✎ Wess-Zumino-Witten Term [The Physics Travel Guide]

You've loaded an old revision of the document! If you save it, you will create a new version with this data.

====== Wess-Zumino-Witten Term ======

<blockquote> 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. [...]

(a) where the W -Z term itself comes from, or (b) why it is like a monopole in $\phi$-space. The short answer to (a) is: from the very fermion determinant which we studied in the previous lecture, but generalized to $SU(3)_f$, i.e. it is
a term in the effective action for the .P fields which arises after integrating over the fermions [22, 23].
This is all very well in its way, but it too is mysterious: //why does such an exotic term get induced in
the boson sector when we integrate out the fermions?// The technical answer to this is that the
underlying fermion theory has anomalies, which can be calculated from single fermion loop
diagrams. These diagrams generate effective vertices in the external fields (<i>a, gauge fields, etc.)
coupled to the fermions. Hence any bosonic action obtained by integrating out the fermions- which
is equivalent to summing all single fermion loop diagrams- must faithfully represent these
anomaly-induced vertices. //The W-Z action precisely encodes these anomalous vertices//: if we only
consider the 'ungauged' W -Z action, which is a function of the $SU(3)_f$ chiral field $\phi$ alone, we are
representing correctly just the $SU(3)_f$ flavour anomalies of the underlying Fermion theory.

<cite>Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison</cite></blockquote>
  
<tabbox Concrete>

<blockquote>**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.

<cite>[[http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cp07-upload.pdf|WZW term in quantum mechanics: single spin]]</cite></blockquote>
 
<tabbox Abstract>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

The chiral [[advanced_notions:quantum_field_theory:anomalies|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.
<cite>https://physics.stackexchange.com/a/34022/37286</cite>
</blockquote>

<blockquote>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 [[advanced_notions:skyrmions|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")

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

</tabbox>

Please solve the following equation to prove you're human. 39 +12 =  Please keep this field empty:

Edit summary

Note: By editing this page you agree to license your content under the following license: CC Attribution-Share Alike 4.0 International

User Tools

Site Tools

Page Tools