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Brauer group of moduli of stable parabolic $\text{SL}(r,\mathbb{C})$ and $\text{PGL}(r,\mathbb{C})$ -connections and Higgs bundles over a curve

This paper computes the Brauer groups of the moduli spaces of stable parabolic $\text{SL}(r,\mathbb{C})$ -connections and Higgs bundles over a compact Riemann surface of genus at least 3, while also proving that the Brauer group of the moduli stack of stable parabolic $\text{PGL}(r,\mathbb{C})$ -connections coincides with that of the smooth locus of its coarse moduli space.

Original authors: Pavan Adroja, Sujoy Chakraborty

Published 2026-03-04

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Pavan Adroja, Sujoy Chakraborty

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

The Big Picture: The "Universal Map" Problem

Imagine you are an architect trying to build a massive library. You want to create a catalog (a moduli space) that lists every possible type of book (mathematical object) you could ever write.

In this paper, the "books" are complex mathematical structures called connections and Higgs bundles on a curved surface (like a donut with many holes, known as a Riemann surface). These structures have special "tags" attached to them at specific points, called parabolic data. Think of these tags as bookmarks with specific colors and weights.

The authors are asking a very specific question about this library: Can we build a "Universal Catalog" that perfectly describes every single book in the library at once?

In mathematics, the answer is often "No." Sometimes, the catalog is so twisted and knotted that you can't lay it out flat without tearing it. The Brauer Group is a mathematical tool that measures exactly how twisted the catalog is.

Brauer Group = 0: The catalog is smooth and flat. You can build a universal map.
Brauer Group > 0: The catalog has a "knot." You cannot build a single universal map that works everywhere; you have to settle for local maps that don't quite fit together perfectly.

The goal of this paper is to calculate the size of this "knot" (the Brauer group) for these specific libraries.

The Cast of Characters

To understand the results, let's meet the main characters using analogies:

The Curve ( $X$ ): A donut-shaped surface with at least 3 holes (genus $\ge$ 3). Think of it as the "stage" where everything happens.
The Parabolic Data ( $S, m, \alpha$ ): Imagine the stage has specific spots (points) where the actors must wear special costumes.
- $S$ : The locations of the spots.
- $m$ (Multiplicity): How many layers of costume the actor wears at that spot.
- $\alpha$ (Weights): The specific "color" or "weight" assigned to each layer.
- Analogy: Think of a parade. The "parabolic" part means that at specific street corners, the floats must have a specific hierarchy of decorations.
The Objects (Connections & Higgs Bundles):
- Connections: Think of these as "flowing rivers" on the stage. They tell you how to move from one point to another without getting lost. The "parabolic" part means the river behaves in a specific, controlled way when it hits the decorated street corners.
- Higgs Bundles: Think of these as "static sculptures" or "frozen rivers." They look similar to connections but represent a different kind of mathematical stability.
The Groups ($SL$ vs. $PGL$):
- $SL(r, \mathbb{C})$ : The "Strict" group. The actors must follow a strict rule: the total volume of their costumes must be exactly 1.
- $PGL(r, \mathbb{C})$ : The "Flexible" group. The actors can change their total volume, as long as they stay in the same "family" of shapes. It's like looking at a sculpture from different angles; the shape is the same, even if the size changes.

The Main Discoveries

The authors, Pavan Adroja and Sujoy Chakraborty, found two major things.

1. The Size of the Knot (The Brauer Group)

For the strict group ($SL$), they calculated exactly how big the "knot" in the catalog is.

The Result: The Brauer group is a cyclic group (a loop) of size $\nu$ .
What is $\nu$ ? It is the Greatest Common Divisor (GCD) of a list of numbers:
- The rank of the bundle ( $r$ ).
- The degree ( $d$ ).
- The multiplicities of the flags at the parabolic points ( $m$ ).
The Analogy: Imagine you are trying to tile a floor with tiles of size $r$ , $d$ , and various $m$ 's. The "knot" in your catalog is determined by the largest tile size that fits perfectly into all of these dimensions. If the numbers share no common factor (GCD = 1), the knot disappears, and the catalog is perfect. If they share a factor of 2, the catalog is twisted in a way that repeats every 2 steps.

Why is this cool? It tells us that the "twist" in the mathematical universe isn't random; it's strictly determined by the basic numbers defining the shapes.

2. The "Stack" vs. The "Space" (The PGL Case)

This is the second major result, dealing with the flexible group ($PGL$).

