Open enumerative geometries for Landau-Ginzburg models

This paper surveys recent progress in defining open enumerative invariants for Landau-Ginzburg models by constructing them as integrals over real orbifolds with corners, while also exploring their connections to topological recursion, integrable hierarchies, and mirror symmetry, and concluding with a list of open problems.

The Gist

The Big Picture: From Closed Rooms to Open Doors

Imagine you are a mathematician trying to count things. In the world of geometry, one of the most famous counting games involves Riemann surfaces. Think of these as flexible, rubbery sheets (like a donut, a sphere, or a pretzel).

For a long time, mathematicians only studied "closed" surfaces. These are like sealed rooms with no doors or windows. You can walk around inside them, but you can't get in or out. Counting the ways to draw lines or place dots on these sealed rooms led to a huge breakthrough (Witten's Conjecture), connecting geometry to physics equations called the KdV hierarchy (think of these as the "rules of the universe" for waves).

This paper is about opening the doors.

The authors are building a new theory for "open" surfaces. Imagine taking that sealed room and cutting a hole in the wall. Now you have a disk or a surface with a boundary. This changes everything. You can't just count things anymore; you have to decide what happens at the edge of the hole. Do the lines stop? Do they bounce off? Do they connect to something outside?

The Characters: Landau-Ginzburg Models

To understand what they are counting, we need to meet the Landau-Ginzburg (LG) models.

• The Metaphor: Imagine a landscape made of hills and valleys (a potential energy surface). In physics, particles roll down these hills.
• The Math: These landscapes are defined by a polynomial equation (like $W = x^5 + y^5$).
• The Goal: The authors want to count the "paths" or "shapes" that exist on these landscapes, but specifically on the open versions of them (surfaces with edges).

The Three Main Challenges (The "Why is this hard?" section)

The paper explains that moving from closed rooms to open doors is incredibly difficult for four reasons:

1. The Edge Problem (Boundaries): In a closed room, you just count. In an open room, you have to set rules for the edge. It's like playing soccer. In a closed room, the ball just stays in play. In an open room, you have to decide: if the ball hits the wall, does it bounce back? Does it disappear? Does it turn into a different color? The authors had to invent a set of "boundary conditions" (rules for the edge) that make mathematical sense.
2. The Orientation Problem (Which way is up?): Closed rooms are "complex" objects, meaning they have a natural "clockwise" direction. Open surfaces are "real" objects; they are flatter and don't have a natural clockwise direction. To count things, you need to know which way is "positive" and which is "negative." The authors had to invent a way to paint an arrow on every surface so everyone agrees on the direction.
3. The "Wall-Crossing" Problem (Changing the Rules): This is the most exciting part. In some cases, the authors found that the answer to "how many shapes are there?" depends on which rules you picked for the edge.
   • Analogy: Imagine you are counting how many ways you can arrange furniture in a room. If you decide the door opens inward, you get one number. If you decide the door opens outward, you get a different number.
   • Usually, this is a bug. But here, the authors realized it's a feature. The different answers are related by a "Wall-Crossing Group." It's like a secret code: if you know the answer for one set of rules, you can mathematically transform it to get the answer for another set.
4. The Missing Blueprint (Virtual Fundamental Class): In the closed world, mathematicians have a magical tool called a "Virtual Fundamental Class" that acts like a perfect blueprint for counting. This tool doesn't exist yet for open surfaces. The authors had to build their counting method from scratch, piece by piece, using "multisections" (which are like multiple overlapping maps of the same territory).

The Solutions: Three Different Ways to Open the Door

The paper surveys three different teams of mathematicians who built different versions of this "Open Door" theory:

1. The PST Team (Pandharipande, Solomon, Tessler): They started with the simplest case (just a disk with a specific type of polynomial). They figured out how to set the boundary rules so that the counting numbers worked perfectly with the KdV wave equation (the "rules of the universe").
2. The BCT Team (Buryak, Clader, Tessler): They expanded this to more complex polynomials ($x^r$). They introduced a "positivity" rule: imagine the edge of the surface has a "positive" side. You can only count shapes that touch the edge in a positive way. This made the math work for more complex shapes.
3. The GKT Team (Gross, Kelly, Tessler): They tackled the hardest case: polynomials with two variables ($x^r + y^s$). Here, they hit the "Wall-Crossing" wall. They realized the numbers change depending on the edge rules. But they found a beautiful pattern: even though the individual numbers change, a specific combination of them stays the same. This invariant combination is the key to unlocking Mirror Symmetry.

The Grand Prize: Mirror Symmetry

Why do we care about all this counting? Because of Mirror Symmetry.

