Elliptic Complexes and Factorization Observables

An elliptic field complex produces local observables by resolving fields, equations, gauge symmetries, and Noether identities in one cochain complex and then taking functions on that derived solution space. Ellipticity supplies local regularity and parametrices; compact support supplies covariant maps under inclusion. Neither ingredient removes global zero modes, chooses a positive Lorentzian representation, or determines boundary conditions.

Required background. Prefactorization and factorization algebras supplies the target structure, and domains, signatures, supports, and regularity supplies the analytic data that an elliptic theorem must state.

Helpful background. Green-hyperbolic operators provide the distinct Lorentzian construction, while microcausal functionals explain how support and wavefront conditions replace ellipticity there.

Elliptic BV data

Let $E=\bigoplus_k E^k\to M$ be a finite-rank graded vector bundle on a Riemannian manifold and

$$Q:\Gamma(M,E^k)\longrightarrow\Gamma(M,E^{k+1}), \qquad Q^2=0,$$

a differential operator. The complex is elliptic when its symbol sequence is exact away from the zero section: for every $x\in M$ and $0\ne\xi\in T_x^*M$, the cohomology of $(E_x,\sigma_Q(x,\xi))$ vanishes. In BV language, positive degrees encode ghosts and gauge relations, degree zero contains physical fields, and negative degrees encode antifields and equations. A degree $-1$ nondegenerate local pairing makes the differential formally skew-adjoint in the graded sense.

For $U\subset M$, compactly supported sections $\mathcal E_c(U)=\Gamma_c(U,E)$ extend by zero. The pairing identifies them with continuous linear observables on unrestricted fields, so a convenient free model is

$$\operatorname{Obs}^{\mathrm{cl}}(U) =\widehat{\operatorname{Sym}}\bigl(\mathcal E_c(U)[1]\bigr), \qquad d_{\mathrm{cl}}=Q,$$

where the completion and shift are part of the chosen BV convention. The pairing induces a degree $+1$ Poisson bracket. Disjoint supports multiply without singular contraction, giving prefactorization maps. The general derived construction and the passage from elliptic moduli problems to classical observables are developed in Costello and Gwilliam 2021, Chs. 3–5.

Why ellipticity helps local-to-global reconstruction

Ellipticity yields a gauge-fixing operator $Q^{\mathrm{GF}}$ for which

$$H=[Q,Q^{\mathrm{GF}}]$$

is a generalized Laplacian. Its heat kernel and parametrices control short-distance behavior and give homotopies between local resolutions. The Atiyah–Bott lemma makes the sheaf of smooth solutions to an elliptic complex locally well behaved, while compactly supported sections form the covariant object needed for observables. A partition of unity decomposes compactly supported inputs, and elliptic homotopies control independence of the decomposition. Under the stated nuclear locally convex completions, the observable complex satisfies Weiss descent. Costello and Gwilliam 2023, Theorem 2 and §4 states the perturbative classical and quantum factorization result in this framework.

The parametrix is not unique. Changing it changes cochain-level representatives and, after quantization, the effective interaction and counterterms; the induced observable theories should be related by a controlled homotopy or renormalization-group map. Claiming literal equality would erase precisely the comparison data.

Maxwell theory and its zero modes

On a closed oriented Riemannian four-manifold $M$, the linear Maxwell BV complex may be displayed, with cohomological shifts suppressed, as

$$\Omega^0(M) \xrightarrow{\mathrm d} \Omega^1(M) \xrightarrow{\mathrm d{*}\mathrm d} \Omega^3(M) \xrightarrow{\mathrm d} \Omega^4(M).$$

Here $A\in\Omega^1$ is the gauge potential, $\mathrm d c$ is an infinitesimal gauge transformation, and $\mathrm d{*}\mathrm dA=0$ is the Euler–Lagrange equation written as a three-form. The consecutive maps compose to zero. After Lorenz gauge fixing, the relevant kinetic operators are Hodge Laplacians, so the symbol complex is elliptic.

On a contractible open ball, closed one-forms are exact and the complex gives the expected local gauge resolution. Globally, however, harmonic one-forms represent $H^1(M;\mathbb R)$ and survive as zero modes. Constant ghosts encode stabilizers. A Gaussian determinant over all modes is therefore singular unless zero modes are separated and the residual gauge volume is treated. The factorization algebra of local observables remains meaningful, but a global partition function needs this additional finite-dimensional analysis.

The physical BRST construction and its grading are developed at The BRST Differential and Gauge-Fixed Complex. The present construction shows what ellipticity licenses: local regularity, a derived gauge resolution, and factorization observables. It does not establish Lorentzian unitarity or a nonperturbative Maxwell measure on arbitrary boundary geometries.

Failure boundaries in geometry and signature

If $M$ has boundary, ellipticity of the interior symbol is insufficient. One must choose elliptic boundary conditions—absolute, relative, Atiyah–Patodi–Singer-type, or a BV–BFV-compatible alternative—and prove that the pairing and differential preserve them. Likewise, replacing the Riemannian metric by the site’s Lorentzian (+---) metric turns the Laplacian into a hyperbolic operator; causal propagators and time-slice properties then replace the elliptic parametrix argument.

Exercises

Verify that the displayed Maxwell sequence is a complex.

Solution

For $c\in\Omega^0$, $(\mathrm d{*}\mathrm d)(\mathrm dc)=\mathrm d{*}\mathrm d^2c=0$. For $A\in\Omega^1$, applying the last map gives $\mathrm d(\mathrm d{*}\mathrm dA)=\mathrm d^2({*}\mathrm dA)=0$. Thus both consecutive compositions vanish.

Why does removing harmonic one-forms not follow from local contractibility?

Solution

Contractibility removes cohomology on each sufficiently small open, but a global harmonic form is assembled from compatible local exact representatives whose transition data carry nontrivial topology. Weiss descent reconstructs that global class; it does not force it to vanish.

References

• Costello, Kevin, and Owen Gwilliam. Factorization Algebras in Quantum Field Theory, Volume 2. Cambridge University Press, 2021. doi:10.1017/9781316678664.
• Costello, Kevin, and Owen Gwilliam. “Factorization Algebra.” Encyclopedia of Mathematical Physics, 2nd ed., 2023. arXiv:2310.06137.