Gluing, Reduction, and Composition Theorems

BV–BFV gluing is a reduction theorem, not the instruction “multiply two path integrals and integrate the cut.” Classically, the two evolution relations must meet cleanly in the common BFV phase space. Perturbatively, complementary polarizations are paired, shared residual modes are reduced exactly once, and the resulting BV pushforward must preserve the modified quantum master equation.

Required background. BV–BFV structures, boundaries, and gluing supplies the relative master equation. Boundary phase spaces, constraints, and the BFV charge supplies coisotropic reduction. Corners, stratification, and higher-codimension data supplies endpoint coherence.

Helpful background. State spaces, cobordisms, and gluing states the physical composition law. Boundaries, gluing, and Jeffrey–Kirwan residues is a different localization mechanism whose measure factors should not be imported here without a theorem.

Classical composition

Suppose $M=M_1\cup_\Sigma M_2$, with $\Sigma$ oppositely oriented as a boundary component of the two pieces. Let

$$L_{M_i}\subset \mathcal F^\partial_{\partial M_i}$$

be the boundary values of bulk solutions. In a regular classical BV–BFV theory, the reduced $L_{M_i}$ are Lagrangian relations. Gluing first forms the fiber product imposing equality of the two $\Sigma$ values and then reduces by the characteristic gauge directions:

$$L_M\simeq \left(L_{M_1}\times_{\mathcal F^\partial_\Sigma} L_{M_2}\right)\big/\!\big/\mathcal G_\Sigma.$$

The intersection must be clean, or else the derived fiber product must be retained. Cattaneo, Mnev, and Reshetikhin give the regular symplectic-Euler–Lagrange construction, tangent sequence, and abelian Chern–Simons example in 2014, §3.5, pp. 19–21. A set-theoretic intersection can lose excess tangent directions and stabilizers.

Quantum state and BV pushforward

Choose transverse polarizations on the two copies of $\mathcal F^\partial_\Sigma$. Their state spaces pair by an integral kernel. If $\widehat\psi_{M_i}$ also depend on residual fields $\mathcal V_{M_i}$, gluing has the form

$$\widehat\psi_M =P_*\left( \widehat\psi_{M_1}*_\Sigma\widehat\psi_{M_2} \right),$$

where $*_\Sigma$ pairs the boundary states and $P_*$ is BV pushforward from the product residual space to a chosen residual model $\mathcal V_M$. Stokes’ theorem for the BV Laplacian implies that the new state satisfies the modified QME, provided the Lagrangian integration cycle has no unaccounted boundary and the boundary BFV operators match Cattaneo, Mnev, and Reshetikhin 2018, §2.4.4, pp. 24–26.

Determinants and torsion factors are part of this formula. They cannot be discarded as normalization if they depend on topology, polarization, or residual fields. Gauge-fixing independence means that a change of induction data changes the state by a combined BV–BFV coboundary; it does not make every representative literally equal.

Abelian BF cylinders

For Abelian BF theory, take $A$-polarization on the outgoing copy of $\Sigma$ and complementary $B$-polarization on the incoming copy. The pairing kernel is the exponential of the canonical boundary pairing,

$$K_\Sigma(A,B) =\exp\left[ \frac{i}{\hbar}\int_\Sigma B\wedge A \right].$$

Residual fields are representatives of the appropriate absolute and relative de Rham cohomology groups. On gluing two cylinders, the common boundary integral identifies the two trace fields. The Mayer–Vietoris sequence identifies the combined cohomology and tells which pair of residual modes becomes a contractible pair. BV pushforward removes that pair and leaves one representative for every cohomology class of the glued cylinder.

The explicit Abelian BF state, its polarizations, residual fields, and gluing reduction are computed in Cattaneo, Mnev, and Reshetikhin 2018, §§3.1–3.6, pp. 27–40. The functorial interpretation belongs at State Spaces, Cobordisms, and Gluing. The calculation is formal perturbative field theory with a finite-dimensional residual sector; it is not a measure construction for arbitrary interacting gauge theories.

Associativity and the failure test

Repeated gluing is associative up to the equivalence declared by the theory when Fubini or its BV analogue applies, the corner data match, and the residual-field contractions compose. Homological perturbation explains the last point: two sequences of contracting acyclic pairs give canonically homotopic effective states when their contraction data are compatible.

Now choose boundary constraint surfaces that meet nontransversely and nevertheless use an ordinary fiber product. Alternatively, include the same harmonic mode in both $\mathcal V_{M_1}$ and $\mathcal V_{M_2}$ but omit the pushforward that removes the duplicate. In one gluing order the mode is integrated twice; in another it remains external. The resulting determinant or degree differs. This invalidates composition. What survives is a pairing before residual reduction, not a state assigned to the glued manifold.

Exercises

Why are opposite polarizations convenient on a cut?

Solution

They provide complementary coordinates on the common symplectic phase space, so the pairing kernel is a nondegenerate Fourier-type integral. Equal polarizations can still be compared through an additional transform, but identifying their coordinates directly generally misses the canonical conjugate integration.

How does the Mayer–Vietoris sequence detect a duplicated zero mode?

Solution

The sequence maps a pair of cohomology classes on $M_1$ and $M_2$ to the difference of their restrictions on $\Sigma$. Its kernel represents a class on $M$, while its image and connecting map identify the complementary acyclic directions. Keeping both representatives without this reduction double-counts the kernel class.

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.