• The Metaphor: Imagine two different worlds.
  • World A (The A-Model): A world of geometry, where you count shapes and paths (what the authors are doing).
  • World B (The B-Model): A world of calculus and oscillating waves (integrals).
• The Magic: Mirror Symmetry says these two worlds are actually the same, just looking at it from a different angle.
• The Breakthrough: In the past, proving this for open surfaces was nearly impossible. The authors showed that if you take their "Open Counting Numbers" and use them to build a specific mathematical formula (a "perturbed potential"), it perfectly matches the "Wave Calculus" of the other world.
  • Simple version: They found a secret dictionary that translates "Open Geometry" directly into "Wave Physics."

The "Wall-Crossing" Group: The Secret Code

The paper introduces a concept called the Wall-Crossing Group.

• Imagine you are standing in a room with a foggy window. You can't see the outside clearly.
• If you move the window slightly (change the boundary condition), the view changes.
• The "Wall-Crossing Group" is the mathematical machine that tells you exactly how the view changes as you move the window. It proves that even though the view changes, the structure of the room remains consistent. This is crucial because it allows mathematicians to pick the "easiest" set of rules to calculate with, and then translate the answer to the "hardest" set of rules.

The Future: Open Questions

The paper ends by listing what is still missing:

• Virtual Tools: We still need a better "blueprint" (Virtual Fundamental Chain) to make the counting easier and more rigorous.
• Higher Dimensions: Can we do this for surfaces with more holes or higher dimensions?
• The Quintic: Can we apply this to the famous "Fermat Quintic" (a 5-dimensional shape) to solve a long-standing physics problem about string theory?
• Tropical Geometry: Can we draw these shapes using straight lines and corners (like a pixelated video game) to understand them better?

Summary

This paper is a survey of a new frontier. It tells the story of how mathematicians took a theory that only worked for sealed, closed rooms and successfully opened the doors. They had to invent new rules for the edges, deal with the fact that the answers change depending on those rules, and discovered that this "instability" is actually the key to connecting geometry to physics. They built a bridge between counting shapes on open surfaces and solving wave equations, proving that even in the messy, open world, there is a beautiful, hidden order.

Technical Summary

1. Problem Statement

The paper addresses the challenge of defining open enumerative invariants for Landau-Ginzburg (LG) models, specifically extending the Fan-Jarvis-Ruan-Witten (FJRW) theory from closed Riemann surfaces to Riemann surfaces with boundary.

While closed FJRW theory is well-established (providing invariants for $(X, G, W)$ where $X$ is a vector space, $G$ a finite group, and $W$ a $G$-invariant polynomial), the open case presents fundamental geometric and topological obstructions:

• Real vs. Complex Moduli: Closed moduli spaces are complex orbifolds, allowing standard intersection theory via cohomology. Open moduli spaces are real orbifolds with corners, making cohomological intersection theory inapplicable.
• Boundary Conditions: To define intersection numbers (Euler numbers) on real manifolds with boundary, one must prescribe boundary conditions for sections of vector bundles. The resulting invariants depend on these choices.
• Orientation: Closed FJRW bundles are complex and canonically oriented. Open bundles are real; defining a canonical orientation requires new geometric structures (liftings/gradings).
• Wall-Crossing: In higher-rank cases, the invariants are not uniquely defined but depend on the choice of boundary conditions, leading to "wall-crossing" phenomena where invariants change discontinuously as parameters vary.
• Lack of Virtual Fundamental Class: There is no open analogue of the virtual fundamental class machinery used in closed Gromov-Witten and FJRW theories, forcing definitions to rely on chain-level constructions and multisections.

2. Methodology

The authors survey and synthesize recent constructions of open FJRW (OFJRW) theories, primarily focusing on Fermat polynomials $W = \sum x_i^{r_i}$. The methodology involves:

A. Geometric Foundations: Graded W-Spin Surfaces

• Objects: They define $W$-spin Riemann surfaces with boundary. These are orbifold Riemann surfaces equipped with an anti-holomorphic involution $\phi$ (defining the boundary) and spin bundles $S_i$ satisfying specific isomorphisms.
• Liftings and Grading: A crucial new ingredient is the lifting, which orients the real sub-bundle of the spin structure on the boundary. A grading is a lifting satisfying specific conditions at special points (Ramond nodes). This allows for the definition of a canonical orientation for the moduli space and the Witten bundle.
• Alternation: The concept of "alternating" boundary points (where the orientation cannot be extended) is introduced to classify boundary conditions.

B. Moduli Spaces

• The moduli spaces $\mathcal{M}^W_{g,k,l}$ are constructed as smooth real orbifolds with corners.
• They are stratified by dual graphs (W-spin graphs) representing the topological type of the surface (genus, number of boundary/internal nodes, and markings).
• The dimension formula is given by $\dim_{\mathbb{R}} = k + 2l + 3g - 3$.

C. Defining Intersection Numbers

Since cohomology is insufficient, invariants are defined as integrals of multisections of the Open Witten Bundle (the real vector bundle of invariant sections of the dual spin bundles) over the moduli space.

