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Open FJRW Theory and Mirror Symmetry

This paper constructs an open enumerative theory for a specific two-dimensional Landau-Ginzburg model, establishing an open mirror symmetry theorem with all descendents and characterizing the wall-crossing phenomena of the resulting invariants through a dedicated LG wall-crossing group.

Original authors: Mark Gross, Tyler L. Kelly, Ran J. Tessler

Published 2026-07-28

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CC BY 4.0

Original authors: Mark Gross, Tyler L. Kelly, Ran J. Tessler

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 the universe not as a collection of solid stars and planets, but as a vast, invisible landscape of pure energy and mathematics. In this landscape, physicists and mathematicians play a game called "Mirror Symmetry." Think of it like looking at a complex sculpture in a funhouse mirror. The reflection might look warped, twisted, and strange, but if you know the rules of the mirror, you can use the reflection to understand the original sculpture better than you could by looking at it directly. Sometimes, the reflection is actually easier to measure.

To do this, scientists use two different "languages" to describe the same reality. One language, called the A-model, counts the shapes and paths things can take, like counting how many ways a string can wrap around a knot. The other language, the B-model, looks at the smooth, flowing waves of energy that ripple through the landscape. For a long time, mathematicians could translate between these two languages for simple, closed loops (like a rubber band with no ends). But when they tried to look at shapes with open ends—like a string with loose tips floating in space—the translation broke down. The "mirror" seemed to shatter, and the rules for counting these open shapes were missing. Without a way to translate the open shapes, a huge part of the mathematical map of the universe remained a blank spot.

This paper, written by Mark Gross, Tyler L. Kelly, and Ran J. Tessler, is about building a new bridge to cross that gap. They have successfully constructed a way to count these "open" shapes in a specific type of mathematical universe (known as a Landau-Ginzburg model) and proved that this new counting method perfectly matches the smooth wave language on the other side.

The Puzzle of the Open String

To understand what the authors did, imagine you are trying to count the number of ways a rubber band can stretch and twist. If the rubber band is a complete circle (closed), it's a well-understood problem. But what if the rubber band has a loose end that you can drag around? This is the "open" problem. In the world of these mathematical models, dragging that loose end changes the rules of the game.

The authors focused on a specific, tricky version of this problem involving two variables (think of them as two different colors of paint, red and blue) that mix together in a specific way. They wanted to count the "open" shapes formed by these paints. The problem was that the answer wasn't unique. Depending on how you decided to "pin down" the loose ends of your rubber bands (a technical choice called "boundary conditions"), you got different numbers. It was like trying to measure a shadow: if you move the light source, the shadow changes shape. In previous attempts, this meant the theory was broken or incomplete.

The Discovery: A Family of Mirrors

The authors didn't just find one answer; they found a whole family of answers and a rulebook for how they relate to each other. Here is the core of their discovery:

The Wall-Crossing Group: They discovered that the different answers you get by changing the "pinning" of the loose ends aren't random errors. Instead, they are connected by a specific mathematical structure they call a "wall-crossing group." Imagine standing in a room with many doors. Each door leads to a different version of the room where the furniture is arranged slightly differently. The authors found the key that unlocks every door and shows you exactly how to move from one arrangement to another. They proved that this group acts like a perfect translator, moving you between all possible versions of the open counting theory.
The New Mirror: Using this new way of counting open shapes, they built a "mirror" potential. In the old days, to get the mirror, you had to guess the shape of the reflection based on combinatorial tricks. Here, they showed that if you take their new open counts and use them as ingredients, you automatically cook up the exact mirror image. This mirror image is a "versal deformation," which is a fancy way of saying it's the most complete, flexible version of the reflection possible.
The Topological Recursion: Perhaps the most exciting part is how they connect the open counts to the closed, smooth waves. They proved a "Topological Recursion." Think of this as a recipe. If you know the counts for a few simple open shapes, this recipe tells you exactly how to calculate the counts for incredibly complex shapes with many twists and turns. It's like having a master formula that generates the entire library of answers from a single seed.

