Non-perturbative topological strings from resurgence

This paper establishes a non-perturbative definition for topological string partition functions on Calabi-Yau threefolds by expressing them as products of resolved conifold partition functions weighted by sheaf invariants, enabling the computation of Borel sums and Stokes jumps that constitute non-perturbative corrections dependent solely on genus zero Gopakumar-Vafa invariants.

The Gist

Imagine you are trying to describe the shape of a complex, multi-dimensional mountain range (a "Calabi-Yau manifold") using a map. In the world of theoretical physics, specifically topological string theory, scientists have a way to calculate the "volume" or "partition function" of this landscape.

For a long time, they could only draw this map using a rough sketch. This sketch was an "asymptotic series"—a mathematical recipe that works well if you take just a few steps, but if you keep adding more and more steps (calculating higher and higher levels of detail), the numbers start to explode and become meaningless. It's like trying to predict the weather by only looking at the last hour; it works for a bit, but eventually, the prediction breaks down.

This paper, by Murad Alim, claims to have found the complete, non-perturbative map. Instead of a rough sketch that eventually fails, the author provides a formula that works perfectly, even when you look at the deepest, most complex details of the landscape.

Here is how the paper achieves this, broken down with simple analogies:

1. The "Lego" Strategy (Building Blocks)

The author's main discovery is that the complex mountain range of any Calabi-Yau shape can be built out of a single, simple Lego brick.

• The Brick: This brick is a specific, well-understood shape called the resolved conifold. Physicists already knew how to calculate the "perfect map" for this specific shape.
• The Construction: The paper proves that the map for any complex shape is just a giant product (a multiplication) of these resolved conifold bricks.
• The Glue: To stick these bricks together correctly, the author uses special numbers called sheaf invariants. Think of these as the "blueprint numbers" that tell you exactly how many bricks you need and how to shift their positions.

2. The "Resurgence" Magic (Fixing the Broken Sketch)

Why was the old sketch broken? Because it was missing "hidden" information. In mathematics, there is a technique called resurgence (think of it as a magical magnifying glass) that looks at the broken parts of a series and finds the hidden patterns that were causing the explosion.

• The author uses this technique to look at the "broken sketch" of the complex shapes.
• By applying the "perfect map" of the simple Lego brick (the resolved conifold) to the complex shape, they can reconstruct the full, non-perturbative answer.
• The Result: They derive a new formula that doesn't just approximate the answer; it is the answer. It is written as a product of analytic functions (smooth, well-behaved mathematical curves) rather than a series that eventually breaks.

3. The Surprise: Simplicity in the Chaos

One of the most surprising findings in the paper is about what drives these complex corrections.

• The Expectation: You might think that to fix the map of a complex mountain, you need to know every single detail about the mountain's geology (all the high-genus invariants).
• The Reality: The author finds that the "corrections" needed to fix the map depend only on the simplest, most basic features of the mountain (the genus-zero invariants).
• The Analogy: Imagine trying to fix a broken radio. You expect you need to replace the whole circuit board. Instead, you discover that the radio only needs a single, tiny screw tightened. The complex math "cancels out" all the messy details, leaving only the simple, fundamental numbers to do the heavy lifting.

4. The "Deformed Prepotential" (The Master Key)

The paper introduces a new mathematical object called a deformed prepotential.

• Think of the "prepotential" as the original, rough blueprint of the mountain.
• The "deformed" version is that blueprint with a special "twist" or "stretch" applied to it.
• This twisted blueprint acts as a master key. It captures all the non-perturbative corrections (the parts that fix the broken sketch) in a single, elegant function.
• Interestingly, this "twisted blueprint" is mathematically identical to a known concept in a different area of physics (the Nekrasov-Shatashvili limit) when applied to the simple Lego brick. This suggests a deep, hidden connection between different ways of looking at the universe.

5. The "Stokes Jumps" (The Hidden Switches)

In the world of these mathematical maps, there are invisible boundaries called Stokes lines. When you cross these lines, the map suddenly "jumps" to a different value.

• The paper calculates exactly how big these jumps are.
• They find that these jumps are determined entirely by the simple numbers mentioned earlier (the genus-zero invariants).
• This means the "glitches" in the mathematical map are not random; they are perfectly organized and predictable, governed by the same simple rules that govern the basic shape of the mountain.

Summary

In essence, Murad Alim's paper says:

"We used to think the map of these complex quantum shapes was too messy to ever be perfect. We found that if you break the shape down into simple Lego bricks (resolved conifolds) and use a special set of counting numbers (sheaf invariants), you can build a perfect, non-perturbative map. Surprisingly, the complex corrections needed to make this map work depend only on the simplest features of the shape, and they can all be captured by a single, elegant mathematical function."

This work bridges the gap between the rough, approximate calculations physicists have used for decades and a precise, complete mathematical description of these topological strings.

Technical Summary

Technical Summary: Non-perturbative topological strings from resurgence

Problem Statement
Topological string theory on Calabi-Yau (CY) threefolds is traditionally defined perturbatively as an asymptotic series in the topological string coupling $\lambda$. This series encodes higher-genus Gromov-Witten (GW) invariants and, via the Gopakumar-Vafa (GV) resummation, integer GV invariants. While the perturbative structure is well-understood through holomorphic anomaly equations and polynomial structures, the non-perturbative completion of the partition function remains elusive for general CY manifolds. Previous resurgence studies have successfully addressed specific geometries, such as the resolved conifold and matrix models, identifying Borel singularities and Stokes jumps. However, a general non-perturbative expression for the partition function on arbitrary CY threefolds, derived directly from the asymptotic series of enumerative invariants, has not been established.