The Problem: In math, there are two ways to look at a collection of objects:
1. The Stack: A hyper-detailed view that remembers every tiny symmetry and rotation of the objects. (Like a 3D hologram).
2. The Coarse Moduli Space: A simplified, blurry view where we just see the shapes, ignoring the tiny rotations. (Like a 2D photograph).
The Question: Does the "knot" (Brauer group) change when we switch from the detailed 3D hologram to the blurry 2D photo?
The Result: No. The authors proved that for the smooth parts of these spaces, the Brauer group of the detailed "Stack" is exactly the same as the Brauer group of the simplified "Coarse Space."
The Analogy: Imagine a spinning top.
- The Stack is the top spinning rapidly; it looks like a blur.
- The Coarse Space is the top when it stops spinning; it's a solid object.
- The paper proves that the "twistiness" of the spinning blur is identical to the "twistiness" of the solid object. You don't lose any topological information by simplifying the view, as long as you stay in the "smooth" (non-broken) regions.

How Did They Do It? (The Method)

The authors used a clever "cut-and-paste" strategy:

The "Good" Part: They identified a huge, smooth, open area of the library where everything works perfectly (no singularities).
The "Bad" Part: They identified the tiny, messy corners where things break (singularities).
The Trick: They proved that the "Bad" part is so small (high codimension) that it doesn't affect the global "knot" of the library. In topology, if you remove a tiny speck from a large room, the room's overall shape doesn't change.
The Connection: They linked the complex world of Connections (flowing rivers) to the simpler world of Vector Bundles (static flags). They showed that the "knot" in the river world is the same as the "knot" in the flag world, which they already knew how to calculate.

Summary in One Sentence

The authors calculated the "twistiness" (Brauer group) of mathematical libraries containing special curved surfaces with decorated points, proving that this twist is determined by the greatest common divisor of the defining numbers, and that simplifying the view of these libraries doesn't change their fundamental twistiness.

Why Should You Care?

While this sounds abstract, it's crucial for Rationality Questions in mathematics.

If a space has a non-trivial Brauer group, it often means the space is irrational.
In simpler terms: You cannot describe the entire library using a simple, rational formula (like a polynomial equation). You need more complex tools.
By calculating the Brauer group, the authors are essentially drawing a map that tells mathematicians: "You can't simplify this problem further; the complexity is built into the very fabric of the numbers."

1. Problem Statement and Context

The paper addresses the computation of the Brauer group (specifically the cohomological Brauer group, $H^2_{\text{\'et}}(Y, \mathbb{G}_m)_{\text{tors}}$ ) for specific moduli spaces arising in the theory of parabolic bundles on a compact Riemann surface $X$ of genus $g \geq 3$ .

The Brauer group is a crucial invariant in algebraic geometry, often interpreted as the obstruction to the existence of a universal bundle over a moduli space. If the Brauer group is non-trivial, a universal bundle does not exist globally.

The authors focus on three specific types of moduli spaces:

Stable Parabolic $SL(r, \mathbb{C})$ -connections: Logarithmic connections with fixed determinant and specific parabolic weights.
Stable Strongly Parabolic $SL(r, \mathbb{C})$ -Higgs bundles: Higgs fields satisfying specific residue conditions relative to the parabolic structure.
The Moduli Stack of Stable Parabolic $PGL(r, \mathbb{C})$ -connections: The stack quotient of the $SL(r, \mathbb{C})$ moduli space by the action of $r$ -torsion line bundles.

The goal is to generalize previous results on parabolic bundles and logarithmic connections to these more complex settings (connections and Higgs fields) and to establish an equality between the Brauer group of a moduli stack and its coarse moduli space.

2. Methodology

The authors employ a combination of geometric invariant theory, deformation theory, and cohomological techniques.

A. Geometric Setup

Parabolic Data: They fix a set of points $S = \{x_1, \dots, x_n\}$ on $X$ , a system of multiplicities $m$ , and a generic system of weights $\alpha$ . "Generic" ensures that parabolic semistability coincides with parabolic stability, avoiding strictly semistable points.
Moduli Spaces: They define $M^{m,\alpha}_{pc}(r, \xi)$ for connections and $M^{m,\alpha}_{\text{Higgs}}(r, \xi)$ for Higgs bundles, where $\xi$ is a fixed determinant line bundle.

B. Dimensional Analysis and Codimension Estimates

A central technique involves analyzing the "bad" locus (the complement of the stable locus) to apply cohomological purity.

Lemma 4.2: The authors prove that the locus $Z$ of connections where the underlying parabolic vector bundle is not stable has codimension at least 2 in the total moduli space (for $g \geq 3, r \geq 2$ ).
Lemma 6.2 & Corollary 6.4: For the $PGL(r, \mathbb{C})$ case, they analyze the fixed point locus of the action of the group of $r$ -torsion line bundles $\Gamma$ . They prove that the union of these fixed point loci has codimension at least 3. This high codimension is critical for the purity arguments used later.

