Locally Constant Factorization Algebras and Eₙ Algebras

A locally constant factorization algebra on framed $n$-space is the geometric form of an $E_n$ algebra: configurations of little $n$-disks supply the operations, isotopies supply their coherent homotopies, and Weiss descent extends the disk data to arbitrary opens. The theorem depends on local constancy, a framed or otherwise specified tangential structure, and a homotopically suitable target. Spacetime dimension alone never produces an $E_n$ algebra.

Required background. Prefactorization and factorization algebras supplies disjoint operations, and Weiss descent supplies the extension from disk configurations to general opens.

Helpful background. Classical Poisson factorization supplies field-theoretic examples, while braided sectors and low-dimensional nets shows a different source of higher multiplication and braiding that should not be identified without a theorem.

Local constancy and disk operations

Let $\mathcal F$ be a factorization algebra on $\mathbb R^n$. It is locally constant if every inclusion of disks $D\hookrightarrow D'$ induces an equivalence

$$\mathcal F(D)\xrightarrow{\simeq}\mathcal F(D').$$

Choose a standard disk $D$. An embedding of $k$ disjoint little disks into $D$ gives

$$\mathcal F(D)^{\otimes k} \longrightarrow \mathcal F(D).$$

As the embedding varies, these maps are parameterized by the little-disks space $E_n(k)$. Composition of configurations is exactly composition of factorization maps, and the symmetric-group action permutes inputs. Local constancy identifies values on differently sized or translated disks; isotopies between embeddings become homotopies between operations. Thus $A=\mathcal F(D)$ is an algebra over the $E_n$ operad.

Conversely, an $E_n$ algebra $A$ assigns $A^{\otimes k}$ to a union of $k$ disks and uses the operad action for inclusions. Homotopy left Kan extension along the inclusion of disks into opens, followed by Weiss descent, produces a locally constant factorization algebra. Under standard presentability and tensor-colimit hypotheses, these constructions give an equivalence of $\infty$-categories. The recognition result appears as Lurie 2017, Theorem 5.4.5.9 and is summarized in Gwilliam and Williams 2025, Theorem 5.10.

Tangential structures are data

The preceding statement is framed: coordinates identify small disks with standard framed disks. On an oriented manifold, an oriented-disk algebra is required; equivalently, one supplies the appropriate homotopy-coherent $SO(n)$ action on the framed $E_n$ data. Spin, unoriented, or stratified theories require their own disk categories. Forgetting this structure can make an algebra definable on $\mathbb R^n$ but not transportable around a general manifold.

Local constancy is also a cohomological statement when values are complexes: inclusions need only be quasi-isomorphisms, not literal equalities. Choosing arbitrary inverses destroys coherence. The $\infty$-categorical formulation retains the entire contractible space of choices and its higher compatibilities.

The hierarchy has concrete algebraic content. An $E_1$ algebra is coherently associative; an $E_2$ algebra carries braiding up to higher homotopy; increasing $n$ provides progressively more room to exchange disks. Only in the stable $E_\infty$ limit does one obtain fully coherent commutativity. Thus it is wrong to replace every $E_n$ product by an ordinary commutative multiplication merely because disjoint regions can be permuted in the ambient symmetric monoidal category.

One-dimensional topological quantum mechanics

Let $A$ be a unital associative dg algebra. For an oriented interval $I$, set $\mathcal F_A(I)=A$. If $I_1,\ldots,I_k$ occur from left to right inside $I$, define

$$\mu_{I_1,\ldots,I_k}(a_1\otimes\cdots\otimes a_k) =a_1a_2\cdots a_k.$$

Reassociating nested interval configurations changes no result because multiplication in $A$ is associative. Orientation fixes the order; reversing it replaces $A$ by the opposite algebra unless an additional involution is supplied. Inclusions of intervals act by the identity up to the chosen coherent model, so the factorization algebra is locally constant. The configuration space of ordered little intervals is contractible in each ordering component, and the resulting $E_1$ structure is precisely associativity up to coherent homotopy.

This construction is the observable side of a one-dimensional topological theory. The physical distinction between topological invariance and metric-dependent dynamics is developed at What Is a Topological Field Theory?. Not every Hamiltonian quantum mechanics is locally constant: nontrivial time evolution must first become cohomologically trivial, as in a topological twist.

Failure and nonconverse tests

A generic relativistic QFT on $n$ dimensions is not locally constant; correlation functions vary with distances, masses, and causal separation. Its factorization algebra therefore need not reduce to one $E_n$ algebra. Conversely, an abstract $E_n$ algebra supplies local topological observables, but not reflection positivity, a stress tensor, a vacuum representation, or a relativistic net. Finally, an $E_n$ algebra on framed disks does not define an oriented theory until the rotation data have been provided. These failures identify the exact missing hypotheses rather than weakening the recognition theorem.

Exercises

Recover associativity from three nested configurations of intervals.

Solution

One nesting first combines the left two intervals and gives $(a_1a_2)a_3$; the other first combines the right two and gives $a_1(a_2a_3)$. Both configurations compose to the same three-interval embedding, so factorization coherence identifies them. This is exactly associativity.

What extra datum is needed to move the framed construction to oriented two-manifolds?

Solution

One needs a homotopy-coherent rotation action compatible with the $E_2$ operations, equivalently an algebra over the oriented two-disk operad. Without it, transporting a framed disk around a loop can change the operation.

References

• Gwilliam, Owen, and Brian R. Williams. “Holomorphic Field Theories and Higher Algebra.” Bulletin of the London Mathematical Society 57 (2025): 2903–2974. arXiv:2508.07443.
• Lurie, Jacob. Higher Algebra. 2017. Author PDF.