Why This Matters

The authors didn't just guess; they proved it. They constructed the mathematical objects (called moduli spaces) where these shapes live, defined the bundles (the mathematical "containers" for the shapes), and showed that the integrals (the counting methods) work perfectly.

They explicitly ruled out the idea that there is just one single, unique answer for these open counts. Instead, they showed that the "correct" answer depends on your choice of boundary conditions, but that all these choices are valid and connected. They also showed that for certain simple cases (like specific types of singularities), the wall-crossing disappears, and the answer becomes unique, which matches what was already known in simpler theories.

In short, this paper solves a long-standing puzzle in the "open" world of mirror symmetry. It provides a rigorous, complete framework for counting open shapes in a two-dimensional mathematical universe. It proves that even though the open shapes seem to behave chaotically, they are actually governed by a strict, beautiful symmetry that links them perfectly to the smooth, closed world on the other side of the mirror. The authors have handed us the keys to a new room in the house of mathematics, one that was previously locked and dark.

Technical Summary of "Open FJRW Theory and Mirror Symmetry"

Problem Statement
The paper addresses the construction of an open enumerative theory for the Landau-Ginzburg (LG) model $(W, G) = (x^r + y^s, \mu_r \times \mu_s)$ . While closed FJRW theory (Fan-Jarvis-Ruan-Witten) provides a robust enumerative framework for the A-model of LG theories, a corresponding open theory (involving boundaries) has been less developed, particularly for rank $a \geq 2$ . Previous work by Buryak, Clader, and Tessler established open $r$ -spin theory (the case $a=1$ ), but extending this to higher rank presents significant geometric and combinatorial challenges. Specifically, the authors aim to define open invariants that satisfy a mirror symmetry theorem relating the open A-model to the B-model (Saito-Givental theory) and to understand the dependence of these invariants on boundary conditions, which leads to wall-crossing phenomena.

Methodology
The authors employ a geometric approach based on moduli spaces of stable W-spin disks, generalizing the construction of open $r$ -spin invariants.

Moduli Spaces and Structures:
- They define graded W-spin disks, which are genus zero orbifold Riemann surfaces with boundary, equipped with an anti-holomorphic involution, spin bundles $S_i$ , and specific twist data at marked points and nodes.
- The moduli spaces $\mathcal{M}^W_\Gamma$ are constructed as real orbifolds with corners, where $\Gamma$ represents a dual graph encoding the degeneration of the disk (nodes, boundary nodes, contracted boundary nodes).
- A crucial technical step involves defining gradings (liftings of the spin bundles) to impose necessary boundary conditions. This includes distinguishing between "alternating" and "non-alternating" boundary points and handling "contracted boundary nodes."
Witten Bundles and Invariants:
- The invariants are defined as integrals of multisections of the descendent Witten bundle $E_\Gamma(d)$ over the moduli space. The bundle is a direct sum of the Witten bundles $W_i$ (related to $H^1(C, S_i)^\vee$ ) and descendent line bundles $L_i$ .
- Because the moduli spaces have corners, invariants are not simply Euler classes. Instead, they are defined as the signed count of zeroes of multisections satisfying specific boundary conditions.
- The authors introduce a notion of positivity for multisections along certain boundary strata (specifically those corresponding to "positive" boundary nodes). This ensures the multisections are non-vanishing on these strata, allowing for a well-defined relative Euler class.
Inductive Structure and Canonical Families:
- To define a consistent system of invariants across all moduli spaces, the authors construct families of canonical multisections. These are defined inductively: boundary conditions on a stratum corresponding to a degeneration into two disks are determined by pulling back multisections from the moduli spaces of the normalized components.
- This inductive structure is essential for establishing relations between invariants, such as topological recursion.
Wall-Crossing and the LG Wall-Crossing Group:
- The authors identify that the choice of boundary conditions is not unique. Different choices lead to different systems of invariants.
- They define a Landau-Ginzburg wall-crossing group ( $G_{r,s}^{AI,sym}$ ), a subgroup of automorphisms of the potential's parameter space. This group acts on the set of all possible systems of open invariants.
- The action is shown to be faithful and transitive, meaning any two valid systems of invariants are related by an element of this group. This group is generated by transformations associated with "critical" boundary strata where the dimension of the moduli space drops by one relative to the bundle rank.
Mirror Symmetry and Topological Recursion:
- Using a system of open invariants, the authors construct a perturbation $W^{\nu, sym}$ of the potential $W = x^r + y^s$ .
- They prove that the oscillatory integrals of $e^{W^{\nu, sym}/\hbar}$ yield generating functions for closed extended FJRW invariants.
- Crucially, they establish Open Topological Recursion. Unlike the closed case or the $r$ -spin case, the recursion does not hold for a single open invariant but for specific polynomial combinations of invariants (denoted $A(I, d)$ ). This is necessitated by the wall-crossing phenomenon.

