Twisted di erential nonabelian cohomology the twisted bundles appearing in twisted K-theory. We explain

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  • Twisted differential nonabelian cohomology

    Twisted (n−1)-brane n-bundles and their Chern-Simons (n+1)-bundles with characteristic (n + 2)-classes

    Hisham Sati∗, Urs Schreiber†, Zoran Škoda‡, Danny Stevenson§

    November 23, 2008

    Abstract

    We introduce nonabelian differential cohomology classifying ∞-bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian n-group Bn−1U(1). Notable examples are String 2-bundles [9] and Fivebrane 6-bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spin-structures to String-structures [13] and further to Fivebrane-structures [133, 52], are abelian Chern- Simons 3- and 7-bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞-Lie- integrating the L∞-algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2- and twisted Fivebrane 6-bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted K-theory. We explain the Green-Schwarz mechanism in heterotic string theory in terms of twisted String 2-bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6-bundles. We close by transgressing differential cocycles to mapping spaces, thereby obtaining their volume holonomies, and show that for Chern-Simons cocycles this yields the action functionals for Chern-Simons theory and its higher dimensional generalizations, regarded as extended quantum field theories.

    Handle with care. This is stuff we are still working on.

    ∗hisham.sati@yale.edu †schreiber@math.uni-hamburg.de ‡zskoda@irb.hr §d.stevenson@maths.gla.ac.uk

    1

  • Contents

    1 Overview 6 1.1 General subject of nonabelian cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    2 Underlying Machinery: Space and Quantity 9 2.1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    2.2.1 ω-Groupoids and crossed complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Weak and surjective equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.3 Cofibrations and pseudo-∞-functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.4 Fibrations and ∞-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.5 The homotopy category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    2.3 Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4 ∞-Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    3 Homotopy and Cohomology 41 3.1 ∞-Stacks and ∞-costacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    3.1.1 ω-Categories and simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.2 Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.3 Codescent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    3.2 Operations on cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.1 Lifts, twisted lifts and obstructions to lifts . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.2 Transgression of cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.3 Construction of cocycles by killing of homotopy groups . . . . . . . . . . . . . . . . . . 57

    3.3 Types of cohomologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.1 Sheaf cohomology / Čech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.2 Equivariant cohomology / group cohomology . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.3 Twisted cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.4 Differential cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    3.4 Characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4.1 Characteristic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4.2 The generalized Chern-Weil homomorphism . . . . . . . . . . . . . . . . . . . . . . . . 67

    4 ∞-Lie theory 68 4.1 Infinitesimal paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    4.1.1 Fundamental L∞-algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.2 The adjunction between Spaces and L∞Algebroids . . . . . . . . . . . . . . . . . . . . 69

    4.2 Finite paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.1 Fundamental ω-groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.2 The adjunction between ωGroupoids and Spaces . . . . . . . . . . . . . . . . . . . . . 73

    4.3 ∞-Lie integration and ∞-Lie differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 L∞-algebraic cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    4.4.1 Integrating L∞-algebraic cocycles to nonabelian cocycles . . . . . . . . . . . . . . . . 76 4.4.2 Lifts, obstructions and twists of L∞-algebraic cocycles . . . . . . . . . . . . . . . . . . 79

    4.5 [variation] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    2

  • 5 Examples and Applications 83 5.1 ω-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    5.1.1 1-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.2 2-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.3 3-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

    5.2 ∞-Lie integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2.1 Integration of Lie 1-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2.2 Integration of bn−1u(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2.3 Integration of string(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2.4 Integration of fivebrane(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2.5 Integration of sugra(11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    5.3 ∞-Lie differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.3.1 L∞-Differentiation of Lie 1-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.3.2 L∞-Differentiation of Lie 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    5.4 Principal ω-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4.1 Principal 1-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4.2 Principal 2-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4.3 Principal 3-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    5.5 Characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5.1 Characteristic classes of principal 1-bundles . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5.2 Characteristic classes of String(n)-principal bundles . . . . . . . . . . . . . . . . . . . 104

    5.6 Chern-Simons ω-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.6.1 Chern-Simons 3-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.6.2 Chern-Simons 7-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.6.3 Chern-Simons 11-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    5.7 Twisted ω-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.7.1 Twisted 1-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.7.2 Twisted 2-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.7.3 Twisted 6-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

    5.8 L∞-integration of L∞-cocycles to nonabelian cocycles . . . . . . . . . . . . . . . . . . . . . . 118 5.8.1 Principal 1-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.8.2 Chern-Simons n-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

    5.9 Transgression to mapping spaces: σ-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.9.1 Chern-Sim