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Every AKSZ Feynman graph is the partition function of a 1D theory

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2026-08-01 16:50 UTC pith:QULJUJNO

load-bearing objection Operator construction is solid; the abstract's general 1d-AKSZ path-integral claim outruns the tentative dequantisation in §4. the 3 major comments →

arxiv 2607.26394 v1 pith:QULJUJNO submitted 2026-07-29 math-ph hep-thmath.MP

Towards First Quantisation Formalism for AKSZ Theories

Leon Menger , Pavel Mnev This is my paper
classification math-ph hep-thmath.MP MSC 81T4553D50
keywords AKSZ theoryfirst quantizationtopological quantum mechanicsBV-BFV formalismcyclic L∞-algebrasymplectic categorycoadjoint orbitChern-Simons theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper claims that any AKSZ field theory—a broad class of topological field theories including Chern-Simons and BF theory—can be rewritten in a 'first quantised' form. The central statement is that for a given AKSZ theory T, one can construct a 1-dimensional AKSZ theory t on metric graphs, coupled to 1d supergravity, whose partition function on any graph Γ equals the Feynman weight of Γ in T. The construction is carried out in the BV-BFV formalism: the graph's edge length and its de Rham differential are interpreted as graviton and gravitino zero modes, so the propagator of T emerges from an interval integral, and vertices are encoded by Lagrangian submanifolds in a symplectic category. If the claim works, the entire Feynman-diagram expansion of T is a single 1d partition function, giving a concrete bridge between AKSZ theories and topological quantum mechanics. The paper works out the examples of Chern-Simons theory, where the vertex Lagrangian is the SU(2) submanifold of triples summing to zero, and BF theory via doubled coadjoint orbits.

Core claim

The paper's central claim is the identity (Equation 5.1): for every connected Feynman graph Γ of an AKSZ theory T, the integral over the positive edge lengths of the 1d theory's pre-amplitude I∞_t(Γ)(a) equals the Feynman weight F_Γ(a). The 1d theory t is itself an AKSZ theory with target T^*(T[1](M×R[1])) extended by a symplectic-category object ν that 'dequantises' the target Y of T. The edge length T and its differential dT are the zero modes of the graviton and gravitino of 1d supergravity; a gauge-fixing of T corresponds to a gauge-fixing of t. The vertex structure of t is classical geometric data: Lagrangian submanifolds L_n in Cartesian powers of the phase space, which by geometric qu

What carries the argument

The load-bearing machinery is the pairing of two constructions. (1) A 1d AKSZ theory t on an interval, with target T^*(T[1](M×R[1])), coupled to 1d supergravity by adding the factor T^*(T[1]R[1]). After integrating out the supergravity sector via BV-pushforward and reducing the boundary states, the interval partition function becomes e^{-T H - dT G}, where G is a degree -1 gauge-fixing operator (a chain homotopy) and H=[d_M,G]. The propagator of T is then the integral of this over edge length T>0. (2) The vertex structure: a tentative definition (Definition 4.4) of a cyclic L∞-algebra in the symplectic category—an object ν with a pairing Lagrangian ρ and interaction Lagrangians L_n, satisfyi

Load-bearing premise

The construction depends on the tentative assumption (Definition 4.4) that every AKSZ target Y admits a 'dequantisation' ν in the symplectic category—an object with Lagrangians whose geometric quantisation reproduces the cyclic L∞ operations of Y—together with the unproved Conjecture 4.9 identifying the vertex Lagrangian for simple Lie algebras via a point-like symplectic reduction.

