Equivariant Cohomotopy and Branes Towards microscopic M … · 1.Super homotopy theory and the Atom...

Urs Schreiber

(New York University, Abu Dhabi & Czech Academy of Science, Prague)

Equivariant Cohomotopy and Branes–

Towards microscopic M-Theory

talk atString and M-Theory: The New Geometry of the 21st Century

Singapore 2018

joint work with

Hisham Sati & V. Braunack-Mayer

based on [FSS13, FSS15, FSS16a, FSS16b, HS17, HSS18, BSS18]

ncatlab.org/schreiber/print/Equivariant+Cohomotopy+and+Branes

Motivation.Nonperturbative QFT and an old Prophecy

Part I.Some M-Theory from Super homotopy theory

Part II.Some corners of M-theory

Motivation

back to Contents

Glaring open problem of contemporary quantum field theory:All non-perturbative physics.

Such as:-quark confinement in hadrons (existence of ordinary matter!)

-quark-gluon plasma & nucleosynthesis (becoming of ordinary matter!)

-Higgs field metastability (existence of vacuum spacetime!)

-QCD cosmology (becoming of vacuum spacetime!)

Important non-answers:

lattice QFT numerics is (great but) not the answer :like Bohr-Sommerfeld’s “old quantization”it allows to compute some numbersbut without conceptual understanding

string theory is (great but) not the answer :string scattering series just as perturbative as Feynman series(vanishing radius of convergence, both)

But string theory is the vehicle with which to glimpse M-Branes...

The emerging answer: Intersecting M-branesWeb of plausibility arguments and consistency checks suggests:

Non-perturbative standard model of particle physics & cosmologyarises on intersecting M-branesat asymptotic boundary of approximately AdS spacetime.

In particularWitten-Sakai-Sugimoto model for QCD:

Nc M5-branes intersecting Nf M9-branesKK-compactified, breaking all supersymmetry, toNc D4-branes intersecting Nf D8-branes

yields QFT at least close to non-perturbative QCDwith transparent interpretation of non-perturbative effects

graphics from Sugimoto 16

However...

Glaring open problem of contemporary M-brane theory:What is it, really?

We still have no fundamental formulation of “M-theory” -

Work on formulating the fundamental principles underlying M-theoryhas noticeably waned. [. . . ]. If history is a good guide, then we shouldexpect that anything as profound and far-reaching as a fully satisfac-tory formulation of M-theory is surely going to lead to new and novelmathematics. Regrettably, it is a problem the community seems tohave put aside - temporarily. But, ultimately,Physical Mathematics must return to this grand issue.

G. Moore, Physical Mathematics and the Future, at Strings 2014

Important non-answer:

BFSS/IKKT matrix model is (great but) not the answer :like lattice QFT numericsit allows to compute some numbersbut without conceptual understanding

What is missing?..

An old prophecy

Back in the ’70s, the Italian physicist, D, Amati reportedly said thatstring theory was part of 21st-century physics that fell by chanceinto the 20th century. I think it was a very wise remark. How wiseit was is so clear from the fact that 30 years later we’re still tryingto understand what string theory really is.

E. Witten, Nova Interview 2003

New development brought by the 21st century:Homotopy theory & higher topos theory (“higher structures”)

physics mathematics

gauge principle homotopy theory

& Pauli exclusion super-geometry

= super homotopy theory

Part I.

Some M-theoryfrom

Super homotopy theory

1. Super homotopy theory and the Atom of Superspace

RationalSuper homotopy theory and the fundamental super p-Branes

Global equivariantSuper homotopy theory and the C-field at singularities

4. Super Cartan geometry and 11d orbifold supergravity

back to Contents

Super homotopy theory

and the Atom of Superspace

back to Part I

Global equivariant Super homotopy theory

Definition. Consider the 2-site

SuperSingularities :=

Rd|N︸︷︷︸super-space

× D︸︷︷︸infinitesimal

× BG︸︷︷︸orbifold

singularity

Global equivariantsuper homotopy theoryis the ∞-stack ∞-toposover SuperSingularities:

H := Sh∞︸︷︷︸generalized

spacesprobe-ableby these

local model spaces

(SuperSingularities

)[Sch13]

& [Rezk14]

