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Holomorphic disks and GIT quotients

This paper establishes a correspondence between the moduli spaces of holomorphic disks bounded by a $G$ -invariant Lagrangian submanifold and its quotient in the GIT quotient, providing a computationally effective formula to derive the disk potential of the quotient from the original via the semistable disk potential.

Original authors: Yoosik Kim

Published 2026-05-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yoosik Kim

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 an architect trying to understand the shape of a complex, multi-dimensional building. You want to count the number of specific "paths" (like holomorphic disks) that can be drawn inside this building, starting and ending on a specific wall (a Lagrangian submanifold).

This paper, by Yoosik Kim, is about a clever shortcut. Instead of trying to map every single path in the massive, complicated original building, the author shows you how to look at a simplified, smaller version of that building and get the exact same count.

Here is the breakdown of the paper's ideas using everyday analogies:

1. The Two Buildings: The Original and the Quotient

The Original Building ( $X$ ): Imagine a huge, intricate sculpture garden ( $X$ ) with a specific wall ( $L$ ) inside it. You want to count all the rubber bands (holomorphic disks) you can stretch from the wall, loop around, and come back to the wall without breaking.
The Symmetry Problem: This garden has a lot of symmetry. If you rotate the whole garden by a certain amount (a group action $G$ ), it looks exactly the same.
The Simplified Building ( $X // G$ ): Because the garden looks the same after rotation, you can "fold" or "squash" the garden down. You take all the points that are just rotations of each other and glue them together into a single point. The result is a smaller, simpler garden ( $X // G$ ). The wall ( $L$ ) also gets squashed into a smaller wall ( $L/G$ ).

2. The Big Question

The paper asks: If I count the rubber bands in the giant garden, will I get the same number as if I count the rubber bands in the tiny, squashed garden?

Usually, this is hard because squashing a garden throws away a lot of space (the "unstable" parts). You might think, "If I throw away space, I must be throwing away some rubber bands, so the numbers won't match."

3. The Main Discovery (The "Magic Bridge")

Kim proves that under the right conditions, the numbers do match perfectly.

The Bridge: The author builds a mathematical bridge between the two gardens. He shows that for every rubber band in the tiny garden, there is a unique, corresponding rubber band in the giant garden that "lifts" up to it.
The "Lifting" Trick: Imagine the tiny garden is a shadow on the floor. The giant garden is the 3D object casting the shadow. The paper proves that if you see a rubber band in the shadow, you can always find the exact 3D rubber band in the air that cast it, provided you follow specific rules.
The Result: The "counting invariant" (the number of rubber bands) is identical in both places. This means you can do the hard math in the giant garden by doing the easy math in the tiny one.

4. The "Filter" (Semistability)

There is a catch. When you squash the garden, you have to throw away the "unstable" parts (areas that don't behave well under rotation).

The Filter: The author introduces a "filter" called the semistable disk potential. Think of this as a sieve.
How it works: You take the list of all possible rubber bands in the giant garden. You run them through the sieve.
- If a rubber band touches the "unstable" area (the part you threw away), it gets filtered out.
- If it stays entirely within the "safe" (semistable) area, it survives.
The Formula: The paper provides a recipe: Take the list of rubber bands in the giant garden, filter out the ones that hit the unstable zones, and then "squash" the remaining variables. The result is the exact formula for the rubber bands in the tiny garden.

5. Real-World Examples in the Paper

The author doesn't just talk theory; he uses this method to solve specific puzzles:

The Projective Plane: He shows how to take a 2D plane, squash it down to a line, and correctly calculate the rubber bands on that line.
The Quadric Hypersurface (The Big Win): This is the paper's main achievement. He calculates the rubber bands for a specific type of curved shape called a Quadric Hypersurface (a higher-dimensional sphere-like object).
- The Trick: He couldn't solve it directly because it was too complex. So, he "deformed" (squished) the shape into a simpler version that did have a lot of symmetry. He used the symmetry to squash it down to a 2D shape (a quadric surface), solved the easy version, and then used his "bridge" to translate the answer back to the original complex shape.

Summary

In simple terms, this paper is a translation guide. It tells mathematicians: "If you are stuck trying to count complex shapes in a high-dimensional space, look for a symmetry, squash the space down to a lower dimension, filter out the 'bad' paths, and you will get the correct answer for the original complex space without doing the hard work."

It turns a massive, impossible calculation into a manageable one by using the geometry of symmetry as a shortcut.

Technical Summary: Holomorphic Disks and GIT Quotients

Problem Statement
This paper investigates the relationship between the symplectic geometry of a smooth algebraic variety $X$ equipped with a Hamiltonian action of a connected compact Lie group $G$ , and that of its reduction. Specifically, the author aims to establish a correspondence between the moduli spaces of holomorphic disks in $X$ with boundary on a $G$ -invariant Lagrangian submanifold $L$ , and the moduli spaces of holomorphic disks in the Geometric Invariant Theory (GIT) quotient $X // G$ with boundary on the quotient Lagrangian $L/G$ .

