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A Higher Stacky Perspective on Chern–Simons Theory

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Mathematical Aspects of Quantum Field Theories

Part of the book series: Mathematical Physics Studies ((MPST))

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. 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. 2.

    Even 1-dimensional Chern–Simons theory exhibits a rich structure once we pass to derived higher gauge groups as in [46]. This goes beyond the present exposition, but see Sect. 5.1 for an outlook and Sect. 5.7.10 of [79] for more details.

  3. 3.

    That is, for the collections of all such bundles, with gauge transformations as morphisms.

  4. 4.

    The reader unfamiliar with the language of higher stacks and simplicial presheaves in differential geometry can find an introduction in [31].

  5. 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. 6.

    See for instance [54] for an original reference on geometric quantization and see [67] for further pointers.

  7. 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. 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. 9.

    This can be traced back to [4]; a nice modern review can be found in Sect. 4 of [6].

  10. 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. 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. 12.

    More detailed discussion of how (quantum) fields generally are maps in slices of cohesive toposes has been given in the lecture notes [80] and in Sects. 1.2.16, 5.4 of [79].

  13. 13.

    A discussion of this and the following can be found in Sects. 3.9.13 and 4.4.19 of [79]; see also [27].

  14. 14.

    This follows for instance as the Lie integration of the result in [5] that the Heisenberg Lie 2-algebra here is the \(\mathfrak {string}\,(\mathfrak {g})\) Lie 2-algebra; see also [27].

  15. 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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Authors and Affiliations

  1. Università degli Studi di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185, Rome, Italy

    Domenico Fiorenza

  2. University of Pittsburgh, 4200 Fifth Avenue, Pittsburgh, PA, 15260, USA

    Hisham Sati

  3. Radboud University Nijmegen, PO Box 9010, 6500 GL, Nijmegen, The Netherlands

    Urs Schreiber

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  1. Domenico Fiorenza
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  1. Université Montpellier 2, Montpellier, France

    Damien Calaque

  2. Université Claude Bernard Lyon 1, Villeurbanne Cedex, France

    Thomas Strobl

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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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