We give an explicit gauge invariant, off-shell, and local double copy construction of gravity from Yang-Mills theory to quartic order. To this end we use the framework of homotopy algebras, and we identify a rich new algebraic structure associated with color-stripped Yang-Mills theory. This algebra, which is a generalization of a Batalin-Vilkovisky algebra, is the underlying structure necessary for double copy. We give a self-contained introduction into these algebras by illustrating them for Chern-Simons theory in three dimensions. We then construct $N=0$ supergravity in the form of double field theory in terms of the algebraic Yang-Mills building blocks to quartic order in interactions. As applications of the same universal formula, we rederive the four-graviton scattering amplitude and compute a chiral form of the Courant algebroid gauge structure of double field theory.

We formulate gauge theories based on Leibniz(-Loday) algebras and uncover their underlying mathematical structure. Various special cases have been developed in the context of gauged supergravity and exceptional field theory. These are based on ‘tensor hierarchies’, which describe towers of p-form gauge fields transforming under non-abelian gauge symmetries and which have been constructed up to low levels. Here we define ‘infinity-enhanced Leibniz algebras’ that guarantee the existence of consistent tensor hierarchies to arbitrary level. We contrast these algebras with strongly homotopy Lie algebras ( $$L_{\infty }$$L ∞ algebras), which can be used to define topological field theories for which all curvatures vanish. Any infinity-enhanced Leibniz algebra carries an associated $$L_{\infty }$$L ∞ algebra, which we discuss.

Conformal higher spin (CHS) fields, yet being non unitary, provide a remarkable example of a consistent interacting higher spin theory in flat space background, that is local to all orders. The non-linear action is defined as the logarithmically UV divergent part of a one-loop scalar effective action. In this paper we take a particle model, that describes the interaction of a scalar particle to the CHS background, and compute its path integral on the circle. We thus provide a worldline representation for the CHS action, and rederive its quadratic part. We plan to come back to the subject, to compute cubic and higher vertices, in a future work.

We show that to cubic order double field theory is encoded in Yang-Mills theory. To this end we use algebraic structures from string field theory as follows: The $L_{\infty }$-algebra of Yang-Mills theory is the tensor product $\mathcal{K}\otimes \mathfrak{g}$ of the Lie algebra $\mathfrak{g}$ of the gauge group and a “kinematic algebra” $\mathcal{K}$ that is a $C_{\infty }$-algebra. This structure induces a cubic truncation of an $L_{\infty }$-algebra on the subspace of level-matched states of the tensor product $\mathcal{K}\otimes \overline{\mathcal{K}}$ of two copies of the kinematic algebra. This $L_{\infty }$-algebra encodes double field theory. More precisely, this construction relies on a particular form of the Yang-Mills $L_{\infty }$-algebra following from string field theory or from the quantization of a suitable worldline theory.

The double-copy program relies crucially on the so-called color-kinematics duality which, in turn, is widely believed to descend from a kinematic algebra possessed by gauge theories. In this paper we construct the kinematic algebra of gauge-invariant and off shell self-dual Yang-Mills theory, up to trilinear maps. This structure is a homotopy algebra of the same type as the ones recently uncovered in Chern-Simons and full Yang-Mills theories. To make contact with known results for the self-dual sector, we show that it reduces to the algebra found by Monteiro and O’Connell upon taking the light cone gauge and partially solving the self-duality constraints. Finally, we test a double-copy prescription recently proposed in Bonezzi et al. [Gauge invariant double copy of Yang-Mills theory: The quartic theory, Phys. Rev. D 107, 126015 (2023)] and reproduce self-dual gravity.

The $$ \mathcal{N} $$ N = 4 supersymmetric spinning particle admits several consistent quantizations, related to the gauging of different subgroups of the SO(4) R-symmetry on the worldline. We construct the background independent BRST quantization for all of these choices which are shown to reproduce either the massless NS-NS spectrum of the string, or Einstein theory with or without the antisymmetric tensor field and/or dilaton corresponding to different restrictions. Quantum consistency of the worldline implies equations of motion for the background which, in addition to the admissible string backgrounds, admit Einstein manifolds with or whithout a cosmological constant. The vertex operators for the Kalb-Ramond, graviton and dilaton fields are obtained from the linear variations of the BRST charge. They produce the physical states by action on the diffeomorphism ghost states.