Fermionic Collective Coordinates Research Articles

Non-Abelian statistics versus the Witten anomaly

This paper is motivated by prospects for non-Abelian statistics of deconfined particlelike objects in $3+1$ dimensions, realized as solitons with localized Majorana zero modes. To this end, we study the fermionic collective coordinates of magnetic monopoles in $3+1$-dimensional spontaneously-broken SU(2) gauge theories with various spectra of fermions. We argue that a single Majorana zero mode of the monopole is not compatible with cancellation of the Witten SU(2) anomaly. We also compare this approach with other attempts to realize deconfined non-Abelian objects in $3+1$ dimensions.

On instanton contributions to anomalous dimensions in [formula omitted] supersymmetric Yang–Mills theory

Instanton contributions to the anomalous dimensions of gauge-invariant composite operators in the N=4 supersymmetric SU( N) Yang–Mills theory are studied in the one-instanton sector. Independent sets of scalar operators of bare dimension Δ 0⩽5 are constructed in all the allowed representations of the SU(4) R-symmetry group and their two-point functions are computed in the semiclassical approximation. Analysing the moduli space integrals the sectors in which the scaling dimensions receive non-perturbative contributions are identified. The requirement that the integrations over the fermionic collective coordinates which arise in the instanton background are saturated leads to non-renormalisation properties for a large class of operators. Instanton-induced corrections to the scaling dimensions are found only for dimension 4 SU(4) singlets and for dimension 5 operators in the 6 of SU(4). In many cases the non-renormalisation results are argued to be specific to operators of small dimension, but for some special sectors it is shown that they are valid for arbitrary dimension. Comments are also made on the implications of the results on the form of the instanton contributions to the dilation operator of the theory and on the possibility of realising its action on the instanton moduli space.

Instantons in four-Fermi term broken supersymmetric quantum mechanics with general potential

We have shown here how to find an integral representation for the solution of the Euclidean equations of motion of a quantum mechanical point particle in a general potential and in the presence of a four-Fermi term. The classical action in this theory depends explicitly on a set of four fermionic collective coordinates. The corrections to the classical action due to the presence of fermions are of topological nature in the sense that they depend only on the values of the fields at the boundary points τ → ±∞. As an application, the quantum mechanical sine-Gordon model with a four-Fermi term is solved explicitly and the corrections to the classical action are computed.

Instanton corrections to circular Wilson loops in Script N = 4 Supersymmetric Yang-Mills

It is argued that whereas supersymmetry requires the instanton contribution to the expectation value of a straight Wilson line in the N=4 supersymmetric SU(2) Yang-Mills theory to vanish, it is not required to vanish in the case of a circular Wilson loop. A non-vanishing value can arise from a subtle interplay between a divergent integral over bosonic moduli and a vanishing integral over fermionic moduli. The one-instanton contribution to such Wilson loops is explicitly evaluated in semi-classical approximation. The method utilizes the symmetries of the problem to perform the integration over the bosonic and fermionic collective coordinates of the instanton. The integral is singular for small instantons touching the loop and is regularized by introducing a cutoff at the boundary of the (euclidean) AdS_5 moduli space. In the case of a circular loop a nonzero finite result is obtained when the cutoff is removed and a perimeter divergence subtracted. This is contrasted with the case of the straight line where the result is zero after subtraction of an identical divergence per unit length. The linear divergence is an artifact of our non supersymmetric regulator that deserves further consideration. The generalization to gauge group SU(N) with arbitrary N is straightforward in the limit of small 't Hooft coupling. The extension to strong 't Hooft coupling is more challenging and only a qualitative discussion is given of the AdS/CFT correspondence

New instantons in the double-well potential

A new solution of the Euclidean equations of motion is found for the quantum-mechanical double-well potential with a four-fermion term. It extends the usual kink-instanton solution in which both the kink field and the fermionic fields contain a finite number of new terms which are polynomial in the fermionic collective coordinates. The solution has finite action, S=− m 3 3λ + 9 140 m g ϵ ijklξ iξ jξ kξ l , where ξ i,i=1,… ,4 , are fermionic collective coordinates and g, m and λ are coupling constants. We explain why in general the instanton action can depend on collective coordinates.

Instantons, Euclidean supersymmetry and Wick rotations

We discuss the reality properties of the fermionic collective coordinates in Euclidean space in an instanton background and construct hermitean actions for N=4 and N=2 supersymmetric Euclidean Yang–Mills theories.

Fermion exchange between D-instantons

We define fermionic collective coordinates for type-HB Dirichlet instantons and discuss some effects of the associated fermionic zero modes within the dilute gas framework. We show that the standard rules for clustering of zero modes in the dilute limit, and the fermion-exchange interactions follow from world-sheet Ward identities. Fermion exchange is strongly attractive at string-scale distances, which makes the short-distance Hagedorn singularity between instantons and anti-instantons even stronger.

Supersymmetry transformations of instantons

Instantons in the simplest supersymmetric Yang-Mills theory are considered. We introduce bosonic and fermionic collective coordinates and study how they change under the supersymmetry transformations. The instanton measure is shown to be explicitly invariant under the transformations. We discuss the relation between quantum anomalies and the functional form of the instanton measure.