[BMSS19]

space-time

SuperManifolds,SuperOrbifoldswith G-structure

(e.g.super-Riemannian-, super-conformal-, ...Spin-, String-, Fivebrane-, ... ([SSS09])structure)

//H oo ? _

higher gauge fields classifying spaces forequivariant, differential

generalized cohomology theories

The Modalities of Super homotopy theory

The terminal functor factors into a system of dualities = adjunctions.

supergeometric∞-groupoids︷︸︸︷

HΓ //

nothing

H oo ∆ ? _

Γ //∗

∅ a ∗

pure being

H oo ? _

oo ∇ ? _

∅ a ∗

discrete

H oo ∆ ? _

oo ? _

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

oo ? _

a ]∨

∅ a ∗

continuous

H oo ? _

oo ∇ ? _

H[oo ? _

oo ? _

S a [∨

a ]∨

∅ a ∗

shaped

HΠ //

oo ∆ ? _

oo ? _

H[oo ? _

oo ? _

S a [∨

a ]∨

∅ a ∗

infinitesimally shaped

HΠ //

oo ∆ ? _

oo ? _

reduced differential-geometric∞-groupoids︷︸︸︷

oo ? _

H[oo ? _

oo ? _

a &∨

S a [∨

a ]∨

∅ a ∗

infinitesimally discrete

oo ∆ ? _

oo ? _

reduced differential-geometric∞-groupoids︷︸︸︷

oo ? _

H[oo ? _

oo ? _

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

reduced

supergeometric︷︸︸︷H

oo ι ? _

oo ? _

reduced differential-geometric︷︸︸︷H<

oo ? _

H[oo ? _

oo ? _

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

bosonic

oo ι ? _

oo ? _

bosonic︷︸︸︷H

oo ? _

reduced︷︸︸︷H<

oo ? _

H[oo ? _

oo ? _

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

rheonomic

oo ? _

oo ∆ ? _

oo ? _

H[oo ? _

oo ? _

⇒ a a

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

super-geometric︷︸︸︷

oo ι ? _

oo ? _

H[oo ? _

oo ? _

⇒ a a

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

super-geometric

oo ? _

H[oo ? _

oo ? _

⇒ a a

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

A1-local

The central modalities are motivic A1-localizations.

oo ? _

Πsup //

oo ∆sup ? _

oo ? _

Πinf//

oo ∆inf? _

oo ∆ ? _

H[oo ? _

oo ? _

⇒ a a

< a =∨

a &∨

R1 a [∨

a ]∨

∅ a ∗

continuum-local

oo ? _

Πsup //

oo ∆sup ? _

oo ? _

Πinf//

oo ∆inf? _

oo ∆ ? _

H[oo ? _

oo ? _

⇒ a a

a &∨

R1 a [∨

a ]∨

∅ a ∗

infinitum-local

oo ? _

Πsup //

oo ∆sup ? _

oo ? _

Πinf//

oo ∆inf? _

oo ? _

H[oo ? _

oo ? _

⇒ a a

a &∨

R1 a [∨

a ]∨

∅ a ∗

superpoint-local

oo ? _

Πsup //

oo ∆sup ? _

oo ? _

H[oo ? _

oo ? _

⇒ a a

a &∨

R1 a [∨

a ]∨

∗⇒ emergence of Atom of Superspace from \nothing

⇒ a a

a &∨

R1 a [∨

a ]∨

∗⇒ emergence of Atom of Superspace from \nothing

now apply the microscope of homotopy theoryto discover what emerges, in turn, out of the superpoint...

RationalSuper homotopy theory

and the fundamental super p-Branes

back to Part I

Higher super Lie theory and Rational homotopy

infinitesimalrational

approximation of super-homotopy by

higher Lie integrationSullivan construction

Definition. FDAs︸︷︷︸terminologycommon in

supergravity([vanNieuwenhuizen82])

:= dgcSuperAlgR,cn︸︷︷︸∞-category of

differentialgraded-commutative

superalgebras

oo CE'

(SuperL∞Algbdnil

R,cn︸︷︷︸∞-category of

nilpotentsuper L∞-algebroids

Theorem. ([BM18], see Sch18)