What would settle it

Find an AKSZ theory whose target Y has a cyclic L∞ structure that cannot be obtained as the geometric quantisation of Lagrangian submanifolds in a symplectic category; or compute the symplectic reduction µ^{-1}(0)/G for the adjoint coadjoint orbit of, say, so(7) or sp(6) and show it is not a point, contradicting Conjecture 4.9 and thus the explicit vertex construction for those structure groups.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If correct, every AKSZ theory admits a first-quantised formulation: the Feynman graph sum is literally the partition function of a 1d AKSZ theory on metric graphs, making the 'second-quantised' vs 'first-quantised' relation explicit.
  • Gauge-fixing equivalence: choosing a gauge-fixing chain homotopy G on forms (e.g., Lorenz gauge G=d*, or a gradient-flow gauge G=i_v+εd*) is the same as choosing the gauge in the 1d theory; so standard gauge choices for Chern-Simons and BF theory reappear as choices of 1d path-integral measure.
  • The quantum master equation for the effective action of T follows from the closedness equation of the 1d theory, giving a structural, graph-combinatorial explanation of why effective actions satisfy the QME.
  • The vertex data gives a geometric quantisation dictionary for cyclic L∞-algebras: the paper's paradigm suggests that the interaction tensors of an AKSZ theory are quantisations of Lagrangian submanifolds in a symplectic category, with the SU(2) case worked out explicitly via the vertex Lagrangian of triples summing to zero.
  • The construction realises a large class of AKSZ theories as 'topological quantum mechanics on graphs', connecting them to the intersection theory of gradient trajectories, as in the gl_N Chern-Simons example with a gradient-flow gauge.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper's vertex construction is explicitly tentative; if it can be made precise, it would supply a functorial version of geometric quantisation for cyclic L∞-algebras, likely with implications for quantising higher Lie algebras (e.g., in the Poisson sigma model).
  • The identity (5.1) suggests a graph-complex interpretation: the integral over edge lengths is reminiscent of graph integrals appearing in deformation quantisation, so the 1d theory may provide a new combinatorial route to the cohomology of moduli spaces of graphs or to finite-type invariants of manifolds.
  • The paper's Conjecture 4.9 (that the symplectic reduction of three adjoint coadjoint orbits is a point for simple Lie algebras other than su(N≥3)) is a concrete, testable geometric statement; if true, it would uniquely pick out the vertex Lagrangian for Chern-Simons theory for any simple compact Lie algebra.
  • For BF theory, the construction uses a doubled coadjoint orbit; this hints at a first-quantised description of BF theory as a 'particle on a doubled group manifold', which might be extended to higher-dimensional AKSZ theories such as the Poisson sigma model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a first-quantisation formalism for AKSZ field theories. For an AKSZ theory T with fields Ω•(M;Y) and target Hamiltonian Θ_Y, the authors construct a one-dimensional theory t on metric graphs. Section 3 builds a 1d AKSZ sigma-model coupled to 1d supergravity, reduces the SUGRA sector by BV pushforward, and shows that a suitable gauge-fixing gives the interval partition function e^{-e^I H - ε^I G}, so that the propagator identity (3.33) reproduces the propagator of T. Section 5.1 gives a Hamiltonian/HTQM presentation in which the pre-amplitude of t integrates to the Feynman weight F_Γ(a) of T, Eq. (5.1). Sections 4 and 5.2 then propose a Lagrangian/path-integral description in which vertices are decorated by Lagrangian submanifolds in a symplectic category, encoded in a tentative cL^{symp}_∞-algebra whose geometric quantisation is supposed to reproduce the cyclic L∞-operations of Y. Worked examples include Chern–Simons theory (including an explicit SU(2)/Wigner Lagrangian), BF theory, and gl(N) Chern–Simons theory in a Fukaya–Morse–Witten gauge.

Significance. If the conjectural part can be completed, the construction would connect BV-BFV quantisation of AKSZ models, Losev's HTQFT, and geometric quantisation, and would give a graph-model repackaging of the perturbative expansion of a general AKSZ theory. The paper is transparent about its tentative assumptions and contains a solid, parameter-free operator-formalism core: the SUGRA reduction and the derivation of the operator e^{-e^I H - ε^I G} in §3, the propagator identity (3.33), and the HTQM amplitude identity (5.1) in §5.1 are explicit and checkable. The main value of the paper lies in this core plus the concrete SU(2) example. The advertised path-integral version of the central claim, however, is conditional on an unproved existence statement about cL^{symp}_∞-algebra dequantisation.

