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Abelian Maurer–Cartan gauge classes are cohomology

Example

Let $(P, q_{1})$ be a graded cochain complex over a characteristic-zero field with $q_{n} = 0$ for $n \geq 2$ , $F^{1} P = P$ and $F^{p} P = 0$ for $p \geq 2$ . These operations form a complete filtered shifted L∞-algebra. Its Maurer–Cartan set is $\ker (q_{1} : P^{0} \to P^{1})$ and its completed polynomial gauge quotient is $H^{0} (P, q_{1}) = \ker q_{1} / im (q_{1} : P^{- 1} \to P^{0})$ .

For a concrete comparison, let $P^{- 1} = k a$ , $P^{0} = k e$ , and all other degrees be zero. In the first structure take $q_{1} = 0$ ; in the second take $q_{1} (a) = e$ , $q_{1} (e) = 0$ . Both have the identical Maurer–Cartan equation $0 = 0$ on $k e$ . The first canonical gauge quotient is $k e$ , while the second has one class. Thus the solution equation on degree-zero elements alone omits data essential to canonical gauge classes.

Facts & Assumptions

Given: The cochain complex, filtration and two specified examples.

[F1]

Completed polynomial gauge paths for filtered Maurer–Cartan elements defines the gauge equation $w^{'} = q_{1}^{w} (a (t))$ and its generated equivalence.

[F2]

Existence and uniqueness of complete filtered Maurer–Cartan gauge flow gives existence, uniqueness and preservation of curvature for these filtered paths.

Verification

1.1givenF1construct

The only nonzero shifted identity is $q_{1}^{2} = 0$ , which is the complex axiom. Its unary operation preserves the two-step filtration, and higher operations vanish, so all additive filtration inclusions hold. The filtration is separated and complete since the quotient by $F^{2}$ is $P$ itself. Its curvature is $C (w) = q_{1} w$ , so the Maurer–Cartan set is $\ker q_{1}$ in degree zero.

2.1F1F2step 1.1algebra

Here $q_{1}^{w} = q_{1}$ independently of $w$ . Any path therefore satisfies $w (1) - w (0) = q_{1} (\int_{0}^{1} a (t) d t)$ , so its endpoints have the same cohomology class. A finite chain has that property as well. Conversely, if $w_{1} - w_{0} = q_{1} b$ with $b \in P^{- 1}$ , take $a (t) = b$ and $w (t) = w_{0} + t q_{1} b$ . This satisfies the path equation and has $q_{1} w (t) = 0$ since $q_{1}^{2} = 0$ ; it is also the unique flow in [F2]. Thus the gauge equivalence classes are precisely the stated cohomology classes.

3.1givenstep 1.1step 2.1algebra∎

In both concrete complexes $q_{1}$ is zero on degree zero, so the curvature equation is identically zero on $k e$ . In the zero complex the image from degree minus one is zero, giving quotient $k e$ . In the second complex that image is all $k e$ , giving a single class. The two quotient computations follow from step 2.1 with identical degree-zero solution loci.

Depends on

Used by

The Maurer–Cartan equation alone does not specify gauge classes Counterexample

Dependency tree · two levels

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Sources

Dolgushev and Rogers, A Version of the Goldman–Millson Theorem for Filtered L-infinity-Algebras, section 2 and Appendix A (standard reference, not scraped)
Manetti, Deformation theory via differential graded Lie algebras, section 5 (standard reference, not scraped)