Fermion zero modes under 1+1 D Higgs spacetime vortex?

Jackiw and Rossi had a classic paper Zero modes of the vortex-fermion system (1981). In that nice-written paper, they found fermionic zero modes of Dirac operator under nontrivial Higgs vortex in 2D space, which is a 2+1D spacetime problem. The winding number n of Higgs vortex corresponds to the number n of fermionic zero modes.

May I please ask: is there known result in the literature where fermion zero mode is formed in 1+1D spacetime under 1+1D spacetime Higgs vortex (i.e. 1D space +1D Euclidean-time vortex formed by a complex scalar Higgs field)?

To clarify, I assume this problem is similar to their 1981's analysis, but the Higgs vortex now I just ask is not a 2D space vortex, but a 1+1D spacetime vortex. (I assume a major difference between these cases should be the number of components of spinor, Jackiw and Rossi had 4-component spinor, here I have 2-component spinor.) Thank you for your time of thoughts and reply.

[Below for the details:]

Here the complex scalar Higgs $\Phi(x,t) = \Phi_{Re}(x,t)+I \Phi_{Im}(x,t)$, with $\Phi_{Re}, \Phi_{Im} \in \mathbb{R}$ which couple to the fermions by Yukawa coupling $\bar{\Psi} \Phi \Psi$:

The full 1+1D action is: $$S=\int dt dx \;\bar{\Psi} (i \not{\partial}+ \Phi_{Re}(x,t)+I \gamma^5 \Phi_{Im}(x,t))\Psi +L_{\text{Higgs}}$$ with $\Psi=(\Psi_L,\Psi_R)$ a 2-component spinor. $L_{\text{Higgs}}=a |\Phi|^2+b |\Phi|^4 \dots$.

The 1+1D spacetime vortex of Higgs can be, for example, written in Euclidean time $t_E=-it$: $$\Phi(z) \equiv \Phi(x,t_E) \simeq \frac{t_E+ix}{|t_E+ix|}=\frac{z}{|z|}$$ with $t_E+ix \equiv z$ as a complex coordinate, which gives 1 winding mode from the homotopy mapping: $$S^1 \text{of} \;z \to S^1 \text{of}\; \Phi(z)$$

We do not consider the gauge field profile here. We only consider the 1+1D spacetime vortex of Higgs (1D space +1D Euclidean-time Higgs vortex).

• Sorry, what do you mean by a vortex in a Minkowskian space? And if it is anything at all, isn't it just a simple analytic continuation of the Euclidean case? In that case, one should say that the fermionic modes are the continuations of the Euclidean ones, too. All such continuations may behave badly in the timelike region. – Luboš Motl Aug 17 '13 at 15:42
• Thank you Luboš for the interest! Would that be possible if one considers vortex in the space-Euclidean-time $(x,t_E)$? I had clarified 1+1D spacetime Higgs vortex as 1D space +1D Euclidean-time Higgs vortex in the text. Also see the equation of $\Phi(z)$. Thank you. – wonderich Aug 18 '13 at 1:00
• To Lubos: How do we argue that the analytic continuation (from Minkowski to an Euclidean instanton) in the time-like region has problem? One certainly can find an analytic solution (I did). Could you explain what did you mean an Euclidean instanton unstable? Thanks. – wonderich Nov 9 '13 at 1:23

There is a possibility that using a Jackiw-Rebbi model in 1+1D, changing their 1D spatial kink solution $\Phi(x)$ to the spacetime vortex $\Phi(x,t)$, by the following way
$$\text{kink}: \Phi(x) \simeq x/|x| \to \text{vortex}: \Phi(x,t) \simeq (t_E+ix)/|t_E+ix|=z/|z|$$ may work. The Higgs profile is in an approximate form at infinity $|x| \to \infty$ and $|z| \to \infty$. This may still provide the zero mode.