### 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, differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted 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 differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-, String(n)- and 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 valued differential form data provided by the differential refinements of these twisted cocycles.

Original language | English (US) |
---|---|

Pages (from-to) | 169-213 |

Number of pages | 45 |

Journal | Communications in Mathematical Physics |

Volume | 315 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1 2012 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*315*(1), 169-213. https://doi.org/10.1007/s00220-012-1510-3

**Twisted Differential String and Fivebrane Structures.** / Sati, Hisham; Schreiber, Urs; Stasheff, Jim.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 315, no. 1, pp. 169-213. https://doi.org/10.1007/s00220-012-1510-3

}

TY - JOUR

T1 - Twisted Differential String and Fivebrane Structures

AU - Sati, Hisham

AU - Schreiber, Urs

AU - Stasheff, Jim

PY - 2012/9/1

Y1 - 2012/9/1

N2 - 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, differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted 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 differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-, String(n)- and 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 valued differential form data provided by the differential refinements of these twisted cocycles.

AB - 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, differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted 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 differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-, String(n)- and 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 valued differential form data provided by the differential refinements of these twisted cocycles.

UR - http://www.scopus.com/inward/record.url?scp=84866155535&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866155535&partnerID=8YFLogxK

U2 - 10.1007/s00220-012-1510-3

DO - 10.1007/s00220-012-1510-3

M3 - Article

AN - SCOPUS:84866155535

VL - 315

SP - 169

EP - 213

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -