Mathematical Physics

The search for algebraic foundations of colour-kinematics duality and the double-copy construction has brought into focus a generalization of Batalin--Vilkovisky algebras, referred to here as coexact BV-algebras and as $\textrm{BV}^\square$-algebras in other sources. While these structures capture the cubic sector, they fail to encode higher-valence phenomena, for which a homotopy-theoretic extension becomes necessary. This work introduces a conceptual notion of homotopy coexact BV-algebra, defined through the homotopical interplay of commutative and BV structures, and provides a concrete model in terms of generators and relations, obtained through an extension the theory of Koszul duality for operads.

Topologically ordered phases of matter have long been regarded as lying beyond the Landau–Ginzburg paradigm of symmetry breaking. Our recent studies of anyon condensation, however, suggest that such phases may, in fact, be brought back within the Landau–Ginzburg framework. In this talk, I will present a framework that brings topological phases back into the Landau–Ginzburg picture by reformulating the string-net model to be a genuine lattice gauge theory coupled to anyonic matter fields. Within this modified model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated as Landau's effective theory of the Higgs mechanism.