[Paper Review] Invariant Effective Actions and Cohomology

Eric D’Hoker|ArXiv.org|May 17, 1995

Advanced Topics in Algebra4 references3 citations

TL;DR

This paper establishes a correspondence between invariant effective actions for Goldstone bosons arising from spontaneous symmetry breaking G → H and non-trivial de Rham cohomology classes of the coset space G/H. It demonstrates that such actions arise from symmetric tensors satisfying specific invariance and trace conditions, yielding either Goldstone-Wilczek-type currents or Wess-Zumino-Witten terms in four dimensions, and Chern-Simons terms in three dimensions that can induce fractional spin on solitons.

ABSTRACT

We review the correspondence between effective actions resulting from non-invariant Lagrangian densities, for Goldstone bosons arising from spontaneous breakdown of a symmetry group G to a subgroup H, and non-trivial generators of the de Rham cohomology of G/H. We summarize the construction of cohomology generators in terms of symmetric tensors with certain invariance and vanishing properties with respect to G and H. The resulting actions in four dimensions arise either from products of generators of lower degree such as the Goldstone-Wilczek current, or are of the Wess-Zumino-Witten type. Actions in three dimensions arise as Chern-Simons terms evaluated on composite gauge fields and may induce fractional spin on solitons. {Contribution to the Proceedings of STRINGS 95, held at University of Southern California, March 13 - 18, 1995.}

Motivation & Objective

• To establish a systematic correspondence between invariant effective actions for Goldstone bosons and non-trivial de Rham cohomology classes of the coset space G/H.
• To characterize the cohomology generators of G/H using symmetric tensors with specific invariance and vanishing properties under G and H.
• To classify effective actions in four and three dimensions arising from these cohomology classes.
• To demonstrate how Chern-Simons-type terms in three dimensions can lead to fractional spin on solitons.
• To unify the construction of Wess-Zumino-Witten and Goldstone-Wilczek-type actions through cohomological methods.

Proposed method

• Construct cohomology generators of G/H using symmetric tensors that are invariant under the full group G and traceless under H.
• Utilize the de Rham cohomology of G/H to classify possible effective actions for Goldstone bosons.
• Derive four-dimensional effective actions as products of lower-degree cohomology generators, such as the Goldstone-Wilczek current.
• Construct three-dimensional actions via Chern-Simons terms evaluated on composite gauge fields built from Goldstone bosons.
• Apply the formalism to show that such Chern-Simons terms can induce fractional spin on solitonic configurations.
• Use the structure of symmetric tensor invariants to ensure gauge invariance and consistency of the effective actions.

Experimental results

Research questions

• RQ1 How can invariant effective actions for Goldstone bosons be systematically derived from the cohomology of the coset space G/H?
• RQ2 What are the necessary and sufficient conditions on symmetric tensors to generate non-trivial de Rham cohomology classes of G/H?
• RQ3 How do Wess-Zumino-Witten-type actions in four dimensions arise from cohomological structures?
• RQ4 In three dimensions, how do Chern-Simons terms built from composite gauge fields lead to fractional spin on solitons?
• RQ5 What is the role of symmetric tensor invariants in ensuring gauge invariance and consistency of the resulting effective actions?

Key findings

• Non-trivial de Rham cohomology classes of G/H correspond precisely to invariant effective actions for Goldstone bosons.
• Cohomology generators are constructed from symmetric tensors that are G-invariant and traceless under H, ensuring consistency with the symmetry breaking pattern.
• Four-dimensional effective actions arise as products of lower-degree cohomology generators, including the Goldstone-Wilczek current.
• Three-dimensional effective actions are of the Chern-Simons type, constructed from composite gauge fields built from Goldstone bosons.
• These Chern-Simons terms can induce fractional spin on solitonic configurations, a hallmark of anyonic statistics.
• The formalism unifies the construction of Wess-Zumino-Witten and Goldstone-Wilczek-type actions under a single cohomological framework.