Completed polynomial gauge paths for filtered Maurer–Cartan elements

Let $W$ be a complete filtered shifted L∞-algebra over a characteristic-zero field $k$. Put $P=F^{1}W$ with its induced filtration. This is closed and complete in each degree: it is the kernel of $W\to W\mathrm{/}{F}^{1}W$, and compatible cosets in $P\mathrm{/}{F}^{p}W$ have their unique $W$-limit in that kernel. All $q_n$ preserve $P$.

The completed polynomial coefficients and completed polynomial differential forms mean

These are compatible polynomials in each quotient; their degrees need not have a common bound. Evaluation at $0,1$, differentiation, and integration from $0$ to $t$ are defined in every quotient by polynomial formulas and commute with reduction; integration uses ${\int }_0^t{u}^{j}du=t^{j+1}\mathrm{/}(j+1)$. Operations on these spaces are extended continuously, with each coefficient of a fixed quotient a finite sum.

Write $C(w)={\sum }_{n\ge 1}{q}_n(w^n)\mathrm{/}n!$ and $q_1^w(a)={\sum }_{n\ge 0}{q}_{n+1}(w^n,a)\mathrm{/}n!$. An elementary gauge path is a pair $w\in P^0[t{]}^{\wedge }$, $a\in P^{-1}[t{]}^{\wedge }$ such that $C(w)=0$ and $w^{\mathrm{\prime }}=q_1^w(a)$. Its endpoints are $w(0),w(1)$. Gauge equivalence is the equivalence relation generated by endpoints of these paths. A constant path has $a=0$; reversal is $(w(1-t),-a(1-t))$; finite chains give transitivity. Polynomial concatenation is not required.

For the polynomial-form interpretation, forms are placed on the right. For homogeneous $x_i\in P$, ${\alpha }_i\in \mathrm{\Omega }$, the extension of $q_n$ has scalar sign $(-1{)}^{{\sum }_{i<j}\mathrm{\mid }{\alpha }_i\mathrm{\mid }\mathrm{\mid }{x}_j\mathrm{\mid }}$; its unary part also contains $(-1{)}^{\mathrm{\mid }x\mathrm{\mid }}x\otimes d\alpha$. The degree-zero form $z=w-adt$ then satisfies $C_{\mathrm{\Omega }}(z)=C(w)+(w^{\mathrm{\prime }}-q_1^w(a))dt$. Thus these paths are precisely Maurer–Cartan elements over the completed polynomial interval. This fixes a gauge construction using the full operations, rather than choosing an arbitrary equivalence relation on a solution set.