4
1 Introduction
Operads related to moduli spaces will not appear in this chapter. Though it seems
natural to consider moduli spaces related to metric graphs, this would take us far
beyond the scope of the present lecture notes.
In Chap. 2, we generalize the above constructions to closed strings. This naturally
leads to the construction of BV algebras and their Maurer–Cartan elements through
the decomposition of the relevant moduli spaces of Riemann surfaces. Combined
with the morphism from the moduli BV algebra to the closed string BV algebra
provided by the conformal field theory, this decomposition gives the vertices of the
corresponding string field theory action. The whole construction is mathematically
interpreted again in terms of a representation of the Feynman transform of the
modular commutative operad described in Part II. We also include a section on
the uniqueness and background independence of the resulting algebraic structure,
which is an important issue in string theory but can be bypassed by readers who are
primarily interested in the operadic/algebraic aspects of string field theory only.
In Chap. 3, we introduce open strings. Here, in analogy with the point particle,
the algebraic structure simplifies to that of a differential graded algebra (DGA).
Nonequivalent deformations of classical open string theory are governed by its
cyclic cohomology. Non-trivial elements are realized as A ∞ -algebras that come
with it. According to general theory described in Part II, A ∞ -algebras and their
cyclic versions are interpreted as representations of the cobar complex of the Assoperad and its cyclic version, respectively.
Part I then closes with Chap. 4, which gives an account of the open-closed
homotopy algebra. The chosen algebraic description is that using involutive bi L ∞ -
(IBL ∞ -) algebras which are revisited in the appendix to this chapter and again, from
an alternative angle and more rigorously, in the last section of Part II. As already
mentioned, this approach would need to introduce properads, which is far beyond
the scope of this introductory lecture notes. The open-closed homotopy algebra
has also an operadic description in terms of a colored operad (open-closed operad
combining the modular commutative and associative operads) which will, however,
also not be covered in this book. This is, however, a straightforward extension.
The main idea behind the presentation and the structure of this book is perhaps
best summarized by the commutative diagram of odd modular operads in Fig. 1.1,
leading to a construction of a quantum string field theory. Let us specify it in
the case of the closed bosonic quantum string field theory, the example discussed
most thoroughly in these lecture notes. Starting from the Feynman transform of
the modular commutative operad, the horizontal arrow determines a Maurer–Cartan
element in the BV algebra of chains on the moduli space of closed Riemann
surfaces with punctures, i.e., a decomposition of the moduli spaces determining
the geometric vertices. The vertical arrow stands for a morphism from the BV
algebra of chains on this moduli space to the endomorphism operad determined by
the closed conformal field theory. In particular, it maps a Maurer–Cartan element
(the geometric vertices) to the Maurer–Cartan element (the algebraic vertices)
that defines a gauge invariant bosonic closed quantum string field theory action.
Hence, the diagonal arrow determines a particular solution to the algebraic Batalin–
Vilkovisky master equation on the BV algebra of functions on the underlying graded
1 Introduction
Operads related to moduli spaces will not appear in this chapter. Though it seems
natural to consider moduli spaces related to metric graphs, this would take us far
beyond the scope of the present lecture notes.
In Chap. 2, we generalize the above constructions to closed strings. This naturally
leads to the construction of BV algebras and their Maurer–Cartan elements through
the decomposition of the relevant moduli spaces of Riemann surfaces. Combined
with the morphism from the moduli BV algebra to the closed string BV algebra
provided by the conformal field theory, this decomposition gives the vertices of the
corresponding string field theory action. The whole construction is mathematically
interpreted again in terms of a representation of the Feynman transform of the
modular commutative operad described in Part II. We also include a section on
the uniqueness and background independence of the resulting algebraic structure,
which is an important issue in string theory but can be bypassed by readers who are
primarily interested in the operadic/algebraic aspects of string field theory only.
In Chap. 3, we introduce open strings. Here, in analogy with the point particle,
the algebraic structure simplifies to that of a differential graded algebra (DGA).
Nonequivalent deformations of classical open string theory are governed by its
cyclic cohomology. Non-trivial elements are realized as A ∞ -algebras that come
with it. According to general theory described in Part II, A ∞ -algebras and their
cyclic versions are interpreted as representations of the cobar complex of the Assoperad and its cyclic version, respectively.
Part I then closes with Chap. 4, which gives an account of the open-closed
homotopy algebra. The chosen algebraic description is that using involutive bi L ∞ -
(IBL ∞ -) algebras which are revisited in the appendix to this chapter and again, from
an alternative angle and more rigorously, in the last section of Part II. As already
mentioned, this approach would need to introduce properads, which is far beyond
the scope of this introductory lecture notes. The open-closed homotopy algebra
has also an operadic description in terms of a colored operad (open-closed operad
combining the modular commutative and associative operads) which will, however,
also not be covered in this book. This is, however, a straightforward extension.
The main idea behind the presentation and the structure of this book is perhaps
best summarized by the commutative diagram of odd modular operads in Fig. 1.1,
leading to a construction of a quantum string field theory. Let us specify it in
the case of the closed bosonic quantum string field theory, the example discussed
most thoroughly in these lecture notes. Starting from the Feynman transform of
the modular commutative operad, the horizontal arrow determines a Maurer–Cartan
element in the BV algebra of chains on the moduli space of closed Riemann
surfaces with punctures, i.e., a decomposition of the moduli spaces determining
the geometric vertices. The vertical arrow stands for a morphism from the BV
algebra of chains on this moduli space to the endomorphism operad determined by
the closed conformal field theory. In particular, it maps a Maurer–Cartan element
(the geometric vertices) to the Maurer–Cartan element (the algebraic vertices)
that defines a gauge invariant bosonic closed quantum string field theory action.
Hence, the diagonal arrow determines a particular solution to the algebraic Batalin–
Vilkovisky master equation on the BV algebra of functions on the underlying graded