Higher-dimensional Chiral Algebras in the Jouanolou Model
Zhengping Gui, Minghao Wang, Brian R. Williams
TL;DR
The paper develops a framework for higher-dimensional chiral algebras by modeling configuration-space cohomology with Jouanolou torsors, yielding a GL$_d$-equivariant dg operad $\mathcal{P}_d$ whose algebras encode higher dimensional chiral operations. A $d$-dimensional homotopy chiral algebra is realized as a map from $\mathcal{L}\!\operatorname{ie}_\infty$ into $\mathcal{P}_d$, generalizing Beilinson–Drinfeld's construction to higher dimensions. The unit chiral algebra is constructed via holomorphic Feynman graph residues, employing Schwinger-space representations to define higher residues and proving $L_\infty$-relations; explicit computations are carried out in dimension two, including a recursive formula for Type I' Laman graphs and loop weights that agree with contemporary physics results. The work also provides concrete examples such as commutative and free ghost chiral algebras and Faonte–Hennion–Kapranov central extensions, together with a higher Virasoro-type structure in dimension two, illustrating rich algebraic and geometric structures arising from higher-dimensional holomorphic QFT data.
Abstract
We appeal to the theory of Jouanolou torsors to model the coherent cohomology of configuration spaces of points in d-dimensional affine space. Using this model, we develop the operadic notion of chiral operations, thus generalizing the notion of chiral algebras of Beilinson and Drinfeld to higher dimensions. To produce examples, we use a higher-dimensional conceptualization of the residue which is inspired by Feynman graph integrals.