BV–BFV Boundaries, Corners, and Gluing

On a manifold with boundary, the BV variational identity contains a surface term. BV–BFV theory promotes that term to boundary phase-space data and specifies the additional structures needed to quantize and glue regions.

Enter this chapter

Begin with the bulk–boundary identity, then boundary constraints. Treat the polarization and residual modes as part of a gluing problem, not as implicit choices. Helpful background. Field variations and boundary terms supplies the variational source of the BFV primitive. Edge modes, subregions, and factorization gives the physical regional problem. State spaces, cobordisms, and gluing gives the functorial comparison.

Topics

1. BV–BFV Structures, Boundaries, and Gluing — Connects the bulk variational defect to boundary cohomological data.
2. Boundary Phase Spaces, Constraints, and the BFV Charge — Resolves boundary constraints and defines their charge.
3. Bulk–Boundary Master Equations and Anomaly Inflow — Relates master-equation defects to boundary terms.
4. Corners, Stratification, and Higher-Codimension Data — Tracks the additional data created at corners.
5. Gluing, Reduction, and Composition Theorems — States the matching, reduction, and pushforward conditions for composition.
6. Edge Modes and Extended Observables at Gauge Boundaries — Distinguishes boundary gauge transformations from physical charged directions.
7. AKSZ Sigma Models as BV–BFV Examples — Provides explicit topological models of the construction.
8. Factorization Algebras with Boundaries and Defects — Organizes observables on stratified regions.
9. Perturbative Boundary Gauge Theory: Scope and Open Problems — Separates established model-specific results from broader gauge-theory questions.

How the ideas fit together

The boundary symplectic form and BFV charge arise from the bulk variation Cattaneo, Mnev, and Reshetikhin 2014, §§ 3–7. Quantum gluing additionally needs compatible polarizations, residual fields, and a BV pushforward Cattaneo, Mnev, and Reshetikhin 2018, §§ 2–4. The smoothened-boundary pAQFT result has its own marked-hypersurface and perturbative scope Rejzner and Schiavina 2026, §§ 4–5, Theorem 5.8 (PDF). These examples do not turn an unspecified functional integral into a general gluing theorem.

Hypotheses and licensed conclusions

Domains, required hypotheses, directional results, and failure tests for BV–BFV boundaries and gluing

Object and domain	Required hypotheses	Licensed result	Excluded converse or upgrade	Adversarial check
Bulk BV theory on $M$ with boundary	Projectable $Q_M$; surjective boundary restriction; exact or prequantum boundary form; controlled variational signs and domains	Boundary BFV phase space, charge, and modified classical master equation	A closed-manifold CME cannot be used after discarding the surface term	Vary the action and retain every integration-by-parts contribution
Boundary constraint system	Regular coisotropic zero locus; declared gauge subgroup; smooth or derived reduction; integrable generators	BFV resolution and reduced boundary observables	A transformation with nonzero surface charge is not automatically gauge	Quotient a Maxwell flux sector by arbitrary boundary parameters and count the lost directions
Boundary condition or polarization	Lagrangianity; $Q^\partial$ tangency; compatible primitive; finite or controlled intersection with evolution relation	Admissible boundary problem and state-space polarization	Dirichlet-looking constraints need not cancel symplectic flux	Evaluate the boundary Green form on allowed variations
Manifold with corners	Iterated restriction; shifted symplectic degrees; compatible corner charges and endpoint polarizations; refinement coherence	Extended BV–BFV data and associative face composition in the proved model	Codimension-one compatibility does not erase corner terms	Compare two orders of gluing three faces
Perturbative quantum state	Gauge fixing; finite residual sector; renormalized BFV operator; local and global anomaly cancellation; mQME	Cohomology class of a formal boundary state	No convergent measure, positivity, or global vacuum follows from a formal QME	Retain a nonzero anomaly class or determinant-line holonomy
Cut $M=M_1\cup_\Sigma M_2$	Clean or derived intersection; complementary polarizations; matching corner data; one copy of each residual mode; BV pushforward	Paired state satisfying the mQME and composition up to the declared equivalence	Multiplication followed by an unspecified functional integral is not a gluing theorem	Double-count a harmonic mode and compare determinant degrees
Boundary model or extrapolation	Named BF, Chern–Simons, 2D Yang–Mills, free stratified, or smoothened Abelian theorem with its geometry and order	Exactly the model-specific classical, perturbative, or gluing statement proved	No general four-dimensional non-Abelian continuum existence or positivity theorem follows	Replace a Gaussian BF cylinder by 4D Yang–Mills and list every missing analytic input

Structured table data (JSON).

Review the chapter

Why must shared residual modes be treated explicitly when two regions are glued?

Solution

Residual modes represent cohomology left after gauge fixing. Counting the same mode independently on both sides duplicates an integration or constraint and changes the resulting state and determinant factors. Gluing must identify the shared sector and perform the prescribed pushforward once.

Where to go next

Continue to conformal nets for operator-algebraic two-dimensional theories, or extended TQFT for categorical gluing.

References

• Cattaneo, Alberto S., Pavel Mnev, and Nicolai Reshetikhin. “Classical BV Theories on Manifolds with Boundary.” Communications in Mathematical Physics 332 (2014): 535–603. DOI; Open PDF.
• Cattaneo, Alberto S., Pavel Mnev, and Nicolai Reshetikhin. “Perturbative Quantum Gauge Theories on Manifolds with Boundary.” Communications in Mathematical Physics 357 (2018): 631–730. DOI; Open PDF.
• Rejzner, Katarzyna, and Michele Schiavina. “Perturbative Algebraic Quantum Field Theory with Smoothened Boundary.” 2026. arXiv:2607.13765.