82
A A ∞ - and L ∞ -Algebras
Analogously to A ∞ -algebras, a Maurer–Cartan element c ∈ A of an L ∞ -algebra
(A, L) is essentially a constant morphism from the trivial L ∞ -algebra k sending
1 ∈ k to c, that is,
L(e
c ) = 0
and
|c| = 0 ,
where the exponential, in the completion of the symmetric algebra, is given by
e
c
=
∞
n=0
1
n!
c
∧n
and satisfies Δ(e c ) = e c ⊗ e c .
Finally, there is also the notion of cyclicity in the context of L ∞ -algebras. Let
(A, L) be a L ∞ -algebra whose underlying vector space is equipped with an odd
symplectic structure ω of degree −1. We call a coderivation D ∈ Coder(SA) cyclic
if the corresponding multilinear map ω( d, −) is graded symmetric, i.e.,
ω(d n (c σ 1 , . . . , c σ n ), c σ n+1 ) = (−1)
ω(d n (c 1 , . . . , c n ), c n+1 ) .
We denote the space of cyclic coderivations by Coder
cycl (SA).
As a simple illustration of L ∞ -morphisms we give a background shift in closed
string field theory. Consider the classical action of closed string field theory, the
theory with genus zero vertices l cl only. The corresponding coderivation L cl defines
an L ∞ -algebra and the action reads
S c,cl = ω c (l cl , −)(e
c ) .
Shifting the background simply means that we expand the string field c around
c rather than around zero. The action in the new background is ω c (l cl , −)(e c +c ).
Hence, the vertices l cl [c ] in the shifted background read
l cl [c
] = l cl ◦ E(c
) ,
where E(c ) is the map defined by
E(c
)(c 1 ∧ · · · ∧ c n ) = e
c ∧ c 1 ∧ · · · ∧ c n .
In the language of homotopy algebras, this shift is implemented by
L cl [c
] = E(−c
) ◦ L cl ◦ E(c
) .
Obviously, E(−c ) is the inverse map of E(c ). Furthermore,
Δ ◦ E(c
) = E(c
) ⊗ E(c
)