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Topological Yang–Mills theory
Topological field theory From Wikipedia, the free encyclopedia
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In gauge theory, topological Yang–Mills theory, also known as the theta term or -term is a gauge-invariant term which can be added to the action for four-dimensional field theories, first introduced by Edward Witten.[1] It does not change the classical equations of motion, and its effects are only seen at the quantum level, having important consequences for CPT symmetry.[2]
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Action
Summarize
Perspective
Spacetime and field content
The most common setting is on four-dimensional, flat spacetime (Minkowski space).
As a gauge theory, the theory has a gauge symmetry under the action of a gauge group, a Lie group , with associated Lie algebra through the usual correspondence.
The field content is the gauge field , also known in geometry as the connection. It is a -form valued in a Lie algebra .
Action
In this setting the theta term action is[3] where
- is the field strength tensor, also known in geometry as the curvature tensor. It is defined as , up to some choice of convention: the commutator sometimes appears with a scalar prefactor of or , a coupling constant.
- is the dual field strength, defined .
- is the totally antisymmetric symbol, or alternating tensor. In a more general geometric setting it is the volume form, and the dual field strength is the Hodge dual of the field strength .
- is the theta-angle, a real parameter.
- is an invariant, symmetric bilinear form on . It is denoted as it is often the trace when is under some representation. Concretely, this is often the adjoint representation and in this setting is the Killing form.
As a total derivative
The action can be written as[3] where is the Chern–Simons 3-form.
Classically, this means the theta term does not contribute to the classical equations of motion.
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Properties of the quantum theory
CP violation
Chiral anomaly
See also
References
External links
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