Perturbative Boundary Gauge Theory: Scope and Open Problems

Boundary gauge theory has several mathematically distinct levels of control: classical BV–BFV compatibility, a perturbative state satisfying a modified master equation, cut-independent gluing in a specific model, renormalized local observables, and a positive nonperturbative quantum theory. Success at one level does not license the next. The present status is strongest for topological, free, and low-dimensional models and remains sharply more limited for four-dimensional non-Abelian Yang–Mills.

Required background. Perturbative gauge QFT: constructions and scope fixes the formal-series ceiling. Bulk–boundary master equations and anomaly inflow supplies the relative obstruction. Gluing, reduction, and composition theorems supplies residual-mode reduction. Factorization algebras with boundaries and defects supplies stratified local observables.

Helpful background. Timelike boundaries, self-adjoint extensions, and AdS boundary conditions gives a different analytic boundary problem. Conformal boundaries and defects supplies physical boundary CFT examples.

A model-by-model comparison

Abelian BF theory has exact classical BV–BFV data in every dimension. After a polarization and cohomological splitting, its perturbative state, modified QME, torsion factor, residual fields, and gluing formula are explicit Cattaneo, Mnev, and Reshetikhin 2018, §3, pp. 26–40. Because the action is quadratic, the nonzero-mode integral is Gaussian; topology remains in finite-dimensional cohomology.

Non-Abelian BF theory and split Chern–Simons theory are treated as formal perturbations of such data. Configuration-space integrals and local counterterms enter, and unimodularity or anomaly conditions matter. The result is a formal state in $\hbar$ and the coupling, not an absolutely convergent measure Cattaneo, Mnev, and Reshetikhin 2018, §§4.1–4.8, pp. 40–61.

Two-dimensional Yang–Mills is unusually tractable. BV–BFV states on disks and cylinders with polarized boundary arcs can be glued across corners; in the compact-group model this reconstruction agrees with the known heat-kernel partition function Iraso and Mnev 2019, §§3–4, pp. 22–60. Its area dependence and two-dimensional structure do not turn that calculation into a theorem for four-dimensional Yang–Mills.

For free bulk–boundary BV systems that are elliptic and topological normal to the boundary, stratified classical and quantum factorization algebras are constructed rigorously, including boundary quasi-isomorphisms Gwilliam, Rabinovich, and Williams 2021, Theorems 4.1–4.2, article pp. 19–23. Interacting, Lorentzian, and general gauge-boundary systems require new analytic work.

A recent Lorentzian advance

A 2026 pAQFT construction formulates smoothened BV–BFV data on globally hyperbolic spacetimes with marked hypersurfaces, builds boundary corrections to the renormalized quantum BV operator, and works out Abelian Yang–Mills on causal cylinders. Its sharp-boundary statement assumes convergence of the smoothing limit, and its direct comparison with the Cattaneo–Mnev–Reshetikhin higher perturbative corrections is not yet general Rejzner and Schiavina 2026, §§4–5 and Theorem 5.8.

This result materially strengthens the Lorentzian perturbative interface: the boundary defect is encoded as curvature in a homotopy dg Lie structure, and the leading Abelian BFV operator is recovered in the appropriate limit. It still does not supply a non-Abelian four-dimensional continuum measure, reflection positivity, arbitrary-corner gluing, or a nonperturbative physical Hilbert space. The paper itself separates the all-orders pAQFT correction from the conjectural direct comparison with topological BV–BFV expansions.

Obligations for four-dimensional Yang–Mills

A full bounded non-Abelian construction would need, at minimum:

• a globally meaningful configuration groupoid with boundary conditions and topological sectors;
• a proper bulk BV complex and compatible boundary BFV complex, including reducible strata;
• renormalized bulk, boundary, and corner time-ordered products satisfying the relative QME;
• cancellation of local and global anomalies;
• polarizations or algebraic substitutes compatible with gluing;
• residual modes and determinants treated without double counting;
• removal of ultraviolet and infrared regulators with uniform estimates;
• positivity or a reconstruction theorem for the physical state space.

Classical first-order Yang–Mills has explicit BV–BFV boundary data Cattaneo, Mnev, and Reshetikhin 2014, §5.2, pp. 31–34. That is an indispensable starting point, not completion of the quantum list.

Adversarial promotion test

Take the Abelian BF cylinder formula, replace the finite-dimensional Lie algebra by $\mathfrak{su}(N)$, add a Yang–Mills kinetic term in four dimensions, and declare the same gluing integral a continuum partition function. The step fails repeatedly: the action is no longer quadratic, boundary composite observables require renormalization, Gribov and stabilizer structure affects the quotient, residual fields no longer exhaust infrared behavior, and no positivity or cutoff-removal theorem has been supplied.

What survives is a formal ansatz for perturbation theory near a selected background and boundary condition. The physical boundary phenomena belong at Boundaries, Interfaces, and Domain Walls; the mathematical claim here must retain its loop order, model, geometry, and unresolved analytic conditions.

Exercises

Why does exact agreement with the two-dimensional Yang–Mills heat kernel not prove convergence of four-dimensional perturbation theory?

Solution

Two-dimensional Yang–Mills has no local propagating gauge degrees of freedom and admits an area-dependent representation-theoretic solution. Four-dimensional Yang–Mills has different ultraviolet scaling, local propagating modes, and an unresolved continuum measure. The gluing categories may look similar while the analytic estimates are entirely different.

Classify the statement “the modified QME holds modulo $\hbar^3$.”

Solution

It is a perturbative quantum statement through two-loop order in the declared renormalization scheme and geometry. It does not assert an all-orders solution, convergence of the series, global anomaly cancellation, regulator removal, or positivity.

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.
• Gwilliam, Owen, Eugene Rabinovich, and Brian R. Williams. “Factorization Algebras and Abelian CS/WZW-Type Correspondences.” 2021. arXiv:2001.07888.
• Iraso, Riccardo, and Pavel Mnev. “Two-Dimensional Yang–Mills Theory on Surfaces with Corners in Batalin–Vilkovisky Formalism.” Communications in Mathematical Physics 370 (2019): 637–702. DOI; Open PDF.
• Rejzner, Katarzyna, and Michele Schiavina. “Perturbative Algebraic Quantum Field Theory with Smoothened Boundary.” 2026. arXiv:2607.13765.