A Noncommutative Geometric Approach
to the Batalin–Vilkovisky Construction
Roberta A. Iseppi
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 anti-ghost 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.
Keywords Noncommutative geometry · Batalin–Vilkovisky construction · Finite
spectral triple · Gauge theory · Matrix models
1 Introduction: Why Noncommative Geometry
Since its early days, noncommutative geometry [9] has shown a reciprocal and
valuable interconnection with several areas of mathematics, such as motivic and
foliation theory, operator algebras, and KK-theory. However, maybe even more
remarkably, noncommutative geometry revealed a deep relation to quantum field
theory and gauge theory in particular. A confirmation of that can be found in a
series of the celebrated results, which began with the pioneering papers by Connes
[8, 10], had a breakthrough in [3, 4, 11], and finally arrived to the key result obtained
by Chamseddine, Connes, and Marcolli [5] of deriving the full Standard Model of
particles, with neutrino mixing and minimally coupled to gravity, from a purely
noncommutative geometrical object. Furthermore, recently new approaches have
been suggested to go beyond the Standard Model, using the framework provided by
R. A. Iseppi ()
Center for Quantum Geometry of Moduli Spaces, Aarhus University, Aarhus, Denmark
e-mail: roberta.iseppi@qgm.au.dk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_23
245
to the Batalin–Vilkovisky Construction
Roberta A. Iseppi
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 anti-ghost 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.
Keywords Noncommutative geometry · Batalin–Vilkovisky construction · Finite
spectral triple · Gauge theory · Matrix models
1 Introduction: Why Noncommative Geometry
Since its early days, noncommutative geometry [9] has shown a reciprocal and
valuable interconnection with several areas of mathematics, such as motivic and
foliation theory, operator algebras, and KK-theory. However, maybe even more
remarkably, noncommutative geometry revealed a deep relation to quantum field
theory and gauge theory in particular. A confirmation of that can be found in a
series of the celebrated results, which began with the pioneering papers by Connes
[8, 10], had a breakthrough in [3, 4, 11], and finally arrived to the key result obtained
by Chamseddine, Connes, and Marcolli [5] of deriving the full Standard Model of
particles, with neutrino mixing and minimally coupled to gravity, from a purely
noncommutative geometrical object. Furthermore, recently new approaches have
been suggested to go beyond the Standard Model, using the framework provided by
R. A. Iseppi ()
Center for Quantum Geometry of Moduli Spaces, Aarhus University, Aarhus, Denmark
e-mail: roberta.iseppi@qgm.au.dk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_23
245