SuperL∞AlgbdnilR,cn '︸︷︷︸higher super Lie integration

rational super-∞-groupoids︷︸︸︷FDAsop

Sullivan rational homotopy theory︷︸︸︷oo O

Spec//

H oo Disc ? _

R0|1︸︷︷︸D = 0, N = 1

supersymmetrysuper Lie algebra

// R0|1︸︷︷︸superpoint

Universal central invariant super-L∞extensions of R0|1: Brane bouquet

the Atom of Superspace

Type I

R0|1+1 oooo R0|1

Type IIB Type I

[HS17]

R2,1|2

vvR0|1+1 oo

oo R0|1

Type IIB Type I

universal central extension: 3d super-Minkowski spacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

Type IIB Type I

R3,1|4

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

Type IIB Type I

universal invariant central extension: 4d super-Minkowski spacetime

R3,1|4+4 oooo R3,1|4

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

Type IIB Type I

R5,1|8

xxR3,1|4+4 oo

oo R3,1|4

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

Type IIB Type I

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

R9,1|16

((R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

R10,1|32

R9,1|16+16 oo

oo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

R10,1|32

R9,1|16+16 oo

oo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

R10,1|32

stringIIB

(([FSS13] stringI

stringIIA

vvR9,1|16+16 oo

oo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

universal higher central invariant extension: stringy extended super-spacetimes

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

d7brane

R10,1|32

d6brane

xxd9brane // stringIIB

(([FSS13] stringI

stringIIA

vvd8braneoo

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

d7brane

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

m5brane

[FSS15]

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

d7brane

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

m5brane

[FSS15]

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

d7brane

branebouquet

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

m5brane

[FSS15]

rr M/IIADuality

[FSS16a]

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

d7brane

branebouquet

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

m5brane

[FSS15]

rr M/IIADuality

[FSS16a]

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

Gaugeenhancement

d7brane

branebouquet

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo [BSS18]

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

m5brane

[FSS15]

qq M/IIADuality

[FSS16a]

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

Gaugeenhancement

d7brane

branebouquet

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo [BSS18]

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

Type IIBdd

T-Duality

::Type I Type IIA [FSS16b]

m5brane

[FSS15]

[FSS18] qq M/IIADuality

[FSS16a]

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

Gaugeenhancement

d7brane

branebouquet

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo [BSS18]

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|32

ooR0|1+1 oo

oo R0|1

Exceptional Type IIBdd

T-Duality

m5brane

[FSS15]

[FSS18] qq M/IIADuality

[FSS16a]

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

Gaugeenhancement

R10,1|32exc

d7brane

branebouquet

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo [BSS18]

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|32

ooR0|1+1 oo

oo R0|1

T-Duality

m5brane

[FSS15]

[FSS18] R10,1|32exc,s

Higher T-duality

qq M/IIA

Duality

[FSS16a]

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

Gaugeenhancement

R10,1|32exc

d7brane

branebouquet

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo [BSS18]

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|32

ooR0|1+1 oo

oo R0|1

T-Duality

m5brane

[FSS15]

[FSS18] R10,1|32exc,s

Higher T-duality

qq M/IIA

Duality

[FSS16a]

m2brane

d5brane

d3brane

d1brane

d0brane

d2brane

d4brane

Gaugeenhancement

R10,1|32exc

d7brane

branebouquet

R10,1|32

d6brane

(([FSS13] stringI

stringIIA

vvd8braneoo [BSS18]

R9,1|16+16 oooo R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

emergentspacetime

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|32

ooR0|1+1 oo

oo R0|1

T-Duality

m5brane

[FSS15]

m2brane

R10,1|32

[FSS13]

R9,1|16

((//// R9,1|16+16

R5,1|8

R5,1|8+8////

R3,1|4+4 oooo R3,1|4

[HS17]

R2,1|2+2 oooo R2,1|2

vvR0|1+1 oo

oo R0|1

emergence of fundamental M-branes from the Atom of Superspace

m5brane

[FSS15]

m2brane

R10,1|32

zoom in on the fundamental M-brane super-extensions

The fundamental M2/M5-brane cocycle

m5brane

[FSS15]

m2brane

hofib(µM2)

