To see Symmetry in a Forest of Trees

The exact symmetry identities among four-point tree-level amplitudes of bosonic open string theory as derived by G. W. Moore are re-examined. The main focuses of this work are: (1) Explicit construction of kinematic configurations and a new polarization basis for the scattering processes. These setups simplify greatly the functional forms of the exact symmetry identities, and help us to extract easily high-energy limits of stringy amplitudes appearing in the exact identities. (2) Connection and comparison between D. J. Gross's high-energy stringy symmetry and the exact symmetry identities as derived by G. W. Moore. (3) Observation of symmetry patterns of stringy amplitudes with respect to the order of energy dependence in scattering amplitudes.Comment: 56 pages; v2. Typos corrected. Minor changes; v3. Reorganized the structure and eliminate verbose expressions. References added. Added words of introduction to each section; v4. Reorganized and streamlined significantly. Version to appear in Nucl.Phys.

Stokes Phenomena and Non-perturbative Completion in the Multi-cut Two-matrix Models

The Stokes multipliers in the matrix models are invariants in the string-theory moduli space and related to the D-instanton chemical potentials. They not only represent non-perturbative information but also play an important role in connecting various perturbative string theories in the moduli space. They are a key concept to the non-perturbative completion of string theory and also expected to imply some remnant of strong coupling dynamics in M theory. In this paper, we investigate the non-perturbative completion problem consisting of two constraints on the Stokes multipliers. As the first constraint, Stokes phenomena which realize the multi-cut geometry are studied in the Z_k symmetric critical points of the multi-cut two-matrix models. Sequence of solutions to the constraints are obtained in general k-cut critical points. A discrete set of solutions and a continuum set of solutions are explicitly shown, and they can be classified by several constrained configurations of the Young diagram. As the second constraint, we discuss non-perturbative stability of backgrounds in terms of the Riemann-Hilbert problem. In particular, our procedure in the 2-cut (1,2) case (pure-supergravity case) completely fixes the D-instanton chemical potentials and results in the Hastings-McLeod solution to the Painlev\'e II equation. It is also stressed that the Riemann-Hilbert approach realizes an off-shell background independent formulation of non-critical string theory.Comment: 71 pages, v3: organization of Sec. 3, Sec. 4, App. C and App. D improved, final version to be published in Nucl. Phys.

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