A primary difficulty in symplectic reduction is that the construction requires restricting to a level set of the moment map $\mu^{-1}_G(0)$ and discarding the unstable locus, making a direct comparison of holomorphic curves before and after reduction non-trivial. While equivariant Floer theory offers a framework for such comparisons, this paper seeks to establish the relationship without relying on equivariant Floer theory, instead utilizing the bridge provided by the Kempf–Ness theorem between GIT quotients and symplectic reductions.

Methodology
The approach relies on the geometric structure of the semistable locus $X^{ss}$ and the holomorphic quotient map $q: X^{ss} \to X // G$ . The paper proceeds in two main steps:

Correspondence via Principal Bundles: The author models the relationship between the semistable locus $X^{ss}$ and the quotient $X // G$ as a holomorphic principal $G$ -bundle $q: P_G \to M$ . By analyzing the moduli spaces of holomorphic disks in the total space $P_G$ (with boundary on $L$ ) and the base $M$ (with boundary on $L/G$ ), the paper establishes a correspondence.
- Lifting Disks: A central technical component is proving that every holomorphic disk in the quotient $M$ admits a holomorphic lift to the total space $P_G$ . This is achieved by constructing a holomorphic trivialization of the pullback bundle over the disk that respects the boundary conditions. The author employs Donaldson's result on the solvability of the Hermitian–Yang–Mills equation with Dirichlet boundary conditions to ensure the existence of a flat metric compatible with the boundary reduction, thereby enabling the construction of the lift.
- Moduli Space Isomorphism: Using the lifting property and the fact that $\pi_2(G, G) = 0$ , the author proves that the induced map between the quotient of the moduli space in the total space and the moduli space in the quotient is an orientation-preserving homeomorphism.
Disk Potential Computation: The paper introduces the concept of "semistable" and "unstable" disk classes. A disk class is semistable if its image lies entirely within $X^{ss}$ ; otherwise, it is unstable.
- Semistable Disk Potential: The author defines a "semistable disk potential" $W^{ss}_L$ by retaining only the terms in the disk potential of $L$ corresponding to semistable classes.
- Elimination of Variables: Under the assumption that every unstable class of Maslov index two is "fully excluded" (i.e., every representative disk intersects the unstable locus), the disk potential of the quotient $L/G$ is computed by restricting $W^{ss}_L$ to the kernel of the induced map on fundamental groups, $q^*: \pi_1(L) \to \pi_1(L/G)$ . This involves eliminating variables in the Laurent polynomial corresponding to the directions of the group action.

Key Contributions and Results

Theorem A (Theorem 3.13): Under suitable regularity and topological assumptions (specifically that $L/G$ is spin and the moduli spaces are compact), the map $\phi: \mathcal{M}_1(P_G, L, J, \beta)/G \to \mathcal{M}_1(X // G, L/G, q^*J, q^*\beta)$ is an orientation-preserving homeomorphism. Consequently, the disk counting invariants $n_\beta$ and $n_{q^*\beta}$ coincide.
Theorem B (Theorem 4.11): For a torus action where $L$ is a positive Lagrangian torus and every unstable Maslov index two class is fully excluded, the disk potential of the quotient $L/G$ is given by the restriction of the semistable disk potential of $L$ to the kernel of $q^*$ :
$W_{L/G}(z) = W^{ss}_L|_{\ker(q^*)}(z)$
This provides a computationally effective formula for deriving the disk potential of the quotient from the prequotient.
Theorem C (Theorem 6.3): The paper applies these methods to compute the disk potential of the Gelfand–Zeitlin (GZ) system on quadric hypersurfaces $Q_n \subseteq \mathbb{CP}^{n+1}$ . By utilizing a degeneration method and a GIT quotient reduction to lower-dimensional quadrics, the author derives an explicit formula for the disk potential of arbitrary torus fibers $L_n(u)$ in $Q_n$ , extending previous results that were limited to monotone fibers. The resulting potential is:
$W_{L_n(u)}(y) = \frac{1}{y_n}T^{n-u_n} + \frac{y_n}{y_{n-1}}T^{u_n-u_{n-1}} + \dots + \frac{y_2}{y_1}T^{u_1-u_2} + 2y_2T^{u_2} + y_1y_2T^{u_1+u_2}$

Significance and Claims
The paper claims that its primary advantage lies in the computational effectiveness of Theorem 4.11. By utilizing the correspondence between GIT quotients and symplectic reductions, the method allows for the explicit computation of disk potentials in settings where direct calculation is difficult. Specifically, it enables the lifting of disk potentials from a base to a total space and the reduction of higher-dimensional problems to lower-dimensional ones.

The author notes that this reduction procedure is expected to provide an effective method for computing disk potentials for higher-dimensional isotropic flag varieties by reducing them to lower-dimensional cases. The work also generalizes previous results on Gelfand–Zeitlin fibers by computing potentials for arbitrary torus fibers, not just monotone ones. The paper emphasizes that these results are derived without the machinery of equivariant Floer theory, relying instead on the geometric properties of principal bundles and the Kempf–Ness theorem.