Derived Intersections, Boundary Conditions, and Correspondences

Cutting a gauge theory produces boundary data that the two pieces must match. If that matching is nontransverse—as it is in the presence of gauge stabilizers or harmonic boundary modes—the correct glued object is a homotopy fiber product. Its cohomology retains both matching solutions and the obstruction to matching. Boundary conditions are therefore best treated as Lagrangian correspondences only after their field complexes, symplectic forms, regularity, and reduction have been specified.

Required background. Derived Critical Loci and Derived Gauge Quotients supplies homotopy fiber products and stabilizer directions. Shifted Symplectic, Poisson, and BV Geometry supplies shifted nondegeneracy and boundary flux. Helpful background. Solitonic, Topological, and Boundary Sectors distinguishes localized charges from boundary sectors. Local Covariance with Boundaries and Background Structures explains why boundary conditions belong to the theory’s input.

Homotopy intersections remember failure of transversality

Let $L_1\to P\leftarrow L_2$ be maps of derived spaces. At a common point, the tangent complex of their homotopy fiber product is

$$\mathbb T_{L_1\times^h_P L_2} \simeq \operatorname{Cone}\!\left( \mathbb T_{L_1}\oplus\mathbb T_{L_2} \longrightarrow\mathbb T_P \right)[-1].$$

When the three tangent objects are ordinary vector spaces in degree zero, $H^0$ is the kernel of the difference of the two tangent maps: it consists of matched infinitesimal deformations. $H^1$ is the cokernel: it measures boundary directions that cannot be reached from either side. A transverse intersection has zero cokernel and is underived. A nontransverse one has higher cohomology that an ordinary intersection discards.

Suppose $P$ is $n$-shifted symplectic and both maps carry Lagrangian structures, meaning specified null-homotopies of the pulled-back form together with quasi-isomorphism conditions on relative tangent complexes. Then $L_1\times_P^hL_2$ is canonically $(n-1)$-shifted symplectic Pantev–Toën–Vaquié–Vezzosi 2013, Theorem 2.9, pp. 38–39. Merely observing that the two images are isotropic is not enough; the Lagrangian nondegeneracy hypotheses do real work in the proof.

Boundary conditions as correspondences

For a region $N$ with boundary $\Sigma$, let $\mathcal F_N$ be its bulk field complex and $\mathcal F_\Sigma$ its boundary phase space. Restriction gives a map $r_N:\mathcal F_N\to\mathcal F_\Sigma$. On solutions, the image $L_N$ is isotropic because the bulk variation reduces to a boundary symplectic potential. Under an explicit regularity assumption, reduction makes it Lagrangian; without that assumption, only isotropy is licensed. Cattaneo, Mnev, and Reshetikhin prove the isotropy statement and identify the extra regularity needed for Lagrangianity 2014, Proposition 3.6 and the discussion following it, p. 12.

A cobordism from $\Sigma_{\mathrm{in}}$ to $\Sigma_{\mathrm{out}}$ therefore determines, in favorable cases, a Lagrangian correspondence

$$\mathcal P_{\Sigma_{\mathrm{in}}} \longleftarrow L_N \longrightarrow \mathcal P_{\Sigma_{\mathrm{out}}}.$$

Gluing cobordisms composes correspondences by a derived intersection over the shared boundary phase space. This composition is associative up to coherent equivalence, not generally by literal equality. Gauge reduction and gluing must also be compared; performing an irregular quotient first can destroy the very stabilizer directions required for the homotopy pullback.

First application: abelian Chern–Simons across a cut

The boundary degrees of freedom discussed in Edge Modes, Subregions, and Factorization appear already in the linear abelian theory. Let a closed oriented three-manifold be cut as

$$M=M_1\cup_\Sigma M_2,$$

with collars chosen near the closed oriented surface $\Sigma$. The abelian Chern–Simons BV complex on a region is $\Omega^\bullet(M_i)[1]$ with differential $d$. The boundary complex is $\Omega^\bullet(\Sigma)[1]$, carrying the BFV pairing induced by integration over $\Sigma$. Restriction from each side has opposite orientation sign.

The glued field complex is modeled by the homotopy fiber

$$\operatorname{Fib}\!\left( \Omega^\bullet(M_1)\oplus\Omega^\bullet(M_2) \xrightarrow{\ r_1-r_2\ } \Omega^\bullet(\Sigma) \right).$$

Using a collar and a partition of unity, the de Rham Mayer–Vietoris map $\Omega^\bullet(M)\to\operatorname{Fib}(r_1-r_2)$ is a quasi-isomorphism. This is an independent check: its long exact sequence in cohomology is the usual Mayer–Vietoris sequence, so it reproduces the global zero modes $H^\bullet(M)$ rather than counting the modes of the pieces independently.

At ghost number zero, the boundary phase space contains one-forms on $\Sigma$ modulo exact forms, and harmonic representatives give $H^1(\Sigma)$. If the two bulk images meet nontransversely along these harmonic directions, the cone retains the resulting kernel and cokernel. The explicit BV–BFV treatment of abelian Chern–Simons identifies its bulk and boundary complexes, reductions, and regularity Cattaneo–Mnev–Reshetikhin 2014, §7.1, pp. 45–47; their general gluing construction compares fields and tangent spaces under cutting §3.5, pp. 19–22.

The correspondence describes classical gluing. Quantization adds polarizations, half-densities or states, determinant factors, and possible boundary anomalies. A classical Lagrangian relation does not automatically define a bounded operator between quantum state spaces.

Failure test: forcing a transverse intersection

Replace the homotopy fiber by the ordinary kernel of $r_1-r_2$. In degree zero this enforces equality of boundary values, but it retains no cokernel. Choose a cut for which $H^1(\Sigma)\neq0$ and a harmonic boundary class not in the sum of the two restriction images. That class is a genuine obstruction to matching; the ordinary kernel makes it disappear. If a harmonic class lies in both images, the nontransverse overlap likewise produces excess cohomology rather than an isolated intersection.

This failure has a precise nonconverse. Agreement of the ordinary glued solution set with the global solution set does not prove that their deformation complexes are quasi-isomorphic. One must compare the full cone, including ghosts and obstruction degrees, and verify the boundary symplectic flux. Similarly, calling a boundary condition “Lagrangian” does not prove that the regularity and nondegeneracy maps needed by the shifted-intersection theorem hold.

Exercises

Let $f:A\to B$ be a linear map between vector spaces regarded as complexes in degree zero. Compute the cohomology of $\operatorname{Cone}(f)[-1]$.

Solution

The homotopy fiber has $A$ in degree zero and $B$ in degree one, with differential $f$. Therefore $H^0=\ker f$ and $H^1=\operatorname{coker}f$. The ordinary fiber retains only the first group.

For $M=S^3$ cut into two three-balls along $S^2$, use Mayer–Vietoris to check that no harmonic one-form survives the gluing.

Solution

Both balls and $S^2$ have vanishing first cohomology. The relevant segment of the long exact sequence therefore gives $H^1(S^3)=0$. The derived gluing complex agrees: there is no degree-one boundary class to supply a kernel or cokernel in this part of the sequence.

References

• Cattaneo, Mnev, and Reshetikhin 2014, Classical BV Theories on Manifolds with Boundary, Communications in Mathematical Physics 332(2), 535–603. Open PDF
• Pantev, Toën, Vaquié, and Vezzosi 2013, Shifted Symplectic Structures, Publications Mathématiques de l’IHÉS 117, 271–328. Open PDF