Holomorphic, Topological, and Mixed Factorization Theories

Holomorphic, topological, and mixed factorization theories are distinguished by which translations act trivially up to coherent homotopy. A topological theory is locally constant in its topological directions. A holomorphic theory retains analytic dependence on holomorphic coordinates while antiholomorphic translations are homotopically trivial. A mixed theory combines de Rham behavior in some real directions with Dolbeault behavior in complex directions. None of these structures implies equivalence to the untwisted unitary theory from which a cohomological model may have arisen.

Required background. Prefactorization and factorization algebras supplies local products and descent, while BV quantization and obstruction complexes supplies the perturbative meaning of a holomorphic anomaly.

Helpful background. Braided sectors and low-dimensional nets provides an operator-algebraic comparison, and analyticity, CPT, and spin–statistics shows why ordinary Lorentzian analyticity is a different condition.

Three kinds of translation invariance

On $\mathbb R^m$, a translation-invariant factorization algebra is topological when each infinitesimal translation $\partial_{x^i}$ is null-homotopic as a derivation. Integration of those homotopies makes disk inclusions quasi-isomorphisms, producing local constancy and an $E_m$ structure.

On $\mathbb C^n$, holomorphic translations $\partial_{z^i}$ may act nontrivially, while the antiholomorphic translations $\partial_{\bar z^i}$ are homotopically trivial. A holomorphic local field complex therefore has the Dolbeault form

$$\bigl(\Omega^{0,\bullet}(X,V),\bar\partial+Q_{\mathrm{hol}}\bigr),$$

with holomorphic differential operators in its brackets and action. Observables vary holomorphically with separated insertion points; they are not locally constant in the complex directions. The exact local-Lie-algebra formulation is given in Gwilliam and Williams 2025, Definitions 2.7–2.8 and §§4–5.

On $\mathbb R^m\times\mathbb C^n$, a topological–holomorphic theory combines $\Omega^\bullet(\mathbb R^m)$ with $\Omega^{0,\bullet}(\mathbb C^n)$. Real translations and antiholomorphic complex translations are homotopically trivial; holomorphic translations survive. This anisotropy must be visible in products and renormalization. It is not accurate to call the whole theory topological merely because some directions are.

Pushing forward along a topological direction can reduce a mixed factorization algebra to holomorphic data with additional operations encoding the compactified direction. The pushforward is defined on an open $V$ of the base by evaluating the original algebra on its inverse image. It preserves factorization only when the geometric map and support conditions make disjoint configurations lift correctly; compactification is therefore a theorem with hypotheses, not deletion of coordinates.

The free βγ system

Let $\Sigma$ be a complex curve and $V$ a finite-dimensional complex vector space. The free $\beta\gamma$ system has fields

$$\gamma\in\Omega^{0,\bullet}(\Sigma,V), \qquad \beta\in\Omega^{1,\bullet}(\Sigma,V^*)[{-1}],$$

with action

$$S(\beta,\gamma)=\int_\Sigma\langle\beta,\bar\partial\gamma\rangle.$$

The shift makes the BV pairing have degree $-1$; sources using another cohomological convention move this shift but preserve the pairing and QME. On a disk, linear currents associated with $X\in\mathfrak{gl}(V)$ are represented by $J_X=\langle\beta,X\gamma\rangle$. The propagator gives the familiar short-distance contraction

$$\beta_i(z)\gamma^j(w)\sim \frac{\hbar\,\delta_i^{,j}}{z-w},$$

which generates the current operations of the holomorphic factorization algebra. The extraction of the $\beta\gamma$ vertex algebra from disk observables is treated in Costello and Gwilliam 2017, Chapter 5, pp. 145–204.

For linear target $V$, the free quantization exists globally on the fixed curve once the relevant bundle data are fixed. The curved $\beta\gamma$ model with target a complex manifold $X$ is subtler: coordinate changes produce a one-loop obstruction represented by $\operatorname{ch}_2(TX)$ (with normalization depending on conventions). A trivialization of that class is the extra datum used to globalize the sheaf of chiral differential operators. This condition should not be retroactively imposed on the linear free model, whose tangent bundle is trivial.

Renormalization and mixed theories

Holomorphic power counting is unusually restrictive, but it does not eliminate anomalies. The anomaly remains a local degree-one BV class even when ultraviolet counterterms vanish. For topological–holomorphic theories on $\mathbb R^m\times\mathbb C^n$, current results prove ultraviolet finiteness under explicit translation-invariant hypotheses and give anomaly-vanishing theorems when there are at least two topological directions Wang and Williams 2024, Theorems 1.1–1.2. These theorems do not state that every one-topological-direction model is anomaly free.

The physical conformal interpretation belongs at Complex Coordinates and Local Conformal Symmetry. The factorization construction isolates chiral observables; it does not reconstruct a modular-invariant full CFT or prove positivity of its state space.

Failure tests

If both $\partial_z$ and $\partial_{\bar z}$ act trivially, the theory is locally constant rather than genuinely holomorphic. If neither action is homotopically trivial, translation invariance alone does not produce holomorphic factorization. If a cohomological twist discards $Q$-exact operators, agreement of its protected observables with the untwisted theory does not imply equivalence of spectra or scattering. Finally, vanishing ultraviolet counterterms does not cancel a nonzero one-loop anomaly class.

Exercises

Check that the $\beta\gamma$ action is invariant under $\gamma\mapsto\gamma+\epsilon X\gamma$ and $\beta\mapsto\beta-\epsilon\beta X$ for constant $X\in\mathfrak{gl}(V)$.

Solution

The variation is $\int\langle-\beta X,\bar\partial\gamma\rangle+\langle\beta,\bar\partial(X\gamma)\rangle$. Since $X$ is constant and acts dually on $\beta$, the two terms cancel.

Why is holomorphic factorization not local constancy?

Solution

Antiholomorphic motion is cohomologically trivial, but holomorphic motion can change an observable by a genuine analytic function such as $(z-w)^{-1}$. Local constancy would trivialize motion in both real directions and erase this dependence.

References

• Costello, Kevin, and Owen Gwilliam. Factorization Algebras in Quantum Field Theory, Volume 1. Cambridge University Press, 2017. doi:10.1017/9781316678626.
• Gwilliam, Owen, and Brian R. Williams. “Holomorphic Field Theories and Higher Algebra.” Bulletin of the London Mathematical Society 57 (2025): 2903–2974. arXiv:2508.07443.
• Wang, Minghao, and Brian R. Williams. “On the Renormalization and Quantization of Topological–Holomorphic Field Theories.” 2024. arXiv:2407.08667.