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Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups
A twist is a datum playing a role of a local system for topological
-theory. In equivariant setting, twists are classified into four types
according to how they are realized geometrically. This paper lists the possible
types of twists for the torus with the actions of the point groups of all the
2-dimensional space groups (crystallographic groups), or equivalently, the
torus with the actions of all the possible finite subgroups in its mapping
class group. This is carried out by computing Borel's equivariant cohomology
and the Leray-Serre spectral sequence. As a byproduct, the equivariant
cohomology up to degree three is determined in all cases. The equivariant
cohomology with certain local coefficients is also considered in relation to
the twists of the Freed-Moore -theory
The formulation of the Chern-Simons action for general compact Lie groups using Deligne cohomology
We formulate the Chern-Simons action for any compact Lie group using Deligne
cohomology. This action is defined as a certain function on the space of smooth
maps from the underlying 3-manifold to the classifying space for principal
bundles. If the 3-manifold is closed, the action is a function with values in
complex numbers. If the 3-manifold is not closed, then the action is a section
of a Hermitian line bundle associated with the Riemann surface which appears as
the boundary.Comment: 16 page
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