40
3 String Theory
of degree 1 on the complex of singular chains. Here a and b denote the punctures
that are being twist sewed.
We can implement the indistinguishability of identical particles already at the
geometric level by requiring the invariance under permutations of punctures. The
complex of chains that are invariant under permutations of punctures is denoted
by C •
inv ( ˆ
P g,n ). The maps a • b and • ab can be lifted to operations on C •
inv ( ˆ
P g,n ).
This gives rise to
{ν g 1 ,n 1 +1 , ν g 2 ,n 2 +1 } =
σ ∈uSh(n 1 ,n 2 )
σ (ν g 1 ,n 1 +1 a • b ν g 2 ,n 2 +1 ),
where uSh(n 1 , n 2 ) is the set of (n 1 , n 2 )-unshuffles, i.e. permutations σ ∈ Σ n 1 +n 2
such that
σ (1) < · · · < σ (n 1 ) and σ (n 1 + 1) < · · · < σ (n 1 + n 2 ),
and
Δν g,n = • ab ν g,n+2 .
It is possible to define a graded commutative, associative product of degree 0 on
the chain complex of moduli spaces of disconnected surfaces by the disjoint union
ν μ of vertices. Alltogether,
(ν μ) ρ = ν (μ ρ) , ν μ = (−1)
|ν||μ| μ ν , |ν μ| = |ν| + |μ| .
With respect to this multiplication, Δ is a nilpotent second order derivation, i.e.
Δ
2
= 0,
Δ(ν μ ρ)−Δ(ν μ) ρ −(−1)
|ν| ν Δ(μ ρ)−(−1)
(|ν|+1)|μ| μ Δ(ν ρ)
+ Δ(ν) μ ρ + (−1)
|ν| ν Δ(μ) ρ + (−1)
|ν|+|μ| ν μ Δ(ρ) = 0 .
We recognize the structure of a BV algebra. The bracket {−, −} expressed in terms
of Δ through
{ν, μ} := (−1)
|ν| Δ(ν μ) − (−1)
|ν| Δ(ν) μ − ν Δ(μ) ,
defines a Gerstenhaber bracket, also called an anti-bracket, with the properties
{ν, μ} + (−1)
(|ν|+1)(|μ|+1)
{μ, ν} = 0 ,
(−1)
(|ν|+1)(|ρ|+1)
{{ν, μ}, ρ} + cycl = 0 .