Twisted differential String and Fivebrane structures

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-infinity-algebra valued differential form data provided by the differential refinements of these twisted cocycles.

Figures (2)

• Figure 1: Topological structures generalizing $\mathrm{Spin}(n)$ structure. In application to effective background field theories appearing in string theory, these bare structures are twisted and moreover refined to differential structures.
• Figure 2: Abelian versus nonabelian cohomology. Since the groups $\mathrm{String}(n)$ as well as $\mathrm{Fivebrane}(n)$ are shifted central extension of nonabelian groups, cohomology with coefficients in these groups has abelian components but also components in nonabelian cohomology Toen2. This appears as abelian cohomology twisted by nonabelian cocycles in a certain way. The Green-Schwarz mechanism implies that two classes in ordinary abelian cohomology, namely in degree four differential integral cohomology, coincide. But these classes are particularly obstruction classes to $\mathrm{String}$-lifts in nonabelian cohomology. The middle part of Figure 1 , labeled "abelian cohomology", identifies the cocycle representative in $H^4(X,\mathbb{Z})$ and the coboundary between them, but does not specify where these cocycles come from. The outer part of the diagram, labeled "nonabelian cohomology" does specify the object whose class is the one identified by the middle part. We can interpret this in ordinary homotopy theory, where it describes topological obstruction theory, but we can also interpret this after differential refinement in the $\infty$-topos Lurie of smooth $\infty$-groupoids nactwist. In any case the morphisms in the above diagram may be interpreted as cocycles. The smooth and differential refinement we discuss in section \ref{['Twisted differential String- and Fivebrane structures']}.

Theorems & Definitions (25)

• Definition 2.1
• Definition 2.4
• Definition 2.5
• Proposition 2.7
• Definition 2.8
• Definition 2.10
• Proposition 2.12
• Definition 2.15
• Definition 2.16
• Definition 2.18