Multi-instantons in 2d string theory

TL;DR

This work resolves a normalization ambiguity for D-instanton effects in 2d string theory by deriving the non-perturbative multipliers $\zeta_n$ from the matrix quantum mechanics dual via an integral representation of the scattering phase and applying median resummation from resurgence theory. It identifies $\zeta_n=\frac{(2n)!}{2^{2n}(n!)^2}$ and shows that the Lorentzian contour prescription is the correct choice for multi-instanton amplitudes, with explicit agreement between compactified 2d string theory and MQM predictions. The non-perturbative free energy, expressed through these amplitudes, matches the D-instanton induced string field theory action, supporting the conjecture that worldsheet topologies with negative Euler number do not contribute in the linear dilaton background. In the decompactification limit, the non-perturbative structure reduces to a Li_2 expression, providing a coherent link between string theory and its matrix-model description and highlighting the central role of resurgence and contour choices in non-perturbative dynamics.

Abstract

Instanton contributions in 2d string theory are known to include subtle numerical factors $ζ_n$ closely related to a contour prescription in multi-instanton string amplitudes. Both ingredients appear to be ambiguous due to a degeneracy between $(1,n)$-ZZ instantons and $n$ (1,1)-ZZ instantons in the linear dilaton background. We resolve this ambiguity using insights from the dual matrix quantum mechanics where the multipliers $ζ_n$ can be derived from an integral representation of the scattering phase and follow from the median resummation prescribed by resurgence theory. We evaluate multi-instanton string amplitudes in the theory compactified on a circle of finite radius for arbitrary number of instantons and show that they reproduce the matrix model predictions provided the Lorentzian contour prescription is used for their evaluation. We also show that the non-perturbative free energy matches the structure of the D-instanton induced string field theory effective action, which suggests the vanishing of contributions from worldsheet topologies of negative Euler number.

Figures (1)

• Figure 1: The diagrams that constitute the $N=3$ contribution to the partition function and are the one-to-one correspondence with the contributions to the representation \ref{['cIfromMQM-set']} of $\mathcal{I}^{(3)}_{{\boldsymbol n}}$. The connected graphs in the first line correspond to contributions with $m=1$ and give the $N=3$ term to the free energy \ref{['Fnp-int']}. The next three terms correspond to contributions with $m=2$ and the last term provides the $m=3$ contribution.