Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Symmetries, Symmetry Breaking, Gauge Symmetries   The concepts of symmetry, symmetry breaking and gauge symmetries are
discussed, their operational meaning being displayed by the observables {\em
and} the (physical) states. For infinitely extended systems the states fall
into physically disjoint {\em phases} characterized by their behavior at
infinity or boundary conditions, encoded in the ground state, which provide the
cause of symmetry breaking without contradicting Curie Principle. Global gauge
symmetries, not seen by the observables, are nevertheless displayed by
detectable properties of the states (superselected quantum numbers and
parastatistics). Local gauge symmetries are not seen also by the physical
states; they appear only in non-positive representations of field algebras.
Their role at the Lagrangian level is merely to ensure the validity on the
physical states of local Gauss laws, obeyed by the currents which generate the
corresponding global gauge symmetries; they are responsible for most
distinctive physical properties of gauge quantum field theories. The
topological invariants of a local gauge group define superselected quantum
numbers, which account for the $\theta$ vacua.

10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
2
The categories of this paper are: math
9
The categories of this paper are: nlin
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The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph