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--- advanced_notions:quantum_field_theory:wess-zumino-witten_term [2018/04/13 15:32]
ellahughes [Why is it interesting?]
+++ advanced_notions:quantum_field_theory:wess-zumino-witten_term [2018/04/13 15:58]
ellahughes [Intuitive]
@@ Line 5: / Line 5: @@
 <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.
+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.
+[...]
+our concern here has been to place the 'monopole' form (5 .15) of $L_{W-Z}$ in the context of an adiabatic
+decoupling problem. From this point of view, the peculiar phase behaviour leading to 'fermion-ness'
+in the $\phi$, sector has arisen as a result of non-trivial structure left behind when the fermion vacuum is
+decoupled adiabatically from the $\phi$'s. If we use only the $\phi$ d.f. 's, and integrate the fermions away, we
+must include a W-Z term which embodies this structure. The ultimate reason that this structure has a
+'monopole' form is to be found in the topological approach to anomalies [27, 28].
  <cite>Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison</cite></blockquote>
@@ Line 49: / Line 70: @@
 <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.
+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>

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