Abstract
In this Thesis we discuss applications of homotopy algebras to several aspect of quantum field theories. In an effort to be self-contained, we start introducing L ∞-, A ∞-, and C ∞-
algebras, and contextualising them in the framework of Batalin{Vilkovisky formalism, that
associates every perturbative Lagrangian field theory to an L ∞-algebra encoding the complete
classical theory. Several instances of field theories are reviewed, and their underlying
homotopy algebras are discussed in detail. The connection between homotopy algebras
and scattering amplitudes are explored, and explicit recursion relations (at tree- and loop level)
are provided and applied to concrete examples. We then apply the homotopy algebra
framework to the study of BCJ colour-kinematic duality and double copy prescription for
Yang-Mills theory. Following a Lagrangian approach and with the help of an appropriate
notion of tensor product for homotopy algebras, we introduce a colour-kinematic factorisation
at the level of the L ∞-algebra associated to the theory. We construct a double
copied Yang-Mills theory, and we show that it is perturbatively quantum equivalent to
N = 0 supergravity, proving the validity of the double copy prescription for Yang-Mills
theory at loop-level.
This Thesis is based on the papers [1-6] that I wrote in collaboration with Leron
Borsten, Branislav Jur_co, Hyungrok Kim, Lorenzo Raspollini, Christian Saemann, and
Martin Wolf.