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User:Zephyr the west wind/Theta term

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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].

Action

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Spacetime and field content

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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

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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

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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.

Properties of the quantum theory

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CP violation

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Chiral anomaly

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References

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  1. ^ Witten, Edward (January 1988). "Topological quantum field theory". Communications in Mathematical Physics. 117 (3): 353–386. Bibcode:1988CMaPh.117..353W. doi:10.1007/BF01223371. ISSN 0010-3616. S2CID 43230714.
  2. ^ Gaiotto, Davide; Kapustin, Anton; Komargodski, Zohar; Seiberg, Nathan (17 May 2017). "Theta, time reversal and temperature". Journal of High Energy Physics. 2017 (5): 91. arXiv:1703.00501. Bibcode:2017JHEP...05..091G. doi:10.1007/JHEP05(2017)091. S2CID 256038181.
  3. ^ a b Tong, David. "Lectures on gauge theory" (PDF). Lectures on Theoretical Physics. Retrieved August 7, 2022.
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