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This study reconstructs topological string partition functions for Calabi-Yau manifolds using quantum curves. These functions are derived from isomonodromic tau-functions, offering insights into string theory and geometry.

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Area of Science:

  • String Theory
  • Mathematical Physics
  • Algebraic Geometry

Background:

  • Topological string theory provides a powerful framework for studying quantum field theories and their dual gravitational descriptions.
  • Calabi-Yau manifolds are central to string theory compactifications, and understanding their properties is crucial.
  • Quantum curves offer a novel approach to quantizing geometric structures.

Purpose of the Study:

  • To reconstruct the topological string partition function for specific local Calabi-Yau manifolds.
  • To establish a connection between quantum curves and topological string theory.
  • To explore the role of isomonodromic tau-functions in this reconstruction.

Main Methods:

  • Quantizing the defining equations of local Calabi-Yau manifolds to obtain quantum curves.
  • Characterizing quantum curves as solutions to Riemann-Hilbert problems.
  • Utilizing isomonodromic tau-functions and their series expansions (generalized theta series) for reconstruction.

Main Results:

  • Successful reconstruction of topological string partition functions for certain local Calabi-Yau manifolds.
  • Demonstration that isomonodromic tau-functions associated with Riemann-Hilbert problems yield these partition functions.
  • Identification of natural normalizations for tau-functions linked to chambers in the Kähler moduli space.

Conclusions:

  • Quantum curves provide a viable method for deriving topological string partition functions.
  • Isomonodromic tau-functions are key ingredients in understanding the string theory landscape on Calabi-Yau manifolds.
  • This work bridges concepts from quantum mechanics, differential equations, and string theory.