New developments in the quantization of supersymmetric solitons (kinks, vortices and monopoles)

Abstract

We discuss the one-loop quantum corrections to the mass M and central charge Z of supersymmetric (susy) solitons: the kink, the vortex and the monopole. Contrary to previous expectations and published results, in each of these cases there are nonvanishing quantum corrections to the mass. For the N = 1 kink and the N = 2 monopole a new anomaly in Z rescues BPS saturation (M = Z); for the N = 2 vortex, BPS saturation is rescued for two reasons: (i) the quantum fluctuations of the Higgs field acquire a nontrivial phase due to the winding of the classical solution, and (ii) a fermionic zero mode used in the literature is shown not to be normalizable. Solitons can be viewed as extended particles ("lumps") which should clearly have finite mass (= finite energy at rest). We shall only consider relativistic field theories. There exist also time-dependent solutions with finite energy (the "breather" solution in the sine-Gordon model, for example), but we consider only time-independent solitons. One can, of course, boost solitons in a relativistic theory, and obtain then moving solitons, but since one can always choose a Lorentz frame in which they are at rest, we restrict our attention to only time-independent solutions.A soliton is closely related to an instanton. The later is defined as follows:Definition 2: an instanton is a nonsingular solution of the classical field equations in Euclidean space with finite action.Because instantons have finite action, they contribute already at the classical level to the path integral. This is the reason for the requirement of finite action. It is clear that a soliton in n + 1 dimensions is an instanton in n dimensions: since the time coordinate plays no role in the soliton solutions, the space integral in Minkowski space can also be viewed as an integral over Euclidean space, and the energyWe shall discuss three solitons:1) the kink in 1+1 dimensions with N = 1 susy 2) the vortex in 2+1 dimensions with N = 2 susy 3) the monopole in 3+1 dimensions with N = 2 susyThere exist also susy extensions of these solitons with more susy (N = 2 for the kink, N = 4 for the monopole) or less susy (N = 1 for the vortex). In addition, discussions have

BPS saturation of the N=4 monopole by infinite composite-operator renormalization

“…where I is the divergent integral appearing in (10). This result is in fact independent of the gauge parameter ξ in (9).…”

“…A new anomalous contribution to the central charge of the 1+1-dimensional N = 1 kink was needed for BPS saturation [4,5,6,7]; using supersymmetry-preserving dimensional regularization, suitably adapted to take solitons into account [8], this anomaly was due to a kink-induced left-right asymmetry of the momenta of the fermion in the extra dimension [9]. For the 2+1-dimensional N = 2 vortex no such anomalies appeared, but a finite quantum correction to the central charge, induced by the winding of the background fields, was essential for BPS saturation [10]. 1 Recently, we studied quantum corrections to the monopole in 3+1-dimensional N = 2 SUSY Yang-Mills theory.…”

Quantum magnetic flux lines, BPS vortex zero modes, and one-loop string tension shifts

“…The matrix PDO K is in turn obtained by adding to the 1D Laplacian along the x 3 -axis times the 4 × 4 unit matrix the old Hessian operator (6) working in the ð2 þ 1ÞD AHM, fully analyzed in previous sections. It is clear that the eigenvalues of the K operator, KF n ðxÞ ¼ ε 2 n F n ðxÞ are of the form ε…”

“…The archive of new data clearly suggests that weak repulsive forces between BPS vortices arise caused by the one-loop vortex fluctuations. In extended N ¼ 2 supersymmetry, however, the one-loop vortex mass shift and the central charge are adjusted in such a way that one may say the BPS bound is preserved at the quantum level, see [6]. Thus, one may conclude that some degree of extended supersymmetry is needed in order to preserve the BPS character of topological solitons in the quantum domain.…”

“…The first successful attempts in this direction were achieved by Vassilevich in [5], and Rebhan et al in [6], by attacking this problem in the N ¼ 2 supersymmetric AHM. Almost simultaneously Bordag and Drozdov in [7] computed the vacuum energy due to purely fermionic fluctuations on a Nielsen-Olesen vortex.…”