Reviews in Mathematical Physics

We discuss general properties of A_∞-algebras and their applications to the theory of open strings. The properties of cyclicity for A_∞-algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A_∞-algebras and cyclic A_∞-algebras and discuss various consequences of it. In particular, it is applied to classical open string field theories and it is shown that all classical open string field theories on a fixed conformal background are cyclic A_∞-isomorphic to each other. The same results hold for classical closed string field theories, whose algebraic structure is governed by cyclic L_∞-algebras.

In a previous article with A. Aftalion and X. Blanc, it was shown that the hypercontractivity property of the dilation semigroup in spaces of entire functions was a key ingredient in the study of the Lowest Landau Level model for fast rotating Bose–Einstein condensates. That former work was concerned with the stationary constrained variational problem. This article is about the nonlinear Hamiltonian dynamics and the spectral stability of the constrained minima with motivations arising from the description of Tkatchenko modes of Bose–Einstein condensates. Again the hypercontractivity property provides a very strong control of the nonlinear term in the dynamical analysis.