Topological Yang–Mills theory

In gauge theory, topological Yang–Mills theory, also known as the theta term or ${\displaystyle \theta }$-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

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 ${\displaystyle G}$, with associated Lie algebra ${\displaystyle {\mathfrak {g}}}$ through the usual correspondence.

The field content is the gauge field ${\displaystyle A_{\mu }}$, also known in geometry as the connection. It is a ${\displaystyle 1}$-form valued in a Lie algebra ${\displaystyle {\mathfrak {g}}}$.

Action

In this setting the theta term action is^[3] $${\displaystyle S_{\theta }={\frac {\theta }{16\pi ^{2}}}\int d^{4}x\,{\text{tr}}(F_{\mu \nu }*F^{\mu \nu })={\frac {\theta }{16\pi ^{2}}}\int \langle F\wedge F\rangle }$$ where

• ${\displaystyle F_{\mu \nu }}$ is the field strength tensor, also known in geometry as the curvature tensor. It is defined as ${\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }+[A_{\mu },A_{\nu }]}$, up to some choice of convention: the commutator sometimes appears with a scalar prefactor of ${\displaystyle \pm i}$ or ${\displaystyle g}$, a coupling constant.
• ${\displaystyle *F^{\mu \nu }}$ is the dual field strength, defined ${\displaystyle *F^{\mu \nu }={\frac {1}{2}}\epsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }}$.
  • ${\displaystyle \epsilon ^{\mu \nu \rho \sigma }}$ is the totally antisymmetric symbol, or alternating tensor. In a more general geometric setting it is the volume form, and the dual field strength ${\displaystyle *F}$ is the Hodge dual of the field strength ${\displaystyle F}$.
• ${\displaystyle \theta }$ is the theta-angle, a real parameter.
• ${\displaystyle {\text{tr}}}$ is an invariant, symmetric bilinear form on ${\displaystyle {\mathfrak {g}}}$. It is denoted ${\displaystyle {\text{tr}}}$ as it is often the trace when ${\displaystyle {\mathfrak {g}}}$ is under some representation. Concretely, this is often the adjoint representation and in this setting ${\displaystyle {\text{tr}}}$ is the Killing form.

As a total derivative

The action can be written as^[3] $${\displaystyle S_{\theta }={\frac {\theta }{8\pi ^{2}}}\int d^{4}x\,\partial _{\mu }\epsilon ^{\mu \nu \rho \sigma }{\text{tr}}\left(A_{\nu }\partial _{\rho }A_{\sigma }+{\frac {2}{3}}A_{\nu }A_{\rho }A_{\sigma }\right)={\frac {\theta }{8\pi ^{2}}}\int d^{4}x\,\partial _{\mu }\epsilon ^{\mu \nu \rho \sigma }{\text{CS}}(A)_{\nu \rho \sigma },}$$ where ${\displaystyle {\text{CS}}(A)}$ 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

CP violation

See also: CP violation

See also: Strong CP problem

Chiral anomaly

See also: Chiral anomaly

See also

• Yang–Mills theory