Factorization Algebras with Boundaries and Defects

A boundary changes factorization observables in two ways. Opens must remember which strata they meet, and boundary-supported observables form a module-like recipient for configurations approaching the boundary. The correct local-to-global condition is therefore stratified descent: a cover that is Weiss in the ambient manifold but blind to the boundary stratum can reconstruct the bulk while losing all endpoint operations.

Required background. Prefactorization and factorization algebras supplies disjoint products and Weiss descent. BV–BFV structures, boundaries, and gluing supplies compatible boundary conditions. Corners, stratification, and higher-codimension data supplies iterated strata.

Helpful background. Interfaces, folding, and fusion gives the physical interface picture. Boundary and defect Weyl anomalies supplies an independent quantum obstruction.

Stratified opens and structure maps

Let $M$ have boundary $D=\partial M$. A stratified prefactorization algebra assigns an object $\mathcal F(U)$ to every open $U\subset M$ and a multiplication

$$\mathcal F(U_1)\otimes\cdots\otimes\mathcal F(U_k) \longrightarrow\mathcal F(V)$$

for pairwise disjoint stratified opens $U_i\subset V$. The isotopy type records whether a component lies in the interior, meets $D$ through a collar, or is contained in a lower stratum. Consequently, operations with several interior disks and one boundary half-disk act on boundary observables.

In the locally constant half-space model, the interior value is an $E_d$ algebra $A$ and the boundary value is an object $M_D$ with compatible half-disk operations. In one dimension this reduces to an associative algebra acting on a module: intervals ordered away from the endpoint multiply in $A$ and then act on $M_D$. In a non-topological field theory the values need not be locally constant, but the same stratified geometry organizes support.

Descent uses covers adapted to finite subsets in every stratum. A factorizing basis contains disks in the interior, half-disks along a boundary face, and quadrant-like neighborhoods at corners. The Čech object must include their intersections and the maps between strata. Ordinary bulk Weiss descent is only the restriction of this stronger statement.

Free bulk–boundary fields

For a free BV theory, suppose the kinetic complex is elliptic and topological in the normal direction near the boundary. Its Green pairing on normal jets is degree-zero symplectic. Choose a local Lagrangian subcomplex $L$ of boundary data preserved by the differential. Compactly supported fields obeying $L$ form a homotopy pullback rather than a naive kernel, so equations and boundary conditions remain derived.

The symmetric algebra on compactly supported linear observables gives the classical factorization algebra; adding the BV Laplacian gives the free quantum version. The bulk restriction recovers the ordinary factorization observables. Along a collar, the boundary algebra is quasi-isomorphic to the observables built from the complementary boundary complex. These identifications are proved for the stated free, normally topological class in Gwilliam, Rabinovich, and Williams 2021, Theorems 4.1–4.2, article pp. 19–23.

The result retains more than two separate restrictions: configurations of bulk opens inside a boundary half-disk define the bulk action on boundary observables. It does not assert that every local boundary condition for an interacting hyperbolic theory satisfies the same theorem.

A scalar on the half-line

Take a free scalar complex on $\mathbb R_{\ge0}$ with a linear boundary condition, such as Dirichlet $\phi(0)=0$ or Neumann $\partial_x\phi(0)=0$, chosen so the Green form vanishes. For an interval $I\subset(0,\infty)$, $\mathcal F(I)$ contains polynomial observables smeared in $I$. For a half-interval $H=[0,\epsilon)$, $\mathcal F(H)$ contains observables compatible with the boundary condition.

If $I_1,\ldots,I_k$ are mutually disjoint and lie inside a larger half-interval $H$, multiplication and extension by zero give

$$\mathcal F(I_1)\otimes\cdots\otimes \mathcal F(I_k)\otimes\mathcal F(H_0) \longrightarrow\mathcal F(H),$$

where $H_0$ is a smaller endpoint neighborhood disjoint from the $I_i$. In the locally constant topological analogue, this is precisely an ordered $A$-module operation. In the metric scalar theory it remains a scale-dependent factorization map, not an $E_1$-module equivalence. The physical boundary and defect interpretation belongs at Conformal Boundaries and Defects, when conformal hypotheses actually hold.

Failure of unstratified descent

Cover the half-line by interior intervals together with sets whose intersections avoid the endpoint, and apply an ordinary Čech reconstruction that never includes a boundary half-interval. Every diagram entry then sees only bulk observables. A boundary-supported linear functional, endpoint field, or module action has no representative in the colimit. The map to $\mathcal F(\mathbb R_{\ge0})$ cannot be a quasi-isomorphism if such a class survives.

The strongest surviving statement is descent for the interior restriction. Recovering the bulk–boundary system requires a stratified factorizing cover and the cross-stratum structure maps. Similarly, a defect cannot be replaced by a measure-zero limit of bulk opens unless a theorem constructs that limit.

Exercises

Why does a boundary value resemble a module in one dimension?

Solution

Disjoint interior intervals inside a half-interval are linearly ordered. Their $E_1$ products combine into an element of the bulk algebra, and the remaining half-interval supplies an action on the endpoint object. Nested configurations give the associativity law for a module.

Give a test that distinguishes Dirichlet from Neumann boundary observables.

Solution

The evaluation observable $\phi(0)$ vanishes in the Dirichlet complex but not generally in the Neumann complex; conversely the normal derivative vanishes in the Neumann complex. A proposed quasi-isomorphism must preserve these boundary cohomology classes and cannot be checked only on interior opens.

References

• Gwilliam, Owen, Eugene Rabinovich, and Brian R. Williams. “Factorization Algebras and Abelian CS/WZW-Type Correspondences.” 2021. arXiv:2001.07888.
• Rabinovich, Eugene. “Factorization Algebras for Classical Bulk–Boundary Systems.” 2020. arXiv:2008.04953.