Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Note on intersecting branes in topological strings
T. Kimura∗1
The intersection of branes is an important object
in string theory, in order to study non-perturbative
aspects of branes, and also its applicatioins to quantum
field theory. In this report we investigate some aspectsof the intersecting branes in topological string theory,
especially through its matrix model description.
We consider the topological B-model on the CalabiYau threefold uv − H(p, x) = 0 with H(p, x) = p2 −W  (x)2 − f (x). This geometry realizes at the large N
limit of the matrix model with the potential function
W (x). There are seemingly two kinds of non-compact
branes in the topological B-model, which correspond tothe characteristic polynomial and the external sourcein the matrix model.1) They play a role of the creation operator of branes for x and p coordinates, respectively. By considering both kinds of the branes
simultaneously, we can discuss intersection of branes
in the B-model. The corresponding matrix model partition function ΨN, M ({aj }; {λα }) is given by
1dX e− gs TrW (X)+TrAX
N ×NM


α=1det(λα − X) .

(1)

This is the M -point function of characteristic polynomials in N × N Hermitian matrix model with external source A. In order to evaluate the partition function (1), we first rewrite it only in terms of eigenvalues
by integrating out the angular part of the matrix X.
Then, after some calculations, we obtain the determinantal expression of the partition function1
Qj−1 (ak ) QN +α−1 (ak )det, (2)
Pj−1 (λβ ) PN +α−1 (λβ )∆(a)∆(λ)
where ∆(x) = i<j (xi − xj ) is the Vandermonde determinant, and Pk (x) = xk + · · · is arbitrary k-thmonic polynomial. The function Qk (a) is the Fourier1(Laplace) transform of Pk (x) e− gs W (x) . Therefore,
from the expression (2), we can see an explicit duality
between a and λ through the Fourier transformation.
In terms of the topological strings, this duality reflects
the symplectic invariance of the canonical pair (p, x)
in the B-model, which is also seen as the open/closed
string duality. We also note that this kind of symplectic invariance appears quite generally in the topologicalexpansion of the spectral curve.
If we apply the Gaussian potential, two functions
Pk (x) and Qk (x) are essentially equivalent, since it isself-dual against the Fourier transformation. In thiscase we can rewrite the partition function in terms ofU(N )×U(M ) bifundamental chiral fermions, which areseen as effective degrees of freedom on the intersecting∗1

RIKEN Nishina Centerbranes. The corresponding effective action is given bygsSeff = ψiα ψ̄jα ψjβ ψ̄iβ + TrAψ α ψ̄ α − TrΛψj ψ̄j , (3)2In this expression the duality between A and Λ is manifest. In this action the full symmetry of U(N )×U(M )is partially broken due to the source terms.
Let us then comment on the integrability of thebrane intersection partition function (2). This kind ofdeterminantal formula generically plays a role as theτ -function,2) and satisfies the Toda lattice equation bytaking the equal parameter limit. To show that, wenow consider the equal position limit of (2) as aj → aand λα → λ. Then we have
M −1
N−1(k−1)(k−1)
1  1
Qj−1 (a) QN +α−1 (a)det.(4)(β−1)(β−1)
j! α=0 α!
Pj−1 (λ) PN +α−1 (λ)
j=0This is just a hybritized version of Wronskian. If weapply the simplest choice Pk (x) = xk , Qk (a) is justgiven as k-th derivative of the generalized Airy function
 k 1dQk (a) =dx e− gs W (x)+ax .
(5)
daApplying the Jacobi identity for determinants to theexpression (4), we obtain the following 3-term reccurence relations
Ψ̃N +1, M · Ψ̃N −1, MN ∂2=
log Ψ̃N, M ,
2M ∂a2Ψ̃N, M∂2
Ψ̃N, M +1 · Ψ̃N, M −1
log Ψ̃N, M .
=

2
∂a∂λ
Ψ̃N, M(6)(7)

where we have rescaled the partition function asΨ̃N, M (a, λ) = e−λ ΨN, M (a, λ). The equations (6) and(7) are the Toda lattice equations in one and two dimensions. This means that the brane intersection partition function is the τ -function for the Toda latticehierarchy in both senses. We can also introduce infinitely many “time” variables for this τ -function inthe Miwa coordinate
11tn = Tr A−n , t̃n = Tr Λ−n .
(8)nn
If we take the large N limit of the matrix model, whichcorresponds to the continuum limit of the Toda lattice
equation, we obtain the KdV/KP integrable equations.
References
1) M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño,C. Vafa, Commun. Math. Phys. 261 (2006) 451–516.
2) A. Morozov, Phys. Usp. 37 (1994) 1–55.

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