Extended higher cup-product Chern–Simons theories

Fortschritte der Physik

Schreiber

2019

Self Cite

We review how core structures of string/M‐theory emerge as higher structures in super homotopy theory; namely from systematic analysis of the brane bouquet of universal invariant higher central extensions growing out of the superpoint. Since super homotopy theory is immensely rich, to start with we consider this in the rational/infinitesimal approximation which ignores torsion‐subgroups in brane charges and focuses on tangent spaces of super space‐time. Already at this level, super homotopy theory discovers all super p‐brane species, their intersection laws, their M/IIA‐, T‐ and S‐duality relations, their black brane avatars at ADE‐singularities, including their instanton contributions, and, last not least, Dirac charge quantization: for the D‐branes it recovers twisted K‐theory, rationally, but for the M‐branes it gives cohomotopy cohomology theory. We close with an outlook on the lift of these results beyond the rational/infinitesimal approximation to a candidate formalization of microscopic M‐theory in super homotopy theory.

“…This is reflected in the fact that non‐homotopy theory has no answer to the evident followup question; if 2‐cocycles classify central extensions, then:

“What do higher cocycles classify?”

For example, the Lie algebra

su (n)

itself (for

n \geq 2

) carries no non‐trivial 2‐cocycle, but it carries, a non‐trivial 3‐cocycle (in fact precisely one, up to rescaling) given on elements

x, y, z \in su (n)

μ_{3} (x, y, z) = true〈 x, [y, z] false⟩,

where

[-, -]

is the Lie bracket, and

⟨ -, - ⟩

Section: Higher Structure From Higher Cocyclesmentioning

confidence: 70%

Section: Higher Structure From Higher Cocyclesmentioning

confidence: 99%

The Rational Higher Structure of M‐theory

Fiorenza

Fortschritte der Physik

Schreiber

2019

Self Cite

“…Similarly, fiber integration of n-connections, hence fiber integration of cocycles in ordinary differential cohomology of degree (p + 2), is equivalently encoded [18] in a morphism of smooth higher stacks of the form…”

Section: Transgression To Ordinary Lie Algebrasmentioning

confidence: 99%

Lie n-algebras of BPS charges

Schreiber²

2017

J. High Energ. Phys.

Self Cite

We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie (p + 1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie (p + 1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.

“…One can also add source terms and also impose self-duality by hand, in which case the action (1.1) might be referred to as a pseudo-action (see [1]). Recent accounts of higher abelian gauge theory in this context, via differential cohomology, are given in [15,16,17,36].…”

Section: Introductionmentioning

confidence: 99%

“…This allows the partition function to be defined as a section of a line bundle over the intermediate Jacobian, and requires a quadratic refinement [39]. Discussions on extension to (higher) differential cohomology and stacks are given in [15,16,17]. The formulation that we propose via noncommutative geometry does not suffer from such an immediate problem, because ultimately H 2p+1 ∧ θ H 2p+1 = 0.…”

mentioning

confidence: 99%

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

Mathai

Journal of Geometry and Physics

2015

Self Cite

We enhance the action of higher abelian gauge theory associated to a gerbe on an M5-brane with an action of a torus T n (n ≥ 2), by a noncommutative T n -deformation of the M5-brane. The ingredients of the noncommutative action and equations of motion include the deformed Hodge duality, deformed wedge product, and the noncommutative integral over the noncommutative space obtained by strict deformation quantization. As an application we then introduce a variant model with an enhanced action in which we show that the corresponding partition function is a modular form, which is a purely noncommutative geometry phenomenon since the usual theory only has a Z 2 -symmetry. In particular, S-duality in this 6-dimensional higher abelian gauge theory model is shown to be, in this sense, on par with the usual 4-dimensional case. *