CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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The Maurer–Cartan equation alone does not specify gauge classes

Statement refuted

For every complete filtered L∞-algebra, its operations and Maurer–Cartan equation alone specify a unique equivalence relation on its Maurer–Cartan solutions.

Facts & Assumptions

Given: A characteristic-zero field $k$ .

[F1]

Maurer–Cartan element of a complete filtered L∞-algebra: Maurer–Cartan solutions are the degree-zero elements satisfying the convergent Maurer–Cartan equation; the definition supplies no equivalence relation.

[F2]

Complete filtered L∞-algebra: the filtration and completeness conditions define the setting for the Maurer–Cartan series.

[F3]

Shifted coalgebra definition of an L∞-algebra: the operations define an L∞-structure by a square-zero coderivation.

[F4]

Field: $0 \neq 1$ in $k$ .

Counterexample

Take $V^{1} = k t$ and all other $V^{j} = 0$ , so $W = V [1]$ has $W^{0} = k t$ . Let every operation $q_{n}$ be zero, and use the filtration $F^{1} W = W$ and $F^{r} W = 0$ for $r \geq 2$ .

1.1givenF2F3construct

The zero coderivation is square-zero, so the all-zero operations define an L∞-structure by [F3]. The filtration is separated. Its compatible cosets are determined by the representative in $W / F^{2} W = W$ , so the natural map in [F2] is bijective; all filtration-preservation inclusions hold because every $q_{n} = 0$ . This is a complete filtered L∞-algebra.

2.1F1step 1.1

Every $w \in W^{0}$ satisfies the Maurer–Cartan equation because every term is zero. Thus the solution set is $W^{0} = k$ .

3.1F1F2F3F4step 2.1algebra∎

Equality and the relation containing every ordered pair are both reflexive, symmetric, and transitive, so each is an equivalence relation on the same solution set. Since $0 \neq 1$ by [F4], their quotients have, respectively, more than one class and exactly one class. No part of [F1]–[F3] selects either relation. Therefore the Maurer–Cartan equation alone does not determine gauge classes or deformation equivalence classes.

Remarks

This counterexample concerns a specification consisting only of a Maurer–Cartan solution set and an unspecified equivalence relation. It does not assert that the full L∞ operations together with a fixed gauge convention leave the relation ambiguous. With the completed polynomial convention of Completed polynomial gauge paths for filtered Maurer–Cartan elements, the displayed zero structure has no degree-minus-one parameters, so its gauge paths are constant and the resulting relation is equality. The indiscrete relation in the counterexample is not that convention's gauge relation. The companion example Abelian Maurer–Cartan gauge classes are cohomology shows how the same solution set can instead acquire distinct canonical gauge quotients when the full unary operation changes.

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Sources

Jean-Louis Loday and Bruno Vallette, Algebraic Operads, §13.2 (standard reference, not scraped)