Chern–Simons action for zero-mode supporting gauge fields in three dimensions

Recent results on zero modes of the Abelian Dirac operator in three dimensions support to some degree the conjecture that the Chern–Simons action admits only certain quantized values for gauge fields that lead to zero modes of the corresponding Dirac operator. Here we show that this conjecture is wrong by constructing an explicit counterexample. )email address: [email protected] )email address: [email protected] )email address: [email protected] In the last few years a considerable amount of interest has been devoted to the study of zero modes of the Abelian Dirac operator in three-dimensional Euclidean space, that is, to square-integrable solutions of the Dirac equation D/Ψ ≡ ~σ(i~ ∂ + ~ A(~x))Ψ(~x) = 0 (1) where ~x = (x1, x2, x3), ~σ are the Pauli matrices, and Ψ is a two-component spinor. In addition, the gauge field ~ A is assumed to obey certain integrability conditions (e.g., square integrability of the related magnetic field ~ B = ~ ∂ × ~ A). On the one hand, such solutions are relevant for the quantum mechanical behaviour of non-relativistic electrons (see e.g., [1]), because solutions to the above equation are, at the same time, solutions to the Pauli equation (the Pauli equation is obtained by just squaring the Dirac operator in the above equation, i.e., D/Ψ = 0). On the other hand, solutions to the Dirac equation are also relevant for (Euclidean) quantum electrodynamics, as was discussed, e.g., in [2, 3]. Some first examples of zero modes were constructed in [4]. In [5] a class of Dirac operators and their zero modes was constructed which depend on a function that is arbitrary up to certain boundary conditions, thereby relating the existence of these zero modes to some topological condition. Some further examples of zero modes were given in [6] and in [7]. In [8, 9] the first examples of Dirac operators with multiple zero modes were given, thereby demonstrating the existence of the phenomenon of zero mode degeneracy. Further, a relation between the number of zero modes and a certain topological linking number (the Hopf index) of the corresponding gauge field was established in [9]. A very detailed and more geometrical discussion of these Dirac operators with multiple zero modes, based on the concept of Riemannian submersions, was given in [10]. In [11] the following two results were proved: i) For the one-parameter family of gauge potentials t ~ A zero modes may exist for at most a finite set of values ti for any t ∈ (t0, t1), and ii) The set of gauge potentials with no zero modes is a dense subset of the set of all gauge potentials (with certain decay properties). Recently, some results on the dimensionality of the space of gauge potential with zero modes were obtained in [12]. There it was proven that locally the space of gauge potentials with (at least) one zero mode is of co-dimension one within the space of all gauge potentials (with certain decay properties). In addition, some results on the dimensionalities of spaces of gauge potentials with multiple zero modes were proven. The above-described results would suit well with the assumption that there exists a certain functional of the gauge potential which may admit only fixed or quantized values for gauge potentials that support zero modes. The simplest functional one can imagine is the Chern–Simons action, which has the additional attractive feature of being a topological invariant (i.e., independent of the metric). Therefore, if the existence and degeneracy of zero modes is related to some topological features, as was speculated, e.g., in [3], the Chern–Simons action would be an obvious candidate. In addition, the assumption of quantized Chern–Simons action for gauge potentials with zero modes is further supported by the results of [9], where a whole class of gauge potentials with an arbitrary number of zero modes was constructed. For all these gauge