Exotic Textures
When a global symmetry group G in a theory of scalar fields is spontaneously broken to a subgroup H, the vacuum manifold (set of field values which minimize the potential energy) is given by the quotient space G/H. A texture is a configuration of scalar fields such that the fields are everywhere in the vacuum manifold, go to a single value at infinity, and wrap around the vacuum manifold in a topologically nontrivial way. In 3+1 dimensions such configurations are characterized by the third homotopy group pi_3(G/H) (although similar configurations in theories with topologically trivial vacuum manifolds may be constructed). They are unstable to collapse (by Derrick's Theorem), and Turok has proposed that they may act as the origin of large-scale structure in the universe.
The simplest models which contain textures are those for which the vacuum manifold is a three-sphere. These include SO(4)/SO(3) and SU(2)/0. There are are large number of additional vacuum manifolds for which pi_3(G/H) is nontrivial; such theories may be said to support "exotic textures."
This paper examines a number of scalar field theories in which a global symmetry group G breaks spontaneously to a subgroup H. For each theory, the lower homotopy groups of the vacuum manifold are computed: pi_1(G/H), characterizing cosmic strings, pi_2(G/H), characterizing monopoles, and pi_3(G/H), characterizing textures.
Abstract from xxx.lanl.gov.
Postscript from xxx.lanl.gov.
Homotopy Groups, including the 1st, 2nd and 3rd homotopy groups of various Lie groups, spheres, and vacuum manifolds G/H.
This paper describes numerical simulations of texture configurations in eight different scalar field theories. Two basic results are deduced. First, the collapse of a single texture configuration depends on whether the theory under consideration admits the existence of cosmic strings or monopoles. Theories which do allow such defects have textures which collapse somewhat more rapidly, and in the process of collapse may nucleate loops of string or monopole/anti-monopole pairs. Second, some theories with nontrivial pi_3(G/H) admit textures with spherically symmetric energy densities, but not texture configurations which are strictly spherically symmetric, in the sense that a spatial rotation of the configuration can be undone by a transformation in the global symmetry group G. Such configurations collapse nonspherically. These two results are related; it is the configurations without true spherical symmetry that nucleate defects during collapse.
Abstract from xxx.lanl.gov.
Postscript from xxx.lanl.gov.
Movies of the collapse of some textures with initially spherically symmetric energy densities.
Models. We list the properties of the models considered in this paper, in which a global symmetry group G is spontaneously broken to a subgroup H. "Fields" is the number of real scalar fields; the twelve real fields in the SU(3) theory are arranged into two complex three-vectors, while the other representations are explicitly real. The lower homotopy groups of the vacuum manifold G/H are listed, as well as the dimensionality of G/H (corresponding to the number of massless Goldstone bosons). The models may be grouped into two classes, depending on whether the model supports strings [pi_1(G/H)] and/or monopoles [pi_2(G/H)].
Energy
density in deformed configurations.
These plots represent energy density
contours in the x-y plane of oblate configurations in
the SO(4)/SO(3) and SO(3)/SO(2) theories. The textures are
obtained from those with spherically symmetric energy densities by
shrinking the configurations by 50% along the x-axis.
The energy density in the SO(4)/SO(3) is simply deformed along
with the coordinates, while in the SO(3)/SO(2) model the maximum
energy density describes a ring in the y-z plane. This behavior
can be traced to the fact that the SO(3)/SO(2) configuration is not
truly spherically symmetric, in the sense explained above.
This is also true for the SO(3)/0 and SO(4)/U(2) configurations.
In these figures, the three-dimensional configuration is obtained by
rotation around the x-axis.
Evolution of
energy densities.
These four sets of plots show the integrated energy densities --
potential, gradient, kinetic, and total -- during the evolution of
collapsing textures in the different models. The simulations shown
are characterized by whether the model admits strings and/or monopoles,
and whether the energy density of the initial configuration was
spherically symmetric or deformed along one axis.
Each form of energy density is integrated over the volume
of the simulation region; the total energy is the sum of the integrated
potential, kinetic and gradient energies. The horizontal axis is
measured in timesteps of the simulations.
Evolution of
quadrupole moments.
Corresponding to the energy density plots portrayed above,
these graphs show the quadrupole moments Q_xx, Q_yy, and Q_zz
of the total energy
density for collapse of the various simulations. Each plot shows
either a single straight line or two curves; the straight
line indicates that the collapse was
spherically symmetric, while in the nonsymmetric collapses two of the
quadrupole moments are always precisely equal, since the collapse has
one preferred axis.
Nucleation
of topological defects.
Three-dimensional contours of total
energy density (cross-hatched regions)
and potential energy (solid regions) at one moment during the
collapse of the SO(3)/SO(2) and SO(3)/0 configurations
with an initially symmetric energy density.
Two monopole/anti-monopole pairs are nucleated in the process of
the SO(3)/SO(2) collapse, while the SO(3)/0 collapse nucleates a
loop of cosmic string.
Email: Jim Bryan (jbryan [at] math.tulane.edu), Sean Carroll (carroll [at] theory.uchicago.edu), Ted Pyne (pyne [at] cfa160.harvard.edu), Andrew Sornborger (ats [at] camelot.mssm.edu).
Topological defects page at DAMTP.
Papers about textures from SPIRES. (The map from this list to papers which are really about textures is neither one-to-one nor onto, but it's a good approximation.)