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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 11th 2013
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   Author: Urs
   Format: MarkdownItexat _[[quantum observable]]_ there used to be just the definition of geometric prequantum observables. I have [added](http://ncatlab.org/nlab/show/quantum+observable#OnASymplecticManifold) a tad more.

   at quantum observable there used to be just the definition of geometric prequantum observables. I have added a tad more.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 20th 2023
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   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Nikolay Bogolyubov]], A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, *Algebra of Observables and State Space* &lbrack;[doi:10.1007/978-94-009-0491-0_6](https://doi.org/10.1007/978-94-009-0491-0_6)&rbrack;, Chapter in: *General principles of quantum field theory*, Mathematical Physics and Applied Mathematics **10**, Kluwer (1990) &lbrack;[doi:10.1007/978-94-009-0491-0](https://doi.org/10.1007/978-94-009-0491-0)&rbrack; here and in related entries (such as at *[[quantum state space]]*) <a href="https://ncatlab.org/nlab/revision/diff/quantum+observable/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/quantum+observable/20">v20</a>, <a href="https://ncatlab.org/nlab/show/quantum+observable">current</a>

   added pointer to:

   • Nikolay Bogolyubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, Algebra of Observables and State Space [doi:10.1007/978-94-009-0491-0_6], Chapter in: General principles of quantum field theory, Mathematical Physics and Applied Mathematics 10, Kluwer (1990) [doi:10.1007/978-94-009-0491-0]

   here and in related entries (such as at quantum state space)

   diff, v20, current

3. • CommentRowNumber3.
   • CommentAuthorperezl.alonso
   • CommentTime11 hours ago
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   Author: perezl.alonso
   Format: MarkdownItexAs discussed elsewhere, one can obtain (higher) topological observables by computing the Pontriagin homology ring. I suppose the particular case of WZW models does not need the space to have a circle factor. To incorporate the gerbe I guess one has to consider fields valued in the higher group extension determined by the gerbe class. But how does one incorporate the kinetic term so as to see the WZW observables?

   As discussed elsewhere, one can obtain (higher) topological observables by computing the Pontriagin homology ring. I suppose the particular case of WZW models does not need the space to have a circle factor. To incorporate the gerbe I guess one has to consider fields valued in the higher group extension determined by the gerbe class. But how does one incorporate the kinetic term so as to see the WZW observables?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTime3 hours ago
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   Author: Urs
   Format: MarkdownItexThe observables obtained as the Pontrjagin algebra of the mapping space to a flux classifying space are to be thought of as the observables on "topological flux sectors" of a higher gauge theory. The WZW model is not directly an example of such theories. But 3D abelian Chern-Simons theory is, with the CS Lagrangian density regarded as the higher flux density $H_3$ of a twisted gerbe, satisfying the Bianchi identity $\mathrm{d} H_3 = F_2 F_2$, $\mathrm{d} F_2 = 0$. An evident choice of flux quantization law for these Bianchi identities is that classified by the 2-sphere $S^2$. And one finds that the Pontrjagin algebras of $\mathrm{Maps}^\ast\big( (\Sigma^2 \times \mathbb{R})_{\cup \{\infty\}}, S^2 \big)$ indeed know about the modular functor of 3D abelian CS theory and as such about the conformal blocks and the modularity of the abelian WZW model! The details of this statement are currently in section 4.2 of the draft with the long title *[[schreiber:Understanding Topological Quantum Gates in FQH Systems|Understanding Topological Quantum Gates in FQH Systems via the Algebraic Topology of exotic Flux Quantization]]*.

   The observables obtained as the Pontrjagin algebra of the mapping space to a flux classifying space are to be thought of as the observables on “topological flux sectors” of a higher gauge theory. The WZW model is not directly an example of such theories.

   But 3D abelian Chern-Simons theory is, with the CS Lagrangian density regarded as the higher flux density $H_3$ of a twisted gerbe, satisfying the Bianchi identity $\mathrm{d} H_3 = F_2 F_2$, $\mathrm{d} F_2 = 0$. An evident choice of flux quantization law for these Bianchi identities is that classified by the 2-sphere $S^2$.

   And one finds that the Pontrjagin algebras of $\mathrm{Maps}^\ast\big( (\Sigma^2 \times \mathbb{R})_{\cup \{\infty\}}, S^2 \big)$ indeed know about the modular functor of 3D abelian CS theory and as such about the conformal blocks and the modularity of the abelian WZW model!

   The details of this statement are currently in section 4.2 of the draft with the long title Understanding Topological Quantum Gates in FQH Systems via the Algebraic Topology of exotic Flux Quantization.