Integrability of type 0A matrix model in the presence of D-brane

Abstract

We consider type 0A matrix model in the presence of spacelike D-brane which is localized in matter direction at any arbitrary point. In string theory, the boundary state, which in matrix model corresponds to the Laplace transform of the macroscopic loop operator, is known to obey the operator constraints corresponding to open string boundary condition. When we analyze MQM as well as the respective collective field theory and compare it with dual string theory, it appears that consistency of the theory requires a condition equivalent to a constraint on the matter part that needed to be imposed in the matrix model. We identified this condition and observed that this results in constraining the macroscopic loop operator so that it projects the Hilbert space generated by the operator to its physical sector at the point of insertion while keeping the bulk matrix model unaffected, thereby describing a situation parallel to string theory. We analyzed the theory with uncompactified time and have shown explicitly that the matrix model predictions are in good agreement with the relevant string theory. Next, we considered the theory with compactified time, analyzed MQM on a circle in the presence of D-brane. We evaluated the partition function along with the constrained macroscopic loop operator in the grand canonical ensemble and showed the free energy corresponds to that of a deformed Fermi surface. We have also shown that the path integral in the presence of D-brane can be expressed as the Fredholm determinant. We have studied the fermionic scattering in a semiclassical regime. Finally, we considered the compactified theory in the presence of the D-brane with tachyonic background. We evaluated the free energy in the grand canonical ensemble. We have shown the integrable structure of the respective partition function and it corresponds to the tau function of Toda hierarchy. We have also analyzed the dispersionless limit.