1. 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

2. 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

3. 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

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

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

6. 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

7. 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

8. 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!

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

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

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

12. W. Bietenholtz, F. Hofheinz, J. Nishimura, hep-lat/0209021

13. Phase of complex Wilson loop

16. 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?