Part I
Physics Preliminaries
The standard formulation of perturbative string theory consists of a set of rules
to compute scattering amplitudes for a collection of n particles—excitations of
a string—typically propagating on a D-dimensional Minkowski space-time M. In
the simplest case of tree level scattering amplitudes, this prescription involves an
integration over the moduli space of spheres with n punctures (or disks with n
punctures on its boundary for open strings) and provides a very efficient procedure
to compute scattering amplitudes. On the other hand, comparing this with the
approach taken for point particles the situation in string theory seems somewhat
incomplete. Indeed, for point particles one starts with an action principle for a set of
fields and then obtains the tree level scattering amplitudes by solving the equations
of motions deriving from this action. Since the various string excitations ought to be
interpreted as particles, one would hope to be able to apply the same formalism for
the scattering of strings. The aim of string field theory is precisely to provide such an
action principle so that the set of rules to compute scattering amplitudes for strings
derive from this action. Since the string consists of an infinite linear superposition
of point particle excitations, one would expect that such an action may be rather
complicated. Yet the first construction of a consistent classical action of interacting
open strings has a remarkably simple algebraic structure of a differential graded
associative algebra (V , ∗, d) together with an odd symplectic form compatible with
d and ∗. Such a structure is called a cyclic differential graded associative algebra.
It turns out that this simple theory does not lead to a consistent quantum
theory without the inclusion of closed strings. However, closed strings require a
more elaborate construction, which nevertheless has a similarly appealing algebraic
structure. For this, one necessarily needs all closed string loops. Therefore, one starts
with a decomposition of the relevant moduli space of closed punctured Riemann
surfaces Σ together with coordinate curves around each puncture into elementary
vertices and propagators. The condition that this moduli space is covered exactly
once by the Feynman diagrams built from this data implies that the geometric
vertices satisfy a Batalin–Vilkovisky (BV) master equation. This equation encodes
information about the decomposition of the relevant moduli space. In a next step
one endows the punctures of Σ with an extra structure by attaching a vector space
Physics Preliminaries
The standard formulation of perturbative string theory consists of a set of rules
to compute scattering amplitudes for a collection of n particles—excitations of
a string—typically propagating on a D-dimensional Minkowski space-time M. In
the simplest case of tree level scattering amplitudes, this prescription involves an
integration over the moduli space of spheres with n punctures (or disks with n
punctures on its boundary for open strings) and provides a very efficient procedure
to compute scattering amplitudes. On the other hand, comparing this with the
approach taken for point particles the situation in string theory seems somewhat
incomplete. Indeed, for point particles one starts with an action principle for a set of
fields and then obtains the tree level scattering amplitudes by solving the equations
of motions deriving from this action. Since the various string excitations ought to be
interpreted as particles, one would hope to be able to apply the same formalism for
the scattering of strings. The aim of string field theory is precisely to provide such an
action principle so that the set of rules to compute scattering amplitudes for strings
derive from this action. Since the string consists of an infinite linear superposition
of point particle excitations, one would expect that such an action may be rather
complicated. Yet the first construction of a consistent classical action of interacting
open strings has a remarkably simple algebraic structure of a differential graded
associative algebra (V , ∗, d) together with an odd symplectic form compatible with
d and ∗. Such a structure is called a cyclic differential graded associative algebra.
It turns out that this simple theory does not lead to a consistent quantum
theory without the inclusion of closed strings. However, closed strings require a
more elaborate construction, which nevertheless has a similarly appealing algebraic
structure. For this, one necessarily needs all closed string loops. Therefore, one starts
with a decomposition of the relevant moduli space of closed punctured Riemann
surfaces Σ together with coordinate curves around each puncture into elementary
vertices and propagators. The condition that this moduli space is covered exactly
once by the Feynman diagrams built from this data implies that the geometric
vertices satisfy a Batalin–Vilkovisky (BV) master equation. This equation encodes
information about the decomposition of the relevant moduli space. In a next step
one endows the punctures of Σ with an extra structure by attaching a vector space