Conformally flat factorization homology in Ind-Hilbert spaces and Conformal field theory

This paper introduces a metric-dependent variant of factorization homology in conformally flat Riemannian geometry, defining conformally flat disk algebras as functors to ind-Hilbert spaces and proving that their left Kan extensions yield symmetric monoidal invariants that reproduce sphere partition functions of conformal field theories, with explicit constructions for $d>2$ derived from unitary representations of $\mathrm{SO}^+(d,1)$.

The Gist

Imagine you are trying to understand the universe by looking at it through a microscope. In physics, there are two main ways to do this:

1. The Geometric Way: You look at the shape of space itself. Is it flat? Is it curved? How do shapes fit together? This is like looking at a map.
2. The Analytic Way: You look at the numbers and probabilities. You calculate the energy of particles, the vibrations of strings, and the "state" of the system. This is like looking at the data on a spreadsheet.

Usually, these two ways of looking at the universe are very hard to combine. The geometric way is great for shapes, but terrible for the messy, infinite numbers of quantum physics. The analytic way is great for numbers, but it gets confused when you try to move those numbers around on a curved shape.

Yuto Moriwaki's paper is about building a bridge between these two worlds.

Here is the story of the paper, broken down into simple analogies.

1. The Problem: The "Unbounded" Monster

In quantum physics, things often get "unbounded." Imagine trying to measure the height of a mountain, but the mountain keeps growing taller the closer you get to it. In math, this is called an "unbounded operator." It's a number-crunching machine that breaks if you feed it the wrong input.

When physicists try to describe a "Conformal Field Theory" (a theory about shapes that can stretch and shrink without tearing), they run into these monsters. If you try to glue two shapes together in a specific way, the math explodes. It's like trying to glue two pieces of paper together, but the glue turns into a black hole that swallows your calculator.

2. The Solution: The "Safe Zone" (Conformally Flat Geometry)

Moriwaki says, "Let's not try to glue any two shapes together. Let's only glue shapes that are 'safe'."

He introduces a special category of shapes called Conformally Flat Manifolds. Think of these as shapes that can be flattened out onto a flat sheet of paper (like a map of the Earth) without distorting the angles.

He creates a new rulebook (a mathematical "operad") that only allows you to glue these safe shapes together.

• The Old Rulebook: Allowed you to glue shapes even if they touched at the very edge. This caused the "unbounded monsters" to appear.
• The New Rulebook: Only allows you to glue shapes if there is a tiny, safe gap between them.

The Magic Trick: Moriwaki proves that if you keep that tiny gap (the "safe zone"), the unbounded monsters disappear! The math that used to break now becomes perfectly stable and bounded. It's like realizing that if you don't press the two pieces of paper too hard, the glue works perfectly fine.

3. The Engine: Factorization Homology

Now, how do we use this? Moriwaki uses a tool called Factorization Homology.

Imagine you have a Lego set (the local rules for how shapes glue together) and a Big Castle (the whole universe).

• Factorization Homology is the instruction manual that tells you: "If you know how to glue two small Legos together, here is how you can build the entire castle."
• In this paper, the "Legos" are small disks, and the "glue" is the new, safe rulebook.
• The "Castle" is the whole universe (like a sphere).

Moriwaki shows that if you follow his safe rules, you can build a mathematical model of the whole universe just by knowing the rules for the tiny disks.

4. The Result: The "Sphere Partition Function"

The ultimate goal of this math is to calculate something called the Partition Function. In physics, this is like the "total energy score" or the "probability weight" of the entire universe.

Moriwaki proves that:

1. You can build this score for a sphere (a perfect ball) using his method.
2. This score matches exactly what physicists have been calculating for decades using other, messier methods.
3. He even builds a specific example using Harmonic Polynomials (a type of smooth, vibrating mathematical wave) and shows that these waves behave exactly like the particles in a "massless free scalar field" (a simple type of quantum field).

5. Why This Matters

This paper is a big deal because it solves a long-standing puzzle: How do we make quantum field theory rigorous using geometry?

• Before: Physicists had to use "hand-waving" or infinite series that didn't always converge to get the right answers.
• Now: Moriwaki provides a solid, geometric foundation. He shows that if you respect the geometry (keep the shapes in the "safe zone"), the physics naturally falls into place without breaking.

