Area-preserving Diffeomorphisms in Gauge Theory on a Non-Commutative Plane: a Lattice Study ANTONIO BIGARINI***-WOLFGANG BIETENHOLZ*** ALESSANDRO TORRIELLI* * * * Dipartimento di Fisica, Universita Á degli Studi di Perugia and INFN, Sezione di Perugia ** Institut fu Èr Physik, Humboldt-Universita Èt zu Berlin, Germany *** John von Neumann Institut fu Èr Computing NIC), Deutsches Elektron Synchrotron DESY) Zeuthen, Germany * * * Center for Theoretical Physics, Laboratory for Nuclear Sciences and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA We consider Yang-Mills theory with the U 1) gauge group on a non-commutative plane. Perturbatively it was observed that the invariance of this theory under area- preservingdiffeomorphismsAPDs)breaksdowntoarigidsubgroup SL2; R).Herewe present explicit results for the APD symmetry breaking at finite gauge coupling and finite non-commutativity. They are based on lattice simulations and measurements of Wilson loops with the same area but with a variety of different shapes. Our results confirm the expected loss of invariance under APDs. Moreover, they strongly suggest that non-perturbatively no residual symmetry persists. 1. Introduction Invarianceunderarea-preservingdiffeomorphismsAPDs)[1]isabasicsymmetry of ordinary Yang-Mills theories in two dimensions. In particular it means that Wilson loopexpectationvaluesonlydependontheorientedareassingledoutonthemanifold. Thanks to this property the theory acquires an almost topological flavour [2] and, as a consequence,itcanbesolvedanalytically.Elegantgroup-theoreticmethods[3]leadto closed expressions for the partition function and a set of observables [4]. The invariance under APDs was initially believed to persist also in U n) gauge theories defined on a non-commutative NC) two-dimensional manifold. It was assumed to play a central role in the large gauge group - characteristic of gauge theories on NC spaces - which merges internal and space-time transformations. A detailedstudyofthenon-commutativegauge-transformationalgebrawasperformedin Ref.[5].IfAPDsymmetryholds,onemighthopetobeabletosolvegaugetheoriesalso on a NC plane by generalising the powerful geometric procedures developed in the commutative case.