major comments (3)
  1. [§5.2.1, Definition 4.4, Remark 4.7] The path-integral presentation of the central claim is conditional on an unproved existence statement. Equations (5.3) and (5.19) are stated for an arbitrary AKSZ target (Y, Θ_Y), but §5.2.1 begins with: 'Let (ν,ρ,{L_n}) be a cL^{symp}_∞-algebra corresponding by geometric quantisation to the target Y'. Definition 4.4 is explicitly labelled tentative, Remark 4.7 defers precision to future work, and no theorem proves that geometric quantisation is surjective onto the cyclic L∞ structures arising from Θ_Y. Consequently, the abstract's claim that t is itself a 1d AKSZ theory whose partition functions reproduce the Feynman graphs of T is established only at the Hamiltonian/HTQM level of §5.1, not for the Lagrangian path integral. The §5.1 result is independent of this and is sound, but the path-integral claim needs either a proof in the stated generality or an explicit restriction to a class
  2. [§4.2, Conjecture 4.9, Appendix C] The Wigner Lagrangian identification, used in Example 5.5 for Chern–Simons theory, rests on Conjecture 4.9: that µ^{-1}(0)/G is a point for simple g ≠ su(N≥3). The evidence in Appendix C is only a dimension count. The remark after Eq. (C.1) correctly notes that the actual dimension of the symplectic reduction is dim_vir + 2k, where k is the stabiliser dimension; for g=so(7), dim_vir=0 but k need not vanish, so the reduction may be positive-dimensional. For g=su(3), dim_vir>0. Thus the conjecture is not supported for general simple g. The explicit su(2) case (Example 4.12) is fine, but the general construction of the Lagrangian for the Lie bracket remains open. This should be flagged as a hypothesis wherever it is used in §5.2.4.
  3. [§4.1, Definition 4.4] Even granting existence, Definition 4.4 imposes the L∞ relations (4.5) only modulo the kernel of the geometric quantisation functor Q. Since Q is not injective on morphisms, the same quantised cyclic L∞ operations c_n can come from inequivalent symplectic data; conversely, a failure of a relation by an element of ker Q still quantises to a valid relation, but the geometric 'dequantisation' is not unique. The paper does not discuss faithfulness or injectivity of the quantisation functor, and it is not shown that the vertex sewing condition is uniquely determined by Θ_Y. This does not affect the soundness of §5.1, but it weakens the dictionary on which the path-integral construction in §5.2 is based.
minor comments (4)
  1. [§5.1, bullet list] The sentence 'The operations with nonzero loop number c ≥1 • are all set to zero' is garbled; it should read 'The operations with nonzero loop number are all set to zero.'
  2. [§3.1, Eq. (3.12)] The formula 'Q_t ec_i = dec+eφ' has an erroneous subscript i; it should be 'Q_t ec = dec+eφ' (or the notation for the superfield should be introduced consistently).
  3. [Appendix C, table header] The column header 'dimO generic = dimg−rankg' is ambiguous; the intended meaning is the dimension of a generic coadjoint orbit, not the dimension of O. Please clarify the notation.
  4. [References] The reference [KW96] lacks publication details (journal, volume, pages, or arXiv identifier). Please complete it.

Circularity Check

0 steps flagged

No circularity: the HTQM/Feynman correspondence is an explicit transcription of the gauge-fixed AKSZ data, and the 1d-AKSZ path-integral claim is conditional on a stated tentative dequantisation assumption, which is a gap, not a circle.

full rationale

The central identity (5.1) is not circular. Section 5.1 explicitly builds the topological quantum mechanics t from the same gauge-fixed data as the AKSZ theory T: V = Ω•(M;Y), Q = d, a self-adjoint chain homotopy G, and vertex operations c^0_n = c^Y_n ⊗ ∫_M. Once the internal-edge integral is evaluated as the propagator (3.33), the HTQM graph amplitude is literally the same contraction as the Feynman weight (1.4). This is a faithful realization/transcription of the Feynman rules, not a prediction of those rules from independent first principles; the paper's own language ('we formulate', 'we associate') presents it as a construction. The only load-bearing gap is the passage to the Lagrangian 1d-AKSZ path integral. In §5.2.1 the paper postulates a cL^{symp}_∞-algebra (ν,ρ,{L_n}) whose geometric quantisation reproduces the cyclic L∞-structure of Y, while Definition 4.4 is explicitly labelled tentative, Remark 4.7 defers precision to future work, and Conjecture 4.9 is supported only by dimension counts in Appendix C. These are unproved existence/identification assumptions, not circular reductions: the path-integral statement is conditional on them, but no step is defined in terms of the conclusion it is used to prove. The self-citations ([BLM24], [CMR18], [Mne15], [CLMY22]) supply background HTQFT, BV-BFV, and Wilson-line technology; they are not used to introduce the claimed equivalence, so they do not create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

