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Moduli spaces of geometric functorial field theories

This paper develops a framework for computing moduli spaces of geometric functorial field theories by defining Cartesian realizations of geometric structures as equivariant simplicial presheaves, thereby presenting these moduli spaces as mapping spaces between such presheaves.

Original authors: Jacek Kenig, Dmitri Pavlov

Published 2026-09-11

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CC BY 4.0

Original authors: Jacek Kenig, Dmitri Pavlov

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

Quantum field theory is the framework physicists use to describe how the smallest particles in the universe behave and interact. For decades, mathematicians have tried to formalize this theory into a precise set of rules, much like writing a rigorous grammar for a language that nature speaks fluently but we struggle to read. Two main approaches have emerged to tackle this problem. One approach, closer to the way we think about forces in a fixed region of space, focuses on the algebra of what can be observed. The other, which this new work embraces, treats the theory as a story of evolution. In this view, the universe is not just a static collection of objects, but a series of transitions. You start with a shape representing a moment in time, and the theory tells you how that shape changes as it moves through space and time, evolving into a new shape. The challenge has always been to capture not just the topological shape of these transitions, but the specific geometric details that matter in the real world, such as the length of a path or the curvature of a surface.

A team of mathematicians, Jacek Kenig and Dmitri Pavlov, has developed a new set of tools to solve a specific problem in this field: how to map out all the possible ways a geometric quantum field theory can exist. Imagine trying to catalog every possible rulebook for how a particle could travel through a curved space. Previously, doing this required navigating a labyrinth of complex, high-dimensional shapes that were incredibly difficult to compute. The researchers have found a way to flatten this landscape. They proved that the vast, complicated space of all possible theories can be reduced to a much simpler, more manageable form. Instead of wrestling with the full complexity of every possible geometric variation, their method allows physicists to calculate the space of theories by looking at how they behave on the simplest possible building blocks: flat, rectangular grids of space.

The core of their discovery lies in a technique they call "Cartesian realization." In the world of quantum field theory, a "bordism" is a geometric object that connects two shapes, representing the history of a system evolving from one state to another. When these objects carry geometric data, like a Riemannian metric which defines distance and angles, the rules for how they can be combined become incredibly intricate. The authors showed that any complex family of these geometric objects can be faithfully represented by a simpler object defined only on Cartesian spaces—essentially, flat Euclidean spaces like a line, a plane, or a cube. They demonstrated that the entire moduli space, which is the mathematical map of all possible theories, can be reconstructed from how these theories behave on these simple, flat families.

This reduction is powerful because it separates the general theory of how shapes evolve from the specific details of the geometry they carry. By translating the problem into the language of "mapping spaces" between these simplified objects, the researchers turned a problem that was previously intractable into one that standard tools of homotopy theory can handle. Homotopy theory is a branch of mathematics that studies shapes by focusing on how they can be continuously stretched or deformed without tearing. By converting the geometric field theories into this language, the authors provided a clear, computational path forward. They showed that the space of all possible theories is equivalent to the space of all possible ways to map a specific, simplified geometric structure into a target category that represents the physical states of the system.

To illustrate the power of this method, the paper applies it to a one-dimensional case, which corresponds to the movement of a particle along a line. In this specific scenario, the geometric structure is simply the length of the path. The researchers showed that their method correctly recovers the known results for this case, proving that the complex machinery they built works. They found that the space of theories for a particle moving on a line is determined by how the theory assigns values to paths of different lengths. This confirms that their approach preserves the essential physical information, such as the fact that a longer path takes more time to traverse, while discarding the unnecessary topological complexity that made previous calculations so difficult.

The significance of this work extends beyond just one dimension. The authors have provided a general framework that can be applied to any dimension and any type of geometric structure, whether it involves Riemannian metrics, bundles, or other geometric data relevant to physics. They have effectively built a bridge between the abstract world of high-dimensional geometry and the concrete world of computation. By showing that the moduli space of these theories can be presented as a mapping space between simpler objects, they have opened the door for physicists to classify and compute these theories in ways that were previously impossible. This does not solve every problem in quantum field theory, but it provides the essential geometric and homotopical foundation needed to tackle the next generation of questions, allowing researchers to finally compute the spaces of theories that govern the behavior of the universe.

Technical Summary: Moduli Spaces of Geometric Functorial Field Theories

Problem Statement
The paper addresses the computational difficulty of determining the moduli spaces of geometric functorial field theories (FFTs). While the Grady–Pavlov framework [GP22, GP23, GP26] successfully formalizes geometric FFTs—incorporating smooth families, descent, and isotopies into the bordism category—the resulting moduli spaces are often presented as derived mapping objects between complex $(\infty, d)$ -categories. Specifically, the space of $d$ -dimensional FFTs with a geometric structure $F$ is defined as the derived mapping space $\text{RMap}(\text{Bord}^F_d, V)$ , where $\text{Bord}^F_d$ is the bordism category equipped with structure $F$ , and $V$ is a target geometric symmetric monoidal $(\infty, d)$ -category.

