250
R. A. Iseppi
where D is a self-adjoint 2 × 2-matrix. By applying Proposition 1, we have that the
above spectral triple induces a gauge theory (X 0 , S 0 ) with
X 0 = {M ∈ M 2 (C) : M
∗
= M}, S 0 [M] = T r(f (M)) and G = U(2),
where f a polynomial in O X 0 . By fixing as basis for X 0 the one given by the Pauli
matrices together with the identity matrix {σ 1 , σ 2 , σ 3 , σ 4 = I d}, we have the following
identifications:
X 0 A
4
R = =M 1 , M 2 , M 3 , M 4 R
S 0 =
r
k=0
(M
2
1 + M
2
2 + M
2
3 )
k g k (M 4 ),
where M a , a = 1, . . . , 4 are real variables and g k (M 4 ) are suitable polynomials only in
M 4 . Given the gauge theory (X 0 , S 0 ) just described, one can verify (cf. [17]) that the
corresponding minimally BV-extended theory has an extended configuration space
whose decomposition as Z-graded vector space is:
X = =E
∗
−3 ⊕ ⊕C
∗
1 , · · · , C
∗
3 −2 ⊕ ⊕M
∗
1 , . . . , M
∗
4 −1 ⊕ X 0 ⊕ ⊕C 1 , · · · , C 3 1 ⊕ ⊕E 2 .
and an extended action
S = S 0 +
i,j,k
ij k M
∗
i M j C k +
i,j,k
C
∗
i (M i E + ij k C j C k ).
(2)
for ij k a totally antisymmetric tensor in the indices i, j, k ∈ {1, 2, 3}, with 123 = 1.
After having determined this BV-extended pair (
X, S), a natural question arises:
indeed, because the pair (X 0 , S 0 ) came as the gauge theory naturally induced by
a finite spectral triple, one might wonder if also the corresponding BV-extended
theory can be encoded in a new BV-spectral triple. In other words, we want to
determine a real spectral triple (A BV , H BV , D BV , J BV ) such that its fermionic action
S ferm coincides with the BV action S BV := S − S 0 of the model. As we explain in
the theorem below, this goal can be reached. However, before constructing all the
elements entering the BV-spectral triple, we remark that the reason why we have to
introduce a real structure and to consider the fermionic action, instead of the spectral
action, is the presence of Grassmannian variables both in
X and S.
We define the following data:
• A BV := M 2 (C);
• H BV :=
M 2 (C)
M
⊕
M 2 (C)
C
⊕
M 2 (C)
E
.
The inner product structure is the Hilbert–Schmidt inner product on each
summand. Moreover, by H BV ,f , we identify the following subspace:
H BV ,f = i · u(2) ⊕ i · su(2) ⊕ u(1)
[M 1 , . . . , M 4 ], [C 1 , . . . , C 3 , 0], [0, . . . , 0, iE]
(3)
R. A. Iseppi
where D is a self-adjoint 2 × 2-matrix. By applying Proposition 1, we have that the
above spectral triple induces a gauge theory (X 0 , S 0 ) with
X 0 = {M ∈ M 2 (C) : M
∗
= M}, S 0 [M] = T r(f (M)) and G = U(2),
where f a polynomial in O X 0 . By fixing as basis for X 0 the one given by the Pauli
matrices together with the identity matrix {σ 1 , σ 2 , σ 3 , σ 4 = I d}, we have the following
identifications:
X 0 A
4
R = =M 1 , M 2 , M 3 , M 4 R
S 0 =
r
k=0
(M
2
1 + M
2
2 + M
2
3 )
k g k (M 4 ),
where M a , a = 1, . . . , 4 are real variables and g k (M 4 ) are suitable polynomials only in
M 4 . Given the gauge theory (X 0 , S 0 ) just described, one can verify (cf. [17]) that the
corresponding minimally BV-extended theory has an extended configuration space
whose decomposition as Z-graded vector space is:
X = =E
∗
−3 ⊕ ⊕C
∗
1 , · · · , C
∗
3 −2 ⊕ ⊕M
∗
1 , . . . , M
∗
4 −1 ⊕ X 0 ⊕ ⊕C 1 , · · · , C 3 1 ⊕ ⊕E 2 .
and an extended action
S = S 0 +
i,j,k
ij k M
∗
i M j C k +
i,j,k
C
∗
i (M i E + ij k C j C k ).
(2)
for ij k a totally antisymmetric tensor in the indices i, j, k ∈ {1, 2, 3}, with 123 = 1.
After having determined this BV-extended pair (
X, S), a natural question arises:
indeed, because the pair (X 0 , S 0 ) came as the gauge theory naturally induced by
a finite spectral triple, one might wonder if also the corresponding BV-extended
theory can be encoded in a new BV-spectral triple. In other words, we want to
determine a real spectral triple (A BV , H BV , D BV , J BV ) such that its fermionic action
S ferm coincides with the BV action S BV := S − S 0 of the model. As we explain in
the theorem below, this goal can be reached. However, before constructing all the
elements entering the BV-spectral triple, we remark that the reason why we have to
introduce a real structure and to consider the fermionic action, instead of the spectral
action, is the presence of Grassmannian variables both in
X and S.
We define the following data:
• A BV := M 2 (C);
• H BV :=
M 2 (C)
M
⊕
M 2 (C)
C
⊕
M 2 (C)
E
.
The inner product structure is the Hilbert–Schmidt inner product on each
summand. Moreover, by H BV ,f , we identify the following subspace:
H BV ,f = i · u(2) ⊕ i · su(2) ⊕ u(1)
[M 1 , . . . , M 4 ], [C 1 , . . . , C 3 , 0], [0, . . . , 0, iE]
(3)