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advanced_notions:quantum_field_theory:wess-zumino-witten_term [2018/04/13 15:33] ellahughes [Why is it interesting?] |
advanced_notions:quantum_field_theory:wess-zumino-witten_term [2018/04/13 15:58] ellahughes [Intuitive] |
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<blockquote> The W-Z term is a generalization, to the configuration space of scalar fields $\phi_a$, of the | <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 | 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> | <cite>Berry Phases, Magnetic Monopoles and {Wess-Zumino} Terms or How the Skyrmion Got Its Spin by I.J.R. Aitchison</cite></blockquote> |