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Register a new userPerturbations of giant magnons and single spikes in $mathbb R times S^2$
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Added by
Soumya Bhattacharya
Publication date
2021
fields
Physics
and research's language is
English
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Perturbations of giant magnons and single spikes in a $2+1$ dimensional $mathbb R times S^2$ background spacetime are analysed. Using the form of the giant magnon solution in the Jevicki-Jin gauge,the well-known Jacobi equation for small normal deformations of an embedded time-like surface are written down. Surprisingly, this equation reduces to a simple wave equation in a Minkowski background. The finiteness of perturbations and the ensuing stability of such giant magnons under small deformations are then discussed. It turns out that only the zero mode has finite deformations and is stable. Thereafter, we move on to explore the single spike solution in the Jevicki-Jin gauge. We obtain and solve the perturbation equation numerically and address stability issues.
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We analyse several physical aspects of the dressed elliptic strings propagating on $mathbb{R} times mathrm{S}^2$ and of their counterparts in the Pohlmeyer reduced theory, i.e. the sine-Gordon equation. The solutions are divided into two wide classes; kinks which propagate on top of elliptic backgrounds and those which are non-localised periodic disturbances of the latter. The former class of solutions obey a specific equation of state that is in principle experimentally verifiable in systems which realize the sine-Gordon equation. Among both of these classes, there appears to be a particular class of interest the closed dressed strings. They in turn form four distinct subclasses of solutions. Unlike the closed elliptic strings, these four subclasses, exhibit interactions among their spikes. These interactions preserve a carefully defined turning number, which can be associated to the topological charge of the sine-Gordon counterpart. One particular class of those closed dressed strings realizes instabilities of the seed elliptic solutions. The existence of such solutions depends on whether a superluminal kink with a specific velocity can propagate on the corresponding elliptic sine-Gordon solution. Finally, the dispersion relations of the dressed strings are studied. A qualitative difference between the two wide classes of dressed strings is discovered. This would be an interesting subject for investigation in the dual field theory.
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We provide a method for fast and exact simulation of Gaussian random fields on spheres having isotropic covariance functions. The method proposed is then extended to Gaussian random fields defined over spheres cross time and having covariance functions that depend on geodesic distance in space and on temporal separation. The crux of the method is in the use of block circulant matrices obtained working on regular grids defined over longitude $times$ latitude.
In the leading order of the large-$N$ approximation, we study the renormalon ambiguity in the gluon (or, more appropriately, photon) condensate in the 2D supersymmetric $mathbb{C}P^{N-1}$ model on~$mathbb{R}times S^1$ with the $mathbb{Z}_N$ twisted boundary conditions. In our large~$N$ limit, the combination $Lambda R$, where $Lambda$ is the dynamical scale and $R$~is the $S^1$ radius, is kept fixed (we set $Lambda Rll1$ so that the perturbative expansion with respect to the coupling constant at the mass scale~$1/R$ is meaningful). We extract the perturbative part from the large-$N$ expression of the gluon condensate and obtain the corresponding Borel transform~$B(u)$. For~$mathbb{R}times S^1$, we find that the Borel singularity at~$u=2$, which exists in the system on the uncompactified~$mathbb{R}^2$ and corresponds to twice the minimal bion action, disappears. Instead, an unfamiliar renormalon singularity emph{emerges/} at~$u=3/2$ for the compactified space~$mathbb{R}times S^1$. The semi-classical interpretation of this peculiar singularity is not clear because $u=3/2$ is not dividable by the minimal bion action. It appears that our observation for the system on~$mathbb{R}times S^1$ prompts reconsideration on the semi-classical bion picture of the infrared renormalon.
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