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ABSTRACT. We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms (Faddeev-Mickelsson cocycle) for the gauge group action. We relate the APS construction to the bundle gerbe approach discussed recently by Carey and Murray, including an explicit computation of the Dixmier-Douady class. An advantage of our method is that it can be applied whenever one has a form of the APS theorem at hand, as in the case of fermions in an external gravitational field. INTRODUCTION.There are subtleties in defining the fermionic Fock spaces in the case of chiral (Weyl) fermions in external vector potentials. The difficulty is related to the fact that the splitting of the one particle fermionic Hilbert space H into positive and negative energies is not continuous as a function of the external field. One can easily construct paths in the space of external fields such that at some point on the path a positive energy state dives into the negative energy space (or vice versa). These points are obviously discontinuities in the definition of the space of negative energy states and therefore the fermionic vacua do not form a smooth vector bundle over the space of external fields. This problem does not arise if we have massive fermions in the temporal gauge A 0 = 0. In that case there is a mass gap [−m, m] in the spectrum of the Dirac hamiltonians and the polarization to positive and negative energy subspaces is indeed continuous.If λ is a real number not in the spectrum of the hamiltonian then one can define a bundle of fermionic Fock spaces F A,λ over the set U λ of external fields A, λ / ∈ Spec(D A ). It turns out that the Fock spaces F A,λ and F A,λ ′ are naturally isomorphic up to a phase. The phase is related to the arbitrariness in filling the Dirac sea between vacuum levels λ, λ ′ . In order to compensate this ambiguity one defines a tensor product F ′ A,λ = F A,λ ⊗ DET A,λ , where the second factor is a complex line bundle over U λ . By a suitable choice of the determinant bundle the

The relation between Kac-Moody groups and algebras and the determinant line bundle of the massless Dirac operator in two dimensions is clarified. Analogous objects are studied in four space-time dimensions and a generalization of Witten's fermionization mechanism is presented in terms of the topology of the Dirac determinant bundle.

In order to construct the quantum field theory in a curved space with no "old" infinities as the curvature tends to zero, the problem of contraction of representations of the corresponding group of motions is studied. The definitions of contraction of a local group and of its representations are given in a coordinate-free manner. The contraction of the principal continuous series of the de Sitter groups S0 0 (n, 1) to positive mass representations of both the Euclidean and Poincare groups is carried out in detail. It is shown that all positive mass continuous unitary irreducible representations of the resulting groups can be obtained by this method. For the Poincare groups the contraction procedure yields reducible representations which decompose into two non-equivalent irreducible representations.