Causal Factorization Algebras and the BV Pushforward: Deriving the L∞ Quasi-Isomorphism from Cohesive Topology
Abstract
The Spectral Realization program establishes a constructive traversal of the differential cohomology hexagon, deriving the Batalin-Vilkovisky (BV) BRST complex from the Lie differentiation of the automorphism groupoid of the synthesized physical state. The present paper proves the foundational theorem that the dynamical chaining mechanism (the cohesive regulator map reg:Γ(♭𝔼̂ω)→(Π𝔼̂ω) is isomorphic, upon Lie differentiation, to the BV pushforward pint of Cattaneo and Mnev as an L∞-quasi-isomorphism.
The proof proceeds in four stages. We first formally define the physical slice as a Causal Factorization Algebra ℱ, with local observables ℱ(I) = Γ(♭𝔼̂ω|I) over each directed causal interval I, and prove that the regulator satisfies the sewing axioms. We then apply the cubical Dold-Kan correspondence to show that the topological Higher-Dimensional Automaton (HDA) of the slice linearizes, under Lie(−), into a tangent L∞-algebra, where commuting 2-cells map to vanishing brackets ℓ2(a,b)=0 and obstructed 2-cells map to curvature brackets ℓ2(a,b)=F. We invoke Lurie's directed ∞-operads to identify the resulting locally constant factorization algebra with a forward-directed E1-algebra (an A∞-algebra), and apply the Costello-Gwilliam theorem to construct its BV quantization. Finally, we construct the strong deformation retract data $(i, p, K)$ identifying the Synthesis lift with the Homological Perturbation Lemma (HPL) inclusion iint, the global sections functor Γ with the HPL projection pint, and the gauge-fixing propagator with the chain homotopy operator K, establishing that Spectral Realization S→ A is the geometric HPL.
Related
BibTeX
@misc{shaw2026causal,
title = {{Causal Factorization Algebras and the BV Pushforward: Deriving the $L_\infty$ Quasi-Isomorphism from Cohesive Topology}},
author = {Barclay Lake Shaw},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23639906},
url = {https://zenodo.org/records/23639906},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Causal Factorization Algebras and the BV Pushforward: Deriving the L_∞ Quasi-Isomorphism from Cohesive Topology
AU - Barclay Lake Shaw
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23639906
UR - https://zenodo.org/records/23639906
AB - The Spectral Realization program establishes a constructive traversal of the differential cohomology hexagon, deriving the Batalin-Vilkovisky (BV) BRST complex from the Lie differentiation of the automorphism groupoid of the synthesized physical state. The present paper proves the foundational theorem that the dynamical chaining mechanism (the cohesive regulator map reg:Γ(♭𝔼̂_ω)→(Π𝔼̂_ω) is isomorphic, upon Lie differentiation, to the BV pushforward p_int of Cattaneo and Mnev as an L_∞-quasi-isomorphism. The proof proceeds in four stages. We first formally define the physical slice as a Causal Factorization Algebra ℱ, with local observables ℱ(I) = Γ(♭𝔼̂_ω|_I) over each directed causal interval I, and prove that the regulator satisfies the sewing axioms. We then apply the cubical Dold-Kan correspondence to show that the topological Higher-Dimensional Automaton (HDA) of the slice linearizes, under Lie(−), into a tangent L_∞-algebra, where commuting 2-cells map to vanishing brackets ℓ_2(a,b)=0 and obstructed 2-cells map to curvature brackets ℓ_2(a,b)=F. We invoke Lurie's directed ∞-operads to identify the resulting locally constant factorization algebra with a forward-directed E_1-algebra (an A_∞-algebra), and apply the Costello-Gwilliam theorem to construct its BV quantization. Finally, we construct the strong deformation retract data $(i, p, K)$ identifying the Synthesis lift with the Homological Perturbation Lemma (HPL) inclusion i_int, the global sections functor Γ with the HPL projection p_int, and the gauge-fixing propagator with the chain homotopy operator K, establishing that Spectral Realization S→ A is the geometric HPL.
ER -