Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vortices and the Ginzburg-Landau phase transition   The methods for studying the role of vortex loops in the phase transition of
the Ginzburg-Landau theory of superconductivity using lattice Monte Carlo
simulations are discussed. Gauge-invariant observables that measure the
properties of the vortex loop distribution are defined. The exact relations
between the lattice and continuum quantities make it possible to extrapolate
the results of the simulations to the continuum limit. The relationship between
spontaneous symmetry breaking and phase transitions is also reviewed, with an
emphasis on the fact that a local symmetry cannot be broken.

24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