Duality Constraints on String Theory: Instantons and spectral networks

We study an implication of $p-q$ duality (spectral duality or T-duality) on non-perturbative completion of $(p,q)$ minimal string theory. According to the Eynard-Orantin topological recursion, spectral $p-q$ 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 $p-q$ 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

Exploring Subexponential Parameterized Complexity of Completion Problems

Let ${\cal F}$ be a family of graphs. In the ${\cal F}$-Completion problem, we are given a graph $G$ and an integer $k$ as input, and asked whether at most $k$ edges can be added to $G$ so that the resulting graph does not contain a graph from ${\cal F}$ as an induced subgraph. It appeared recently that special cases of ${\cal F}$-Completion, the problem of completing into a chordal graph known as Minimum Fill-in, corresponding to the case of ${\cal F}=\{C_4,C_5,C_6,\ldots\}$, and the problem of completing into a split graph, i.e., the case of ${\cal F}=\{C_4, 2K_2, C_5\}$, are solvable in parameterized subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$. The exploration of this phenomenon is the main motivation for our research on ${\cal F}$-Completion. In this paper we prove that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms by showing that: - The problem Trivially Perfect Completion is solvable in parameterized subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$, that is ${\cal F}$-Completion for ${\cal F} =\{C_4, P_4\}$, a cycle and a path on four vertices. - The problems known in the literature as Pseudosplit Completion, the case where ${\cal F} = \{2K_2, C_4\}$, and Threshold Completion, where ${\cal F} = \{2K_2, P_4, C_4\}$, are also solvable in time $2^{O(\sqrt{k}\log{k})} n^{O(1)}$. We complement our algorithms for ${\cal F}$-Completion with the following lower bounds: - For ${\cal F} = \{2K_2\}$, ${\cal F} = \{C_4\}$, ${\cal F} = \{P_4\}$, and ${\cal F} = \{2K_2, P_4\}$, ${\cal F}$-Completion cannot be solved in time $2^{o(k)} n^{O(1)}$ unless the Exponential Time Hypothesis (ETH) fails. Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of ${\cal F}$-Completion problems for ${\cal F}\subseteq\{2K_2, C_4, P_4\}$.Comment: 32 pages, 16 figures, A preliminary version of this paper appeared in the proceedings of STACS'1