On the emergence of an almost-commutative spectral triple from a geometric construction on a configuration space
Johannes Aastrup, Jesper M. Grimstrup
TL;DR
The paper proposes a fundamental framework where the geometry of the configuration space of gauge connections gives rise to both a Yang–Mills–Dirac quantum field theory and the structure of an almost-commutative spectral triple in a semi-classical limit. By building a holonomy-diffeomorphism algebra on the configuration space and introducing a gauge-fixed Dirac operator, the authors show how unitary fluctuations generate a YM Hamiltonian together with a fermionic sector, which can be recast in terms of spinor fields. In the semi-classical regime, the $\mathbf{HD}$-algebra collapses to an almost-commutative algebra with a finite factor that depends on the HD-representation and localization, while the spatial Dirac operator intertwines with this finite part, producing a spectral-triple-like data set on $M$. A notable feature is a double fermionic structure akin to the standard-model doubling, suggesting a deeper connection between dynamical configuration-space geometry and noncommutative geometric formulations of particle physics. The work outlines a path toward embedding quantum field theory and gravity within this emergent framework, while acknowledging substantial open questions about connecting to the full spectral standard model and KO-dimension, requiring further development of real structures, path-integral formulations, and the role of triad fields.
Abstract
We show that the structure of an almost-commutative spectral triple emerges in a semi-classical limit from a geometric construction on a configuration space of gauge connections. The geometric construction resembles that of a spectral triple with a Dirac operator on the configuration space that interacts with the so-called $\mathbf{HD}$-algebra, which is an algebra of operator-valued functions on the configuration space, and which is generated by parallel-transports along flows of vector-fields on the underlying manifold. In a semi-classical limit the $\mathbf{HD}$-algebra produces an almost-commutative algebra where the finite factor depends on the representation of the $\mathbf{HD}$-algebra and on the point in the configuration space over which the semi-classical state is localized. Interestingly, we find that the Hilbert space, in which the almost-commutative algebra acts, comes with a double fermionic structure that resembles the fermionic doubling found in the noncommutative formulation of the standard model. Finally, the emerging almost-commutative algebra interacts with a spatial Dirac operator that emerges in the semi-classical limit. This interaction involves both factors of the almost-commutative algebra.