Schreiber Twisted Differential String and Fivebrane Structures

An article we once wrote:

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• Hisham Sati, Urs Schreiber, Jim Stasheff,

  Twisted differential string and fivebrane structures

  Communications in Mathematical Physics

  October 2012, Volume 315, Issue 1, pp 169-213

  (arXiv:0910.4001, doi:10.1007/s00220-012-1510-3)

on differential string structures and differential fivebrane structures and their interpretation in terms of the Green-Schwarz mechanism in heterotic string theory. The cohomological theory is part of the story at differential cohomology in a cohesive topos.

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Abstract In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of, twisted differential string- andtwisted differential Fivebrane-structures_ that generalize the notion of Spin structure and Spin lifting gerbes and their differential refinement to smooth $\mathrm{Spin}$-connections. We show that all these structures can be encoded in terms of nonabelian cohomology and twisted nonabelian cohomology and twisted differential nonabelian cohomology, extending the differential generalized abelian cohomology as developed by Hopkins and Singer and shown by Freed to formalize the global description of anomaly cancellation problems in higher gauge theories arising in string theory. We demonstrate that the Green-Schwarz mechanism for the $H_3$-field, as well as its magnetic dual version for the $H_7$-field define cocycles in twisted differential nonabelian cohomology that may be called, respectively, differential twisted $\mathrm{Spin}(n)$-, $\mathrm{String}(n)$- and $\mathrm{Fivebrane}(n)$- structures on target space, where the twist in each case is provided by the obstruction to lifting the classifying map of the gauge bundle through a higher connected cover of $U(n)$ or $O(n)$. We show that the twisted Bianchi identities in string theory can be captured by the (nonabelian) L-∞ algebra (L-∞ algebroid) valued differential form data provided by the differential refinements of these twisted cocycles.

Related projects

• differential cohomology in a cohesive topos

  • Synthetic Quantum Field Theory

    • Quantum gauge field theory in Cohesive homotopy type theory

    • Type-semantics for quantization

  • cohomology

    • Principal ∞-bundles – theory, presentations and applications

    • Fivebrane structures

  • ∞-Chern-Weil theory

    • L-∞ algebra connections

    • Cech cocycles for differential characteristic classes

    • Twisted differential structures

      • Twisted Differential String and Fivebrane Structures

      • The moduli 3-stack of the C-field

  • ∞-Chern-Simons theory

    • A higher stacky perspective on Chern-Simons theory

    • Higher Chern-Weil Derivation of AKSZ Sigma-Models

    • nonabelian 7d Chern-Simons theory and the 5-brane

    • Extended higher cup-product Chern-Simons theories

  • ∞-Wess-Zumino-Witten theory

    • The brane bouquet

    • Obstruction theory for parameterized higher WZW terms

  • Higher geometric prequantum theory

    • Classical field theory via higher differential geometry

    • Local prequantum field theory

    • L-∞ algebras of local observables from higher prequantum bundles

    • Higher extensions of mapping class groups

  • Motivic quantization of local prequantum field theory

    • Geometric quantization of symplectic and Poisson manifolds

    • Cohomological quantization of local prequantum boundary field theory

References

For a comprehensive list of references see differential cohomology in a cohesive topos – references.

Talk notes about this subject are in

• Urs Schreiber, Chern-Simons terms on higher moduli stacks, Talk at Hausdorff Institute, Bonn (2011) (pdf)