[Paper Review] From quantum curves to topological string partition functions II

Ioana Coman, Pietro Longhi+1 more|arXiv (Cornell University)|Jan 1, 2020

Black Holes and Theoretical Physics111 references27 citations

TL;DR

This paper proposes a geometric characterization of topological string partition functions on local Calabi-Yau manifolds via quantum curves and isomonodromic tau-functions. By quantizing the classical geometry of the Calabi-Yau, the authors derive partition functions as expansions of generalized theta series associated with Fock-Goncharov or Fenchel-Nielsen-type coordinates on the moduli space of quantum curves. The key result is a canonical holomorphic line bundle over the moduli space, whose sections encode the partition functions, with normalization jumps across Stokes lines defined by exact WKB methods.

ABSTRACT

We propose a geometric characterisation of the topological string partition functions associated to the local Calabi-Yau (CY) manifolds used in the geometric engineering of $d=4$, $\mathcal{N}=2$ supersymmetric field theories of class $\mathcal{S}$. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated to the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalisation of the tau-functions associated to these jumps define a natural line bundle playing a key role in the geometric characterisation of the topological string partition functions proposed here.

Motivation & Objective

• To provide a non-perturbative, geometric definition of topological string partition functions for local Calabi-Yau manifolds used in class S theory.
• To establish a correspondence between topological string partition functions and coordinates on the moduli space of quantum curves.
• To characterize the partition functions via isomonodromic tau-functions and generalized theta series expansions.
• To identify a natural holomorphic line bundle over the moduli space of quantum curves that governs the normalization and transformation properties of the partition functions.

Proposed method

• Quantize the classical algebraic curve defined by the local Calabi-Yau geometry to obtain a quantum curve as a differential operator.
• Represent the quantum curve via holomorphic connections and the Riemann-Hilbert problem for monodromy data.
• Construct free fermion partition functions as tau-functions of the isomonodromic deformation problem.
• Expand these tau-functions as generalized theta series in Fock-Goncharov or Fenchel-Nielsen-type coordinates on the moduli space.
• Use the exact WKB method to define preferred coordinates on the moduli space, which exhibit Stokes phenomena and normalization jumps.
• Derive the transformation laws of the tau-functions under changes of pants decomposition or coordinate systems, identifying a holomorphic line bundle encoding the normalization shifts.

Experimental results

Research questions

• RQ1 How can topological string partition functions be geometrically characterized on local Calabi-Yau manifolds via quantum curves?
• RQ2 What is the role of isomonodromic tau-functions in encoding the partition functions, and how do they relate to theta series expansions?
• RQ3 How do Fock-Goncharov and Fenchel-Nielsen-type coordinates on the moduli space of quantum curves relate to the partition functions?
• RQ4 What is the geometric meaning of the normalization jumps in the tau-functions across Stokes lines in the moduli space?
• RQ5 How does the exact WKB method provide a canonical coordinate system that organizes the partition function expansions?

Key findings

• The topological string partition functions are shown to be in one-to-one correspondence with preferred coordinates on the moduli space of quantum curves, defined via the exact WKB method.
• The partition functions arise as coefficients in generalized theta series expansions of isomonodromic tau-functions, with the coordinates being Fock-Goncharov or Fenchel-Nielsen-type variables.
• Normalization jumps of the tau-functions across Stokes lines in the moduli space are governed by Voros symbols and define a holomorphic line bundle over the moduli space.
• The transformation laws between different coordinate systems (e.g., Fock-Goncharov to Fenchel-Nielsen) are encoded in difference generating functions derived from the Riemann-Hilbert problem.
• The partition functions for specific cases, such as $C_{0,4}$ and $C_{0,2}$, are computed explicitly via strong coupling expansions and theta series, confirming consistency with known results.
• The holomorphic line bundle constructed from normalization shifts is conjectured to be related to the geometry of hypermultiplet moduli spaces and NS5-brane corrections in type II string theory.