String field theory

This thesis discusses several aspects of string field theory. The
first issue is bosonic open-closed string field theory and its
associated algebraic structure -- the quantum open-closed homotopy
algebra. We describe the quantum open-closed homotopy algebra in
the framework of homotopy involutive Lie bialgebras, as a morphism
from the loop homotopy Lie algebra of closed string to the
involutive Lie bialgebra on the Hochschild complex of open strings.
The formulation of the classical/quantum open-closed homotopy
algebra in terms of a morphism from the closed string algebra to
the open string Hochschild complex reveals deformation properties
of closed strings on open string field theory. In particular, we
show that inequivalent classical open string field theories are
parametrized by closed string backgrounds up to gauge
transformations. At the quantum level the correspondence is
obstructed, but for other realizations such as the topological
string, a non-trivial correspondence persists. Furthermore, we
proof the decomposition theorem for the loop homotopy Lie algebra
of closed string field theory, which implies uniqueness of closed
string field theory on a fixed conformal background. Second, the
construction of string field theory can be rephrased in terms of
operads. In particular, we show that the formulation of string
field theory splits into two parts: The first part is based solely
on the moduli space of world sheets and ensures that the
perturbative string amplitudes are recovered via Feynman rules. The
second part requires a choice of background and determines the real
string field theory vertices. Each of these parts can be described
equivalently as a morphism between appropriate cyclic and modular
operads, at the classical and quantum level respectively. The
algebraic structure of string field theory is then encoded in the
composition of these two morphisms. Finally, we outline the
construction of type II superstring field theory. Specific features
of the superstring are the appearance of Ramond punctures and the
picture changing operators. The sewing in the Ramond sector
requires an additional constraint on the state space of the world
sheet conformal field theory, such that the associated symplectic
structure is non-degenerate, at least on-shell. Moreover, we
formulate an appropriate minimal area metric problem for type II
world sheets, which can be utilized to sketch the construction of a
consistent set of geometric vertices. The algebraic structure of
type II superstring field theory is that of a $\mathcal{N}=1$ loop
homotopy Lie algebra at the quantum level, and that of a
$\mathcal{N}=1$ homotopy Lie algebra at the classical level.