Fermionic Coset Models as Topological Models

By considering the fermionic realization of G/H coset models, we show that the partition function for the U(1)/U(1) model defines a Topological Quantum Field Theory and coincides with that for a 2dimensional Abelian BF system. In the non-Abelian case, we prove the topological character of G/G coset models by explicit computation, also finding a natural extension of 2-dimensional BF systems with non-Abelian symmetry. Partially supported by CONICET, Argentina Fellow of the J.S.Guggenheim Memorial Foundation, USA. Investigador CICBA, Argentina Postal address: C.C. 67, (1900) La Plata, Argentina In the last few years coset models [1]-[2] raised much interest in the study of conformally invariant two-dimensional theories particularly in connection with String theories and with Statistical Mechanics models.[3] G/H coset models can be realized by gauging a subgroup H of: (i) a Wess-Zumino-Witten (WZW) model with the basic field taking values on a Lie group G [4]-[5] or, alternatively, (ii) a free fermionic model with fermions in the fundamental representation of G [6]-[7]. Recently, Witten [8] analysed the holomorphic factorization of G/H models (in its bosonic realization) showing in particular that the G/G model defines a Topological field theory (i.e. a quantum field theory with metric independent partition function). We discuss this issue in the present note, by considering the fermionic realization of coset models. The coset construction based on fermionic models goes as follows [7]. One starts with two-dimensional free Dirac fermions in the fundamental representation of G, with Lagrangian: L0 = ψ̄i6∂ψ (1) Calling t the generators for H ⊂ G one constructs the associated currents: j μ = ψ̄t γμψ (2) Then, one imposes the condition that physical states |phys > are singlet under these currents: j μ|phys >= 0 (3) This is achieved in the path-integral formulation by introducing Lagrange multipliers Aμ which play the role of gauge fields in the Lie algebra of H . The partition function for the resulting constrained model reads: