252
R. A. Iseppi
Hence, we are constructing a real spectral triple of mixed KO-dimension.
Theorem 1 The data (A BV , H BV , D BV , J BV ) define a real spectral triple (with
mixed KO-dim.), whose fermionic action coincides with the BV action in (2):
S BV =
1
2
BV (ϕ), D BV ϕ
with ϕ ∈ H BV ,f .
where M a , E and C ∗
j have to be treated as real variables while M ∗
a and C j behave as
Grassmannian variables.
Note It can be checked that the algebra A BV is the largest unital algebra that
complete the triple (H BV , D BV , J BV ) defined above to a real spectral triple.
Because the theorem is proved by directly checking all the requirements
appearing in the definition of a spectral triple (cf. [18]), we prefer not to present the
details but to conclude with few remarks on how the physical properties of the BVextended theory get translated in the noncommutative geometrical language. Indeed,
we have that, while the anti-fields/anti-ghost fields M ∗
a and C ∗
j appear as entries of the
operator D BV , the fields/ghost fields M a , C j , and E determine the elements in H BV ,f .
Moreover, the new phenomena of a spectral triple of mixed-KO dim. accounts for
the presence of bosonic and fermionic fields both
X and S.
5 Conclusions and Outlooks
The construction presented in the above section can be applied also to the more
general case of U(n)-gauge theories, naturally induced by finite spectral triples on
the algebra M n (C). What made the above construction possible was the fact that the
extended action S was precisely linear in the anti-fields/anti-ghost fields: indeed,
this property allows to rewrite the BV action as a fermionic action, having all the
anti-fields/anti-ghost fields as entries of the operator D BV . However, this linearity
condition holds also for U(n)-theories, with n > 2. Even though in principle a BV
action obtained by applying the algebraic BV construction could contain higher
other terms in anti-fields/anti-ghost fields, the fact that for this class of models the
algebra of gauge transformation is closed under commutations on the critical locus
X crit ⊂ X 0 of the action functional S 0 ensures the appearance of only linear terms
in the anti-fields/anti-ghost fields. Hence the structure found for this U(2)-model
perfectly replicates for the whole class of U(n)-theories.
References
1. I.A. Batalin, G.A. Vilkovisky, Gauge algebra and quantization. Phys. Lett. B102, 27–31 (1981)
2. I.A. Batalin, G.A. Vilkovisky, Quantization of gauge theories with linearly dependent
generators. Phys. Rev. D28, 2567–2582 (1983). Erratum D30, 508 (1984)
R. A. Iseppi
Hence, we are constructing a real spectral triple of mixed KO-dimension.
Theorem 1 The data (A BV , H BV , D BV , J BV ) define a real spectral triple (with
mixed KO-dim.), whose fermionic action coincides with the BV action in (2):
S BV =
1
2
BV (ϕ), D BV ϕ
with ϕ ∈ H BV ,f .
where M a , E and C ∗
j have to be treated as real variables while M ∗
a and C j behave as
Grassmannian variables.
Note It can be checked that the algebra A BV is the largest unital algebra that
complete the triple (H BV , D BV , J BV ) defined above to a real spectral triple.
Because the theorem is proved by directly checking all the requirements
appearing in the definition of a spectral triple (cf. [18]), we prefer not to present the
details but to conclude with few remarks on how the physical properties of the BVextended theory get translated in the noncommutative geometrical language. Indeed,
we have that, while the anti-fields/anti-ghost fields M ∗
a and C ∗
j appear as entries of the
operator D BV , the fields/ghost fields M a , C j , and E determine the elements in H BV ,f .
Moreover, the new phenomena of a spectral triple of mixed-KO dim. accounts for
the presence of bosonic and fermionic fields both
X and S.
5 Conclusions and Outlooks
The construction presented in the above section can be applied also to the more
general case of U(n)-gauge theories, naturally induced by finite spectral triples on
the algebra M n (C). What made the above construction possible was the fact that the
extended action S was precisely linear in the anti-fields/anti-ghost fields: indeed,
this property allows to rewrite the BV action as a fermionic action, having all the
anti-fields/anti-ghost fields as entries of the operator D BV . However, this linearity
condition holds also for U(n)-theories, with n > 2. Even though in principle a BV
action obtained by applying the algebraic BV construction could contain higher
other terms in anti-fields/anti-ghost fields, the fact that for this class of models the
algebra of gauge transformation is closed under commutations on the critical locus
X crit ⊂ X 0 of the action functional S 0 ensures the appearance of only linear terms
in the anti-fields/anti-ghost fields. Hence the structure found for this U(2)-model
perfectly replicates for the whole class of U(n)-theories.
References
1. I.A. Batalin, G.A. Vilkovisky, Gauge algebra and quantization. Phys. Lett. B102, 27–31 (1981)
2. I.A. Batalin, G.A. Vilkovisky, Quantization of gauge theories with linearly dependent
generators. Phys. Rev. D28, 2567–2582 (1983). Erratum D30, 508 (1984)