Roland Bittleston

In this Letter, we show that the one-loop QCD β function can be obtained from an index theorem on twistor space. This is achieved by recalling that the θ angle of self-dual gauge theory flows according the one-loop β function. Rewriting self-dual gauge theory as a holomorphic theory on twistor space this flow can be computed as the anomaly to scale invariance. The one-loop Weyl anomaly coefficient a − c can be recovered similarly.

Celestial holography has led to the discovery of new symmetry algebras arising from the study of collinear limits of perturbative gravity amplitudes in flat space. We explain from the twistor perspective how a non-vanishing cosmological constant Λ naturally modifies the celestial chiral algebra. The cosmological constant deforms the Poisson bracket on twistor space, so the corresponding deformed algebra of Hamiltonians under the new bracket is automatically consistent. This algebra is equivalent to that recently found by Taylor and Zhu. We find a number of variations of the deformed algebra. We give the Noether charges arising from the expression of this algebra as a symmetry of the twistor action for self-dual gravity with cosmological constant.

We consider holomorphic Poisson-BF theory on twistor space. Classically, this describes self-dual Einstein gravity on space-time, but at the quantum level it is plagued by an anomaly. The anomaly corresponds to the fact that integrability of the self-dual vacuum Einstein equations does not survive in self-dual quantum gravity. We compute the anomaly polynomials in the Poisson-BF theory, as well as in this theory coupled to a holomorphic BF theory on twistor space describing self-dual Yang-Mills. We show that all anomalies may be cancelled by further coupling to a twistor field representing a type of axion on space-time. When the twistor anomalies are cancelled, all $n\geq4$-pt amplitudes vanish and integrability is restored.

We consider the twistor description of classical self-dual Einstein gravity in the presence of a defect operator wrapping a certain ℂℙ¹. The backreaction of this defect deforms the flat twistor space to that of Eguchi-Hanson space. We show that the celestial chiral algebra of self-dual gravity on the Eguchi-Hanson background is likewise deformed to become the loop algebra of a certain scaling limit of the family of W(μ)-algebras, where the scaling limit is controlled by the radius of the Eguchi-Hanson core. We construct this algebra by computing the Poisson algebra of holomorphic functions on the deformed twistor space, and check this result with a space-time calculation of the leading contribution to the gravitational splitting function. The loop algebra of a general W(μ)-algebra (away from the scaling limit) similarly arises as the celestial chiral algebra of Moyal-deformed self-dual gravity on Eguchi-Hanson space. We also obtain corresponding results for self-dual Yang-Mills.

The question of whether the holomorphic collinear singularities of graviton amplitudes define a consistent chiral algebra has garnered much recent attention. We analyse a version of this question for infinitesimal perturbations around the self-dual sector of 4d Einstein gravity. The singularities of tree amplitudes in such perturbations do form a consistent chiral algebra, however at 1-loop its operator products are corrected by the effective graviton vertex. We argue that this chiral algebra can be interpreted as the universal holomorphic surface defect in the twistor uplift of self-dual gravity, and show that the same correction is induced by an anomalous diagram in the bulk-defect system. The 1-loop holomorphic collinear singularities do not form a consistent chiral algebra. The failure of associativity can be traced to the existence of a recently discovered gravitational anomaly on twistor space. It can be restored by coupling to an unusual 4th-order gravitational axion, which cancels the anomaly by a Green-Schwarz mechanism. Alternatively, the anomaly vanishes in certain theories of self-dual gravity coupled to matter, including in self-dual supergravity.