Abstract: This article presents how the BV formalism naturally inserts in the framework of noncommutative geometry for gauge theories induced by fi- nite spectral triples. Reaching this goal entails that not only all the steps of the BV construction, from the introduction of ghost/anti-ghost fields to the construction of the BRST complex, can be expressed using noncommuta- tive geometric objects, but also that the method to go from one step in the construction to the next one has an intrinsically noncommutative geometric nature. Moreover, we prove that both the classical BV and BRST complexes coincide with another cohomological theory, naturally appearing in noncommutative geometry: the Hochschild complex of a coalgebra. The construction is presented in detail for 𝑈 (𝑛)-gauge theories induced by spectral triples on the algebra 𝑀𝑛 (C).
Abstract: After arguing why the Batalin-Vilkovisky (BV) formalism is expected to find a natural description within the framework of noncommutative geometry, we explain how this relation takes form for gauge theories induced by finite spectral triples. In particular, we demonstrate how the two extension procedures appearing in the BV formalism, that is, the initial extension via the introduction of ghost/anti-ghost fields and the further extension with auxiliary fields, can be described in the language of noncommutative geometry, using the notions of BV spectral triple and total spectral triple, respectively. The construction is presented in details for all 𝑈(2)-gauge theories induced by spectral triples on the algebra 𝑀2(C). Indications are given on how to extend the results to 𝑈(𝑛)-gauge theories, for 𝑛 > 2.
Abstract: In this paper we argue why noncommutative geometry offers a natural geometrical framework to describe the Batalin- Vilkovisky construction for gauge theories over algebraic spaces. A key role is played by the notion of BV-spectral triple, which encodes all the elements of a BV-extended theory within a purely noncommutative geometrical object. An interesting aspect of this approach is that it provides all physical properties, like being a ghost field or antighost field, with a geometrical interpretation. We present our results for the case of 𝑈(2)-matrix models. However, indications are given on how to perform the construction in the general setting of 𝑈(𝑛)-theories.
Abstract: The BRST complex, naturally appearing in the BV construction for gauge theory, plays an interesting role, allowing to recall some physical information about the gauge theory we are analysing. It was know that, in the irreducible case, this BRST complex coincides with the Lie algebra cohomology but what would it happen in the case of reducible theories, where ghosts for ghosts appear? By introducing a new and generalized version of the Lie algebra cohomology, I succeeded in describing the BRST complex for a 𝑈(2)-model in terms of this new cohomology, discovering a double complex structure, not visible at the BRST level.
Abstract: In this article, I present a new method to perform the BV construction in the finite dimensional case. In particular, I describe how to explicitly perform the construction without having to introduce an infinite number of ghosts and ghostfor- ghost fields. The second part of this article is devoted to the application of this method to a matrix model endowed with a 𝑈(2)-gauge symmetry, explicitly determining the corresponding extended theory and finding the general exact solution of the classical master equation for the model.
Abstract: In this publication, for the first time, the BV extension of a gauge theory is described in terms of noncommutative geometry. We analyze a 𝑈(2)-matrix model derived from a finite spectral triple. To describe the BV formalism in the context of noncommutative geometry, we define two finite spectral triples: the BV spectral triple and the BV auxiliary spectral triple. These are constructed from the gauge fields, ghost fields and anti-fields that enter the BV construction. We show that the fermionic actions of these spectral triples add up precisely to the BV action of the theory. This approach allows for a (noncommutative) geometric description of the ghost fields and their properties in terms of the BV spectral triple.
Abstract: Analysis of gauge theories and mathematical aspects of quantum field theory (more precisely, Batalin-Vilkovisky construction and BRST-quantization procedure), using techniques coming from noncommutative geometry and algebraic geometry. In my thesis, I focused on a family of gauge theories induced by finite spectral triples. For these theories I have first of all computed the BV-extended theory associated, by determining the associated ghost sector and a family of solution to the classical master equation. Then I have computed and determined the associated BV and BRST cohomology complex, the last one obtained from the first by gauge fixing procedure. Finally, I constructed two new spectral triples, one encoding the BV extended theory associated to the gauge theories under analysis and the second accounting for the associated fields required for the performing of a gauge fixing procedure. Finally, an explicit relation between the BRST complex and a new, generalized notion of graded Lie algebra cohomology has been determined and analysed in details. My thesis aimed to settle the basis for the construction of a BV formalism in the context of spectral triples, within the framework of noncommutative geometry.
Roberta Iseppi 