6
Operads
In this chapter we recall various versions of operads required in this book. The
standard references are [9] or [12], plus the original sources [2, 3] and [4].
6.1
Cyclic Operads
Consider the cobweb in Fig. 6.1 consisting of white blobs symbolizing correlation
functions, and propagators represented by edges connecting some outputs of the
blobs. We want to understand which abstract properties of the contractions along
the propagators guarantee that the result of multiple contractions would not depend
on the order in which the contractions are performed.
To be more specific, assume that the inputs of the blobs are labeled by elements of
some finite sets, as {b, u, c, v} in case of the blob y of the figure, or by {0, 1, 2, 3, 4}
in the case of the blob w. We will denote, e.g., by x a ◦ b y the result of the contraction
along the edge e connecting the input of x labeled by a with the input of y labeled
by b, see Fig. 6.1 again.
As the first step of abstraction, we want to interpret the blobs as elements of some
abstract dg-vector spaces, for instance,
x ∈ P
{p, q, r, a}
, y ∈ P
{b, c, u, v}
, etc.
The contraction x a ◦ b y along the edge e is the value of a morphism
a ◦ b : P
{p, q, r, a}
⊗ P
{b, c, u, v}
−→ P
{p, q, r, c, u, v}
,
or, with the indexing sets conveniently decomposed,
a ◦ b : P
{p, q, r} ∪ {a}
⊗ P
{c, u, v} ∪ {b}
−→ P
{p, q, r, c, u, v}
.
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_6
95
Operads
In this chapter we recall various versions of operads required in this book. The
standard references are [9] or [12], plus the original sources [2, 3] and [4].
6.1
Cyclic Operads
Consider the cobweb in Fig. 6.1 consisting of white blobs symbolizing correlation
functions, and propagators represented by edges connecting some outputs of the
blobs. We want to understand which abstract properties of the contractions along
the propagators guarantee that the result of multiple contractions would not depend
on the order in which the contractions are performed.
To be more specific, assume that the inputs of the blobs are labeled by elements of
some finite sets, as {b, u, c, v} in case of the blob y of the figure, or by {0, 1, 2, 3, 4}
in the case of the blob w. We will denote, e.g., by x a ◦ b y the result of the contraction
along the edge e connecting the input of x labeled by a with the input of y labeled
by b, see Fig. 6.1 again.
As the first step of abstraction, we want to interpret the blobs as elements of some
abstract dg-vector spaces, for instance,
x ∈ P
{p, q, r, a}
, y ∈ P
{b, c, u, v}
, etc.
The contraction x a ◦ b y along the edge e is the value of a morphism
a ◦ b : P
{p, q, r, a}
⊗ P
{b, c, u, v}
−→ P
{p, q, r, c, u, v}
,
or, with the indexing sets conveniently decomposed,
a ◦ b : P
{p, q, r} ∪ {a}
⊗ P
{c, u, v} ∪ {b}
−→ P
{p, q, r, c, u, v}
.
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_6
95