Fun with Computational Physics: Non-commutative Geometry on the Lattice Alberto de Campo 1, Wolfgang Frisch 2, Harald Grosse 3, Natascha Hörmann 2, Harald Markum 2 1 Institut für Elektronische Musik und Akustik, Kunstuniversität Graz 2 Atominstitut, TU Wien 3 Institut für Theoretische Physik, Universität Wien

Coordinates become Hermitean operators Space-time is deformed deformation parameter antisymmetric tensor Non-commutative geometry in continuum: product between fields is replaced by the Moyal product

Weyl Operator Weyl operator Functions are mapped on operators All functions under consideration are assumed to have a represention as Fourier transforms function in the continuum operator

Non-commutative Yang-Mills Theory U(N) gauge field with generating field A i (x) Yang-Mills action

Non-commutative Torus Periodic boundary conditions expressed by matrix Momentum space is discretized

Operators of coordinates Coordinate operators fulfill the following commutation relations dimensionless tensor for non-commutativity in space-time Non-commutative torus is discretized by introducing a shift operator which performs translations of one lattice spacing a

Solution of equation For a constant and rational there exist finite-dimensional matrices with the condition in 2 dimensions The construction scheme of the matrices is known Those matrices fulfill the Weyl-‘t Hooft algebra

The coordinate operators and the shift operators can be constructed from the matrices with the possible choice The gauge fields (links) can be expanded in a Fourier series U(1) Gauge Theory on a 2-dimensional non- commutative Torus expansion coefficient The dimension N of the matrices corresponds to an NxN lattice!

One-Plaquetten action Partition function Monte Carlo simulations of non-commutative geometry possible

One-Plaquette Action corresponds to the action of the Twisted Eguchi-Kawai model (TEK) continuum limit unitary NxN matrix twist constraint: finite in continuum

string tension Observables Polyakov lines Wilson loops area law for Wilson loops

W. Bietenholtz, F. Hofheinz, J. Nishimura, hep-lat/

Phase of complex Wilson loop

Current Studies Check of the phase structure from Polyakov lines Definition of the topological content via monopole and charge density formulation on the twisted Eguchi-Kawai model Possible Investigations Topological structure of U(1) theory on a 4-dimensional non-commutative torus Topology of Yang-Mills theory on the fuzzy sphere?