The Big Picture Analogy

Imagine you are trying to bake a cake (the Universe) using a recipe (Quantum Field Theory).

• The Problem: The recipe says "mix the ingredients," but if you mix them too fast, the batter explodes (the unbounded operators).
• Moriwaki's Insight: He realizes that if you only mix the ingredients when they are in a specific, gentle motion (the conformally flat geometry), the batter stays smooth.
• The Result: He writes a new, foolproof recipe (the Left Kan Extension) that takes your small mixing bowl (the local disk) and tells you exactly how to bake the whole cake (the sphere) without it ever exploding.

In short: This paper builds a sturdy bridge between the shape of space and the numbers of quantum physics, proving that if you respect the geometry, the math works perfectly.

Technical Summary

1. Problem Statement and Motivation

The Core Problem:
Quantum Field Theory (QFT) possesses two distinct mathematical faces:

1. Geometric/Algebraic: Described by factorization algebras (Beilinson–Drinfeld, Costello–Gwilliam), which provide a framework for local-to-global constructions but are often perturbative or topological.
2. Analytic/Operator-Theoretic: Described by Hilbert spaces and unitary representations (Gårding–Wightman axioms), where quantum fields are typically unbounded operators.

The challenge lies in unifying these perspectives. Specifically, can one construct a "metric-dependent" factorization homology (a geometric theory) that naturally incorporates the analytic structure of Conformal Field Theory (CFT), where operators must be bounded to fit into a categorical framework (specifically, the category of Ind-Hilbert spaces)?

The Obstacle:
In standard conformal geometry, the category of conformal embeddings allows configurations where the boundaries of embedded disks touch or intersect. In the context of free scalar fields, such configurations lead to unbounded operators on Hilbert spaces, making it impossible to define a functor into the category of bounded operators (or Ind-Hilbert spaces) in a straightforward way.

The Goal:
To define a metric-dependent variant of factorization homology for $d \geq 2$ in conformally flat Riemannian geometry, taking values in the category of Ind-Hilbert spaces ($\text{IndHilb}$), such that:

• It yields symmetric monoidal invariants of conformally flat manifolds.
• It reproduces the sphere partition function of the associated CFT.
• It provides explicit, non-perturbative examples for $d \geq 3$.

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2. Methodology and Framework

The author constructs a new geometric framework to resolve the unboundedness issue by refining the domain category.

A. Refined Geometric Categories

• $\text{Mfld}^{CO}_d$: A category of "germs" of Riemannian manifolds. Objects are triples $(U, g, M)$, where $M$ is a compact core (possibly with boundary) and $U$ is an open neighborhood. Morphisms are conformal open embeddings defined on neighborhoods of the cores.
• $\text{Disk}^{CO}_d$: The full monoidal subcategory generated by the unit disk $D^d$.
• The Key Constraint: The author distinguishes between the "naive" operad of conformal embeddings ($CE^{emb}_d$) and a refined operad ($CE_d$).
  • In $CE^{emb}_d$, embeddings can have boundaries that touch ($\partial f_i(D) \cap \partial f_j(D) \neq \emptyset$).
  • In the refined $CE_d$, embeddings must have disjoint closures ($f_i(D) \cap f_j(D) = \emptyset$).
  • Crucial Insight: For $d \geq 3$, the condition of disjoint closures is the necessary and sufficient condition for the associated operators to be bounded.

B. Target Category: Ind-Hilbert Spaces

• The target category is $\text{IndHilb}$, the ind-category of separable Hilbert spaces.
• Conformally Flat $d$-Disk Algebras: Defined as symmetric monoidal functors $A: \text{Disk}^{CO}_d \to \text{IndHilb}$.
• Hilbert Space Filtration: The algebra $A$ is equipped with a filtration $V = \bigcup H_k$ where each $H_k$ is a Hilbert space. This allows the definition of continuous states and ensures that the operators defining the algebra structure are bounded on the completed tensor products.

