Level in Chern-Simons Theory
Marathe, Kishore B.
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The Chern-Simons theory is parametrized by
a real number k called the level of the theory. In many
applications one is required to restrict the level to take
integral values. Formulas involving level usually consider
only the positive integral values. We discuss the significance
of level in different applications of the Chern-Simons theory
and extend the formulas with positive integral values of k
to negative integral values of k. The shift in k by the dual
Coxeter number of the gauge group must also be taken into
account for negative k. In Witten's derivation of the
skein relations for the family of two variable Jones
polynomials by using topological quantum field theory,
the negative values of the level k (suitably shifted by the
dual Coxeter number of SU(n)) give the missing half of this
family which contains the skein relation characterizing the
original single variable Jones polynomials.
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