arXiv subjects

Tommaso Macrelli

Publications and source records attributed to Tommaso Macrelli.

12 recordsLinked to original sources

arXiv2023

Double-Copying Self-Dual Yang-Mills Theory to Self-Dual Gravity on Twistor Space

We construct a simple Lorentz-invariant action for maximally supersymmetric self-dual Yang-Mills theory that manifests colour-kinematics duality. We also show that this action double copies to a known action for maximally supersymmetric self-dual gravity. Both actions live on twistor space and illustrate nicely the homotopy algebraic perspective on the double copy presented in arXiv:2307.02563. This example is particularly interesting as the involved Hopf algebra controlling the momentum dependence is non-commutative and suggests a generalisation to gauged maximally supersymmetric self-dual gravity.

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arXiv2023

Kinematic Lie Algebras From Twistor Spaces

We analyze theories with color-kinematics duality from an algebraic perspective and find that any such theory has an underlying BV${}^{\color{gray} \blacksquare}$-algebra structure, extending the ideas of arXiv:1912.03110. Conversely, we show that any theory with a BV${}^{\color{gray} \blacksquare}$-algebra features a kinematic Lie algebra that controls interaction vertices, both on- and off-shell. We explain that the archetypal example of a theory with BV${}^{\color{gray} \blacksquare}$-algebra is Chern-Simons theory, for which the resulting kinematic Lie algebra is isomorphic to the Schouten-Nijenhuis algebra on multivector fields. The BV${}^{\color{gray} \blacksquare}$-algebra implies the known color-kinematics duality of Chern-Simons theory. Similarly, we show that holomorphic and Cauchy-Riemann (CR) Chern-Simons theories come with BV${}^{\color{gray} \blacksquare}$-algebras and that, on the appropriate twistor spaces, these theories organize and identify kinematic Lie algebras for self-dual and full Yang-Mills theories, as well as the currents of any field theory with a twistorial description. We show that this result extends to the loop level under certain assumptions.

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arXiv2023

Double Copy from Tensor Products of Metric BV${}^{\color{gray} \blacksquare}$-algebras

Field theories with kinematic Lie algebras, such as field theories featuring colour-kinematics duality, possess an underlying algebraic structure known as BV${}^{\color{gray} \blacksquare}$-algebra. If, additionally, matter fields are present, this structure is supplemented by a module for the BV${}^{\color{gray} \blacksquare}$-algebra. We explain this perspective, expanding on our previous work and providing many additional mathematical details. We also show how the tensor product of two metric BV${}^{\color{gray} \blacksquare}$-algebras yields the action of a new syngamy field theory, a construction which comprises the familiar double copy construction. As examples, we discuss various scalar field theories, Chern-Simons theory, self-dual Yang-Mills theory, and the pure spinor formulations of both M2-brane models and supersymmetric Yang-Mills theory. The latter leads to a new cubic pure spinor action for ten-dimensional supergravity. We also give a homotopy-algebraic perspective on colour-flavour-stripping, obtain a new restricted tensor product over a wide class of bialgebras, and we show that any field theory (even one without colour-kinematics duality) comes with a kinematic $L_\infty$-algebra.

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arXiv2023

Tree-Level Color-Kinematics Duality from Pure Spinor Actions

We prove that the tree-level scattering amplitudes for (super) Yang-Mills theory in arbitrary dimensions and for M2-brane models exhibit color-kinematics (CK) duality. Our proof for Yang-Mills theory substantially simplifies existing ones in that it relies on the action alone and does not involve any computation; the proof for M2-brane models establishes this result for the first time. Explicitly, we combine the facts that Chern-Simons-type theories naturally come with a kinematic Lie algebra and that both Yang-Mills theory and M2-brane models are of Chern-Simons form when formulated in pure spinor space, extending previous work on Yang-Mills currents arXiv:2108.11708. Our formulation also provides explicit kinematic Lie algebras for the theories under consideration in the form of diffeomorphisms on pure spinor space. The pure spinor formulation of CK-duality is based on ordinary, cubic vertices, but we explain how ordinary CK-duality relates to notions of quartic-vertex 3-Lie algebra CK-duality for M2-brane models previously discussed in the literature.

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arXiv2023

Tree-Level Color-Kinematics Duality Implies Loop-Level Color-Kinematics Duality up to Counterterms

Color-kinematics (CK) duality is a remarkable symmetry of gluon amplitudes that is the key to the double copy which links gauge theory and gravity amplitudes. Here we show that the complete Yang-Mills action itself, including its gauge-fixing and ghost sectors required for quantization, can be recast to manifest CK duality using a series of field redefinitions and gauge choices. Crucially, the resulting loop-level integrands are automatically CK-dual, up to potential Jacobian counterterms required for unitarity. While these counterterms may break CK duality, they exist, are unique and, since the tree-level is unaffected, may be deduced from the action or the integrands. Consequently, CK duality is a symmetry of the action like any other symmetry, and it is anomalous in a controlled and mostly harmless sense. Our results apply to any theory with CK-dual tree-level amplitudes. We also show that two CK duality-manifesting parent actions may be factorized and fused into a consistent quantizable offspring, with the double copy as the prime example. This provides a direct proof of the double copy to all loop orders.

