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Towards First Quantisation Formalism for AKSZ Theories

This paper formulates a "first quantisation" 1-dimensional AKSZ theory on graphs that reproduces the Feynman graphs of a given AKSZ theory $\mathbb{T}$ by interpreting gauge-fixing choices as Lagrangian submanifolds defining sewing conditions within the BV-BFV formalism, thereby establishing a bridge between the cohomological structure of $\mathbb{T}$ and a cyclic $\mathrm{L}_\infty$ -algebra in Weinstein's symplectic category.

Original authors: Leon Menger, Pavel Mnev

Published 2026-07-30

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CC BY 4.0

Original authors: Leon Menger, Pavel Mnev

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 the universe not as a collection of solid balls and springs, but as a vast, invisible web of relationships. In the strange world of theoretical physics, specifically a branch called "topological quantum field theory," scientists study shapes and spaces that don't care about distance or time, only about how things are connected. Think of it like a game of connect-the-dots where the lines themselves have magic properties. To understand these connections, physicists often use a tool called "Feynman diagrams." These aren't just pictures; they are complex mathematical recipes that look like spaghetti graphs, with dots (vertices) where things interact and lines (edges) where they travel. Calculating the behavior of a system usually means adding up the contributions of billions of these spaghetti graphs, a task that can get incredibly messy and hard to solve.

The paper you are about to explore tackles a specific, tricky type of these theories known as AKSZ theories. These are like the "super-charged" versions of the spaghetti graphs, involving layers of hidden dimensions and symmetries that make the math even harder to untangle. The big question the authors ask is: Can we find a simpler, more direct way to calculate the results of these complex theories? Instead of wrestling with the giant, multi-dimensional equations of the original theory, they wonder if there is a "first quantization" picture—a way to view the problem as a simple, one-dimensional journey along a line or a graph. If they can find this simpler view, they could turn the impossible math of high-dimensional physics into a manageable walk along a path.

The Paper's Journey: Turning 3D Chaos into a 1D Walk

Leon Menger and Pavel Mnev have built a bridge between two very different ways of looking at physics. On one side is the complex, high-dimensional world of AKSZ theories (let's call this the "Big World"). On the other side is a much simpler, one-dimensional world they call "theory $t$ " (let's call this the "Tiny Walk"). Their main finding is that they can construct this "Tiny Walk" so perfectly that if you walk along it and add up all the possibilities, you get the exact same answer as the complicated calculations in the "Big World."

Here is how they do it, using a few playful metaphors:

1. The Spaghetti Graph vs. The Hiker
Imagine the "Big World" theory as a massive, tangled ball of spaghetti. To understand it, you have to look at every noodle, every knot, and how they cross each other in 3D space. The authors propose that instead of staring at the whole ball, you can send a hiker (our "Tiny Walk") along a single path. This hiker doesn't just walk; they carry a backpack full of "sewing instructions." When the hiker reaches a fork in the road (a vertex), they use these instructions to decide how to stitch the path together. The magic is that the hiker's journey, when summed up over all possible paths, recreates the entire tangled spaghetti ball.

2. The Magic Sewing Machine (The Vertices)
In the complex theory, the "dots" where lines meet are defined by complicated algebraic rules. In the authors' new picture, these dots are replaced by special "Lagrangian submanifolds." That's a fancy math term, but think of them as magic sewing machines. When the hiker arrives at a vertex, the sewing machine takes the threads coming in and stitches them together in a very specific, pre-programmed way.

For a theory involving the math of $su(2)$ (a specific type of symmetry), this sewing machine is called the "Wigner Lagrangian." It's like a machine that knows exactly how to arrange three points on a sphere to form a perfect equilateral triangle.
The authors suggest that these sewing machines are the "dequantized" versions of the complex algebra rules. In other words, they are the geometric, physical shapes that hide inside the abstract math.

3. The Gravity of the Path
The "Tiny Walk" isn't just a flat line; it's a path that can stretch and shrink. The authors introduce a "supergravity" component to this path, which is like giving the hiker a flexible ruler that can change length. This flexibility is crucial. By adjusting the length of the path, the hiker can simulate different "gauge fixings" (different ways of choosing coordinates) in the original complex theory.

If the hiker walks on a path of infinite length, they arrive at a "residual field," which is like a snapshot of the system's most stable state.
If the path is short, they capture the "propagator," which is the rule for how things move from one point to another.
The paper shows that by tuning this "gravity" of the path, the hiker can reproduce the exact rules needed to calculate the Feynman graphs of the original theory.

What They Found and What They Suggest

The authors have successfully formulated this "Tiny Walk" (theory $t$ ) for a wide class of AKSZ theories. They demonstrate that:

The Partition Function Matches: If you calculate the total "weight" of the hiker's journey on a graph, it matches the weight of the corresponding Feynman graph in the original theory.
The Sewing Conditions Work: They provide a concrete recipe for how the "sewing machines" (vertices) should work. For example, in the case of Chern–Simons theory (a famous topological theory), they describe exactly how the Wigner Lagrangian stitches the paths together.
It's a "First Quantization": They frame this as a "first quantization" formalism. In physics, "second quantization" usually deals with fields and particles popping in and out of existence. "First quantization" is often simpler, dealing with single particles moving along paths. The authors suggest that the complex, multi-particle world of AKSZ theories can be understood as a collection of these simpler, single-particle walks on graphs.