R10,1|32

µM2 &&

µM2 = dLWZWM2

(ψΓa1a2ψ

)∧ ea1 ∧ ea2

the WZW-curvature of the Green-Schwarz-type sigma-model super-membrane

m5brane

hofib(µM5)

[FSS15]

m2brane

hofib(µM2)

µM5 // S7

R10,1|32

µM2 &&

µM5 = dLWZWM5

(ψΓa1···a5ψ

)∧ ea1 ∧ · · · ∧ ea5 + c3 ∧ i

(ψΓa1a2ψ

)∧ ea1 ∧ ea2

the WZW-curvature of the Green-Schwarz-type sigma-model super-fivebrane

m5brane

hofib(µM5)

[FSS15]

m2brane

hofib(µM2)

µM5 // S7

hofib(c2)

R10,1|32

µM2 &&

c2xxB4R

the quaternionic Hopf fibration (in rational homotopy theory)

m5brane

hofib(µM5)

[FSS15]

m2brane

hofib(µM2)

µM5 // S7

hofib(c2)

R10,1|32

µM2/M5 //

µM2 &&

c2xxB4R

the unified M2/M5-cocycle

[FSS15]

R10,1|32µM2/M5 // S4

the unified M2/M5-cocycle is in rational Cohomotopy in degree 4

[FSS15]

R10,1|32µM2/M5 // S4

(ψΓa1a2ψ

)∧ ea1 ∧ ea2 G4

(ψΓa1···a5ψ

)∧ ea1 · · · ea5 G7

Sullivan model: O(S4R)' R[G4, G7]/

(dG4=0

dG7=− 12G4∧G4

= 11d supergravity equations of motion of the C-field ([Sati13, Sect. 2.5])

[FSS15]

R10,1|32µM2/M5 // S4

the unified M2/M5-cocycle

[FSS15]

R10,1|32µM2/M5 //

double dimensional reduction & gauge enhancement

R9,1|16+16µF1/D2p

// kuB2R

D-brane charge in twisted K-theory, rationally

[BSS18]

The rational conclusion.

infinitesimal

rational

approximation

brane charge quantization follows from first principles

and reveals this situation:

brane speciescohomology theory

of charge quantization

D-branes twisted K-theoryM-branes Cohomotopy in degree 4

The rational conclusion.

infinitesimal

rational

approximation

brane charge quantization follows from first principles

and reveals this situation:

brane speciescohomology theory

of charge quantization

D-branes twisted K-theoryM-branes Cohomotopy in degree 4

Lift beyond

infinitesimal

rational

approximation is not unique

but one lift of rational Cohomotopy is minimal (in number of cells):

actual Cohomotopy represented by the actual 4-sphere

rationalization

cocycle inactual Cohomotopy

cocycle inrational cohomotopy

Towards microscopic M-theory1. Construct

differential equivariant Cohomotopy S4

of 11d super-orbifold spacetimes X2. lifting super-tangent-space-wise

the fundamental M2/M5-brane cocycle.

3. Compare the resulting observables onM-brane charge quantized supergravity field moduliwith expected limiting corners of M-theory

Global equivariantSuper homotopy theory

and the C-field at singularities

back to Part I

orbifolded

Global equivariantSuper homotopy theory

and the C-field at singularities

back to Part I

The modalities of global equivariant homotopy theory

< a ⊂ a ≺

singular smoothorbi-

singular

possibly singular/orbifoldedsuper-

geometric︷︸︸︷H

oo ? _

H[oo ? _

oo ? _

super-geometric︷︸︸︷H ⊂

oo ? _

bosonic︷︸︸︷H ⊂

oo ? _

reduced︷︸︸︷H ⊂

oo ? _

smooth = non-singular

< a ⊂ a ≺

singular

oo ? _

H[oo ? _

oo ? _

< a ⊂ a ≺

singular

oo ? _

H[oo ? _

oo ? _

< a ⊂ a ≺

singular

oo ? _

H[oo ? _

oo ? _

Super-Orbifolds – Abstract definition

Let V︸︷︷︸tangentspacemodel

, G︸︷︷︸generic

singularitytype

∈ Grp(H) be group objects.