No fitted numerical parameters: the main free inputs are gauge choices (metric, Morse functions, spins j_i, ε). The central new object ν (the dequantised target) is conjectural for general Y; two axioms (Definition 4.4 existence and Conjecture 4.9) are explicitly labelled tentative. The remaining axioms are standard geometric-quantisation and BV perturbation theory.

axioms (5)
  • domain assumption Good gauge-fixing datum: G is a square-zero degree -1 operator on Ω•(M)⊗Y with H=[d,G] diagonalisable, non-negative real spectrum, and G vanishing on ker(H).
    Section 3.3.1; the propagator identity (3.33) and generalised Hodge decomposition (3.32) require this, so the 1d edge theory reproduces the T-propagator only in such gauges.
  • standard math Geometric quantisation Q:Symp+ → grVect_C exists with monoidal/duality properties and maps Bohr–Sommerfeld Lagrangians to states; [Q,R]=0 for coadjoint orbits.
    Section 4.1 (eq. 4.1) and Section 4.2; used to map vertex Lagrangians to vertex tensors. Standard but invoked as a black box.
  • ad hoc to paper For any target (Y, Θ_Y) of an AKSZ theory there exists a cL^{symp}_∞ structure (ν,ρ,L_n) whose quantisation is the cyclic L∞ algebra of Θ_Y.
    Section 4.1 Definition 4.4 and Remark 4.7; explicitly tentative, no existence proof; needed for the general claim in the abstract.
  • ad hoc to paper Conjecture 4.9: for simple g with g ≠ su(N≥3), the symplectic reduction µ^{-1}(0)/G of O×O×O is a point.
    Section 4.2; used to single out the Wigner Lagrangian L_W=µ^{-1}(0) for su(2); only dimension-count evidence in Appendix C, no proof.
  • standard math The effective action of T has the connected Feynman expansion (1.3)–(1.4) with weights F_Γ.
    Section 1, eqs. (1.3)–(1.4); standard BV perturbation theory, used as the target that t must reproduce.
invented entities (1)
  • cL^{symp}_∞-algebra dequantisation data (ν, ρ, L_n) in Weinstein's symplectic category no independent evidence
    purpose: Classical/geometric input whose geometric quantisation gives the vertex tensors of T; used as sewing conditions at graph vertices.
    Definition 4.4 is proposed as tentative; no independent tests or falsifiable predictions; only examples (su(2), gl(N), BF) are given.

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read the original abstract

\noindent Given an AKSZ theory $\mathbb{T}$ on a manifold $M$, with target a graded vector space $Y$, we formulate a 1-dimensional theory $\mathbb{t}$ on graphs (the ``first quantisation picture for $\mathbb{T}$''), whose partition functions reproduce the Feynman graphs of $\mathbb{T}$. More precisely, the theory $\mathbb{t}$ is itself a 1d AKSZ theory with the target built out of $M$, and involving a coupling to 1d supergravity. It yields a form on the space of metric graphs (with length $T$ of an edge and its de Rham differential $\mathrm{d} T$ interpreted as the zero-modes of the graviton and gravitino, respectively); its integral yields the sum of Feynman graphs of $\mathbb{T}$. We study the theory $\mathbb{t}$ in the BV-BFV formalism; a gauge-fixing of $\mathbb{T}$ corresponds to a gauge-fixing of $\mathbb{t}$. At the classical level, $\mathbb{t}$ assigns to vertices certain Lagrangian submanifolds $L_k$ in Cartesian powers $\Phi^{\times k}$ of the phase space $\Phi$ of $\mathbb{t}$. These submanifolds can be thought of as defining a cyclic $\mathrm{L}_\infty$-algebra in Weinstein's symplectic category (``dequantising'' the cohomological vector field on the target %target AKSZ dg structure of $\mathbb{T}$). In the path integral construction of $\mathbb{t}$, Lagrangians $L_k$ determine sewing conditions for fields on the incident edges at a $k$-valent vertex. We give examples of this paradigm, such as when $\mathbb{t}$ on edges is the Witten-Morse supersymmetric quantum mechanics (which corresponds to a particular type of gauge-fixing for $\mathbb{T}$ and $\mathbb{t}$). In the example where $\mathbb{T}$ is the non-abelian Chern--Simons theory with structure Lie algebra $\mathfrak{su}(2)$, we describe the vertex Lagrangian $L_{\mathrm{W}}$ (the ``Wigner Lagrangian'' ).

