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

Fiorenza, Domenico; Sati, Hisham; Schreiber, Urs

doi:10.48550/arxiv.2009.11909

Cited by 5 publications

(47 citation statements)

References 82 publications

(115 reference statements)

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“…• For A = K(R, n) an Eilenberg-MacLane space, this construction (1) is ordinary abelian cohomology as computed by singular-or Čech-cochains (see [FSS20c,Ex. 2.2] for pointers); specifically as computed by (PL-)differential forms in the case that R = R (see [FSS20c,§3,Ex. 4.9] for pointers).…”

Section: M-brane Charge and Borsuk-spanier Cohomotopymentioning

confidence: 99%

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M/F-Theory as Mf-Theory

Sati¹,

Schreiber²

2021

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In the quest for mathematical foundations of M-theory, the Hypothesis H, that fluxes are quantized in Cohomotopy theory, implies, on flat but possibly singular spacetimes, that M-brane charges locally organize into equivariant homotopy groups of spheres. Here we show how this leads to a correspondence between phenomena conjectured in M-theory and fundamental mathematical concepts/results in stable homotopy, generalized cohomology and Cobordism theory Mf :-stems of homotopy groups correspond to charges of probe p-branes near black b-branes; -stabilization within a stem is the boundary-bulk transition; -the Adams d-invariant measures G 4 -flux; -trivialization of the d-invariant corresponds to H 3 -flux; -refined Toda brackets measure H 3 -flux; -the refined Adams e-invariant sees the H 3 -charge lattice; -vanishing Adams e-invariant implies consistent global C 3 -fields; -Conner-Floyd's e-invariant is the H 3 -flux seen in the Green-Schwarz mechanism; -the Hopf invariant is the M2-brane Page charge ( G 7 -flux); -the Pontrjagin-Thom theorem associates the polarized brane worldvolumes sourcing all these charges. In particular, spontaneous K3-reductions with 24 branes are singled out from first principles:-Cobordism in the third stable stem witnesses spontaneous KK-compactification on K3-surfaces; -the order of the third stable stem implies the 24 NS5/D7-branes in M/F-theory on K3. Finally, complex-oriented cohomology emerges from Hypothesis H, connecting it to all previous proposals for brane charge quantization in the chromatic tower: K-theory, elliptic cohomology, etc.:-quaternionic orientations correspond to unit H 3 -fluxes near M2-branes; -complex orientations lift these unit H 3 -fluxes to heterotic M-theory with heterotic line bundles. In fact, we find quaternionic/complex Ravenel-orientations bounded in dimension; and we find the bound to be 10, as befits spacetime dimension 10+1. S 3 S 3 S 3 K3\(24 • D 4

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Section: M-brane Charge and Borsuk-spanier Cohomotopymentioning

confidence: 99%

“…The corresponding rational charges are reflected in the periods of these differential forms, expressing the total flux through any closed hypersurfaces. On these fields, a charge quantization law (see [Fr00][Sa10] [FSS20c]) is a non-abelian cohomology theory A(−)…”

Section: Introductionmentioning

confidence: 99%

M/F-Theory as Mf-Theory

Sati¹,

Schreiber²

2021

Preprint

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“…Here we replace the notion of a rational Sullivan minimal model of a topological space with a less common notion of a real Sullivan minimal model, given that real coefficients of physical fields could be a bit more natural than rational ones (see the discussion in [FSS20]). We will therefore assume that our algebraic models are defined over the reals R (see [BS95][GM13]).…”

Section: The Real Sullivan Minimal Model and M-theory Dynamicsmentioning

confidence: 99%

“…The de Rham algebra is, in fact, a real homotopy model of the manifold Y 11 , and this model could be different from the Sullivan minimal model. Rational (or, actually, real [FSS20]) homotopy theory provides a canonical continuous map…”

Section: M-theory Dynamics and S 4 Via Hypothesis Hmentioning

confidence: 99%

“…This is the version in which rational (more precisely, real) homotopy theory connects to differential geometry (e.g. [FOT08]), since the smooth de Rham complex is not defined over Q but over R. This is favorable when dealing with differential forms Ω • dR and fields -see [FSS20] for an extensive discussion and applications. Even though we are working over R for the most part, we also do at times consider rational and integral coefficients, for instance when dealing with algebraic groups.…”

Section: Introductionmentioning

confidence: 99%

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Mysterious Triality and Rational Homotopy Theory

Sati¹,

Voronov²

2021

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Mysterious duality has been discovered by Iqbal, Neitzke, and Vafa [INV02] as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of M-theory and del Pezzo surfaces, both governed by the root system series E k .It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics which gives rise to the same E k symmetry pattern. We present a sequence of topological spaces, starting with the four-sphere S 4 , and then forming its iterated cyclic loop spaces L k c S 4 , within which we discover the E k symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its E k symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space L k c S 4 is naturally related to the compactification of M-theory on the k-torus via identification of the equations of motion of (11 − k)-dimensional supergravity as the defining equations of the Sullivan minimal model of L k c S 4 . This gives an explicit duality between algebraic topology and physics.Thereby, we extend Iqbal-Neitzke-Vafa's mysterious duality between algebraic geometry and physics into a triality, also involving algebraic topology. Via this triality, duality between physics and mathematics is demystified, and the mystery is transferred to the mathematical realm as duality between algebraic geometry and algebraic topology. Now the question is: is there an explicit relation between the del Pezzo surfaces B k and iterated cyclic loop spaces of S 4 which would explain the common E k symmetry pattern?

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Principal $$\infty $$-bundles and smooth string group models

Bunk

2022

Math. Ann.

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We provide a general, homotopy-theoretic definition of string group models within an $$\infty $$ ∞ -category of smooth spaces and present new smooth models for the string group. Here, a smooth space is a presheaf of $$\infty $$ ∞ -groupoids on the category of cartesian spaces. The key to our definition and construction of smooth string group models is a version of the singular complex functor, which assigns to a smooth space an underlying ordinary space. We provide new characterisations of principal $$\infty $$ ∞ -bundles and group extensions in $$\infty $$ ∞ -topoi, building on work of Nikolaus, Schreiber and Stevenson. These insights allow us to transfer the definition of string group extensions from the $$\infty $$ ∞ -category of spaces to the $$\infty $$ ∞ -category of smooth spaces. Finally, we consider smooth higher-categorical group extensions that arise as obstructions to the existence of equivariant structures on gerbes. These extensions give rise to new smooth models for the string group, as recently conjectured in joint work with Müller and Szabo.

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

Cited by 5 publications

References 82 publications

M/F-Theory as Mf-Theory

M/F-Theory as Mf-Theory

Mysterious Triality and Rational Homotopy Theory

Principal $$\infty $$-bundles and smooth string group models

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