nLab adjusted connection

Idea

A notion of connection for principal ∞-bundles that does not impose the fake flatness condition.

Related concepts

• adjusted Weil algebra

References

The construction of non-fake flat principal ∞-connections originates with

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, p. 77 and p. 80 of: Čech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Physics, 16 1 (2012) 149-250 [arXiv:1011.4735, doi:10.1007/BF02104916]

based on the adjusted Weil algebras discussed earlier in

• Hisham Sati, Urs Schreiber, Jim Stasheff, Def. 23 on p. 47 with Prop. 21 on p. 48 in: $L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport, in Quantum Field Theory, Birkhäuser (2009) 303-424 [arXiv:0801.3480, doi:10.1007/978-3-7643-8736-5_17]

• Hisham Sati, Urs Schreiber, Jim Stasheff: middle boxes on p. 28 and p. 32 of: Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics 315 1 (2012) 169-213 [arXiv:0910.4001, doi:10.1007/s00220-012-1510-3))]

As the title of FSS 12 indicates, this procedure constructs Čech cohomology cocycles for non-fake flat higher connections in the style of the cocycles in

• Jean-Luc Brylinski, D. A. McLaughlin. Čech cocycles for characteristic classes, Comm. Math. Phys., 178 1 (1996) 225–236 [doi:10.1007/BF02104916]

for the underlying bundles.

This construction is based on the Lie integration of L-infinity algebras by the “path method” and as such works generally but produces very “large” cocycle data, in a sense.

A variant construction tailored towards “smaller” cocycles for low-degree Lie n-algebras was later proposed in:

• Dominik Rist, Christian Saemann, Martin Wolf, Explicit Non-Abelian Gerbes with Connections, arXiv:2203.00092

Examples for T-duality:

• Hyungrok Kim, Christian Saemann, T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections, arXiv:2303.16162

Further references for T-duality:

• Konrad Waldorf, Geometric T-duality: Buscher rules in general topology, arXiv:2207.11799.

• Thomas Nikolaus, Konrad Waldorf, Higher geometry for non-geometric T-duals, arXiv:1804.00677.