geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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).
equivariant K-theory, orbifold K-theory
equivariant differential K-theory, orbifold differential K-theory
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:
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):
Last revised on March 8, 2021 at 14:27:33. See the history of this page for a list of all contributions to it.