Multi-Instanton Calculus in c = 1 String Theory
We formulate a strategy for computing the complete set of non-perturbative corrections to closed string scattering in $c=1$ string theory from the worldsheet perspective. This requires taking into account the effect of multiple ZZ-instantons, including higher instantons constructed from ZZ boundary conditions of type $(m,1)$, with a careful treatment of the measure and contour in the integration over the instanton moduli space. The only a priori ambiguity in our prescription is a normalization constant ${\cal N}_{m}$ that appears in the integration measure for the $(m,1)$-type ZZ instanton, at each positive integer $m$. We investigate leading corrections to the closed string reflection amplitude at the $n$-instanton level, i.e. of order $e^{-n/g_s}$, and find striking agreement with our recent proposal on the non-perturbative completion of the dual matrix quantum mechanics, which in turn fixes ${\cal N}_{m}$ for all $m$.