nLab equivariant ordinary differential cohomology

Idea

Equivariant ordinary differential cohomology or equivariant Deligne cohomology is the equivariant cohomology-enhancement of ordinary differential cohomology (Deligne cohomology), equivalently the differential cohomology-enhancement of ordinary equivariant cohomology.

There is one evident general abstract definition of what this means concretely: Given a $G$-action on some $X$, the (Borel-)$G$-equivariant ordinary differential cohomology of $X$ in degree $n$ is the (∞,1)-categorical hom-space in the cohesive (∞,1)-topos of Smooth∞Groupoids from the quotient stack $X \sslash G$ to the Deligne complex $\mathbf{B}^n U(1)_{conn}$.

That various definitions in the literature coincide with this one (in particular Kübel-Thom 15, and for finite groups also Lupercio-Uribe 01, Gomi 05) is discussed in (Park-Redden 19).

Related concepts

• discrete torsion

• equivariant de Rham cohomology

• equivariant K-theory, orbifold K-theory

  equivariant differential K-theory, orbifold differential K-theory

• twisted ordinary differential cohomology

References

The main definitions can be found in

• Andreas Kübel, Andreas Thom, Equivariant Differential Cohomology, Transactions of the American Mathematical Society (2018) (doi:10.1090/tran/7315, arXiv:1510.06392)

• Corbett Redden, Differential Borel equivariant cohomology via connections, New York Journal of Mathematics, Volume 23 (2017) 441-487 (nyjm:23-20, arXiv:1602.06921)

An earlier definition thst works (only) for finite groups can be found in:

• Kiyonori Gomi, Equivariant smooth Deligne cohomology, Osaka J. Math. Volume 42, Number 2 (2005), 309-337 (euclid:ojm/1153494380)

Other references:

• Ernesto Lupercio, Bernardo Uribe, Deligne Cohomology for Orbifolds, discrete torsion and B-fields, in: Geometric and topological methods for quantum field theory Proceedings, Summer School, Villa de Leyva, Colombia, July 9-27, 2001 (arXiv:hep-th/0201184, spire:582101)

• Cheng-Yong Du Xiaojuan Zhao, Spark and Deligne-Beilinson cohomology on orbifolds, Journal of Geometry and Physics Volume 104, June 2016, Pages 277-290 (doi:10.1016/j.geomphys.2016.02.011)

• Corbett Redden, An alternate description of equivariant connections, Differential Geometry and its Applications Volume 56, February 2018, Pages 81-94 (doi:10.1016/j.difgeo.2017.11.003 arXiv:1608.01297)

• Byungdo Park, Corbett Redden, A classification of equivariant gerbe connections, Communications in Contemporary MathematicsVol. 21, No. 02, 1850001 (2019) (doi:10.1142/S0219199718500013, arXiv:1709.06003)

• Cheng-Yong Du, Lili Shen, Xiaojuan Zhao, Spark complexes on good effective orbifold atlases categorically, Theory and Applications of Categories, 33(26):784-812, 2018 (tac:33-26, arXiv:1708.07551)

Relation to action functionals for topological field theories (such as Chern-Simons theory):

• Joe Davighi, Ben Gripaios, Oscar Randal-Williams, Differential cohomology and topological actions in physics (arXiv:2011.05768)