Kevin Costello

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.

Form factors of self-dual gauge theory are equal to correlators of an (extended) celestial chiral algebra. This suggests that these form factors can be computed using the “bootstrap” method familiar from 2d CFTs. The method can also be applied to certain QCD amplitudes, which are built from form-factors of self-dual gauge theory. In this paper this bootstrap method is applied to compute two-loop all + QCD amplitudes, for SU(N) gauge theory with certain special matter content. A closed formula is presented for all single-trace amplitudes.

A bstract: We derive and test a novel holographic duality in the B-model topological string theory. The duality relates the B-model on certain Calabi-Yau three-folds to two-dimensional chiral algebras defined as gauged βγ systems. The duality conjecturally captures a topological sector of more familiar AdS ₅ /CFT ₄ holographic dualities.

We study ’t Hooft lines in four-dimensional holomorphic-topological Chern-Simons theory. We relate them to Q-operators in the theory of integrable systems. We give a physical interpretation of the fundamental TQ and QQ relations satisfied by Q-operators and conventional transfer matrices.

A bstract: We elaborate on various aspects of our top-down celestial holographic duality wherein the semiclassical bulk spacetime is a 4d asymptotically flat, self-dual Kähler geometry known as Burns space. The bulk theory includes an open string sector comprising a 4d WZW model and a closed string sector called “Mabuchi gravity” capturing fluctuations of the Kähler potential. Starting with the type I topological B-model on the twistor space of flat space, we obtain the twistor space of Burns space from the backreaction of a stack of N coincident D1 branes, while the chiral algebra is obtained from (a twist of) the brane worldvolume theory. One striking consequence of this duality is that all loop-level scattering amplitudes of the theory on Burns space can be expressed as correlation functions of an explicit 2d chiral algebra. We also present additional large- N checks, matching several 2 and 3-point amplitudes and their collinear expansions in the WZW ₄ sector, and the mixed WZW ₄ -Mabuchi sector, of the bulk theory to the corresponding 2 and 3-point vacuum correlators and operator product expansions in the dual chiral algebra. Key features of the duality, along with our main results, are summarized in the introduction.

There has been recent interest in the question of whether QCD collinear singularities can be viewed as the operator product expansion of a two-dimensional conformal field theory. We analyze a version of this question for the self-dual limit of pure gauge theory (incorporating states of both helicities). We show that the known one-loop collinear singularities do not form an associative chiral algebra. The failure of associativity can be traced to a novel gauge anomaly on twistor space. We find that associativity can be restored for certain gauge groups if we introduce an unusual axion, which cancels the twistor space anomaly by a Green-Schwarz mechanism. Alternatively, associativity can be restored for some gauge groups with carefully chosen matter.

We propose a new program for computing a certain integrand of scattering amplitudes of four-dimensional gauge theories which we call the form factor integrand, starting from 6d holomorphic theories on twistor space. We show that the form factor integrands can be expressed as sums of products of 1.) correlators of a 2d chiral algebra, related to the algebra of asymptotic symmetries uncovered recently in the celestial holography program, and 2.) OPE coefficients of a 4d non-unitary CFT. We prove that conformal blocks of the chiral algebras are in one-to-one correspondence with local operators in 4d. We use this bijection to recover the Parke-Taylor formula, the CSW formula, and certain one-loop scattering amplitudes. Along the way, we explain and derive various aspects of celestial holography, incorporating techniques from the twisted holography program such as Koszul duality. This perspective allows us to easily and efficiently recover the infinite-dimensional chiral algebras of asymptotic symmetries recently extracted from scattering amplitudes of massless gluons and gravitons in the celestial basis. We also compute some simple one-loop corrections to the chiral algebras and derive the three-dimensional bulk theories for which these 2d algebras furnish an algebra of boundary local operators.