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Mock modularity of log Gromov--Witten Invariants: the mirror to $\mathbb{P}^2$

The paper proves that the generating series of certain logarithmic Gromov–Witten invariants of the rational elliptic surface mirror to $\mathbb{P}^2$ are mock modular forms by establishing a correspondence between these invariants and the mock modular Vafa–Witten invariants of $\mathbb{P}^2$ .

Original authors: Hülya Argüz

Published 2026-02-10

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Hülya Argüz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery involving two different worlds that seem completely unrelated: one world is a vast, complex ocean of waves (representing physics and gauge theory), and the other is a delicate, intricate lace pattern (representing geometry and curves).

This paper, written by Hülya Argüz, is about discovering that these two worlds are actually looking at the exact same thing from different angles.

1. The Two Worlds

World A: The Vafa–Witten Ocean (Physics)
In theoretical physics, there is a concept called "Vafa–Witten theory." Imagine an ocean where particles (called "sheaves") are moving around. Scientists want to count how many ways these particles can arrange themselves. When you organize these counts into a mathematical list (a "generating series"), physicists predict that the list should follow a very strict, beautiful rhythm—like a complex piece of music that repeats its patterns in predictable ways. This rhythm is called "Mock Modularity."

World B: The Log Gromov–Witten Lace (Geometry)
In geometry, there is a different way of counting. Instead of particles in an ocean, imagine you are looking at a piece of lace. You want to count how many tiny, delicate threads (called "curves") can be woven through specific holes in the lace without breaking the pattern. This is "Log Gromov–Witten theory."

2. The Mystery: The "Mock" Rhythm

The paper focuses on a specific kind of rhythm called "Mock Modularity."

Think of a standard Modular Form like a perfect, crystal-clear bell. When you strike it, the sound is pure and follows perfect mathematical laws.

A Mock Modular Form, however, is like a bell that sounds slightly "off" or "fuzzy." It doesn't seem to follow the perfect rules at first glance. But, the math tells us that if you add a very specific "correction note" (a non-holomorphic term), the fuzziness disappears, and the bell suddenly rings with perfect, mathematical clarity. The mystery is: Why does this "fuzzy" rhythm exist in geometry?

3. The Breakthrough: The Mirror

The author uses a concept called Mirror Symmetry. In mathematics, "Mirror Symmetry" is like finding out that a beautiful painting of a mountain is actually just a reflection in a lake. The mountain (the physics side) and the reflection (the geometry side) look different, but they are mathematically identical.

What the paper actually does:

The Connection: The author uses a previously established "bridge" (a correspondence) to show that counting the "particles in the ocean" (Vafa–Witten invariants) is mathematically the same as counting the "threads in the lace" (Log Gromov–Witten invariants).
The Proof: Because we already knew the "ocean" side had that "fuzzy" (mock modular) rhythm, the author proves that the "lace" side must also have that same rhythm.
The Result: She proves that for a specific geometric shape (the mirror to a space called $\mathbb{P}^2$ ), the way these curves are woven follows this exact, sophisticated "mock" musical pattern.

Summary in a Nutshell

If you count the ways particles can dance in a specific physical field, you get a "fuzzy" musical pattern. This paper proves that if you instead count the ways tiny curves can be drawn on a specific geometric surface, you will get the exact same "fuzzy" musical pattern.

She has shown that the "music" of physics and the "weaving" of geometry are playing the same song.

Technical Summary: Mock Modularity of Log Gromov–Witten Invariants

Title: Mock Modularity of Log Gromov–Witten Invariants: The Mirror to $\mathbb{P}^2$
Author: Hülya Argüz

1. The Problem

The paper addresses a fundamental question in enumerative geometry and mathematical physics: the modularity properties of generating series of logarithmic Gromov–Witten (log GW) invariants.

While it is well-established that generating series of Gromov–Witten invariants for elliptic curves are quasi-modular forms (Okounkov–Pandharipande), and predictions exist for elliptic fibrations, the behavior of logarithmic invariants—which allow for prescribed tangency conditions along singular divisors—remains largely unexplored. Specifically, the author investigates whether the generating series of log GW invariants of an elliptic fibration relative to a singular fiber exhibit mock modularity. Mock modularity is a more complex phenomenon where modularity is only recovered after adding a specific non-holomorphic correction term.

2. Methodology

The author employs a mirror symmetry approach to bridge two seemingly different mathematical theories:

Logarithmic Gromov–Witten Theory: Counting curves in a log Calabi–Yau pair $(Y, D)$ with specific contact orders along a divisor $D$ .
Vafa–Witten Theory: Studying the invariants of semistable rank $r$ coherent sheaves on a surface $S$ (specifically $S = \mathbb{P}^2$ ), which originate from $N=4$ super Yang–Mills theory.

The core methodological tool is a correspondence theorem (established in joint work with Bousseau) that relates the Vafa–Witten invariants of $\mathbb{P}^2$ to the log GW invariants of its mirror, a specific rational elliptic surface $(Y, D)$ . By transporting the known (or conjectured) modularity properties from the Vafa–Witten side to the log GW side via this correspondence, the author can deduce the modularity of the latter.

3. Key Contributions and Results

The paper provides both a broad conjecture and a specific, proven case:

Conjecture 1.1 (General Case): The author conjectures that for any elliptic fibration $\phi: Y \to B$ and a normal crossing divisor $D$ (where the generic fiber of $D$ is a cycle of $\mathbb{P}^1$ s or a nodal elliptic curve), the generating series of log GW invariants is a component of a higher-depth vector-valued mock modular form.
Theorem 5.7 (The Correspondence): The author proves a precise equivalence between the Vafa–Witten generating series of $\mathbb{P}^2$ and the log GW generating series of the rational elliptic surface $(Y, D)$ mirror to $\mathbb{P}^2$ . Specifically, for $r > 0$ and $-1 < c_1/r \leq 0$ :
$h^{GW}_{r,c_1}(\tau) = h^{VW}_{r,c_1}(\tau)$
Corollary 5.8 & Conjecture 5.9 (Proven/Conjectured Modularity):
- Proven: For $1 \leq r \leq 3$ , the vector of log GW generating series for the mirror to $\mathbb{P}^2$ is a vector-valued mock modular form of weight $-3/2$ and depth $r$ . This is proven by leveraging existing results for Vafa–Witten invariants of $\mathbb{P}^2$ for these ranks.
- Conjectured: For any $r > 0$ , the generating series is a vector-valued mock modular form of weight $-3/2$ and depth $r$ .

4. Significance

This work is significant for several reasons:

Bridging Fields: It provides a rigorous mathematical link between the enumerative geometry of curves (Gromov–Witten theory) and the gauge-theoretic invariants of sheaves (Vafa–Witten theory) through the lens of mirror symmetry.
Advancing Mock Modularity: It extends the study of mock modularity from the realm of "open" Gromov–Witten theory into the "logarithmic" setting, providing a new context for these complex analytic functions.
Mirror Symmetry Validation: The results support the Gross–Siebert program and the Strominger–Yau–Zaslow (SYZ) conjecture by demonstrating that the expected modular properties predicted by physics (S-duality) have a concrete manifestation in the geometry of the mirror manifold.