A Matrix Model Solution of the Hirota Equation

Казаков, Владимир

doi:10.1007/0-306-47075-6_7

Cited by 4 publications

(6 citation statements)

References 45 publications

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“…It is well known that the partition function of the one (multi) matrix model with the Penner type potential satisfies the discrete time Toda equation. [10][15] [16] We select the flows l andt 3 where l is the multiple of 3 of discrete time lattice which belongs to the 2d discrete time Toda equation. If we select the multiple of k of discrete time lattice andt k , we can obtain the discrete time k-hungry Volterra equation.…”

Section: Discussionmentioning

confidence: 99%

Hungry Volterra Equation, Multi Boson KP Hierarchy and Two Matrix Models

Hisakado

1999

J. Phys. Soc. Jpn.

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We consider the hungry Volterra hierarchy from the view point of the multi boson KP hierarchy. We construct the hungry Volterra equation as the Bäcklund transformations (BT) which are not the ordinary ones. We call them "fractional " BT. We also study the relations between the (discrete time) hungry Volterra equation and two matrix models. From this point of view we study the reduction from (discrete time ) 2d Toda lattice to the (discrete time ) hungry Volterra equation.

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Section: Discussionmentioning

confidence: 99%

Hungry Volterra Equation, Multi Boson KP Hierarchy and Two Matrix Models

Hisakado

1999

J. Phys. Soc. Jpn.

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“…In applications the implementation of boundary conditions and Green's function varies depending on the situation. For example, in investigations of the continuous Hirota equation [6] one is interested in the case of time-dependent potentials V(x, t) leading to non-trivial Green's functions. Here the initial and final velocities are specified and the initial and final eigenvalue distributions are identified (ρ 0 = ρ T ) so that (2.13) takes on the form of a non-linear integral equation for ρ 0 .…”

Section: Collective Field Theory To Quantum Mechanicsmentioning

confidence: 99%

“…Examples include two-dimensional Yang-Mills theory on the cylinder and sphere [4,5], matrix model approaches to the continuous Hirota equation [6] and lattice QCD [7] and, the case we will focus on here, linear chains of interacting Hermitean matrices.…”

Section: Introductionmentioning

confidence: 99%

“…There arise circumstances though where the time evolution of a system governed by an action over matrix variables is relevant, especially if there are definite initial and final conditions on the state of the system. Examples include two-dimensional Yang-Mills theory on the cylinder and sphere [4,5], matrix model approaches to the continuous Hirota equation [6] and lattice QCD [7] and, the case we will focus on here, linear chains of interacting Hermitean matrices.…”

Section: Introductionmentioning

confidence: 99%

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Eigenvalue dynamics and the matrix chain

Paniak

1999

Nuclear Physics B

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We introduce a general method for transforming the equations of motion following from a Das-Jevicki-Sakita Hamiltonian, with boundary conditions, into a boundary value problem in one-dimensional quantum mechanics. For the particular case of a onedimensional chain of interacting N × N Hermitean matrices, the corresponding large N boundary value problem is mapped into a linear Fredholm equation with HilbertSchmidt type kernel. The equivalence of this kernel, in special cases, to a second order differential operator allows us recover all previously known explicit solutions for the matrix eigenvalues. In the general case, the distribution of eigenvalues is formally derived through a series of saddle-point approximations. The critical behaviour of the system, including a previously observed Kosterlitz-Thouless transition, is interpreted in terms of the stationary points. In particular we show that a previously conjectured infinite series of sub-leading critical points are due to expansion about unstable stationary points and consequently not realized.

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“…Before starting with the discussion in detail let us briefly explain what we mean by the correspondence between the D-brane dynamics and integrable systems. There have been some papers [12], which studied the Matrix model in connection with equation (4). Our approach, however, will be different from them.…”

Section: Introductionmentioning

confidence: 99%

D-brane correlators as solutions of Hirota-Miwa equation

Sato

Saito

2004

J. High Energy Phys.

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We present a tachyon field, which simply connects to the calculus of the tachyon condensation. The tachyon field acts on any bare boundary state: the Neumann or the Dirichlet state, and generates in a special case the boundary state suggested by S.P. de Alwis, which leads to the correct ratio between D-brane tensions. On the other hand we generalize the Hirota-Miwa equation to the case where there are two boundaries. We show that correlation functions made from only the integrand of the tachyon field satisfy the generalized Hirota-Miwa equation. Using the formulation based on this evidence, we suggest that the coordinates in which there exist tachyons on an unstable D-brane be identified with the soliton coordinates in integrable systems. We also evaluate these correlation functions, which have not yet been integrated, to obtain the local information about the tachyons on unstable D-branes. We further see how these amplitudes are affected through the tachyon condensation by integrating the correlators over the tachyon momenta and taking the on-shell limit.

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A Matrix Model Solution of the Hirota Equation

Cited by 4 publications

References 45 publications

Hungry Volterra Equation, Multi Boson KP Hierarchy and Two Matrix Models

Hungry Volterra Equation, Multi Boson KP Hierarchy and Two Matrix Models

Eigenvalue dynamics and the matrix chain

D-brane correlators as solutions of Hirota-Miwa equation

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