Gabriella Tarantello, Università di Roma "Tor Vergata"¶
formulated with abelian Maxwell-Higgs Gauge Field theory
a manifestation of Meissner effect.
presence of 2D vortices.
The only satisfactory results so far are at the critical coupling regime: BPS (Bogolmin-Prasad-Sommerfield) vortices This regime is set up in such a way that the vortices don't really interact.
The electromagnetic coupling will be replaced by a more complicated non-abelian coupling.
This is not yet clear in the Chern-Simons setting.
elerically + magnetically charged MCS vortices
formulated within nonabelian GFT
at critical coupling
non-abelian GFT $\longmapsto \mathcal{N} =2$ SuSy
use "duality pair"
The domain walls are not really understood but, through the theory of supersymmetry, they appear to be some kind of "dual phenomenon" to the Meissner effect.
After Jaffe-Tauges: Vortices and Monopolse 1980
$\exists$ self-dual vortices $\iff$ solvability of elliptic PDES with exponential nonlinearities.
Jackiw-Weinberg-Hong-kim-Pac (1990)
$$ \mathcal{L}_{CS} (A, \phi) = - \frac{\kappa}{4} \epsilon^{\alpha \beta \gamma} A_\alpha F_{\beta \gamma} + D_\alpha \phi {\overline{D^\alpha \phi}} - V(|\phi|^2). $$The first term is the Chern-Simons term.
determined by the usual variational calculations
$$ D_\alpha D^\alpha \phi = -2 \frac{\partial V}{\partial {\overline{\phi}} } $$$$ \frac{\kappa}{4} \epsilon^{\mu \alpha \beta} F_{\alpha \beta} = \frac{i}{2}({\overline{\phi}} D^\mu \phi - {\overline{D^\mu} \phi} \phi ) = J^\mu$$Static solution of the field equations with finite energy.
Gauss law governing vortices.
This simple model has interesting features. There are solutions that are charged both electically and magnetically.
$$ \kappa F_{12} = 2 A_0 |\phi|^2 ~(= 2 \rho_0 ~{\mbox{charge density}}) $$$\implies$ electrically + magnetically charged vortices.
The advantage of the Chern-Simons term: it produces charged vortices!
Analytical difficulties (typical of Gauge Field Theories). The theory is formulated with respect to terms that you can't measure. Everything is defined only up to gauge equivalence.
Gauge Transformations
Here $\omega: \mathbb{R}^{1+2} \longmapsto \mathbb{R}$ is smooth.
Question: What is the good choice of gauge to suite the CS term?
Chen-Guo-Spirn-Yuo
(constraint) Energy Minimizers
The static case has the $D^0$ term vanishing in the energy.
A specific choice of potential and a "gauss law constraint" $$V(|\phi|^2) = \frac{1}{\kappa^2} |\phi|^2 (1 - |\phi|^2)^2$$ leads to some nice properties for the static problem.
Q: Is this the "critical coupling"?
A: Yes. There is an analogy with Ginzburg-Landau and there will be some discussion later regarding limiting behavior in the parameter $\kappa$.
The calculation leads to a reduced "decoupled" set of equations for energy minimizers (at fixed flux) called the BPS-soliton equation.
$$D_1 \phi \pm i D_2 \phi = 0.$$$$ F_{12} = \pm \frac{2}{\kappa^2} |\phi|^2 (1 - |\phi|^2)$$$$A_0 = \mp \frac{1}{\kappa} (|\phi|^2 - 1)$$In mathematics, these solutions are called "self-dual". The language borrows from some similarities with corresponding objects in the Yang-Mills theory. In physics, these are called the BPS-solitons.
Recall $D_1$ and $D_2$ are covariant derivatives so the $A$ is in there. The idea to approach this equation was pioneered by Taubes.
$\implies ~\phi$ holomorphic (up to gauge transformations). You can use the "Poincaré Lemma" aka "the d-bar lemma"
We know that $\phi$ must of have a finite collection of zeros $p_1, p_2, \dots, p_N$ repeated with multiplicity. (vortex points). Here $N$ is the vortex number.
I introduce a new variable $u$ such that $$ e^u = |\phi (z) |^2$$ Therefore $u$ must have a singularity at the points $p_j$. $$ u(z) = 2 \log |z - p_j| + O(1), ~{\mbox{as}}~ z \rightarrow p_j.$$
You take this ansatz and plug back into the equation. This is a nice exercise. You will find that you can interpret the covariant d-bar equation in a new way. You will find that you can solve for $A_1$ and $A_2$ in terms of $u$. You will also find that $A_1$ and $A_2$ extend smoothly across the vortex points $p_j$. These calculations eventually produce an elliptic equation
$$ -\Delta u = 2 F_{12} - 4 \pi \sum_{j=1}^N \delta_{p_j}$$The process can be reversed. Anytime you find a solution $u$ from this equation, you can go the other way to define a complete solution to the BPS-soliton equation.
$$F_{12} = \frac{2}{\kappa^2} |\phi|^2 (1 - |\phi|^2)$$(2nd BPS equation)
A system generalization of previous work is then displayed.