C. Left Kan Extension

The global theory is constructed via the Left Kan extension of the local algebra $A$ along the inclusion $\text{Disk}^{CO}_d \hookrightarrow \text{Mfld}^{CO}_d$.
$$\text{Lan}_A: \text{Mfld}^{CO}_d \to \text{IndHilb}$$
This functor assigns a Hilbert space (or Ind-Hilbert object) to any conformally flat manifold, effectively "gluing" local data into global invariants.

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3. Key Contributions and Results

1. Formulation of Conformally Flat Disk Algebras

The paper rigorously defines $CE_d$-algebras in $\text{IndHilb}$. It establishes that the left Kan extension of such an algebra yields a symmetric monoidal functor, providing metric-dependent invariants for conformally flat manifolds.

2. The Boundedness Criterion (Theorem 2.31)

This is the paper's most significant technical result.

• Result: For $d \geq 3$, the operator associated with a pair of conformal embeddings $(f_1, f_2)$ is bounded if and only if the images of the disks have disjoint closures (i.e., $(f_1, f_2) \in CE_d(2)$).
• Implication: If the closures touch (as allowed in the naive category $CE^{emb}_d$), the operator becomes unbounded. This geometric constraint naturally selects the correct category for a rigorous operator-algebraic formulation of CFT.

3. Construction of Explicit Examples (Theorem 2.33)

For $d \geq 3$, the author constructs explicit examples of $CE_d$-algebras using harmonic polynomials and reproducing kernel Hilbert spaces.

• Construction:
  • Start with the space of harmonic polynomials on $\mathbb{R}^d$.
  • Equip it with a specific inner product derived from the representation theory of the conformal group $SO^+(d, 1)$.
  • Complete this space to a Hilbert space $H$.
  • Define the algebra on the symmetric algebra $V = \text{Sym}(H)$.
• Properties:
  • The monoid action satisfies Reflection Positivity (Unitarity condition U) and Dilation conditions (Condition D).
  • The higher operations (Wick contractions) are constructed using the Green's function of the Laplacian and proven to be bounded operators due to the disjoint closure condition.
  • These algebras correspond to the CFT of the massless free scalar field.

4. Partition Functions and States (Theorem 3.22)

The paper analyzes the "states" of the theory, which correspond to linear functionals on the factorization homology (partition functions).

• Result: Under positivity and continuity assumptions, the space of conformally invariant continuous states on the standard sphere $S^d$ is one-dimensional.
• Significance: This unique state corresponds to the sphere partition function of the associated CFT. The left Kan extension successfully reproduces the expected physical quantity.

5. Simplicity and Ideals

The paper introduces the notion of simple $CE_d$-algebras (those with no non-trivial closed ideals). It shows that for the constructed examples, the vacuum expectation value is non-degenerate, implying the algebra is simple. This parallels the role of simple Vertex Operator Algebras (VOAs) in 2D CFT.

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4. Significance and Impact

• Bridging Geometry and Analysis: The paper successfully places a metric-dependent QFT (specifically CFT) into a categorical framework (factorization homology) that respects the analytic constraints of unbounded operators. It resolves the tension between the geometric flexibility of conformal maps and the analytic rigidity required for bounded operators.
• Non-Perturbative Construction: Unlike many factorization algebra constructions that rely on formal power series (perturbation theory), this work provides non-perturbative, explicit examples in dimensions $d \geq 3$ using unitary representations.
• Geometric Criterion for Boundedness: The discovery that "disjoint closures" is the precise geometric condition for boundedness offers a new perspective on the structure of CFTs and suggests a deep link between the geometry of the configuration space and the analytic properties of the theory.
• Generalization of VOAs: The framework generalizes the algebraic structures of 2D VOAs to higher dimensions in a way that is compatible with the conformal group $SO^+(d, 1)$ and the geometry of Riemannian manifolds, moving beyond the holomorphic constraints of 2D.
• Foundation for Future Work: The paper sets the stage for relating this framework to bordism categories (Segal's axioms) and exploring the relationship with the universal Teichmüller space in dimension 2.

In summary, Moriwaki provides a rigorous mathematical foundation for "metric-dependent factorization homology," demonstrating that by carefully restricting the geometric category to ensure operator boundedness, one can construct a robust, non-perturbative theory of conformal field theories in dimensions $d \geq 3$ valued in Ind-Hilbert spaces.