C. Torsor and Affine Bundle Structures

Theorem 4.4 (Connections): They utilize the forgetful morphism $\pi: U \to \mathcal{N}^{m,\alpha}(r, \xi)$ , where $U$ is the open subset of connections with stable underlying bundles. They show that $U$ is a $T^*\mathcal{N}$ -torsor (an affine bundle modeled on the cotangent bundle of the moduli space of stable parabolic bundles).
Theorem 5.2 (Higgs): Similarly, they show that the moduli space of Higgs bundles contains the cotangent bundle of the moduli space of parabolic bundles as a Zariski open subset with a complement of codimension $\geq 2$ .

D. Group Actions and Quotients

Section 6: They construct an action of the finite abelian group $\Gamma = \{L \in \text{Pic}(X) \mid L^{\otimes r} \cong \mathcal{O}_X\}$ on the moduli space of $SL(r, \mathbb{C})$ connections.
They define the moduli stack of $PGL(r, \mathbb{C})$ connections as the quotient stack $[\mathcal{M}/\Gamma]$ .
Using the codimension $\geq 3$ result for the fixed point locus, they apply the cohomological purity theorem (which states that removing a closed subset of codimension $\geq 2$ (for Brauer groups) or $\geq 3$ (for certain stack properties) does not change the Brauer group).

3. Key Contributions and Results

Result 1: Brauer Group of $SL(r, \mathbb{C})$ Connections and Higgs Bundles

Theorem 1.1 (Theorems 4.4 & 5.2):
For a generic system of weights $\alpha$ , the Brauer groups of the moduli spaces of stable parabolic $SL(r, \mathbb{C})$ -connections ( $M^{m,\alpha}_{pc}$ ) and stable strongly parabolic Higgs bundles ( $M^{m,\alpha}_{\text{Higgs}}$ ) are isomorphic to the cyclic group:
$\text{Br}(M) \cong \mathbb{Z}/\nu\mathbb{Z}$
where
$\nu = \gcd(r, d, m^1_1, m^1_2, \dots, m^{\ell_1}_1, \dots, m^{\ell_n}_n)$
Here, $d$ is the degree of the determinant, and $m^i_j$ are the multiplicities of the flags at the parabolic points.

Significance: This extends previous results (e.g., from [BBS, BD]) from vector bundles to connections and Higgs bundles, showing that the Brauer group depends only on the topological data ( $r, d$ ) and the parabolic multiplicities, not on the specific weights (as long as they are generic).

Result 2: Brauer Group of the $PGL(r, \mathbb{C})$ Moduli Stack

Theorem 1.2 (Theorem 7.3):
Let $\mathcal{N}^{m,\alpha}_{pc}(r, i)$ be the moduli stack of stable parabolic $PGL(r, \mathbb{C})$ -connections, and let $\mathcal{N}^{m,\alpha}_{pc}(r, i)^{\text{sm}}$ be the smooth locus of its coarse moduli space. Then:
$\text{Br}\left(\mathcal{N}^{m,\alpha}_{pc}(r, i)\right) \cong \text{Br}\left(\mathcal{N}^{m,\alpha}_{pc}(r, i)^{\text{sm}}\right)$

Significance: This establishes that the Brauer group of the stack (which accounts for automorphisms) is identical to the Brauer group of the smooth locus of the coarse moduli space. This generalizes a known result for parabolic bundles (from [BCD, BC]) to the setting of connections. The proof relies heavily on the fact that the singular locus (fixed points of the $\Gamma$ action) has codimension $\geq 3$ .

4. Technical Significance

Extension of Hodge Theory: The paper bridges the gap between the moduli of parabolic bundles, connections, and Higgs bundles (related via the Hodge moduli space of $\lambda$ -connections) by showing they share the same Brauer group structure.
Purity in Stacks: The work provides a robust application of cohomological purity to Deligne-Mumford stacks. By proving the fixed point locus of the torsion action has high codimension, the authors demonstrate that the "stacky" nature of the moduli space does not alter the Brauer group compared to the coarse space.
Universal Bundles: The non-triviality of the Brauer group (when $\nu > 1$ ) implies that no universal parabolic connection or Higgs bundle exists over the entire moduli space. This is a fundamental obstruction in the geometric quantization and gauge theory of these spaces.
Methodological Rigor: The paper provides detailed dimension counts and codimension estimates for the non-stable loci and fixed point sets, which are essential for the validity of the purity arguments in higher genus ( $g \geq 3$ ).

Conclusion

Adroja and Chakraborty successfully compute the Brauer groups for moduli spaces of parabolic connections and Higgs bundles, showing they are cyclic groups determined by the greatest common divisor of the rank, degree, and parabolic multiplicities. Furthermore, they prove that for the associated $PGL(r, \mathbb{C})$ moduli stacks, the Brauer group is invariant under the passage from the stack to the smooth locus of the coarse moduli space, reinforcing the stability of this invariant under the specific geometric operations involved.