Key Contributions and Results

Construction of Open FJRW Invariants: The paper successfully constructs open FJRW invariants for the rank-2 Fermat LG model $(x^r + y^s, \mu_r \times \mu_s)$ . These are defined as integrals of multisections over real orbifolds with corners.
Existence of Canonical Families: Theorem 0.1 establishes the existence of families of canonical transverse multisections, and symmetric versions thereof, proving that open invariants exist.
Wall-Crossing Group: The paper identifies and characterizes the LG wall-crossing group. It proves that this group acts transitively on the set of all possible systems of open invariants (Theorem 0.7). This provides a complete classification of the ambiguity in the definition of open invariants.
Open Mirror Symmetry Theorem: Theorem 0.3 proves an LG/LG open mirror symmetry theorem. It states that if the open invariants are used to construct the deformed potential $W^{\nu, sym}$ , the resulting oscillatory integrals generate the closed extended FJRW invariants with descendents. This establishes a direct link between the open A-model and the closed B-model without relying on combinatorial constructions like Berglund-Hübsch duality.
Open Topological Recursion: Theorem 0.5 establishes a new form of topological recursion. Due to wall-crossing, the recursion relates polynomials of open invariants rather than single invariants. This recursion is the key tool used to prove the mirror symmetry theorem.
Open-Closed Correspondence: Corollary 0.6 provides a precise correspondence linking polynomials of open invariants to closed extended invariants.

Significance and Claims
The authors claim that this work provides the first open enumerative theory for a Landau-Ginzburg model in dimension two (rank $a=2$ ) with all descendents.

Geometric Naturalness: The construction relies on geometric boundary conditions (positivity) rather than ad-hoc choices, allowing for a "matching" of the A- and B-models "on the nose."
Resolution of Ambiguity: By introducing the wall-crossing group, the paper clarifies that the non-uniqueness of open invariants is a structural feature governed by a specific group action, rather than a defect in the theory.
Generalization of Previous Work: The theory generalizes the open $r$ -spin theory of Buryak, Clader, and Tessler to the W-spin case, while highlighting new features (such as the necessity of polynomial recursions) that arise in higher rank.
Mirror Symmetry: The paper demonstrates that open Gromov-Witten-style invariants (in the LG context) naturally generate the mirror potential, extending the philosophy of Cho-Oh, Gross, and Fukaya-Oh-Ohta-Ono to Landau-Ginzburg models.

The paper does not claim to solve the problem for arbitrary potentials or higher ranks immediately, but establishes the framework and proves the results for the specific case of Fermat polynomials in two variables. It notes that for simple singularities (specific cases of $r, s$ ) without descendents, the invariants are well-defined (no wall-crossing), but wall-crossing is inevitable when descendents are included.

The Puzzle of the Open String

The Discovery: A Family of Mirrors

Why This Matters

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