2
1 Introduction
from the Feynman transform of the modular commutative operad is the loop
homotopy algebra (aka quantum L ∞ -algebra). Such an algebra can equivalently
be characterized by a solution to the quantum Batalin–Vilkovisky (BV) master
equation, i.e., by Maurer–Cartan elements of the corresponding BV algebra. The
origin of the BV operator and the BV bracket can be traced back to the operadic selfcompositions and compositions, respectively. Obviously, this is one of the reasons
why operads and the corresponding algebras are relevant to physics. More generally,
a morphism from the Feynman transform of any modular operad to a general odd
modular operad can be described by a solution to a proper generalization of the
quantum BV equation.
All of these constructions contain a genus zero part, due to the forgetful functor
from the category of modular operads to the category of cyclic operads. One simply
forgets about loops. Hence, the dg-vector spaces associated with corollas with nonzero genus are trivial as well as the operations corresponding to self-contractions
of corollas. This functor has an adjoint, the modular completion (or envelope),
which roughly speaking freely adds loops. The resulting (cyclic) homotopy algebras
(e.g., L ∞ - or A ∞ -algebras) can then be described by solutions to the corresponding
classical master equations.
All this is nicely illustrated by string field theory. For simplicity, let us consider
closed strings. One starts with the modular commutative operad (the modular
envelope of the cyclic commutative operad) and its Feynman transform. The theory
of closed strings provides us with two natural odd modular operads. One is formed
by the singular chain complex on the moduli space of closed Riemann surfaces with
punctures and with the operadic compositions induced from (twisted) sewing and
self-sewing of surfaces. The other one is the endomorphism operad over the state
space of the (first quantized) closed string with the differential given by the BRST
(Becchi-Rouet-Stora-Tyutin) operator and the odd symplectic form induced by the
BPZ (Belavin-Polyakov-Zamolodchikov) pairing.
A morphism from the Feynman transform of the modular commutative operad
to the first one gives a BV structure on the moduli space of Riemann surfaces
and a solution to the corresponding quantum BV equation. This solution describes
the decomposition of the moduli space and hence tells us what the geometric
vertices are. This is the geometric background independent part of the construction
of closed string field theory in Zwiebach’s approach. It is then composed with a
morphism of odd modular operads, provided by conformal field theory, going from
the moduli space operad to the endomorphism operad associating algebraic vertices
to the geometric ones. This is the background dependent part of the construction.
This composition describes a particular morphism from the Feynman transform of
the modular commutative operad to the endomorphism operad, i.e., a solution to
the quantum BV equation, describing the algebraic vertices forming the quantum
BV action of the closed string field theory. Obviously, the vertices are labeled by
their valence n and the genus g. The algebraic vertices are degree zero graded
symmetric functions on the string state space. Using the odd symplectic form,
one can equivalently use the string operations, i.e., degree one graded symmetric
multilinear maps from the state space to itself. Hence, we have a description in
1 Introduction
from the Feynman transform of the modular commutative operad is the loop
homotopy algebra (aka quantum L ∞ -algebra). Such an algebra can equivalently
be characterized by a solution to the quantum Batalin–Vilkovisky (BV) master
equation, i.e., by Maurer–Cartan elements of the corresponding BV algebra. The
origin of the BV operator and the BV bracket can be traced back to the operadic selfcompositions and compositions, respectively. Obviously, this is one of the reasons
why operads and the corresponding algebras are relevant to physics. More generally,
a morphism from the Feynman transform of any modular operad to a general odd
modular operad can be described by a solution to a proper generalization of the
quantum BV equation.
All of these constructions contain a genus zero part, due to the forgetful functor
from the category of modular operads to the category of cyclic operads. One simply
forgets about loops. Hence, the dg-vector spaces associated with corollas with nonzero genus are trivial as well as the operations corresponding to self-contractions
of corollas. This functor has an adjoint, the modular completion (or envelope),
which roughly speaking freely adds loops. The resulting (cyclic) homotopy algebras
(e.g., L ∞ - or A ∞ -algebras) can then be described by solutions to the corresponding
classical master equations.
All this is nicely illustrated by string field theory. For simplicity, let us consider
closed strings. One starts with the modular commutative operad (the modular
envelope of the cyclic commutative operad) and its Feynman transform. The theory
of closed strings provides us with two natural odd modular operads. One is formed
by the singular chain complex on the moduli space of closed Riemann surfaces with
punctures and with the operadic compositions induced from (twisted) sewing and
self-sewing of surfaces. The other one is the endomorphism operad over the state
space of the (first quantized) closed string with the differential given by the BRST
(Becchi-Rouet-Stora-Tyutin) operator and the odd symplectic form induced by the
BPZ (Belavin-Polyakov-Zamolodchikov) pairing.
A morphism from the Feynman transform of the modular commutative operad
to the first one gives a BV structure on the moduli space of Riemann surfaces
and a solution to the corresponding quantum BV equation. This solution describes
the decomposition of the moduli space and hence tells us what the geometric
vertices are. This is the geometric background independent part of the construction
of closed string field theory in Zwiebach’s approach. It is then composed with a
morphism of odd modular operads, provided by conformal field theory, going from
the moduli space operad to the endomorphism operad associating algebraic vertices
to the geometric ones. This is the background dependent part of the construction.
This composition describes a particular morphism from the Feynman transform of
the modular commutative operad to the endomorphism operad, i.e., a solution to
the quantum BV equation, describing the algebraic vertices forming the quantum
BV action of the closed string field theory. Obviously, the vertices are labeled by
their valence n and the genus g. The algebraic vertices are degree zero graded
symmetric functions on the string state space. Using the odd symplectic form,
one can equivalently use the string operations, i.e., degree one graded symmetric
multilinear maps from the state space to itself. Hence, we have a description in