Noncommutative geometry and the BV formalism: application to a matrix model (Q2410644)

The BV (Batalin-Vilkovisky) formalism has been invented with the aim of solving the problem of quantizing a nonabelian gauge theory using the path integral approach. The quantization of non-abelian gauge theories is an interesting subject both from a mathematical and from a physical point of view. The importance of having a precise formulation of a procedure for quantizing gauge theories comes from the fact that all known fundamental interactions appearing in nature are governed by gauge theories. The BV formalism, as a far-reaching extension of BRST formalism, is a powerful method to analyze the nature of the gauge symmetries in gauge theories. The paper under review applies the BV formalism to a \(U(2)\)-matrix model, derived from a finite spectral triple on the algebra \(M_2(\mathbb{C})\). It is seen that the gauge structure of this model is richer than expected. The general form of the extended action that solves the classical master equation is stated and the BV auxiliary pairs (essential for performing a gauge-fixing procedure) are determined. The BV spectral triple and the BV auxiliary spectral triple are defined to obtain a noncommutative geometric description of the BV formalism for this particular model. These are constructed from the gauge fields, ghost fields and anti-fields that enter the BV construction. The construction proposed in the paper under review, fits nicely with previous studies of the BV formalism applied to gauge models derived from noncommutative geometry. This is the first description of the BV formalism completely in terms of noncommutative geometric data (spectral triples). This approach allows for a geometric description of the ghost fields and their properties in terms of the BV spectral triple.