion matrices, respectively. Actually, when the Gaussian potential is assigned, one can obtain a similar duality formula [29,22,30], which claims that and also an M5-brane, appearing in its M-theory lift [31]. In this case, since the geometry of this M5-brane indicates the Seiberg-Witten curve of the corresponding N = 2 theory, positions of branes are directly related to the gauge theory dynamics. Thus it is expected that a nontrivial gauge theory duality is derived from the duality between the two coordinates of branes. Actually a similar duality is discussed along this direction [32].
From the matrix model perspective, it is interesting to consider the ratio of the characteristic polynomials in the presence of the external source [28], and its interpretation in terms of topological strings. The characteristic polynomial in the numerator and the denominator plays a role of the creation operator for the brane and anti-brane. Thus the ratio should describe the pair creation and annihilation of branes. In particular it is expected that the scaling limit of the ratio extracts some interesting features of the tachyon condensation in topological strings. This kind of problem is also interesting in the context of the matrix model itself, because one can often find an universal property of the matrix model in such a scaling limit.

A A formula for determinants
In order to obtain (3.20) and (3.22), it is convenient to use the following relation where we have introduced two M × M matrices and x N (0) · · · x N +M −2 (0) x N +M (0) x N (1) · · · x N +M −2 (1) x N +M (1) . . . Here we denote (x j ) (l) = (d/dx) l x j and so on. Using the Jacobi identity (3.19) for det A M with i = k = M and j = l = M − 1, we have It is convenient to rewrite this relation as This is interpreted as a remnant of the Toda lattice equation [6].
What we have to do next is to evaluate det A M . To obtain this, we consider a ratio of determinants, and then take the equal parameter limit, x α → x for all α = 1, · · · , M ,