Abstract
The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern–Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern–Simons theory is naturally localized (“extended”, “multi-tiered”) to a map on the universal moduli stack of principal connections, a map that itself modulates a circle-principal 3-connection on that moduli stack, and how the iterated transgressions of this extended Lagrangian unify the action functional with its prequantum bundle and with the WZW-functional. In the second part we provide a brief review and outlook of the higher prequantum field theory of which this is a first example. This includes a higher geometric description of supersymmetric Chern–Simons theory, Wilson loops and other defects, generalized geometry, higher Spin-structures, anomaly cancellation, and various other aspects of quantum field theory.
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Notes
- 1.
We are using the term “gauge group” to refer to the structure group of the theory. This is not to be confused with the group of gauge transformations.
- 2.
- 3.
That is, for the collections of all such bundles, with gauge transformations as morphisms.
- 4.
The reader unfamiliar with the language of higher stacks and simplicial presheaves in differential geometry can find an introduction in [31].
- 5.
It is noteworthy that this indeed is a stack on the site \({\mathrm {CartSp}}\). On the larger but equivalent site of all smooth manifolds it is just a prestack that needs to be further stackified.
- 6.
- 7.
The existence and functoriality of the path \(\infty \)-groupoids is one of the features characterizing the higher topos of higher smooth stacks as being cohesive, see [79].
- 8.
That is, when written in local coordinates \((u, \sigma )\) on \(U \times \varSigma _2\), then \(A=A_i(u, \sigma ) du^i + A_j (u, \sigma ) d\sigma ^j\) reduces to the second summand.
- 9.
- 10.
This means that here we are secretly moving from the topos of (higher) stacks on smooth manifolds to its arrow topos, see Sect. 5.2.
- 11.
See [13] for a comprehensive treatment of the étale site of smooth manifolds and of the higher topos of higher stacks over it.
- 12.
- 13.
- 14.
- 15.
The notion of \((\mathbf {B}U(n))\)-fiber 2-bundle is equivalently that of nonabelian \(U(n)\)-gerbes in the original sense of Giraud, see [64]. Notice that for \(n = 1\) this is more general than then notion of \(U(1)\)-bundle gerbe: a \(G\)-gerbe has structure 2-group \(\mathbf {Aut}(\mathbf {B}G)\), but a \(U(1)\)-bundle gerbe has structure 2-group only in the left inclusion of the fiber sequence \(\mathbf {B}U(1) \hookrightarrow \mathbf {Aut}(\mathbf {B}U(1)) \rightarrow \mathbb {Z}_2\).
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Acknowledgments
D.F. thanks ETH Zürich for hospitality. The research of H.S. is supported by NSF Grant PHY-1102218. U.S. thanks the University of Pittsburgh for an invitation in December 2012, during which part of this work was completed.
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Fiorenza, D., Sati, H., Schreiber, U. (2015). A Higher Stacky Perspective on Chern–Simons Theory. In: Calaque, D., Strobl, T. (eds) Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-09949-1_6
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