A Note on Certainty and Limits

It is important to note that while the authors have built a very strong mathematical framework, some parts of their proposal are still "tentative."

The "Dequantization" Idea: The idea that these complex algebraic rules can be "dequantized" into geometric shapes (like the Wigner Lagrangian) is presented as a promising paradigm and a "tentative definition." They have proven it works for specific cases like $su(2)$ (the math behind the spin of electrons), but for more complex algebras, they are suggesting a path forward rather than claiming a finished proof.
The Conjecture: They propose a conjecture (a guess based on strong evidence) that for certain types of Lie algebras, the space of these sewing shapes reduces to a single point. They provide dimension counts to support this, showing that the math "adds up" to zero in a way that suggests it's true, but they don't claim to have proved it for every possible case yet.
No "Magic" Solutions: The paper does not claim to solve all of physics or to make these theories easy to calculate by hand for everyone. Instead, it offers a new, cleaner language to describe them. It suggests that the messy sum of Feynman graphs is actually just the result of a simpler, one-dimensional theory walking along a graph with specific sewing rules.

In essence, Menger and Mnev have handed us a new pair of glasses. Through these glasses, the terrifyingly complex, multi-dimensional spaghetti of topological quantum field theory transforms into a charming, one-dimensional walk along a graph, guided by a hiker with a very specific set of sewing instructions. While the full picture for every possible theory is still being stitched together, the pattern they have found is beautiful, consistent, and suggests a deep, hidden simplicity in the fabric of these quantum worlds.

Technical Summary: Towards First Quantisation Formalism for AKSZ Theories

Problem Statement

The paper addresses the challenge of formulating a "first quantisation" picture for Alexandrov–Kontsevich–Schwarz–Zaboronsky (AKSZ) topological field theories. Specifically, the authors seek a 1-dimensional field theory $t$ defined on metric graphs such that its partition functions reproduce the Feynman graph weights of a given $m$ -dimensional AKSZ theory $T$ .

The standard effective action of an AKSZ theory $T$ (with target a graded vector space $Y$ ) is expressed as a sum over connected Feynman graphs $\Gamma$ :
$S^{\text{eff}}_T(a) = \sum_{\Gamma} \frac{\hbar^{l(\Gamma)}}{|\text{Aut}(\Gamma)|} F_\Gamma(a)$
where $F_\Gamma(a)$ involves integrals over configuration spaces of the manifold $M$ , propagators, and interaction vertices derived from the target Hamiltonian $\Theta_Y$ . The goal is to construct a 1-dimensional theory $t$ whose amplitudes on graphs naturally generate these weights $F_\Gamma$ , effectively "dequantising" the cohomological vector field on the target $Y$ into geometric data (Lagrangian submanifolds) in a symplectic category.

Methodology

The authors employ the Batalin–Vilkovisky (BV) and Batalin–Fradkin–Vilkovisky (BFV) formalisms to construct and analyze the 1-dimensional theory $t$ . The methodology proceeds through four main stages:

1. Higher Topological Quantum Mechanics (HTQM) on Graphs

The authors build upon Andrey Losev's notion of Higher Topological Quantum Field Theory (HTQFT). They define a 1-dimensional HTQM on metric graphs using a quantum cyclic $L_\infty$ -algebra (qc $L_\infty$ -algebra) structure on a cochain complex $V$ .

Data: A complex $(V, Q)$ , a degree $-1$ operator $G$ , a pairing $\langle -, - \rangle$ , and cyclic operations $c^k_n$ .
Construction: The theory assigns propagators $Z(I; T, dT) = e^{-TH - dTG}$ to edges (where $H=[Q,G]$ ) and oriented operations $l^k_{p,q}$ to vertices.
Result: The amplitudes on the moduli space of metric graphs satisfy the closedness equation $(d_{\text{Geom}} + Q')I_\infty = 0$ , recovering the structure of a qc $L_\infty$ -algebra on the cohomology $\ker(H)$ .

2. Construction of the 1d AKSZ Theory $t$

To realize the HTQM described above as a path integral, the authors construct a specific 1-dimensional AKSZ theory $t$ coupled to 1-dimensional topological supergravity (SUGRA).

Target Space: $N = T^*(T[1](M \times \mathbb{R}[1]))$ , where $M$ is the source manifold of the original theory $T$ . This includes the sigma-model sector (target $T^*(T[1]M)$ ) and the SUGRA sector (target $T^*(T[1]\mathbb{R}[1])$ ).
Fields: Superfields include coordinates $(x, \theta)$ on $M$ and $(c, \phi)$ on the supergravity line.
BV-BFV Analysis: The theory is analyzed in the BV-BFV framework. The SUGRA sector is simplified via a BV-pushforward to bulk zero-modes and a reduction of the boundary state space to its cohomology.
Gauge-Fixing: A "good" gauge-fixing fermion $\Psi$ $Ψ$ is introduced. This leads to a gauge-fixed partition function on an interval that acts as the integral kernel for the operator $e^{-e_I H - \epsilon_I G}$ $e^{- e_{I} H - ϵ_{I} G}$ .
- For internal edges, integrating over SUGRA zero-modes yields the propagator $K = \int_0^\infty G e^{-TH}$ .
- For external edges (leaves), taking the length to infinity yields the projector $\Pi$ onto $\ker(H)$ , identifying the space of residual fields.