The challenge lies in the fact that the source $\text{Bord}^F_d$ can possess a complicated homotopy type, making direct computation of these mapping spaces intractable using standard homotopy-theoretic methods. The authors seek to reduce this problem to a form amenable to explicit calculation by expressing the moduli space as a mapping space between $O(d)$ -equivariant simplicial presheaves on the site of Cartesian spaces ( $\mathbb{R}^n$ ), thereby isolating the geometric structure from the general bordism theory.

Methodology
The authors develop a toolkit based on enriched model category theory, specifically utilizing:

Sites of Fiberwise Embeddings: They work with the site $\mathcal{C}^\infty\text{Embd}$ of smooth families of $d$ -manifolds (fiberwise embeddings) and its enriched counterpart $\mathcal{C}^\infty\text{Embd}$ (incorporating isotopies).
Isotopification: They utilize the isotopification functor $\text{Id}$ , which maps geometric structures without isotopies to those with isotopies. This functor is shown to be a left Quillen functor for Čech-local injective model structures.
Cartesian Restriction: A key step involves restricting geometric structures from the full site of fiberwise embeddings to the "Cartesian subsite" $\mathcal{C}^\infty\text{EmbCart}_d$ , consisting of families isomorphic to projections $\mathbb{R}^d \times U \to U$ . The authors prove that this restriction induces a Quillen equivalence, meaning the local homotopical information is preserved.
Cartesian Realization: They define a "Cartesian realization" functor $C_d$ , which transforms a geometric structure (restricted to the Cartesian subsite) into an $O(d)$ -equivariant simplicial presheaf on $\mathcal{C}^\infty\text{Cart}$ . This is achieved via a weighted colimit (derived coend) using a specific $O(d)$ -equivariant Cartesian coefficient $K_d$ .
Equivariant Mapping Spaces: The target of the moduli space is transformed using the restriction functor $\rho^*_{\infty, d}$ , which evaluates the field stack on standard Cartesian families and encodes the $O(d)$ -action induced by fiberwise orthogonal maps.

Key Contributions and Results
The central result is Theorem 4.2.9, which establishes a natural weak equivalence between the moduli space of $d$ -dimensional functorial field theories with geometric structure $F$ and a mapping space of equivariant presheaves:

$\text{FFT}^{\text{Id}F}_{d,V} \simeq C^\infty\text{sSet}\left( \lambda^\infty_{O(d)} C_d q^*(F), \rho^*_{\infty, d} q^* \text{FFT}_d(V) \right)$

Where:

$\text{FFT}^{\text{Id}F}_{d,V}$ is the smooth simplicial set of $d$ -dimensional FFTs with structure $F$ valued in $V$ .
$q^*$ denotes restriction to the Cartesian subsite.
$C_d$ is the derived Cartesian realization functor.
$\lambda^\infty_{O(d)}$ regards a simplicial presheaf as a smooth simplicial presheaf constant in the smooth direction, equipped with an $O(d)$ -action.
$\rho^*_{\infty, d}$ is the restriction to standard Cartesian families, yielding an $O(d)$ -equivariant presheaf.
$\text{FFT}_d(V)$ is the field stack of $V$ -valued theories.

This theorem effectively separates the computation of the moduli space into two parts:

The Source: A computable $O(d)$ -equivariant presheaf derived solely from the geometric structure $F$ .
The Target: A presheaf derived from the target category $V$ . If $V$ has duals, the geometric framed cobordism hypothesis implies the target is weakly equivalent to the core $V^\times$ (objects and equivalences), potentially simplifying the target further.

Specific Application (Dimension 1)
In Corollary 4.2.11, the authors apply this framework to 1-dimensional oriented Riemannian field theories ( $F = R_M$ ) valued in a target $V$ with duals. They show:
$\text{FFT}^{I_1 R_M}_{1,V} \simeq C^\infty\text{sSet}\left( \lambda^\infty N(R\text{Path}^g_M), V^\times \right)$
Here, $R\text{Path}^g_M$ is the Riemannian path category. This result reduces the moduli space to maps from the nerve of the Riemannian path category to the core of the target, explicitly handling the smooth variation of lengths and the $O(1)$ -action (orientation reversal).

Significance and Claims
The paper claims to provide a "simpler form" of the moduli space that is "more amenable to the usual computational methods of homotopy theory." By reducing the problem to mapping spaces between $O(d)$ -equivariant simplicial presheaves on Cartesian spaces, the authors separate the general bordism theory from structure-specific computations.

The authors explicitly state that this work supplies the "geometric and homotopical input common to later problems," specifically:

Constructing analytically appropriate geometric target categories (e.g., suitable versions of Hilbert spaces).
Computing the derived mapping spaces to classify FFTs for specific targets.
Developing quantization procedures compatible with the functorial framework.

The paper does not perform these subsequent computations or construct the specific analytic targets (like rigged Hilbert spaces) itself, noting these as directions for future work. It focuses strictly on establishing the homotopical machinery required to present the moduli spaces in a computable form.

Technical Summary: Moduli Spaces of Geometric Functorial Field Theories

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