1 Introduction
3
terms of a collection of n − 1-ary brackets of degree one labeled by genus g. The
genus zero part of the construction describes the classical closed string field theory.
This picture remains valid for the full quantum open-closed string field theory.
One has to use a two-colored modular operad combining the modular commutative
and associative operads, consider the moduli space of bordered Riemann surfaces
with punctures in the interior as well as on the boundaries, and the open-closed
conformal field theory.
When described in terms of higher brackets, a loop homotopy algebra can be
viewed as a particular example of an IBL ∞ (homotopy involutive Lie bi-) algebra.
For this, we view the BV operator Δ as an operation with zero input and two
outputs, i.e., as a particular cobracket. The algebraic properties of the brackets and
the cobracket are (after a dualization) encoded in the square-zero condition on the
full quantum BRST operator Δ + {S, −}. Here, S is the quantum BV action and
{−, −} the BV bracket. IBL ∞ -algebras are algebras over the cobar complex of the
terminal properad. Roughly speaking, we consider corollas as before but now their
legs are oriented, they can be either outputs or inputs. The operations can now be
pictured by simultaneously joining several outputs of one corolla with inputs of an
another one followed by a contraction of new edges created in the process. Since
the simultaneous joining of legs of two corollas increases the genus, we do not need
an analogue of the self-composition from modular operads any more. In this book
we describe IBL ∞ -algebras, but refrain from introducing properads. The related
constructions are analogous to the case of operads but more complex. Nevertheless,
the approach using the language of IBL ∞ -algebras is fruitful and can be used for
an algebraic description of the open-closed string theory, leading to its formulation
analogous to Kontsevich’s formality.
We start with an introduction of the concepts relevant to the BV action as it
arises in string field theory in Part I, while in Part II we focus on the theory
of (cyclic, modular, odd modular) operads and morphisms between these which,
in turn, correspond to the homotopy algebras introduced in the physics context
of Part I. Also, a concise mathematical description of IBL ∞ -algebras is given in
Part II. A more informal parallel description of these in a form directly used in
our description of the open-closed string field theory is provided in the appendix to
Part I. Whenever possible we will comment on the connection between these two
parts at the end of the respective sections.
Structure of the Book In the first chapter, we formulate the relativistic scalar field
within the BV formalism starting from the world-line formulation for a relativistic
particle. This serves primarily as a toy model and a motivation for the analogous
construction in string field theory. In this approach, the appearance of differential
graded algebras and homotopy algebras within the BV formalism becomes very
natural. As explained in detail in Part II, the homotopy algebras of quantum field
theory and string field theory are representations of operads. For example, as already
mentioned above, loop homotopy algebras which are directly related to the usual BV
formalism are described as operad morphisms from the Feynman transform of the
modular envelope of the cyclic commutative operad to the endomorphism operad.
3
terms of a collection of n − 1-ary brackets of degree one labeled by genus g. The
genus zero part of the construction describes the classical closed string field theory.
This picture remains valid for the full quantum open-closed string field theory.
One has to use a two-colored modular operad combining the modular commutative
and associative operads, consider the moduli space of bordered Riemann surfaces
with punctures in the interior as well as on the boundaries, and the open-closed
conformal field theory.
When described in terms of higher brackets, a loop homotopy algebra can be
viewed as a particular example of an IBL ∞ (homotopy involutive Lie bi-) algebra.
For this, we view the BV operator Δ as an operation with zero input and two
outputs, i.e., as a particular cobracket. The algebraic properties of the brackets and
the cobracket are (after a dualization) encoded in the square-zero condition on the
full quantum BRST operator Δ + {S, −}. Here, S is the quantum BV action and
{−, −} the BV bracket. IBL ∞ -algebras are algebras over the cobar complex of the
terminal properad. Roughly speaking, we consider corollas as before but now their
legs are oriented, they can be either outputs or inputs. The operations can now be
pictured by simultaneously joining several outputs of one corolla with inputs of an
another one followed by a contraction of new edges created in the process. Since
the simultaneous joining of legs of two corollas increases the genus, we do not need
an analogue of the self-composition from modular operads any more. In this book
we describe IBL ∞ -algebras, but refrain from introducing properads. The related
constructions are analogous to the case of operads but more complex. Nevertheless,
the approach using the language of IBL ∞ -algebras is fruitful and can be used for
an algebraic description of the open-closed string theory, leading to its formulation
analogous to Kontsevich’s formality.
We start with an introduction of the concepts relevant to the BV action as it
arises in string field theory in Part I, while in Part II we focus on the theory
of (cyclic, modular, odd modular) operads and morphisms between these which,
in turn, correspond to the homotopy algebras introduced in the physics context
of Part I. Also, a concise mathematical description of IBL ∞ -algebras is given in
Part II. A more informal parallel description of these in a form directly used in
our description of the open-closed string field theory is provided in the appendix to
Part I. Whenever possible we will comment on the connection between these two
parts at the end of the respective sections.
Structure of the Book In the first chapter, we formulate the relativistic scalar field
within the BV formalism starting from the world-line formulation for a relativistic
particle. This serves primarily as a toy model and a motivation for the analogous
construction in string field theory. In this approach, the appearance of differential
graded algebras and homotopy algebras within the BV formalism becomes very
natural. As explained in detail in Part II, the homotopy algebras of quantum field
theory and string field theory are representations of operads. For example, as already
mentioned above, loop homotopy algebras which are directly related to the usual BV
formalism are described as operad morphisms from the Feynman transform of the
modular envelope of the cyclic commutative operad to the endomorphism operad.