Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · hep-lat/9604012

Lattice Gauge Fields and Noncommutative Geometry

A. P. Balachandran·G. Bimonte·G. Landi·F. Lizzi·P. Teotonio-Sobrinho
View original source ↗Read source document ↗Export metadata ↓

Abstract

Conventional approaches to lattice gauge theories do not properly consider the topology of spacetime or of its fields. In this paper, we develop a formulation which tries to remedy this defect. It starts from a cubical decomposition of the supporting manifold (compactified spacetime or spatial slice) interpreting it as a finite topological approximation in the sense of Sorkin. This finite space is entirely described by the algebra of cochains with the cup product. The methods of Connes and Lott are then used to develop gauge theories on this algebra and to derive Wilson's actions for the gauge and Dirac fields therefrom which can now be given geometrical meaning. We also describe very natural candidates for the QCD theta term and Chern-Simons action suggested by this algebraic formulation. Some of these formulations are simpler than currently available alternatives. The paper treats both the functional integral and Hamiltonian approaches.

Explore related subjects

hep-latgr-qchep-th

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

A. P. Balachandran, G. Bimonte, G. Landi, F. Lizzi, P. Teotonio-Sobrinho. 1996-12-27. Lattice Gauge Fields and Noncommutative Geometry. https://doi.org/10.1016/s0393-0440(97)00017-x

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Exact and Near-Exact Inverse Renormalization Group with Constrained Monte Carlo

We establish conditions for exactness of the inverse renormalization group via constrained Markov processes, a class of Monte Carlo simulations which sample the degrees of freedom within a block on a fine lattice under the constraint that they remain consistent with the corresponding degree of freedom on the coarse lattice according to a forward renormalization group transformation. We show analytically, for an arbitrary transformation, that constrained Markov processes correspond to an exact inversion of the renormalization group at the fixed point action, and to a near-exact inversion in the vicinity of the fixed point. We then optimize renormalization group transformations directly on inversely renormalized configurations to effectively map the action onto itself and ensure computational stability in the generation of configurations for high-dimensional systems. Starting from a set of uncorrelated configurations for the three-dimensional $ϕ^{4}$ quantum field theory at lattice volume $V=16^{3}$, we invert the renormalization group with constrained Monte Carlo to construct a set of uncorrelated, near-equilibrium configurations for $V=128^{3}$ and extract two critical exponents of the system. The inversion exhibits no critical slowing down, with dynamic critical exponent $z=0.03(2)$.

hep-lat↗
arXiv2026

First Overlap Formulation and Perturbative Results for Supersymmetric QCD

We present a lattice formulation of N=1 supersymmetric QCD (SQCD) based on overlap fermions, providing a discretization consistent with the Ginsparg-Wilson relation. Particular emphasis is placed on the construction of Yukawa terms that preserve a lattice-modified chiral symmetry involving overlap quarks and gluinos. To realize this modified chiral symmetry in the Yukawa sector, we follow Lüscher's strategy and introduce auxiliary Dirac and Majorana fermionic fields for the quark and gluino fields, respectively. After functional integration over these auxiliary fields, ultralocal interaction terms arise as additional contributions to the lattice action. Within this framework, and using the Wilson gauge action for gluons, we also carry out one-loop perturbative calculations. In particular, we compute the self-energies of the quark, gluon, gluino, and squark fields in the presence of quark and squark masses, enabling an unambiguous determination of the corresponding field and mass counterterms on the lattice. This work provides a robust foundation for future perturbative determinations of the fine-tunings of the gauge, Yukawa, and quartic couplings, and already shows, as expected, a reduction in the number of required counterterms relative to Wilson-type fermions.

hep-lat↗
arXiv2026

Quark-mass dependence of two-nucleon scattering

Understanding how the properties of the visible Universe depend on the parameters of the Standard Model of particle physics requires \emph{ab initio} theoretical computations. To that end, we investigate the quark-mass dependence of low-energy nucleon-nucleon scattering in the deuteron and dineutron channels. We perform lattice-QCD calculations at pion masses $M_π\approx 200$ and $280$ MeV and compare their results to phenomenology at the physical point and to a previous calculation at $M_π\approx 710$ MeV. Scattering amplitudes, phase shifts, and threshold parameters are determined from finite-volume spectra using quantization conditions that explicitly incorporate long-distance effects from one-pion exchange. We analyze the pion-mass dependence of scattering lengths and effective ranges using chiral effective field theory with Weinberg and Kaplan-Savage-Wise power countings. From this, we estimate the pion-mass derivatives of the inverse scattering lengths and binding momentum, at the physical point. Both power countings indicate that the deuteron unbinds near $M_π\approx 220$ MeV.

hep-lat↗