Topological Yang–Mills theory explained

In gauge theory, topological Yang–Mills theory, also known as the theta term or

\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).

G , with associated Lie algebra ak{g} through the usual correspondence.

The field content is the gauge field

A _{\mu} , also known in geometry as the connection. It is a 1 -form valued in a Lie algebra ak{g} .

Action

In this setting the theta term action is^[3] $$S\_\backslash theta\; =\; \backslash frac\backslash int\; d^4x\; \backslash ,\; \backslash text(F\_*F^)\; =\; \backslash frac\backslash int\; \backslash langle\; F\; \backslash wedge\; F\; \backslash rangle$$where

F _{\mu\nu} is the field strength tensor, also known in geometry as the curvature tensor. It is defined as 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 \pm i or g , a coupling constant. *F ^{\mu\nu} is the dual field strength, defined *F ^{\mu\nu} = (1)/(2) \epsilon ^{\mu\nu\rho\sigma} F _{\rho\sigma} .\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 *F is the Hodge dual of the field strength F .\theta is the theta-angle, a real parameter.tr is an invariant, symmetric bilinear form on ak{g} . It is denoted tr as it is often the trace when ak{g} is under some representation. Concretely, this is often the adjoint representation and in this setting tr is the Killing form.

As a total derivative

The action can be written as^[3] $$S\_\backslash theta\; =\; \backslash frac\; \backslash int\; d^4\; x\; \backslash ,\; \backslash partial\_\backslash mu\; \backslash epsilon^\; \backslash text\backslash left(A\_\backslash nu\; \backslash partial\_\backslash rho\; A\_\backslash sigma\; +\; \backslash fracA\_\backslash nu\; A\_\backslash rho\; A\_\backslash sigma\backslash right)=\; \backslash frac\; \backslash int\; d^4\; x\; \backslash ,\; \backslash partial\_\backslash mu\; \backslash epsilon^\; \backslash text(A)\_,$$where

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

External links

• nLab

Notes and References

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