• Canonical Boundary Conditions: To ensure well-definedness, sections must satisfy specific boundary conditions on the codimension-1 strata (boundaries of the moduli space).
• Types of Theories: The paper distinguishes between different constructions based on allowed boundary twists and conditions:
  1. PST (Pandharipande-Solomon-Tessler): $W=x^2$, all twists 0, alternating boundaries.
  2. BCT (Buryak-Clader-Tessler): $W=x^r$, all boundary twists $r-2$, alternating. Uses forgetful and positivity boundary conditions.
  3. GKT (Gross-Kelly-Tessler): General Fermat $W=\sum x_i^{r_i}$. Handles rank 2 ($W=x^r+y^s$) and higher. Introduces compatibility conditions for boundary sections.
  4. TZ (Tessler-Zhao): Generalizes BCT to allow lower boundary twists using point insertion techniques (gluing moduli spaces along equivalent strata).

D. Handling Wall-Crossing

In the GKT theory (rank $\ge 2$), invariants depend on the choice of boundary conditions. The authors show that while individual invariants vary, specific polynomial combinations of them (denoted $\mathcal{A}$) are invariant. This variation is controlled by a wall-crossing group acting on the space of potentials.

3. Key Contributions and Results

A. Construction of Open Invariants

The paper provides a rigorous framework for defining open FJRW invariants $\langle \dots \rangle^o$ for various LG models. It establishes the existence of global multisections satisfying canonical boundary conditions (forgetful, positivity, or point insertion) that are nowhere vanishing on the boundary, ensuring the intersection numbers are well-defined (up to wall-crossing in higher ranks).

B. Topological Recursion Relations (TRR)

The authors derive Open Topological Recursion Relations for genus 0 and 1:

• PST/BCT: Satisfy relations analogous to Solomon's Open WDVV equations, allowing the computation of all genus 0, 1 invariants.
• TZ: Satisfy a new, universal form of TRR based on point insertion, distinct from Open WDVV.
• GKT: Due to wall-crossing, the TRR applies to the invariant polynomial combinations $\mathcal{A}$, not the raw invariants.

C. Mirror Symmetry (Open-Closed Correspondence)

A major result is the proof of Open Mirror Symmetry for rank 2 Fermat polynomials ($W=x^r+y^s$):

• The generating function of closed FJRW invariants (the Givental $J$-function) is recovered from oscillatory integrals of a perturbed potential $W^s$.
• Crucially, $W^s$ is constructed using open FJRW invariants as coefficients.
• The perturbed potential $W^s$ depends on the choice of boundary conditions (multisections), but the resulting oscillatory integral (the closed invariant) is independent of this choice.
• This provides an effective, explicit proof of mirror symmetry, bypassing the difficult construction of Frobenius manifold isomorphisms required in the closed theory.

D. Integrable Hierarchies

The paper connects open invariants to integrable systems:

• r-KdV Hierarchy: For $r$-spin theory, the generating function of open invariants is related to the wave function of the $r$-KdV hierarchy.
• Conjecture: It is conjectured that the all-genus open potential yields the $r$-KdV wave function (generalizing Witten's conjecture to the open setting).
• String/Dilaton Equations: Open string and dilaton equations are proven for PST and BCT theories, consistent with the hierarchy structure.

E. Wall-Crossing Group

For rank 2 theories, the authors define a wall-crossing group (a pro-nilpotent Lie group) that acts on the space of open potentials. This group explains how different choices of boundary conditions (different "slices" of the moduli space) relate to one another, ensuring that the physical closed invariants remain invariant.

4. Significance

• Foundational Breakthrough: This work establishes the first rigorous foundations for open enumerative geometry in the Landau-Ginzburg context, overcoming the lack of virtual fundamental classes by using real geometry and multisections.
• Unification of Mirror Symmetry: It demonstrates that open invariants are not just auxiliary tools but are essential for constructing the mirror potential. The "open" side effectively generates the "closed" mirror symmetry via oscillatory integrals.
• Resolution of Ambiguity: By characterizing the wall-crossing phenomenon, the paper turns the non-uniqueness of open invariants into a structured feature (controlled by a group action) rather than a defect.
• Integrable Systems: It extends the deep connection between enumerative geometry and integrable hierarchies (KdV, KP) to the open setting, suggesting a rich algebraic structure underlying open LG models.
• Future Directions: The paper outlines critical open problems, including the construction of a virtual fundamental chain for $g>1$, extending these theories to non-Fermat potentials, and proving the open LG/Calabi-Yau correspondence for the quintic.

In summary, this survey synthesizes a rapidly developing field, providing the geometric machinery, computational tools (TRR), and structural insights (Mirror Symmetry, Integrable Hierarchies, Wall-Crossing) necessary to understand and compute open invariants for Landau-Ginzburg models.