FOOO single-object filtered curved A-infinity algebra

For the actual rational chain model specified below, assume the Full Axiom of Choice. Its uses are the coherent sequence of finite perturbation choices across energy cutoffs and the degreewise rational Hamel splittings in the transfer to the full continuous-singular-chain complex. These are conditional construction data; they do not assert that an arbitrary pointwise-tame $J$ admits the required virtual procedure.

Fix a compact relatively spin Lagrangian $L$ in a smooth symplectic manifold $(M,\omega )$ with a chosen smooth $\omega$-tame almost-complex structure $J$. Use the relative-spin data and stable disk domains defined on this page. Take the $\mathbb{Z}\mathrm{/}2$-graded source convention without the auxiliary formal degree variable $e$, over the positive-energy Novikov subring ${\mathrm{\Lambda }}_{\nu ,\mathbb{Q}}^0$ with $\mathrm{\Gamma }=\mathbb{R}$ and $\nu (\lambda )=\lambda$. Let $n=\dim L$ and let $C_{\ast }(L;\mathbb{Q})$ be the underlying singular chain complex generated in HOMOLOGICAL degree $r$ by continuous maps $\sigma :{\mathrm{\Delta }}^r\to L$, with differential the alternating sum of face restrictions. Use the source COHOMOLOGICAL regrading $C^q(L;\mathbb{Q})=C_{n-q}(L;\mathbb{Q})$ before taking parity, extension of coefficients and completion; its differential has degree $+1$. Let $C$ be the resulting cohomologically graded completion after extension of coefficients to ${\mathrm{\Lambda }}_{\nu ,\mathbb{Q}}^0$ in the energy filtration, so $F^{\lambda }C$ consists of series whose nonzero energy terms all have exponent at least $\lambda$. The operations depend on a specified coherent orientation and chain-valued virtual procedure. In the source geometrically tame regime and admissible auxiliary almost-complex data, Geometrically tame disk compactness and actual real-nodal atlas and Rational collared disk virtual correspondences and curved Morse operations construct such disk correspondences. Here geometrically tame means the complete bounded-geometry metric and uniform taming bounds defined in the former supplier; it is stronger than pointwise tameness of an arbitrary chosen J. The present Definition still allows the originally specified smooth pointwise-tame J as data conditional on its chosen virtual procedure, and does not assert that every such J is admissible.

For $k\ge 0$ and a relative class $\beta$, the source operation $m_{k,\beta }$ is the evaluation push-forward of the virtual-chain fiber product of the stable disk moduli ${\mathcal{M}}_{k+1}(\beta ;L,J)$ with its $k$ input chains along $e{v}_1,\dots ,e{v}_k$, using $e{v}_0$ as output; this phrase denotes its chain-model correspondence, with an explicit choice of comparison. One actual rational choice is the finite Morse/tree virtual model from Rational collared disk virtual correspondences and curved Morse operations, followed by the rational filtered curved transfer in Rational filtered curved transfer to all continuous singular chains to ALL continuous singular chains with their original boundary differential. Its transferred operations are filtered A-infinity equivalent to the geometric evaluation correspondence; literal simplexwise equality to an unspecified smooth fiber integral is not asserted. The choice of chain comparison is part of the virtual procedure. The source's relative class is defined across its stable-domain strata and supplies the energy $E_{\omega }(\beta )$; the fixed-domain relative-area lemma is supplemented by the explicit node-collapse homotopy of Geometrically tame disk compactness and actual real-nodal atlas, defining the global free relative disk class by oriented rooted-tree collapse and proving its homotopy compatibility across strata. Fixed-domain component annotations are retained. Its integral homology image is additive, including torsion; homotopy classes are not silently replaced by homology. Weight it by $T^{E_{\omega }(\beta )}$ and set $m_k={\sum }_{\beta }{m}_{k,\beta }{T}^{E_{\omega }(\beta )}$, interpreted as the source's energy-filtration sum when it converges. The curvature term $m_0$ is retained. The shift is $C[1{]}^q=C^{q+1}$, so an element of unshifted cohomological degree $d$ has shifted degree $d-1$; it reverses parity. For insertion before the $i$th input the Koszul sign is $(-1{)}^{{\sum }_{j<i}(d_j-1)}$, equivalently $(-1{)}^{{\sum }_{j<i}{d}_j+i-1}$. Here $d_j$ always denotes the unshifted COHOMOLOGICAL degree, not singular-chain dimension. The shift $C[1]$ therefore uses this regrading; the bar coalgebra is the algebraic direct sum $B(C[1])={⨁}_{r\ge 0}(C[1]{)}^{\otimes r}$, with $B_0(C[1])={\mathrm{\Lambda }}_{\nu ,\mathbb{Q}}^0$ and deconcatenation coproduct. The coefficient chain module $C$ is completed in the energy filtration; no completion in tensor length is asserted. On the bar coalgebra, each $m_k$ extends as the source coderivation ${\widehat{m}}_k$ of degree $1$. Define $F^{>0}C={\bigcup }_{\lambda >0}{F}^{\lambda }C$. The package is a filtered curved A-infinity algebra when each $m_k$ is continuous and filtration-preserving, $m_0\in F^{>0}C[1]$, and $D={\sum }_{k\ge 0}{\widehat{m}}_k$ satisfies $D^2=0$.

This is a single-object algebra interface; it is not a multi-object Fukaya category. In §3 the cited article first describes its moduli and operations for a chosen compatible almost-complex structure, then Theorem 3.8 states the identity $D^2=0$ for an arbitrary compact relatively spin $L$ in an arbitrary tame symplectic manifold, without separately quantifying over every chosen tame $J$. It refers the complete proof to the FOOO trilogy. Thus the local definition permits a chosen tame $J$ as data, but it does not strengthen that source-reported theorem to every fixed tame $J$, nor assert the geometric construction for inadmissible chosen data. The attached suppliers give a local construction in that source geometrically tame regime, and the identity theorem below applies to its chosen rational comparison model.

Symplectic and almost-complex curve data

Relatively spin Lagrangian data

Stable J-holomorphic disks with boundary markings

Relative disk area is invariant under relative homotopy

Novikov convolution is a filtered ring operation

Tame J-holomorphic area equals metric Dirichlet energy

Geometrically tame disk compactness and actual real-nodal atlas Rational collared disk virtual correspondences and curved Morse operations Rational filtered curved transfer to all continuous singular chains

The Axiom of Choice