We study an implication of pβq duality (spectral duality or T-duality) on
non-perturbative completion of (p,q) minimal string theory. According to the
Eynard-Orantin topological recursion, spectral pβq duality was already
checked for all-order perturbative analysis including instanton/soliton
amplitudes. Non-perturbative realization of this duality, on the other hand,
causes a new fundamental issue. In fact, we find that not all the
non-perturbative completions are consistent with non-perturbative pβq
duality; Non-perturbative duality rather provides a constraint on
non-perturbative contour ambiguity (equivalently, of D-instanton fugacity) in
matrix models. In particular, it prohibits some of meta-stability caused by
ghost D-instantons, since there is no non-perturbative realization on the dual
side in the matrix-model description. Our result is the first quantitative
observation that a missing piece of our understanding in non-perturbative
string theory is provided by the principle of non-perturbative string duality.
To this end, we study Stokes phenomena of (p,q) minimal strings with spectral
networks and improve the Deift-Zhou's method to describe meta-stable vacua. By
analyzing the instanton profile on spectral networks, we argue the duality
constraints on string theory.Comment: v1: 84 pages, 43 figures; v2: 86 pages, 43 figures, presentations are
improved, references added; v3: 126 pages, 69 figures: a solution of local
RHP, physics of resolvents, commutativity of integrals are newly added;
organization is changed and explanations are expanded to improve
representation with addition of review, proofs and calculations; some
definitions are changed; references adde