Figures

Figures reproduced from arXiv: 2607.26394 by Leon Menger, Pavel Mnev.

Figure 1
Figure 1. Figure 1: Decoration of a Feynman graph appearing in the effective action of non-abelian Chern– Simons theory with coefficients in a quadratic Lie algebra g. Internal edges are decorated by a propagator K, interaction vertices (vertex tensors) are given by wedging forms and taking the Lie bracket in g. The plan of the paper is as follows: Section 2: We review Andrey Losev’s notion of HTQFT [Los19; BLM24] which serve… view at source ↗
Figure 2
Figure 2. Figure 2: Example of a Y-shaped graph with the only vertex being of weight k. Note that for an oriented graph Γ with Nin incoming exterior edges and Nout outgoing exterior edges and IE(Γ) internal edges, the contraction above is an element in Hom(V ⊗Nin ,(V [1])⊗Nout ) ⊗ Ω • (R IE(Γ) + ). (2.21) Changing the orientation of internal edges of the graph leaves the contraction invariant, while changing the orientation o… view at source ↗
Figure 3
Figure 3. Figure 3: Decoration of three 1-cells by graphs with one internal edge and ordered inputs, glued along a 0-cell decorated by a graph with no internal edges. All vertices are assumed to be of weight 0 and we introduce an arbitrary orientation from left to right. The graph decorating the 0-cell corresponds to c 0 4 and vanishes in the above example, making I∞ non-distributional. Collapsing the internal edges of the gr… view at source ↗
Figure 4
Figure 4. Figure 4: Example of replacing a vertex v of weight k in a graph Γ by either (middle graph) two vertices connected through a marked edge e and with same total loop number k ′ + k ′′ = k or by (right graph) a single vertex with loop number decreased by one and connected via e to itself. The cancellation occurs by virtue of structure relations (2.17) on c k n : 1 n! Qck n | {z } from (a) + X p+q=n+2 X k ′+k ′′=k 1 p!q… view at source ↗
Figure 5
Figure 5. Figure 5: The relevant 0- and 1-cells for MG0 3,1 . Similarly for the 1-loop relation for A 1 1,0 we get Q ′A 1 1,0 = l 1 1,0 ◦ Q ◦ ı − Z ∂(MG1,top 1,0 ) (I∞) 1 1,0 . (2.37) The ultraviolet contributions from the r.h.s. (cf [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The relevant 0- and 1-cells for MG1 1,0 . 18 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A triple of points in O spanning an equilateral triangle through the origin. The space of such triples/triangles defines LW. In this case we can also explicitly write (up to scaling) |LW⟩ = (z1 − z2)(z2 − z3)(z3 − z1) = −z 2 1 z2 + z 2 1 z3 − z 2 2 z3 + z1z 2 2 − z1z 2 3 + z2z 2 3 (4.18) = − |1, 0, −1⟩ + |1, −1, 0⟩ − |−1, 1, 0⟩ + |0, 1, −1⟩ − |0, −1, 1⟩ + |−1, 0, 1⟩. (4.19) The last line is written abstrac… view at source ↗

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Reference graph

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