Geometrical interpretation of the topological recursion, and integrable string theories

Abstract

Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative geometry like maps, partitions, Hurwitz numbers, intersection numbers, Gromov-Witten invariants... The problem is thus to understand what they count, or in other words, given a spectral curve, construct an enumerative geometry problem. This is what we do in a semi-heuristic approach in this article. Starting from a spectral curve, i.e. an integrable system, we use its flat connection and flat coordinates, to define a family of worldsheets, whose enumeration is indeed solved by the topological recursion and symplectic invariants. In other words, for any spectral curve, we construct a corresponding string theory, whose target space is a submanifold of the Jacobian.

57 pages, many figures

References in corpus (13)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• A matrix model for simple Hurwitz numbers, and topological recursion
• Holomorphic anomaly and matrix models
• Symplectic invariants, Virasoro constraints and Givental decomposition
• Local Mirror Symmetry for One-Legged Topological Vertex
• Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity
• Hurwitz numbers, matrix models and enumerative geometry
• Invariants of algebraic curves and topological expansion
• Topological expansion of the Bethe ansatz, and quantum algebraic geometry
• Bouchard-Klemm-Marino-Pasquetti Conjecture for ${\mathbb{C}}^3$
• Local Mirror Symmetry for the Topological Vertex
• The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers
• Topological Strings and (Almost) Modular Forms

Cited by in corpus (9)

• What is the Simplest Gauge-String Duality?
• Stokes Phenomena and Non-perturbative Completion in the Multi-cut Two-matrix Models
• Stokes Phenomena and Quantum Integrability in Non-critical String/M Theory
• Open String Invariants and Mirror Curve of the Resolved Conifold
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Algebro-Geometric Solutions of the Generalized Virasoro Constraints
• Fractional-Superstring Amplitudes, Multi-Cut Matrix Models and Non-Critical M Theory
• Duality Constraints on String Theory: Instantons and spectral networks
• Wronskians, dualities and FZZT-Cardy branes