Rolling tachyon solution of two-dimensional string theory

TL;DR

This work reframes the c=1 matrix model as a noncommutative field theory with noncommutativity set by the string coupling $g_s$ and constructs a concrete classical solution that tracks the rolling tachyon on an unstable D0 brane. The solution decomposes the phase-space density into a depleted Fermi-sea background $u_0$ and a localized wave-packet $u_1$, whose evolution yields a finite-energy decay into closed-string tachyons with total energy $E_{tot}=1/g_s$. At early times the configuration represents a D0 brane atop the Fermi sea; at late times $u_1$ becomes a near-Fermi-surface fluctuation that maps to tachyon modes, avoiding BCFT divergences thanks to noncommutativity. The construction generalizes to multiple D0 branes and highlights a coherent open/closed string duality in this two-dimensional setting.

Abstract

We consider a classical (string) field theory of $c=1$ matrix model which was developed earlier in hep-th/9207011 and subsequent papers. This is a noncommutative field theory where the noncommutativity parameter is the string coupling $g_s$. We construct a classical solution of this field theory and show that it describes the complete time history of the recently found rolling tachyon on an unstable D0 brane.

Figures (2)

• Figure 1: The classical solution $u_1$ represents a localized Gaussian solution in phase space far from the Fermi level. At late times, it can be identified with ripples "near" the Fermi surface; the process can be interpreted as conversion into closed string modes.The support of the Gaussian wave packet is non-zero because of uncertainty principle, with $\hbar \sim g_s$, giving a finite decay amplitude.
• Figure 2: Same phenomenon in coordinate space. The finite localization in space implies finite width in energy.