Methodology
The paper employs the theory of resurgence, developed by Écalle, to analyze the asymptotic series of topological string partition functions. The methodology proceeds through the following steps:

1. Decomposition via Sheaf Invariants: The author utilizes sheaf invariants $\Omega_{\beta,n}$, defined by Maulik and Toda in terms of higher-genus GV invariants, to restructure the GW generating function. This allows the free energy of an arbitrary CY threefold to be expressed as a linear combination of the free energy of the resolved conifold with shifted arguments ($t_\beta + n\check{\lambda}$).
2. Resurgence of the Building Block: The analysis relies heavily on the exact Borel resummation results for the resolved conifold established in prior work [25]. These results provide explicit analytic functions (involving triple sine functions and polylogarithms) that represent the Borel sums of the conifold's asymptotic series, including their Stokes jumps.
3. Construction of the Non-perturbative Partition Function: By substituting the resolved conifold's non-perturbative Borel sums into the sheaf-invariant decomposition, the author constructs a candidate non-perturbative partition function for general CY manifolds.
4. Borel Analysis: The asymptotic series of the general theory is re-expressed in terms of sheaf invariants. The Borel transform of this series is computed and analytically continued to a meromorphic function. The poles of this function in the Borel plane are identified, and the corresponding Stokes jumps (discontinuities across Stokes rays) are calculated.

Key Contributions and Results

• Product Formula for the Partition Function: The paper proves that the perturbative partition function (excluding constant map contributions at genus 0 and 1) can be written as an infinite product of the resolved conifold partition function with shifted arguments, raised to the power of sheaf invariants. This leads to the following non-perturbative expression for the partition function $Z_{\text{top,np}}$:
  Z_{\text{top,np}}(\lambda, t) = Z_{c,\text{np}}(\lambda) \prod_{n \in \mathbb{Z}} \prod_{\beta > 0} \left( Z_{\text{con}}_{\text{np}}(\lambda, t_\beta + n\check{\lambda}) \right)^{\Omega_{\beta,n}}
  where Z_{\text{con}}_{\text{np}} is the non-perturbative partition function of the resolved conifold.

• Dependence on Genus Zero Invariants: A significant result is that the non-perturbative corrections (Stokes jumps) depend only on the genus zero GV invariants $[GV]_{\beta,0}$. This occurs due to cancellations where the sum of sheaf invariants over the index $n$ for a fixed curve class $\beta$ equals the genus zero GV invariant: $\sum_{n \in \mathbb{Z}} \Omega_{\beta,n} = [GV]_{\beta,0}$.

• Borel Transform and Singularities: The Borel transform of the higher-genus asymptotic series is shown to be a meromorphic function with poles at $\xi = 2\pi i m (t_\beta + k)$ for $m \in \mathbb{Z} \setminus \{0\}$ and $k \in \mathbb{Z}$. The Stokes jumps associated with these singularities are explicitly computed.

• Deformation of the Prepotential: The sum of the Stokes jumps, representing the non-perturbative corrections to the free energy, is expressed in terms of a single function defined as an $\epsilon$-deformation of the genus zero prepotential:
  $$\tilde{F}^0_{\text{def}}(\epsilon, t) := \frac{1}{2} \sum_{k, \beta > 0} \frac{[GV]_{\beta,0}}{k^2} \frac{Q^{k\beta}}{\sin(k\epsilon/2)}$$
  The non-perturbative correction term is shown to be proportional to $\partial_\lambda (\lambda \tilde{F}^0_{\text{def}})$.

• Connection to Refined Topological Strings: In the specific case of the resolved conifold, the proposed deformation of the prepotential coincides with the refined topological string partition function in the Nekrasov-Shatashvili (NS) limit.

Significance and Claims
The paper claims to provide the first general expression for the non-perturbative topological string partition function on any Calabi-Yau threefold (compact or non-compact) in the holomorphic limit, derived directly from the asymptotic series of enumerative invariants.

The significance lies in:

1. Universality: It generalizes previous results limited to the resolved conifold or matrix models to arbitrary CY geometries by utilizing the structural decomposition via sheaf invariants.
2. Simplification of Non-perturbative Data: It demonstrates that the complex non-perturbative structure (Stokes jumps) is governed entirely by genus zero GV invariants, despite the perturbative series involving all genera.
3. Analytic Continuation: It offers a concrete analytic form (involving triple sine functions and polylogarithms) for the Borel sums, which are well-behaved in the limit of vanishing Kähler parameters ($t \to 0$), unlike the perturbative series which diverges.

The author notes modestly that while the expression captures the non-perturbative corrections in the holomorphic limit, the global properties of the partition function on the full moduli space (beyond the large radius limit) remain an open question. Furthermore, the current results focus on the holomorphic limit and do not yet incorporate the non-holomorphic corrections governed by the holomorphic anomaly equations, which may introduce additional non-perturbative effects (such as those related to NS5-branes). The work suggests that the deformed prepotential plays a role analogous to the NS limit in the Topological String/Spectral Theory correspondence, potentially canceling poles in the free energy.