# $c=1$ string There is a triality among three theories: the $c=1$ string, the double-scaled $(0+0)$-dimensional matrix integral, and the $c=1$ matrix quantum mechanics. ## Worldsheet description The worldsheet CFT consists of a timelike free boson (which has $c=1$) coupled to a $c=25$ [[0562 Liouville theory|Liouville CFT]], with the usual $bc$ ghosts. ## Target space picture The target space is ($1+1$)-dimensional. There is an S-matrix because the string is weakly coupled in the region far away from the Liouville wall, making the asymptotic states well-defined. ## MQM dual The $c=1$ string has long been conjectured to be dual to a single $N\times N$ Hermitian matrix with an inverted harmonic oscillator potential. ## Operator weights Vertex operators have worldsheet dimension $(1,1)$. In this case,$h_{\text{Liouville}}=1-p^2,\quad h_{\text{boson}}=p^2.$Tachyon vertex operators:$\mathcal{T}_{p_j=\frac{i}{2} \omega_j} \simeq \mathrm{e}^{i \omega_j X^0} V_{p_j=\frac{i}{2} \omega_j}.$Notice that these are not symmetric under $p\to-p$, unlike in [[0650 Complex Liouville string|CLS]]. ## Refs - holographic dual as a matrix QM - [[1991#Moore, Plesser, Ramgoolam]] - triality - [[2026#Collier, Eberhardt, Rodriguez]]