Abstract
We study the scattering of long strings in c = 1 string theory, both in the worldsheet description and in the non-singlet sector of the dual matrix quantum mechanics. From the worldsheet perspective, the scattering amplitudes of long strings are obtained from a decoupling limit of open strings amplitudes on FZZT branes, which we compute by integrating Virasoro conformal blocks along with structure constants of boundary Liouville theory. In particular, we study the tree level amplitudes of (1) a long string decaying by emitting a closed string, and (2) the scattering of a pair of long strings. We show that they are indeed well defined as limits of open string amplitudes, and that our results are in striking numerical agreement with computations in the adjoint and bi-adjoint sectors of the dual matrix model (based on proposals of Maldacena and solutions due to Fidkowski), thereby providing strong evidence of the duality.
Highlights
Scaling limit of a U(N ) gauged Hermitian matrix model, which we refer to as the c = 1 matrix model
We study the scattering of long strings in c = 1 string theory, both in the worldsheet description and in the non-singlet sector of the dual matrix quantum mechanics
The scattering amplitudes of long strings are obtained from a decoupling limit of open strings amplitudes on FZZT branes, which we compute by integrating Virasoro conformal blocks along with structure constants of boundary Liouville theory
Summary
Despite that the long string has infinite energy, most of it lies in the region of space where the string coupling is exponentially suppressed, and one may anticipate well defined scattering amplitudes of the long strings with one another and with closed strings. · · · Liouville stands for the disc correlator in the c = 25 Liouville theory subject to FZZT boundary condition It can be computed as an integral of the boundary Liouville structure constants multiplied by the relevant Virasoro conformal block, ψωs,1s(0)ψωs,2s(x)Vω3/2(i/2) Liouville = 22h1 |x − i/2|−2h2 x − i/2 −h1 x + i/2. Such divergences in the moduli integration are familiar in string perturbation theory, and is usually regularized by suitable analytic continuation in the external momenta In our case, such an analytic continuation is inaccessible as we would like to evaluate the amplitude (3.2) numerically at physical energies and compare directly with results in the dual matrix model.
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