Corners, Stratification, and Higher-Codimension Data

When boundary faces meet, the variation of a face action generally has its own boundary term on the corner. Discarding that codimension-two term breaks compatibility under iterated restriction and can make two orders of gluing disagree. An extended BV–BFV theory assigns graded symplectic and cohomological data to each stratum, with degrees shifted by codimension and with restriction maps satisfying the same relative Hamiltonian equation at every step.

Required background. BV–BFV structures, boundaries, and gluing supplies the first bulk–boundary step. Boundary phase spaces, constraints, and the BFV charge supplies reduction on each face.

Helpful background. Charge algebras, central terms, and corners gives the physical charge interpretation. Fusion, junctions, and endpoints gives the corresponding defect geometry.

Iterated BV–BFV data

Let $M^{(r)}$ denote a codimension- $r$ stratum of a $d$ -manifold with corners. A $k$ -extended BV theory assigns, for $0\le r<k$ , a field space $\mathcal F_{M^{(r)}}$ , a cohomological vector field $Q_{M^{(r)}}$ , a primitive $\alpha_{M^{(r)}}$ , an action or charge $S_{M^{(r)}}$ , and a restriction

\pi_r:\mathcal F_{M^{(r)}}\longrightarrow \mathcal F_{M^{(r+1)}}.

The degrees shift as

\deg\omega_{M^{(r)}}=r-1, \qquad \deg S_{M^{(r)}}=r, \qquad \deg Q_{M^{(r)}}=1,

and the compatibility equation becomes

\iota_{Q_{M^{(r)}}}\omega_{M^{(r)}} =(-1)^{d-r}\delta S_{M^{(r)}} +\pi_r^*\alpha_{M^{(r+1)}}.

The restriction maps intertwine the $Q$ fields. Thus the failure of the codimension- $r$ action to be Hamiltonian is not discarded; it becomes the primitive for the next stratum. This hierarchy is formalized in Cattaneo, Mnev, and Reshetikhin 2014, Definitions 3.26–3.27, pp. 21–22.

Maximal extension is a theorem for particular models, not a default. AKSZ theories and two-dimensional Yang–Mills admit the hierarchy in the cited setting, while ordinary Yang–Mills above two dimensions is assigned length one there. A codimension-two charge algebra may exist physically even when a full smooth higher BFV reduction has not been constructed.

A rectangular region

Consider a gauge theory on a rectangle $R=[0,L]\times[0,T]$ , or on a higher-dimensional region with this rectangle as its normal geometry. Decompose its boundary into four oriented faces $F_i$ . The first variation has the schematic form

\delta S_R=\mathrm{EL}_R+\sum_i\alpha_{F_i}.

If a face is treated as a cobordism, varying its BFV charge produces endpoint contributions

\delta S_{F_i} =\iota_{Q_{F_i}}\omega_{F_i} -\sum_{v\in\partial F_i}\epsilon_{iv}\alpha_v.

At a vertex $v=F_i\cap F_j$ , the signs $\epsilon_{iv}$ depend on the induced orientations. Compatible corner data make the two face contributions cancel or combine into the prescribed codimension-two charge. Merely summing four closed-boundary formulas misses these endpoints.

For first-order Yang–Mills, the face primitive contains $\int_{F_i}\operatorname{tr}(B\wedge\delta A+A^\dagger\wedge\delta c)$ . An integration by parts in the face BFV charge leaves a corner term involving the restricted $B$ field and ghost $c$ . The corresponding codimension-two primitive in the classical extended model is $\int_v\operatorname{tr}(B\,\delta c)$ , with the form degree understood for the ambient dimension Cattaneo, Mnev, and Reshetikhin 2014, §5.2.1, pp. 32–33. Physical corner-charge algebras are treated at Charge Algebras, Central Terms, and Corners.

A stratification may be refined by splitting a face or inserting an intermediate corner. The extended data must commute with this refinement up to the declared equality, canonical transformation, or coherent homotopy. In perturbative quantization, a corner also changes the boundary state space: intervals with distinct endpoint polarizations carry bimodule-like data, and gluing an interval entails a pairing or Fourier transform.

Two-dimensional Yang–Mills supplies a controlled example. Surfaces are cut into disks and cylinders with polarized arcs; corner state spaces and the modified quantum master equation are constructed so that interval gluing recovers the known partition function Iraso and Mnev 2019, §§4.1–4.2, pp. 35–51. This result does not establish the same analytic gluing theorem for four-dimensional Yang–Mills.

Failure under a missing corner

Glue three rectangles first as $(R_1\cup R_2)\cup R_3$ and then as $R_1\cup(R_2\cup R_3)$ . If the common endpoints are discarded, the first construction integrates face variables before imposing the second endpoint matching condition, while the second does the reverse. A leftover corner phase or determinant can differ. The disagreement diagnoses missing codimension-two data; it is not repaired by declaring gluing associative.

The strongest surviving result is facewise BV–BFV compatibility away from the endpoints. A global composition theorem requires the corner primitive, compatible polarizations, residual modes, and a coherent pairing.

Exercises

Why does the symplectic degree increase by one on each successive stratum?

Solution

Transgression over a manifold lowers the degree of a target form by the dimension integrated. Replacing a stratum by its boundary lowers that dimension by one, so the transgressed symplectic form gains one degree. The bulk degree $-1$ becomes degree zero on a face and degree one on a corner.

What sign test should be made at a corner shared by two oriented faces?