[Paper Review] Strings from Tachyons

John McGreevy, Herman Verlinde|CERN Bulletin|Apr 28, 2003

Black Holes and Theoretical Physics45 references131 citations

TL;DR

This paper proposes that the c=1 matrix model describes the worldline theory of N unstable D-particles, with the hermitian matrix representing the non-abelian open string tachyon. It establishes a quantitative match between closed string emission from a rolling tachyon and that from a rolling eigenvalue in the matrix model, identifying the double-scaling limit as a decoupling limit and suggesting 1+1-d string theory arises as the near-horizon limit of a dense gas of IIB D-particles.

ABSTRACT

We propose a new interpretation of the c=1 matrix model as the world-line theory of N unstable D-particles, in which the hermitian matrix is provided by the non- abelian open string tachyon. For D-particles in 1+1-d string theory, we find a direct quantitative match between the closed string emission due to a rolling tachyon and that due to a rolling eigenvalue in the matrix model. We explain the origin of the double-scaling limit, and interpret it as an extreme representative of a large equivalence class of dual theories. Finally, we define a concrete decoupling limit of unstable D-particles in IIB string theory that reduces to the c=1 matrix model, suggesting that 1+1-d string theory represents the near-horizon limit of an ultra-dense gas of IIB D-particles.

Motivation & Objective

• To provide a physical interpretation of the c=1 matrix model as the worldline theory of N unstable D-particles.
• To establish a quantitative match between closed string emission from a rolling tachyon and from a rolling eigenvalue in the matrix model.
• To clarify the physical origin of the double-scaling limit in the matrix model as a decoupling limit.
• To propose a decoupling limit of IIB D-particles that reduces to the c=1 matrix model, embedding 1+1-d string theory in a consistent framework.

Proposed method

• Models the dynamics of N unstable D-particles in 1+1-d string theory, focusing on non-abelian open string tachyon degrees of freedom.
• Uses the rolling tachyon solution $ T_{\rm roll}(X^0) = \lambda \exp X^0 $ as an exact boundary CFT solution to describe tachyon condensation.
• Computes the closed string emission amplitude via the annulus amplitude in the Liouville CFT, using Ishibashi states and modular invariance.
• Performs a Fourier transform in the modular parameter $ \tilde{q} $ to extract the on-shell closed string response, identifying the physical tachyon background shift.
• Relies on the contour integral $ \mathcal{C} = -iQ/2 + \mathbb{R} $ to compute the Liouville path integral and extract the physical amplitude.
• Matches the resulting tachyon background shift $ \delta \mathcal{T}(\varphi) \propto e^{2\varphi} $ to the matrix model's eigenvalue dynamics, confirming quantitative equivalence.

Experimental results

Research questions

• RQ1 Can the c=1 matrix model be interpreted as the worldline theory of unstable D-particles?
• RQ2 Is there a quantitative match between closed string emission from a rolling tachyon and from a rolling eigenvalue in the matrix model?
• RQ3 What is the physical origin of the double-scaling limit in the c=1 matrix model?
• RQ4 Can 1+1-d string theory emerge as a decoupling limit of a consistent supersymmetric string theory?
• RQ5 How does the non-perturbative instability of the matrix model relate to tachyon condensation?

Key findings

• A direct quantitative match is found between the closed string emission from a rolling tachyon and that from a rolling eigenvalue in the c=1 matrix model.
• The double-scaling limit is interpreted as a decoupling limit, selecting an extreme representative of a large equivalence class of dual theories with D-brane back-reaction.
• The matrix model's non-perturbative instability against eigenvalue tunneling corresponds to tachyon condensation toward a potential unbounded from below.
• The static shift in the tachyon background is $ \delta \mathcal{T}(\varphi) \propto e^{2\varphi} $, matching the asymptotic behavior of the matrix model's eigenvalue potential.
• The amplitude for closed string emission reduces to a massless propagator $ \int_0^1 \frac{d\tilde{q}}{\tilde{q}} \tilde{q}^{P^2 - \omega^2} = \frac{1}{P^2 - \omega^2} $, with all $ \eta(\tilde{q}) $ factors canceling.
• The shift in the tachyon background at the dilaton wall ($ \varphi = 0 $) is of order one, implying a $ \delta\mu \propto (\log \mu)^{-1} $ shift in the effective parameter $ \mu $, consistent with the matrix model's level density $ \rho(\mu) \simeq -\frac{2}{\pi} \log \mu $.