Topology on the lattice; 2d Yang-Mills theories with a theta term

Details Topology on the lattice; 2d Yang-Mills theories with a theta term Topology on the lattice; 2d Yang-Mills theories with a theta term

1995-05-02

arxiv.org/abs/hep-th/9505005v1

Nucl.Phys. B454 (1995) 45-58

Physics

High Energy Physics

High Energy Physics - Theory

19 pages LaTeX

Scientific paper

10.1016/0550-3213(95)00440-4

We study two-dimensional U($N$) and SU($N$) gauge theories with a topological term on arbitrary surfaces. Starting from a lattice formulation we derive the continuum limit of the action which turns out to be a generalisation of the heat kernel in the presence of a topological term. In the continuum limit we can reconstruct the topological information encoded in the theta term. In the topologically trivial cases the theta term gives only a trivial shift to the ground state energy but in the topologically nontrivial ones it remains to be coupled to the dynamics in the continuum. In particular for the U($N$) gauge group on orientable surfaces it gives rise to a phase transition at $\theta= \pi$, similar to the ones observed in other models. Using the equivalence of 2d QCD and a 1d fermion gas on a circle we rewrite our result in the fermionic language and show that the theta term can be also interpreted as an external magnetic field imposed on the fermions.

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