L∞ Algebras, Formal Moduli, and Field Equations

An $L_\infty$ algebra packages a nonlinear equation, its infinitesimal symmetries, and all higher compatibility identities into one square-zero coderivation. Its Maurer–Cartan elements describe solutions only in a formal or nilpotent neighborhood, and its natural output is an infinity-groupoid rather than a bare set. The decisive equivalence is therefore a filtered $L_\infty$ quasi-isomorphism under completeness hypotheses—not an arbitrary identification of classical solutions.

Required background. Derived Critical Loci and Derived Gauge Quotients explains why tangent cohomology carries stabilizers and obstructions. BV Quantization and Obstruction–Deformation Complexes supplies the BV origin of the controlling complex. Helpful background. Domains, Signatures, Supports, and Regularity supplies the analytic qualifications. Local and Microcausal Functionals with Peierls Brackets gives the observable setting in which such formal deformations are later used.

Maurer–Cartan equations encode nonlinear field equations

Use cohomological grading and let $L$ carry multilinear brackets

$$\ell_n:L^{\otimes n}\longrightarrow L[2-n],\qquad n\geq1,$$

graded antisymmetric and satisfying the higher Jacobi identities. The unary bracket $\ell_1$ is a differential. For $a\in L^1$, the Maurer–Cartan curvature is

$$\mathcal F(a)=\sum_{n\geq1}\frac{1}{n!}\ell_n(a,\ldots,a).$$

This sum is automatically finite for a nilpotent $L_\infty$ algebra. More generally, take a complete descending filtration $L=F_1L\supset F_2L\supset\cdots$ with $L\simeq\varprojlim L/F_rL$ and

$$\ell_n(F_{r_1}L,\ldots,F_{r_n}L)\subseteq F_{r_1+\cdots+r_n}L.$$

Then the series converges in the filtration topology. A Maurer–Cartan element satisfies $\mathcal F(a)=0$. Twisting by it produces new brackets

$$\ell_n^a(x_1,\ldots,x_n) =\sum_{k\geq0}\frac{1}{k!}\ell_{n+k}(a^{\otimes k},x_1,\ldots,x_n),$$

and $\ell_1^a$ is the linearized differential at that solution. Its cohomology recovers infinitesimal automorphisms, deformations, and obstructions in the relevant shifted degrees. Getzler gives the nilpotent Maurer–Cartan construction and twisting formulas explicitly Getzler 2009, Definition 4.3 and Proposition 4.4, pp. 283–284.

Gauge equivalence is not enough to capture higher stabilizers. Tensoring $L$ with polynomial differential forms on simplices produces a simplicial Maurer–Cartan set $\operatorname{MC}_\bullet(L)$. Under nilpotence it is a Kan complex, so paths, homotopies between paths, and higher homotopies are all retained Getzler 2009, Theorem 5.4, p. 291.

The formal-moduli comparison theorem

Let $L$ and $\widetilde L$ be complete filtered $L_\infty$ algebras, and let $U:L\rightsquigarrow\widetilde L$ be a filtration-compatible $L_\infty$ morphism. Assume its linear term induces a quasi-isomorphism $F_rL\to F_r\widetilde L$ for every $r$. Then

$$\operatorname{MC}_\bullet(U): \operatorname{MC}_\bullet(L)\longrightarrow \operatorname{MC}_\bullet(\widetilde L)$$

is a weak equivalence. This is the filtered Goldman–Millson theorem in the precise form of Dolgushev–Rogers 2015, Theorem 2.2, p. 8. Its proof passes to the nilpotent quotients $L/F_rL$, establishes equivalences stage by stage, and then controls the inverse limit of the resulting tower of fibrant simplicial sets.

The theorem licenses an equivalence of formal deformation infinity-groupoids. It does not assert that the underlying field theories are globally isomorphic, that their actions converge, or that they have equivalent Hilbert-space representations. In characteristic zero, model-categorical versions identify appropriate homotopy categories of differential graded Lie and strong-homotopy Lie deformation theories Pridham 2010, Theorem 4.55 and Corollary 4.57, pp. 817–818. The qualifications “appropriate,” “formal,” and “characteristic zero” are part of the conclusion.

