Algebras over not too little discs

By the introduction of locally constant prefactorization algebras at a fixed scale, we show a mathematical incarnation of the fact that observables at a given scale of a topological field theory propagate to every scale over euclidean spaces. The key is that these prefactorization algebras over $\mathbb{R}^n$ are equivalent to algebras over the little $n$-disc operad. For topological field theories with defects, we get analogous results by replacing $\mathbb{R}^n$ with the spaces modelling corners $\mathbb{R}^p\times\mathbb{R}^{q}_{\geq 0}$. As a toy example in $1d$, we quantize, once more, constant Poisson structures.

💡 Research Summary

The paper introduces a mathematically precise formulation of a well‑known physical intuition: in a topological quantum field theory (TQFT) the observables defined at a fixed length scale can be uniquely extended to all scales. The authors achieve this by working with locally constant prefactorization algebras (also called factorization algebras) that are only required to be defined on Euclidean discs whose radii exceed a chosen positive number R.

The central object is the operad D_Rⁿ, the full sub‑operad of the usual disc operad Disc(ℝⁿ) spanned by open discs of radius > R. A D_Rⁿ‑algebra in a symmetric monoidal ∞‑category V is precisely a prefactorization algebra at scale R. The main theorem (Theorem 2.3) states that for any V, the pull‑back functor induced by the canonical map γ:D_Rⁿ→𝔈ₙ gives an equivalence between the ∞‑category of 𝔈ₙ‑algebras and the ∞‑category of locally constant D_Rⁿ‑algebras. In other words, a topological theory is completely determined by its observables at any single scale R.

The proof proceeds in two stages. First, the authors study “fat‑configuration” spaces D_R Confₘ(ℝⁿ), consisting of collections of disjoint closed discs of radius > R together with their embeddings. Lemma 2.8 shows that evaluation at the centers yields a homotopy equivalence D_R Confₘ(ℝⁿ) ≃ Confₘ(ℝⁿ). The key construction is an “inflation” map that uniformly rescales each disc so that all radii become at least a fixed ε > R, thereby providing a homotopy inverse to the inclusion of the ordinary configuration space. This topological result is the geometric backbone for the operadic localization argument.

Second, the authors replace both D_Rⁿ and 𝔈ₙ by convenient cubical models and analyse the ∞‑operadic localization directly. Theorem 2.4 asserts that γ exhibits 𝔈ₙ as the ∞‑localization of D_Rⁿ at all unary operations. Unlike the classical Lurie‑Harpaz approach, which relies on the notion of weak approximation, the present morphism fails to be a weak approximation (Appendix A). Consequently, the authors develop a new argument based on the homotopy equivalence of configuration spaces and explicit control of the operadic composition.

Section 3 extends the framework to theories with defects. By replacing ℝⁿ with manifolds of the form ℝᵖ×ℝᵩ_{≥0} (modeling corners), the authors construct higher‑dimensional Swiss‑cheese operads and prove analogous equivalences (Theorems 3.10 and 3.25). This shows that even in the presence of linear or corner defects, a topological field theory is determined by its observables at any fixed scale.

Section 4 provides a concrete application: the quantization of a constant Poisson structure in one dimension. Instead of using analytic infinite‑dimensional techniques, the authors work with a discretized version of the compactly supported de Rham complex. They build a locally constant D_{1/2}¹‑algebra that encodes the Weyl algebra at scale R = 1/2. When the mesh of the discretization is finer than the chosen scale, the locally constant property is lost, but Theorem 2.3 restores the full 𝔈₁‑algebra structure, yielding the expected quantization.

Overall, the paper delivers a clear and robust mathematical statement: for topological (and defect‑enhanced) field theories, the data of observables at any single, non‑zero scale suffices to reconstruct the full ∞‑categorical algebraic structure governing the theory. The work bridges operadic homotopy theory, factorization algebras, and quantum field theoretic renormalization, and it opens the door to discrete or combinatorial models of TQFTs that avoid traditional analytic difficulties.