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Graded q-Differential Algebra Approach to Chern-Simons Form |
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PP: 29-38 |
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Author(s) |
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Viktor Abramov,
Olga Liivapuu,
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Abstract |
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| In the present paper we develop noncommutative approach to a connection which is based on a notion of graded qdifferential
algebra, where q is a primitive Nth root of unity. We define the curvature of connection form and prove Bianchi identity.
We construct a graded q-differential algebra to calculate the curvature of connection for any integer N 2. Making use of Bianchi
identity we introduce the Chern character form of connection form and show that this form is closed. We study the case N = 3 which
is the first non-trivial generalization because in the case N = 2 we have a classical theory. We calculate the curvature of connection
form and show that it can be expressed in terms of graded q-commutators, where q is a primitive cubic root of unity. This allows us to
prove an infinitesimal homotopy formula, and making use of this formula we introduce the Chern-Simons form. |
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