NCG and the Batalin–Vilkovisky Construction
249
X = X 0 ∪ { ghost & anti-ghost fields } S = S 0 + terms in ghost/anti-ghosts.
However, conditions need to be imposed on how we perform this extension, on the
number and type of ghost and anti-ghost fields we have to introduce and on the
properties satisfied by the extended action S (cf. [15, 17]).
Definition 7 Let the pair (X 0 , S 0 ) be a gauge theory. Then an extended theory
associated with (X 0 , S 0 ) is a pair (
X, S) where
X = ⊕ i∈Z [
X] i is a super graded vector
space suitable to be decomposed as
X ∼ = F ⊕ F
∗
[1],
with [
X]
0
= X 0
(1)
for F = ⊕ i0 F i a graded locally free O X 0 -module with homogeneous components
of finite rank, and S ∈ [O
X ] 0 is a regular function on
X, with S| X 0 = S 0 , S = S 0 and
such that it solves the classical master equation, i.e.,
{ S, S} = 0,
where {−, −} denotes the graded Poisson structure on the algebra O
X .
Note The condition in (1) enforces the prescription of the BV formalism of
introducing all anti-fields/anti-ghost fields corresponding to the fields/ghost fields
in
X. In particular, F describes the fields/ghost fields in the extended theory while
F ∗ [1] denotes the shifted dual module of the anti-fields/anti-ghost fields:
F
∗
[1] = ⊕ i∈Z
F
∗
[1]
i
with
F
∗
[1]
i =
F
∗
i+1 .
Moreover, the Poisson structure on O
X is completely determined by requiring that,
on the generators of
X, it satisfies the following conditions:
β i , β j
= 0,
β
∗
i , β j
= δ ij
and
β
∗
i , β
∗
j
= 0
for β i ∈ F p and β ∗
i ∈
F ∗ [1]
−p−1 , p ∈ Z , while its value on any other possible
combination of fields/ghost fields/anti-fields and anti-ghost fields is equal to zero.
4 BV-Spectral Triple: The Notion and the Relevance
In this section we present how the BV construction for gauge theories on affine
spaces can be encoded in the framework of noncommutative geometry. For simplicity, we focus on a U(2)-gauge theory, induced by the finite spectral triple
(M 2 (C), C
2 , D),
249
X = X 0 ∪ { ghost & anti-ghost fields } S = S 0 + terms in ghost/anti-ghosts.
However, conditions need to be imposed on how we perform this extension, on the
number and type of ghost and anti-ghost fields we have to introduce and on the
properties satisfied by the extended action S (cf. [15, 17]).
Definition 7 Let the pair (X 0 , S 0 ) be a gauge theory. Then an extended theory
associated with (X 0 , S 0 ) is a pair (
X, S) where
X = ⊕ i∈Z [
X] i is a super graded vector
space suitable to be decomposed as
X ∼ = F ⊕ F
∗
[1],
with [
X]
0
= X 0
(1)
for F = ⊕ i0 F i a graded locally free O X 0 -module with homogeneous components
of finite rank, and S ∈ [O
X ] 0 is a regular function on
X, with S| X 0 = S 0 , S = S 0 and
such that it solves the classical master equation, i.e.,
{ S, S} = 0,
where {−, −} denotes the graded Poisson structure on the algebra O
X .
Note The condition in (1) enforces the prescription of the BV formalism of
introducing all anti-fields/anti-ghost fields corresponding to the fields/ghost fields
in
X. In particular, F describes the fields/ghost fields in the extended theory while
F ∗ [1] denotes the shifted dual module of the anti-fields/anti-ghost fields:
F
∗
[1] = ⊕ i∈Z
F
∗
[1]
i
with
F
∗
[1]
i =
F
∗
i+1 .
Moreover, the Poisson structure on O
X is completely determined by requiring that,
on the generators of
X, it satisfies the following conditions:
β i , β j
= 0,
β
∗
i , β j
= δ ij
and
β
∗
i , β
∗
j
= 0
for β i ∈ F p and β ∗
i ∈
F ∗ [1]
−p−1 , p ∈ Z , while its value on any other possible
combination of fields/ghost fields/anti-fields and anti-ghost fields is equal to zero.
4 BV-Spectral Triple: The Notion and the Relevance
In this section we present how the BV construction for gauge theories on affine
spaces can be encoded in the framework of noncommutative geometry. For simplicity, we focus on a U(2)-gauge theory, induced by the finite spectral triple
(M 2 (C), C
2 , D),