The character map in (twisted differential) non-abelian cohomology

Abstract

We extend the Chern character on K-theory, in its generalization to the Chern-Dold character on generalized cohomology theories, further to (twisted, differential) non-abelian cohomology theories, where its target is a non-abelian de Rham cohomology of twisted L-infinity algebra valued differential forms. The construction amounts to leveraging the fundamental theorem of dg-algebraic rational homotopy theory to a twisted non-abelian generalization of the de Rham theorem. We show that the non-abelian character reproduces, besides the Chern-Dold character, also the Chern-Weil homomorphism as well as its secondary Cheeger-Simons homomorphism on (differential) non-abelian cohomology in degree 1, represented by principal bundles (with connection); and thus generalizes all these to higher (twisted, differential) non-abelian cohomology, represented by higher bundles/higher gerbes (with higher connections). As a fundamental example, we discuss the twisted non-abelian character map on twistorial Cohomotopy theory over 8-manifolds, which can be viewed as a twisted non-abelian enhancement of topological modular forms (tmf) in degree 4. This turns out to exhibit a list of subtle topological relations that in high energy physics are thought to govern the charge quantization of fluxes in M-theory.

arXiv comment124 pages; v2: one more example and further references added; v3: several improvements and more details, expanded introduction

References in corpus (72)

• Deformation quantization of Poisson manifolds, I
• Eleven-Dimensional Supergravity on a Manifold with Boundary
• Introduction to sh Lie algebras for physicists
• On Flux Quantization In M-Theory And The Effective Action
• Rational Maps, Monopoles and Skyrmions
• Twisted K-theory and K-theory of bundle gerbes
• Equivalences of monoidal model categories
• Heterotic Line Bundle Standard Models
• Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds
• The Uncertainty of Fluxes
• Algebraic topology and modular forms
• Čech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction
• Integrating L-infinity algebras
• Differential cohomology in a cohesive infinity-topos
• 2-Group Symmetries of 6d Little String Theories and T-duality
• Some Relations between Twisted K-theory and E8 Gauge Theory
• Unoriented WZW Models and Holonomy of Bundle Gerbes
• On Ramond-Ramond fields and K-theory
• Twisted equivariant K-theory with complex coefficients
• Twistor theory at fifty: from contour integrals to twistor strings
• Homotopy of Rational Maps and the Quantization of Skyrmions
• Units of ring spectra and their traces in algebraic K-theory
• Perturbative algebraic quantum field theory
• Axiomatic Characterization of Ordinary Differential Cohomology
• Infinite loop spaces, and coherence for symmetric monoidal bicategories
• Extended higher cup-product Chern-Simons theories
• Units of ring spectra and Thom spectra
• L-infinity algebras of local observables from higher prequantum bundles
• Holomorphic SCFTs with small index
• Equivariant Cohomotopy implies orientifold tadpole cancellation
• Rational sphere valued supercocycles in M-theory and type IIA string theory
• Principal infinity-bundles - Presentations
• Higher bundle gerbes and cohomology classes in gauge theories
• On a generalized Connes-Hochschild-Kostant-Rosenberg theorem
• What Does(n't) K-theory Classify?
• The $\infty$-Categorical Eckmann-Hilton Argument
• Differential K-theory. A survey
• The Rational Higher Structure of M-theory
• Structured vector bundles define differential K-theory
• Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino term of the M5-brane
• Type II string theory and modularity
• Twisted Cohomotopy implies twisted String structure on M5-branes
• Gravity with a cosmological constant from rational curves
• Twisted Cohomotopy implies M5-brane anomaly cancellation
• Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation
• An approach to anomalies in M-theory via KSpin
• Super-exceptional embedding construction of the heterotic M5: Emergence of SU(2)-flavor sector
• A Higher Twist in String Theory
• Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers
• Twisted bundles and twisted K-theory
• Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory
• Twisted differential cohomology
• Lift of fractional D-brane charge to equivariant Cohomotopy theory
• Higher T-duality in M-theory via local supersymmetry
• Algebraic aspects of higher nonabelian Hodge theory
• Primary operations in differential cohomology
• Simplicial principal bundles in parametrized spaces
• M-Theory with Framed Corners and Tertiary Index Invariants
• Higher order Whitehead products and $L_{\infty }$ structures on the homology of a DGL
• Combinatorics and formal geometry of the master equation
• Anomalies of E8 gauge theory on String manifolds
• Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models
• Higher T-duality of super M-branes
• A noncommutative model for higher twisted K-Theory
• Twisted smooth Deligne cohomology
• Classifying theory for simplicial parametrized groups
• Regulators of canonical extensions are torsion: the smooth divisor case
• An introduction to Bousfield localization
• The model structure for chain complexes
• Commutative ring spectra
• On the homotopy theory of $\mathbf{G}$ - spaces
• Quantization of Skyrmions

Cited by in corpus (15)

• M/F-Theory as Mf-Theory
• Flux Quantization on 11-dimensional Superspace
• Flux Quantization on M5-branes
• Topological Quantum Gates in Homotopy Type Theory
• Flux Quantization on Phase Space
• Higher Topos Theory in Physics
• Cyclification of Orbifolds
• Quantum Observables of Quantized Fluxes
• Anyons on M5-Probes of Seifert 3-Orbifolds via Flux Quantization
• The Smith Fiber Sequence and Invertible Field Theories
• The Quantum Monadology
• Engineering of Anyons on M5-Probes via Flux Quantization
• Gauge potentials on the M5 brane in twisted equivariant cohomotopy
• Quantum and Reality
• Mysterious Triality and the Exceptional Symmetry of Loop Spaces