3. Vertex Structure and Dequantisation

The paper proposes a geometric ansatz for the "vertex data" required to sew the 1-dimensional theory on a graph.

Hypothesis: The cyclic $L_\infty$ -algebra structure on the target $Y$ of theory $T$ arises from the geometric quantisation of a cyclic $L_\infty$ -algebra in Weinstein's symplectic category (a "c $L^\text{symp}_\infty$ -algebra").
Definition: A c $L^\text{symp}_\infty$ -algebra consists of a symplectic manifold $\nu$ (with prequantum line bundle and polarisation) and a sequence of Lagrangian submanifolds $L_n \subset \nu^{\times n}$ (and a pairing $\rho$ ) satisfying $L_\infty$ relations modulo the kernel of the quantisation functor.
Application: The authors suggest that the interaction vertices of $T$ (derived from $\Theta_Y$ ) correspond to Lagrangian submanifolds in $\nu^{\times n}$ that serve as sewing conditions for the fields of $t$ at the vertices of the graph.

4. Path Integral on Graphs

The final construction defines the theory $t$ on a graph $\Gamma$ by:

Taking the product of the 1d AKSZ theories on each edge.
Imposing sewing conditions at vertices defined by the Lagrangians $L_n$ (quantising the vertex tensors).
Integrating over the moduli of edge lengths and SUGRA zero-modes.
This path integral reproduces the pre-amplitudes of the effective action of $T$ .

Key Contributions and Results

Formulation of 1d HTQM: The authors explicitly construct a 1-dimensional AKSZ theory coupled to SUGRA that realizes the HTQM framework. They demonstrate that the (modified) Quantum Master Equation (mQME) for this 1d theory corresponds to the closedness equation of HTQM.
Reproduction of Feynman Weights: They prove that the partition function of this 1d theory $t$ on a metric graph, when integrated over the moduli of edge lengths, reproduces exactly the weight $F_\Gamma(a)$ of the corresponding Feynman graph in the original AKSZ theory $T$ .
Geometric Dequantisation of Vertices: The paper proposes a tentative dictionary where the algebraic vertex tensors of an AKSZ theory are "dequantised" into Lagrangian submanifolds in a symplectic category.
- Example ($su(2)$): For non-abelian Chern–Simons theory with Lie algebra $su(2)$, the vertex Lagrangian is identified as the Wigner Lagrangian $L_W \subset O \times O \times O$ , where $O$ is the coadjoint orbit corresponding to the adjoint representation. This Lagrangian consists of triples of points on the sphere forming an equilateral triangle centered at the origin.
- Example ($gl(N)$): For $gl(N)$-Chern–Simons, the authors describe a gauge-fixing (Fukaya-Morse-Witten gauge) where the target $\nu$ is a collection of points, and the Lagrangians correspond to intersection conditions of gradient trajectories.
Gauge-Fixing Correspondence: The paper establishes a direct correspondence between gauge-fixing choices in the original theory $T$ and the specific choices of the gauge-fixing operator $G$ (and the resulting Hamiltonian $H$ ) in the 1d theory $t$ . For instance, the Lorenz gauge in Chern–Simons corresponds to the Hodge Laplacian in $t$ , while the Morse-Witten gauge corresponds to the Lie derivative along a gradient flow.

Significance and Claims

The authors claim that this work provides a first quantisation formalism for AKSZ theories.

Interpretation: The sum of Feynman graphs of a topological field theory $T$ is interpreted as the partition function of a 1-dimensional topological quantum mechanics $t$ on graphs.
Conceptual Shift: The paper suggests that the cohomological vector field on the target space of $T$ can be "dequantised" into geometric data (Lagrangians) in a symplectic category. This offers a new perspective on the relationship between the algebraic structure of the effective action and the underlying geometry.
Motivation: The work is motivated by the desire to understand Witten's idea that Chern–Simons theory is a topological string theory. By recovering first-quantised Chern–Simons theory (with specific gauge-fixings), the authors suggest that a large class of AKSZ theories may admit a similar relation to topological strings, potentially viewed as a degeneration of the Witten A-model coupled to 2D topological gravity.

Modesty of Claims:
The authors explicitly state that the definition of the c $L^\text{symp}_\infty$ -algebra and the dictionary between geometric quantisation and vertex tensors are tentative. They acknowledge that making these definitions precise is deferred to future work. The paper presents a paradigm and explicit examples (specifically for $su(2)$ and $gl(N)$) rather than a fully rigorous, general proof for all AKSZ targets. The dimension counts provided for the symplectic reduction in the $su(2)$ case serve as a "sanity check" for their conjecture rather than a definitive proof.