Chiral superconducting strings and Nambu–Goto strings in arbitrary dimensions

Abstract

We present general solutions to the equations of motion for a superconducting relativistic chiral string that satisfy the unit magnitude constraint in terms of products of rotations. From this result we show how to construct a general family of odd harmonic superconducting chiral loops. We further generalise the product of rotations to an arbitrary number of dimensions.PACS numbers: 11.27.+d,98.80.Cq I. PRELIMINARIESParticle physics models where symmetry breaking is involved predict, in many cases, the existence of topological defects, which are formed when the topology of the vacuum manifold of the low energy theory is non-trivial [1]. Cosmic strings, in particular, are line-like objects that are formed when the the vacuum manifold contains unshrinkable loops. For a review see [2].In [3] it was shown that cosmic strings can be superconducting. In the case when the charge carriers on the string are not coupled to a gauge field the action for the string and the current can be taken to bewhere µ is the mass per unit length of the string, γ ab is the induced metric on the string worldsheet and φ is the field of the charge carriers living on the string. These strings were shown in [4] and [5] to have solutions in the case when γ ab φ ,a φ ,b = 0 of the formfor the string position andfor the field living on the string with the constraintsandwhere u = σ − τ and v = σ + τ and σ and τ are space-like and time-like parameters respectively that parametrise the string world-sheet. These strings are called chiral because the current only moves in one direction on the string.Comparing this to the usual Nambu-Goto case one can see that f (v) acts like a fourth component of the threevector b(v), making chiral superconducting strings behave like Nambu-Goto ones with chiral excitations in an extra fifth dimension. Indeed, this property was used in [6] in an investigation of the properties of superconducting cosmic string cusps.The right-and left-moving excitations, a ′ and b ′ , on a regular Nambu-Goto string in Minkowski space-time are arbitrary functions that satisfy the unit magnitude constraint, |a ′ | = |b ′ | = 1. Expressions for these functions are often given as Fourier sums and the unit magnitude constraint generally gives a non-linear set of equations involving the vector coefficients of the Fourier expansion. As a result, parametrising strings beyond the first few harmonics proves to be a difficult task. Fortunately, in that case, there exists a method to generate strings involving products of rotation matrices [7] that act on a starting unit vector so that the unit magnitude constraint is satisfied trivially.In a recent study of the properties of chiral cosmic strings [8] it was assumed that the current is constant. The work in [9] assumed that the current takes a very simple non-constant form. As was pointed out in the latter work one could expect to have loops with varying currents if the loops are formed by intersections involving different strings or if different segments of the loop or string were at some point i...

A note on tubular brane dynamics

“…There are of course many other solutions apart from the simple cases we presented here that would be simultaneously rotating and oscillating. In fact, it is possible to use the methods developed in [15] to generate solutions that we could readily interpret as solutions of the tubular brane ansatz.…”

“…(15,16) and (17,18) that we can interpret these solutions for the transverse brane motion as a tubular brane propagating in one more dimension, where the field θ parametrizes the position of the string along the extra dimension. In fact this interpretation clarifies the origin of the relation (10) as a duality transformation between the tubular D2-brane solution with an electromagnetic field on its worldvolume and a M2-brane propagating in one extra compact dimension [9].…”

“…In fact, it is possible to use the methods developed in [15] to generate solutions that we could readily interpret as solutions of the tubular brane ansatz.…”