First application: Chern–Simons near a flat connection

The BV interpretation developed in Master Equations and BV Gauge Fixing becomes concrete for Chern–Simons theory. Let $M$ be a closed oriented three-manifold, $G$ a Lie group with an invariant nondegenerate bilinear form on $\mathfrak g$, and $A_0$ a smooth flat connection on $P\to M$. The differential graded Lie algebra is

$$L=\Omega^\bullet(M,\operatorname{ad}P),\qquad \ell_1=d_{A_0},\qquad \ell_2(\alpha,\beta)=[\alpha\wedge\beta],$$

with higher brackets zero. For a degree-one element $a\in\Omega^1(M,\operatorname{ad}P)$,

$$\ell_1a+\tfrac12\ell_2(a,a) =d_{A_0}a+\tfrac12[a\wedge a] =F_{A_0+a}.$$

Thus the Maurer–Cartan equation is exactly nonlinear flatness. Degree-zero elements generate infinitesimal gauge transformations, negative-degree cohomology after the conventional shift measures stabilizers, and $H^2_{A_0}$ contains obstructions.

To obtain a formal problem, introduce a parameter ideal $\mathfrak m$—for example in an Artin local algebra—and use $L\widehat\otimes\mathfrak m$. Nilpotence of $\mathfrak m$ makes the Maurer–Cartan sum finite. A filtered quasi-isomorphism of such completed models preserves the full local solution infinity-groupoid. As an independent check, linearizing at $a=0$ gives $d_{A_0}a=0$ and quotienting by $a=d_{A_0}\epsilon$ yields $H^1_{A_0}$, agreeing with the tangent calculation on the preceding page.

Failure test: leaving the completed neighborhood

Suppose a formal $L_\infty$ quasi-isomorphism is given by a series $U_1(a)+\frac12U_2(a,a)+\cdots$. Apply it to a finite displacement $a$ that is not nilpotent and does not tend to zero in the chosen filtration. The series need not converge, so it may define no field configuration at all. Even if two series happen to converge on selected configurations, the filtered theorem supplies neither a global bijection of flat-connection moduli spaces nor control across components separated by large gauge transformations.

This gives a sharp nonconverse: equivalent formal neighborhoods need not imply globally equivalent theories. Conversely, a coincidence of classical moduli sets does not determine the higher paths and stabilizers in $\operatorname{MC}_\bullet$. One must specify the filtration, completion, base category, and weak-equivalence notion before saying that an $L_\infty$ model preserves a field equation.

Exercises

Show that the Maurer–Cartan equation in the Chern–Simons differential graded Lie algebra is equivalent to $F_{A_0+a}=0$.

Solution

The curvature expansion is $F_{A_0+a}=F_{A_0}+d_{A_0}a+\frac12[a\wedge a]$. Since $A_0$ is flat, the first term vanishes. The remaining two terms are exactly $\ell_1a+\frac12\ell_2(a,a)$.

Let $L$ be abelian, so $\ell_n=0$ for $n\geq2$. Identify its Maurer–Cartan elements and infinitesimal gauge equivalences.

Solution

The equation reduces to $\ell_1a=0$ for $a\in L^1$. A degree-zero parameter $\epsilon$ changes $a$ by $\ell_1\epsilon$. Therefore connected components are $H^1(L)$, while closed degree-zero elements give stabilizers. The simplicial Maurer–Cartan object retains these stabilizers rather than collapsing everything to $H^1$ as a set.

References

• Dolgushev and Rogers 2015, A Version of the Goldman–Millson Theorem for Filtered L∞-Algebras, Journal of Algebra 430, 260–302. Open PDF
• Getzler 2009, Lie Theory for Nilpotent L∞-Algebras, Annals of Mathematics 170(1), 271–301. Open PDF
• Pridham 2010, Unifying Derived Deformation Theories, Advances in Mathematics 224(3), 772–826. Open PDF