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R. A. Iseppi
noncommutative geometry as starting point to search for a new spectral model of
gravity coupled with matter [6, 7, 13].
Therefore, as confirmed by the quoted monumental results, it is natural to
investigate whether noncommutative geometry could provide a new mathematical
description of other constructions developed in the context of gauge theories. In
this article we focus in particular on the Batalin–Vilkovisky approach to the BRST
construction. After a concise recall of the needed notions from the noncommutative
geometric world (cf. Sect. 2), we will briefly outline the algebraic approach to the
Batalin–Vilkovisky construction for gauge theories with an affine configuration
space (cf. Sect. 3). Then, Sect. 4 is devoted to present our main result: focusing on
a U(2)-matrix model, we present how all the elements playing a role in the BV
construction can be successfully encoded in a purely noncommutative geometrical
object, the so-called BV-spectral triple. Finally, in Sect. 5 we explain how the
problem can be approached and solved in the general context of finite spectral triples
on the algebra M n (C).
2 Finite Spectral Triples and Induced Gauge Theories
Without any doubt, the notion of spectral triple plays a central role in contemporary
noncommutative geometry. In its full generality, a spectral triple can be viewed as
a noncommutative version of the classical concept of Riemannian spin manifold.
However, conversely to what happens in the classical setting, a spectral triple
presents a very rich and interesting structure also when the underline topological
space is 0-dimensional, and hence the corresponding spectral triple is finite dimensional. Even more, it is precisely a finite spectral triple that, in the description of
the full Standard Model as induced by an almost commutative spectral triple [5],
encodes the particle content of the theory. Therefore, we briefly recall the needed
notions in this finite dimensional context, where also our construction will take place
(cf. [9, 12]).
Definition 1 A spectral triple (A, H, D) consists of an involutive unital algebra A,
faithfully represented as operators on a Hilbert space H, together with a self-adjoint
operator D on H, with a compact resolvent, such that the commutators [D, a] are
bounded operators for each a ∈ A. A spectral triple (A, H, D) is finite if the Hilbert
space H and hence the algebra A are finite dimensional.
Given a spectral triple (A, H, D), its structure can be further enriched via the
introduction of a real structure, determining a real spectral triple (A, H, D, J ).
Definition 2 A real structure of odd KO-dimension n (mod 8) on a spectral triple
(A, H, D) is an anti-linear isometry J : H → H that satisfies
J
2
= and J D =
DJ.
R. A. Iseppi
noncommutative geometry as starting point to search for a new spectral model of
gravity coupled with matter [6, 7, 13].
Therefore, as confirmed by the quoted monumental results, it is natural to
investigate whether noncommutative geometry could provide a new mathematical
description of other constructions developed in the context of gauge theories. In
this article we focus in particular on the Batalin–Vilkovisky approach to the BRST
construction. After a concise recall of the needed notions from the noncommutative
geometric world (cf. Sect. 2), we will briefly outline the algebraic approach to the
Batalin–Vilkovisky construction for gauge theories with an affine configuration
space (cf. Sect. 3). Then, Sect. 4 is devoted to present our main result: focusing on
a U(2)-matrix model, we present how all the elements playing a role in the BV
construction can be successfully encoded in a purely noncommutative geometrical
object, the so-called BV-spectral triple. Finally, in Sect. 5 we explain how the
problem can be approached and solved in the general context of finite spectral triples
on the algebra M n (C).
2 Finite Spectral Triples and Induced Gauge Theories
Without any doubt, the notion of spectral triple plays a central role in contemporary
noncommutative geometry. In its full generality, a spectral triple can be viewed as
a noncommutative version of the classical concept of Riemannian spin manifold.
However, conversely to what happens in the classical setting, a spectral triple
presents a very rich and interesting structure also when the underline topological
space is 0-dimensional, and hence the corresponding spectral triple is finite dimensional. Even more, it is precisely a finite spectral triple that, in the description of
the full Standard Model as induced by an almost commutative spectral triple [5],
encodes the particle content of the theory. Therefore, we briefly recall the needed
notions in this finite dimensional context, where also our construction will take place
(cf. [9, 12]).
Definition 1 A spectral triple (A, H, D) consists of an involutive unital algebra A,
faithfully represented as operators on a Hilbert space H, together with a self-adjoint
operator D on H, with a compact resolvent, such that the commutators [D, a] are
bounded operators for each a ∈ A. A spectral triple (A, H, D) is finite if the Hilbert
space H and hence the algebra A are finite dimensional.
Given a spectral triple (A, H, D), its structure can be further enriched via the
introduction of a real structure, determining a real spectral triple (A, H, D, J ).
Definition 2 A real structure of odd KO-dimension n (mod 8) on a spectral triple
(A, H, D) is an anti-linear isometry J : H → H that satisfies
J
2
= and J D =
DJ.