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arXiv2022

Colour-kinematics duality, double copy, and homotopy algebras

Colour-kinematics duality is a remarkable property of Yang-Mills theory. Its validity implies a relation between gauge theory and gravity scattering amplitudes, known as double copy. Albeit fully established at the tree level, its extension to the loop level is conjectural. Lifting the on-shell, scattering amplitudes-based description to the level of action functionals, we argue that a theory that exhibits tree-level colour-kinematics duality can be reformulated in a way such that its loop integrands manifest a generalised form of colour-kinematics duality. Moreover, we show how the structures of higher homotopy theory naturally describe this off-shell reformulation of colour-kinematics duality.

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arXiv2021

Double Copy from Homotopy Algebras

We show that the BRST Lagrangian double copy construction of $\mathcal{N}=0$ supergravity as the `square' of Yang-Mills theory finds a natural interpretation in terms of homotopy algebras. We significantly expand on our previous work arguing the validity of the double copy at the loop level, and we give a detailed derivation of the double copied Lagrangian and BRST operator. Our constructions are very general and can be applied to a vast set of examples.

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arXiv2021

BRST-Lagrangian Double Copy of Yang-Mills Theory

We show that the double copy of gauge theory amplitudes to $\mathcal{N}=0$ supergravity amplitudes extends from tree level to loop level. We first explain that color-kinematic duality is a condition for the Becchi-Rouet-Stora-Tyutin operator and the action of a field theory with cubic interaction terms to double copy to a consistent gauge theory. We then apply this argument to Yang-Mills theory, where color-kinematic duality is known to be satisfied onshell at tree level. Finally, we show that the latter restriction can only lead to terms that can be absorbed in a sequence of field redefinitions, rendering the double copied action equivalent to $\mathcal{N}=0$ supergravity.

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arXiv2020

Scattering Amplitude Recursion Relations in BV Quantisable Theories

Tree-level scattering amplitudes in Yang-Mills theory satisfy a recursion relation due to Berends and Giele which yields e.g. the famous Parke-Taylor formula for MHV amplitudes. We show that the origin of this recursion relation becomes clear in the BV formalism, which encodes a field theory in an $L_\infty$-algebra. The recursion relation is obtained in the transition to a smallest representative in the quasi-isomorphism class of that $L_\infty$-algebra, known as a minimal model. In fact, the quasi-isomorphism contains all the information about the scattering theory. As we explain, the computation of such a minimal model is readily performed in any BV quantisable theory, which, in turn, produces recursion relations for its tree-level scattering amplitudes.

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arXiv2020

Perturbative Quantum Field Theory and Homotopy Algebras

We review the homotopy algebraic perspective on perturbative quantum field theory: classical field theories correspond to homotopy algebras such as $A_\infty$- and $L_\infty$-algebras. Furthermore, their scattering amplitudes are encoded in minimal models of these homotopy algebras at tree level and their quantum relatives at loop level. The translation between Lagrangian field theories and homotopy algebras is provided by the Batalin--Vilkovisky formalism. The minimal models are computed recursively using the homological perturbation lemma, which induces useful recursion relations for the computation of scattering amplitudes. After explaining how the homological perturbation lemma produces the usual Feynman diagram expansion, we use our techniques to verify an identity for the Berends--Giele currents which implies the Kleiss--Kuijf relations.

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arXiv2020

Loop Amplitudes and Quantum Homotopy Algebras

We derive a recursion relation for loop-level scattering amplitudes of Lagrangian field theories that generalises the tree-level Berends-Giele recursion relation in Yang-Mills theory. The origin of this recursion relation is the homological perturbation lemma, which allows us to compute scattering amplitudes from minimal models of quantum homotopy algebras in a recursive way. As an application of our techniques, we give an alternative proof of the relation between non-planar and planar colour-stripped scattering amplitudes.

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arXiv2019

$L_\infty$-Algebras, the BV Formalism, and Classical Fields

We summarise some of our recent works on $L_\infty$-algebras and quasi-groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of $L_\infty$-algebras, we discuss their Maurer-Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin-Vilkovisky formalism. As examples, we explore higher Chern-Simons theory and Yang-Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of $L_\infty$-quasi-isomorphisms, and we propose a twistor space action.

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