theories, which feature these line-like configurations as topological defects, briefly discussed.

Lattice Field Theory {#sec:lft}
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Central to our discussion is the O($N$) symmetric relativistic quantum field theory formulated on a hypercubic lattice in $d$ Euclidean spacetime dimensions, with the time coordinate analytically continued to the imaginary axis $\tau$. As an illustration, the cubic lattice in Fig. \[fig:cubic\] represents discretized spacetime in one time and two space dimensions $x_1$ and $x_2$.

\[t\]\[t\]\[.8\]\[0\][$x_1$]{} \[t\]\[t\]\[.8\]\[0\][$x_2$]{} \[t\]\[t\]\[.8\]\[0\][$\tau$]{}

![Cubic lattice representing three-dimensional discretized spacetime. \[fig:cubic\]](cubic.eps){width=".3\textwidth"}

The theory, describing self-interacting scalar particles of mass $m$, is specified by the (Euclidean) partition function $$\label{Z}
Z = \mathrm{Tr} \, \mathrm{e}^{-S},$$ where $S$ is the lattice action $$\label{Sor}
S = a^d \sum_x \left[ \frac{1}{2a^2} \sum_\mu \left(\varphi_{x+a
      \hat{\mu}} - \varphi_x\right)^2 + \frac{m^2}{2}
      \varphi^2_x + \frac{g}{4!}
      \varphi^4_x \right],$$ with $g$ the coupling constant and $a$ the lattice spacing. The real scalar field $\varphi_x$ defined on each lattice site $x$ of the spacetime box has $N$ components $\varphi_x = \varphi_x^\alpha =
(\varphi^1_x, \varphi^2_x, \cdots , \varphi^N_x)$, with the index $\alpha=1,2 \cdots, N$ labeling the field components and $\varphi_x^4
\equiv (\varphi_x \cdot \varphi_x)^2$, where the dot product implies a summation over the field components $\varphi_x \cdot \varphi_x =
\sum_{\alpha=1}^N \varphi_x^\alpha \varphi_x^\alpha$. Lattice 