Intuitive

The W-Z term is a generalization, to the configuration space of scalar fields $\phi_a$, of the charge-monopole interaction term in ordinary configuration space for particles. It acts like a monopole in $\phi$-space.

Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison

Concrete

Properties of WZ terms

• (i)are metric independent
• (ii) are imaginary in Euclidian formulation
• (iii) do not contribute to stress-energy tensor (and to Hamiltonian).
• (iv) do not depend on m – the scale, below which an effective action is valid (but do depend on sgn (m))
• (v) are antisymmetric in derivatives with respect to different space-time coordinates (contain !µνλ…)
• (vi) are written as integrals of (D+1)-forms over auxiliary (D+1)-dimensional space - disk $D^{D+1}$ such that ∂$D^{D+1}$ = $S^D$ - compactified space-time
• (vii) are multi-valued functionals. Multi-valuedness results in quantization of coupling constants (coefficients in front of WZ terms)
• (viii) do change equations of motion by changing commutation relation between fields (Poisson’s brackets) not by changing Hamiltonian
• (ix) might lead to massless excitations with “half-integer spin”
• (x) describe boundary theories of models with θ-terms
• (xi) being combined (see the spin chains chapter) produce θ-terms
• (xii) can be calculated by gradient expansion of the variation of fermionic determinants
• (xiii) produce θ terms as a reduction of target space

Among the listed properties the first five (i)-(v) are the properties of all topological terms while the others are more specific to WZ terms.

WZW term in quantum mechanics: single spin

Abstract

Why is it interesting?

The chiral anomaly can be corrected by adding a Wess-Zumino term to the Lagrangian, but this term is not perturbatively renormalizable, thus does not solve the nonrenormalizability problem. https://physics.stackexchange.com/a/34022/37286

Witten [8, 9] showed that the Wess-Zumino (W-Z) term [8, 9, 10] in the action for the scalar fields $\phi_a$ (whose solitons are Skyrmions) actually determines how these solitons are to be quantized. He obtained the remarkable result that the Skyrmion is a fermion if N, is odd, and a boson if N, is even: furthermore, the W-Z term also determines the pattern of spin-SU(3) multiplets $([1/2^+,8],[3/2^+,10],\ldots) in the baryon spectrum [9, 11, 12]. Though obviously correct mathematically, these results were nevertheless still hard to explain in physical terms, especially to anyone who did not know what a W -Z term was- and even to those who did")

Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison