BV–BFV Structures, Boundaries, and Gluing

A gauge theory on a manifold with boundary is not an ordinary BV theory plus a condition imposed afterward. Integration by parts changes the master identity itself: the bulk BV vector field fails to be Hamiltonian by the pullback of a boundary one-form, and that one-form induces a degree-zero BFV symplectic structure and charge. BV–BFV theory records this compatibility before a polarization, state, or gluing formula is attempted.

Required background. The BV complex and classical master equation supplies the degree $-1$ bulk symplectic geometry. Presymplectic systems and the covariant phase-space ambiguity map supplies the boundary term and its reduction.

Helpful background. State spaces, cobordisms, and gluing gives the functorial target that perturbative BV pushforward approximates in controlled models.

The bulk–boundary compatibility equation

Let $M$ be a compact oriented $d$-manifold with boundary. In the sign convention used here, an exact classical BV–BFV theory consists of bulk data

$$(\mathcal F_M,\omega_M,Q_M,S_M),\qquad \deg\omega_M=-1,\quad Q_M^2=0,$$

boundary data $(\mathcal F^\partial_{\partial M},\omega^\partial_{\partial M},Q^\partial_{\partial M},S^\partial_{\partial M})$, and a surjective restriction map $\pi_M:\mathcal F_M\to\mathcal F^\partial_{\partial M}$. The boundary form has degree zero and a primitive,

$$\omega^\partial_{\partial M}=(-1)^{d-1}\delta\alpha^\partial_{\partial M}.$$

Compatibility means $\delta\pi_M(Q_M)=Q^\partial_{\partial M}$ and

$$\iota_{Q_M}\omega_M =(-1)^d\delta S_M+\pi_M^*\alpha^\partial_{\partial M}.$$

For a closed manifold the last term vanishes and the usual Hamiltonian BV equation returns. With a boundary, taking another contraction gives a modified classical master equation whose defect is the pullback of the BFV charge. The exact formulas, including the grading and sign convention, are Cattaneo, Mnev, and Reshetikhin 2014, Definition 3.1 and Proposition 3.1, pp. 8–10.

The equation is more informative than the slogan “boundary terms must cancel.” It states which bulk fields restrict to which boundary phase space, which cohomological vector field is projectable, and which primitive is used. Changing the action by a boundary functional changes the primitive and hence the polarization data, even when the boundary symplectic form is unchanged.

Abelian BF theory

Take the BV superfields

$$\mathcal A\in\Omega^\bullet(M)[1], \qquad \mathcal B\in\Omega^\bullet(M)[d-2],$$

with

$$\omega_M=\int_M\delta\mathcal A\wedge\delta\mathcal B, \qquad S_M=\int_M\mathcal B\wedge d\mathcal A, \qquad Q_M\mathcal A=d\mathcal A,\quad Q_M\mathcal B=d\mathcal B.$$

Varying $S_M$ and applying Stokes’ theorem produces the boundary primitive

$$\alpha^\partial_{\partial M} =\int_{\partial M}\mathcal A\wedge\delta\mathcal B, \qquad \omega^\partial_{\partial M} =\int_{\partial M}\delta\mathcal A\wedge\delta\mathcal B.$$

The boundary charge generates the de Rham differential on both restricted superfields. Direct substitution verifies the compatibility equation; this is the finite calculation underlying the general construction Cattaneo, Mnev, and Reshetikhin 2014, §5.4.1, pp. 36–38.

At ghost number zero, boundary values of bulk solutions are closed forms that extend through $M$. After reduction by exact forms, their image in boundary cohomology is Lagrangian by Poincaré–Lefschetz duality. This statement already displays the role of residual fields: relative and absolute cohomology record zero modes not removed by a local propagator. The corresponding physical BV formulation is developed at Master Equations and BV Gauge Fixing.

From compatibility to gluing

Classically, cutting $M=M_1\cup_\Sigma M_2$ identifies the two copies of the BFV data on $\Sigma$ with opposite orientations and forms a derived or clean fiber product of solution relations. Quantum mechanically, one must also choose complementary polarizations, a boundary state space, residual fields, a measure or half-density, and a BV pushforward. In the perturbative formalism the state obeys

$$\left(\hbar^2\Delta_{\mathcal V_M} +\Omega^\partial_{\partial M}\right)\widehat\psi_M=0,$$

where $\mathcal V_M$ is the finite-dimensional residual-field space and $\Omega^\partial$ quantizes the BFV charge. The construction is measure-theoretic in finite dimension; for field spaces it is a formal Feynman expansion whose identities must be checked model by model Cattaneo, Mnev, and Reshetikhin 2018, §§2.3–2.4, pp. 18–25.

Failure boundary

A boundary condition is admissible only if it is a Lagrangian submanifold tangent to $Q^\partial$, with the selected primitive vanishing there, or if an explicitly generalized construction replaces those conditions. An arbitrary Dirichlet-looking restriction can leave symplectic flux. Likewise, deleting cohomological zero modes because they do not appear in a local Gaussian determinant changes the BV pushforward.

For Abelian BF theory on a cylinder, omit the harmonic mode dual to the cut circle. Gluing then integrates over too small a space: the result depends on which cylinder decomposition was chosen. The strongest surviving statement is a local nonzero-mode calculation, not a cut-independent partition function.

Exercises

Show that reversing the orientation of a glued hypersurface changes the sign of its BFV symplectic form.

Solution

The form is an integral over the oriented hypersurface. Replacing $\Sigma$ by $\Sigma^{\mathrm{opp}}$ multiplies the integral by $-1$. Thus the two boundary factors in a gluing problem carry opposite symplectic forms, and their diagonal is Lagrangian.

Why can a residual field not simply be included in the propagator?

Solution

A residual field represents cohomology, so the kinetic differential has no inverse on that direction. A propagator is a chain homotopy on a chosen complement. Treating a zero mode as invertible either makes the propagator undefined or silently removes a physical finite-dimensional integration variable.

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.