216
8 Structures Relevant to Physics
strings. The geometric vertices of the closed string theory ν
g
n define a solution to
the master equation (3.17) on the space of (invariant under the permutations of
punctures) chains on the moduli space of closed Riemann surfaces with punctures.
They are the same thing as a particular morphism from the Feynman transform of the
modular commutative operad to the odd modular operad of the chain complex on
the moduli space. A two-colored version of that construction combining modular
commutative and associative operads would give geometric vertices for the openclosed string field theory.
The BV formalism, as it is know from textbooks, can be recognized as an
instance of Proposition 7.2 applied to a representation of the Feynman transform
of the modular commutative operad in Sect. 8.2. In this case, we obtain a loop
homotopy algebra as described in Theorem 8.3. In the physics language, fields
are coordinate functionals on the dg-vector space (V , d). These would correspond
to a full BV theory. The fields of degree 0 correspond to the original fields of
the starting quantum field theory, fields being of positive degrees, anti-field of
negative ones. Interaction vertices in the quantum BV action are the degree 0 totally
(graded) symmetric functions f
g
n . The quantum BV master equation is recognized
in (8.37). Here, d corresponds to the BRST operator. The role of degree one d-closed
symmetric element
s
i ⊗ s
i in (8.37) is twofold. In the first term on the right-hand
side of the equation, it gives the BV operator Δ. In the second one, it gives the BV
bracket {−, −}. It is the same quantum BV master equation as the equation satisfied
by the closed string field theory action in Part I, Sect. 3.4.
Obviously, a variant of Theorem 8.3 can be given in the case of modular
associative operad and the quantum open-closed operad, the latter leading to the
open-closed quantum BV master equation for action (5.1).
It should be now obvious, by looking at the g = 0 part of the above construction,
that representations of the cobar construction for cyclic operads give the ordinary
homotopy algebras. For example, in the case of the cyclic commutative operad
we get cyclic L ∞ -algebras whereas in the associative case we get cyclic A ∞ -
algebras. The corresponding algebraic structures are equivalent to solutions to the
corresponding classical BV master equations. These are the algebraic structures
discussed in relation to the point particle in Sect. 2.3 and in relation to classical
string field theories, for instance, in Sects. 3.6 and 4.2, cf. also Sect. 5.2 for the
classical open-closed case.
Finally, in Sect. 8.3, we reviewed I BL ∞ -algebras, a common generalization
of loop homotopy algebras and Lie bialgebras. They can also be understood as
algebras over the cobar construction of properads. We however limited ourselves to
their direct algebraic description without going into their properadic origins. They
are generalizations of loop homotopy algebras in the following sense: in place of
the second order, degree one, nilpotent differential operator ¯
hΔ + {S, −}, now we
consider higher order, degree one, nilpotent differential operators. Correspondingly
we have brackets with several inputs and outputs. In particular, we can have an
ordinary bracket and cobracket as in a Lie bialgebra. The I BL ∞ -algebras were
relevant to our discussion of the open-closed string field theory in Sect. 5.2.
8 Structures Relevant to Physics
strings. The geometric vertices of the closed string theory ν
g
n define a solution to
the master equation (3.17) on the space of (invariant under the permutations of
punctures) chains on the moduli space of closed Riemann surfaces with punctures.
They are the same thing as a particular morphism from the Feynman transform of the
modular commutative operad to the odd modular operad of the chain complex on
the moduli space. A two-colored version of that construction combining modular
commutative and associative operads would give geometric vertices for the openclosed string field theory.
The BV formalism, as it is know from textbooks, can be recognized as an
instance of Proposition 7.2 applied to a representation of the Feynman transform
of the modular commutative operad in Sect. 8.2. In this case, we obtain a loop
homotopy algebra as described in Theorem 8.3. In the physics language, fields
are coordinate functionals on the dg-vector space (V , d). These would correspond
to a full BV theory. The fields of degree 0 correspond to the original fields of
the starting quantum field theory, fields being of positive degrees, anti-field of
negative ones. Interaction vertices in the quantum BV action are the degree 0 totally
(graded) symmetric functions f
g
n . The quantum BV master equation is recognized
in (8.37). Here, d corresponds to the BRST operator. The role of degree one d-closed
symmetric element
s
i ⊗ s
i in (8.37) is twofold. In the first term on the right-hand
side of the equation, it gives the BV operator Δ. In the second one, it gives the BV
bracket {−, −}. It is the same quantum BV master equation as the equation satisfied
by the closed string field theory action in Part I, Sect. 3.4.
Obviously, a variant of Theorem 8.3 can be given in the case of modular
associative operad and the quantum open-closed operad, the latter leading to the
open-closed quantum BV master equation for action (5.1).
It should be now obvious, by looking at the g = 0 part of the above construction,
that representations of the cobar construction for cyclic operads give the ordinary
homotopy algebras. For example, in the case of the cyclic commutative operad
we get cyclic L ∞ -algebras whereas in the associative case we get cyclic A ∞ -
algebras. The corresponding algebraic structures are equivalent to solutions to the
corresponding classical BV master equations. These are the algebraic structures
discussed in relation to the point particle in Sect. 2.3 and in relation to classical
string field theories, for instance, in Sects. 3.6 and 4.2, cf. also Sect. 5.2 for the
classical open-closed case.
Finally, in Sect. 8.3, we reviewed I BL ∞ -algebras, a common generalization
of loop homotopy algebras and Lie bialgebras. They can also be understood as
algebras over the cobar construction of properads. We however limited ourselves to
their direct algebraic description without going into their properadic origins. They
are generalizations of loop homotopy algebras in the following sense: in place of
the second order, degree one, nilpotent differential operator ¯
hΔ + {S, −}, now we
consider higher order, degree one, nilpotent differential operators. Correspondingly
we have brackets with several inputs and outputs. In particular, we can have an
ordinary bracket and cobracket as in a Lie bialgebra. The I BL ∞ -algebras were
relevant to our discussion of the open-closed string field theory in Sect. 5.2.