Equivariant principal infinity-bundles (2112.13654v3)

Abstract: In this book we prove unified classification results for equivariant principal bundles when the topological structure group is truncated. The conceptually transparent proof invokes a smooth Oka principle, which becomes available after faithfully embedding traditional equivariant topology into the singular-cohesive homotopy theory of globally equivariant higher smooth stacks. This works for discrete equivariance groups acting properly on smooth manifolds with resolvable singularities, whence we are equivalently describing principal bundles on good orbifolds. In preparation, we re-develop the theory of equivariant principal bundles from scratch by systematic use of Grothendieck's internalization method. In particular we prove that all equivariant local triviality conditions considered in the literature are implied by regarding G-equivariant principal bundles as principal bundles internal to the BG-slice of the ambient cohesive infinity-topos. Generally we find that the characteristic subtle phenomena of equivariant classifying theory all reflect basic "modal" properties of singular-cohesive homotopy theory. Classical literature has mostly been concerned with compact Lie structure groups. Where these are truncated, our classification recovers and generalizes results of Lashof, May, Segal and Rezk. A key non-classical example is the infinite projective unitary structure group, in which case we are classifying degree-3 twists of equivariant KU-theory, recovering results of Atiyah, Segal, Lueck and Uribe. Our theorem enhances this to conjugation-equivariance, where we are classifying the geometric twists of equivariant KR-theory, restricting on "O-planes" to the geometric twists of KO-theory. This is the generality in which equivariant K-theory twists model quantum symmetries of topological phases and the B-field in string theory on orbi-orientifolds.

• The paper introduces a cohesive homotopy theoretic framework that unifies the classification of stable equivariant principal bundles with truncated structure groups.
• It employs smooth and singular cohesive homotopy theory, including the smooth Oka principle, to extend classical bundle theory to orbifolds with singularities.
• The findings have significant applications in quantum symmetries, topological phases of matter, and string theory, suggesting new directions for future research.

Equivariant Principal $\infty$-Bundles

The paper "Equivariant Principal $\infty$-Bundles" by Hisham Sati and Urs Schreiber addresses the classification of equivariant principal bundles under the lens of homotopy theory and $\infty$-stacks. It provides a mathematical framework for understanding these objects in the context of truncated structure groups. The authors develop their theory within the mathematical framework of smooth and singular-cohesive homotopy theory, which provides a robust foundation for extending classical bundle theory to the equivariant setting.

Unified Classification Results

The central contribution of this work is the unified classification of stable equivariant $G$-principal bundles when the structure group $G$ exhibits truncated homotopy types. Such bundles are understood through the homotopy-theoretic perspective, where $G$-spaces are deconstructed into non-abelian group cohomology of the equivariance group and coefficient space $G$. This is further explored through several theorems, notably Theorems 4.2.7 and 4.3.24, which detail how equivariant higher non-abelian gerbes simultaneously classify $\infty$-bundles and equivariant $n$-group $G$-bundles.

Advances and Methodology

The methodology employs concepts such as Grothendieck’s internalization and the application of a smooth Oka principle within the cohesive homotopy theory. This theoretical advancement allows the embedding of classical equivariant topology into a more global framework, thereby facilitating the classification of bundles on orbifolds with resolvable singularities. The discussions are grounded heavily in existing literature and extend multiple results, including classifying spaces with compact Lie structure groups, to broader contexts.

Examples and Applications

The paper discusses a range of characteristic phenomena and potential applications including projective unitary structures and their connections to 3-twisted equivariant $KU$-theory and $KR$-theory. The implications are profound: for example, these results hold significance for quantum symmetries in topological phases of matter and for modeling aspects of string theory such as the B-field on orbi-orientifolds.

Implications and Future Directions

The theoretical implications extend to foundational aspects of equivariant homotopy theory. One notable conjecture posits that classifying spaces might reflect quantum symmetries through generalized cohomology as equivariant bundles. Practically, this could transform the understanding of string theoretic models, providing a refined view on B-field descriptions.

Looking towards future directions, the potential to adapt these ideas beyond the discussed cases, to more complex scenarios involving non-simply connected spaces and rational homotopy-theoretic settings in twisted complex K-theory, could further enhance both theoretical and applied aspects of mathematical physics.

Conclusion

In its entirety, this work provides a cohesive homotopy theoretic framework for understanding equivariant principal bundles. While maintaining an analytical depth, the theorems elucidate new classifications, link classical structure with $\infty$-toposes, and propose future lines of inquiry with a solid mathematical foundation. The integration of global homotopy theoretic contexts offers a modern, comprehensive perspective on bundles, potentially catalyzing further research in related fields.