Fermionic Zero Modes Research Articles

Superinstanton modes

We construct the superfield version of the instanton and investigate the zero modes using the background superfield method. A general method of generating covariant expressions for the discrete zero modes is described. It is found that the linearly independent set of superfield zero modes is not just the 8 bosonic and 4 fermionic zero modes which were expected, but also 4 further fermionic zero modes corresponding to anomalous supergauge modes — making the super-instanton measure manifestly supersymmetric.It is shown that the continuous supergauge zero modes and the positive-frequency modes can be parameterised through antichiral eigenstates of background □. In particular, this implies that background-chiral, background-antichiral, and vector superfield modes have the same spectrum of non-zero eigenvalues. These modes allow us to construct the full Green functions in the background field and projection operators onto the corresponding spaces.

On the interpretation of fermionic zero-modes

We examine the effect of fermionic zero modes on tunneling amplitudes within some simple quantum mechanical models. It is shown that the fermionic zero modes do not cause a total suppression of the tunneling, although it may be reduced. With a θ term present, due to a non-trivial topology, it is shown that the θ dependence is not eliminated by the zero modes. Instead the non-trivial topology introduces a “twist” in the fermionic coordinates which breaks a symmetry of the hamiltonian.

New gravitational index theorems and super theorems

Gravitational index theorems relate the number of zero eigenvalue modes of various differential operators to the gravitational topological invariants: the Euler number and the Pontryagin number. In order to apply these theorems to quantum theories such as supergravity, which involve operators of spin-0, 1 2 , 1, 3 2 and 2 , we derive index theorems for fields of arbitary spin. In this way, we discover new higher-spin theorems in addition to recovering the well-known low-spin theorems as special cases (i.e., the spin- 1 2 Atiyah-Singer theorem, the Hirzebruch signature theorem and the Gauss-Bonnet theorem). All these theorems relate operators differring by integral spin, i.e., boson to boson and fermion to fermion. We calculate en route, for fields of arbitary spin in an arbitary gravitational field, the coefficients B 4 which appear in the asymptotic expansion of the heat kernel of the corresponding differential operators. The B 4 coefficients are of interest in their own right, since they determine the one-loop counterterms, the stress tensor conformal anomalies, and the axial current anomalies. Next, we look for “super theorems” which relate the number of boson zero-modes to the number of fermion zero-modes. Such theorems may indeed be derived in the case that the gravitational field is self-dual. This in turn reveals an interesting property of supergravity: beyond one loop the theory is finite when the background fields satisfy certain self-duality conditions.

Fermions in the background pseudoparticle field in an O(5) formulation

The eigenvalue equation for a fermion in the field of the pseudoparticle of the Yang-Mills theory is solved using the formalism of the four dimensional hypersphere. All the fermionic zero modes for a SU(2) gauge group belong to a singly nondegenerate O(5) representation. In the case of isospin n their multiplicity is given by 2 3 n (n+1)(2n+1) . If the fermions are in the adjoint representation the eigenfunctions of the zero modes are directly obtainable from considerations pertaining to the supersymmetry of the Yang-Mills Lagrangian.