Definition. A G-orbi V -fold is

an object X ∈ H/BG ≺

which is

1. 0-truncated: τ0(X ) ' X2. orbi-singular: ≺(X ) ' X

3. a V -fold: there exists a V -atlasU

V X ⊂

(a) pX is a covering: (τ−1)/X (pX) ' ∗(b) pX is a local diffeomorphism: =/X (pX) ' pX(c) pV is a local diffeomorphism: =/V (pV ) ' pV

The global equivariant 4-sphere

In the following G := Pin(5)[

the unoriented spin group in 5d, regarded as geometrically discrete.

This unifiesADE-singularitieswithO-plane singularities

[HSS18]︷︸︸︷ADE-singularity

[MFFGME09, MFF10]︷︸︸︷(GADE ×Z G′ADE)×

O-plane

︷︸︸︷Z2

Spin(4) × O(1) //Pin(5)

Write S4 ∈ SmoothManifolds→ H

for the smooth 4-sphere.

with S4 := S(S4)∈ ∞Groupoids its shape.

≺ :=(S4 [Pin(5)

≺ ∈ H/BPin(5)[

≺is a Pin(5)[-orbi R4-fold

≺ := S(S4 Pin(5)[

is its shape orbi-space

Equivariant Cohomotpy of Super-orbifolds

R10,1|32 ∈ Grp(H) D = 1, N = 1 translational supersymmetry

X ∈ H/BPin(5)[

a Pin(5)[-orbi R10,1|32-fold.

Definition.The cocycle space of equivariant Cohomotopy of X is

H/BPin(5)[≺

(X , S4

cocycle inequivariant Cohomotopy

BPin(5)[≺

and so the cohomology set is

H(X , S4

):= π0H/BPin(5)[

(X , S4

Differential Equivariant Cohomotopy of Super-Orbifolds

Definition. Ωflat

(−, lS4

)∈ H/B[Pin(5)

is the universal moduli space of

Rd|N × D×

B[Pin(5)

flat L∞-algebra valuedsuper-differential forms︷︸︸︷

Ωflat

(Rd|N × D, l

(S4)K︸︷︷︸

fixed point sphere︸︷︷︸L∞-algebra dual to itsminimal Sullivan model

Claim: S(Ωflat

(−, lS4

≺))'(S4

Definition. The differential equivariant 4-sphere is

≺ := S4

≺ ×(S4

Ωflat

(−, lS4

)Hence differential equivariant Cohomotopy in degree 4 is

H(X , S4

):= π0H/BPin(5)[

(X , S4

M-brane charge quantized C-field

C-fieldelectromagnetic field

(“A-field”)

flux forms

X (G4,G7)//Ωflat

(−, lS4

)Faraday tensor

XF2 //Ωflat

(− lBU(1)

)︸︷︷︸=Ω2

clsd(−)

M-brane charge quantizedC-field

(G4,G7)//Ωflat

(−, lS4

Dirac charge quantizedelectromagnetic field

BU(1)conn

//Ω2clsd

Super Cartan geometry

and 11d orbifold supergravity

back to Part I

Cartan geometry formalizes Einstein principle of equivalence

Spacetime is locally equivalent to Minkowski spacetime,namely in the infinitesimal neighbourhood of every point

We now generalize thisfrom manifolds to super-orbifolds...

G-Structures on orbi V -folds ([Wellen17, Sch13] )

Def.: infinitesimal disk around origin: DV := V ×=(V )e → V

Prop.:every orbi V -fold X carriesits canonical V -frame bundle

frame−→ BAut(DV)

Def.:for G

homom.−→ Aut(DV)

a G-structure is a lift(E is the vielbein)

frame &&

BAut(DV)Erz

Prop.:V itself carriescanonical G-structuregiven by left translation

frame &&

BAut(DV)Eli

Def.:a G-structure is torsion-free and flatif it coincides with this canonical oneon each infinitesimal disk

E|DVx ' (Eli)|DVe

11d Supergravity from Super homotopy theory

Consider now V = R10,1|32 and X an orbi R10,1|32-fold.

Claim:

G := Aut Grp

(R10,1|32

)' Spin(10, 1)

G-structure on X ' super-vielbein on X' metric/field of gravity

G-structure is torsion-free: ⇔ super-torsion on X vanishes

⇔[CaLe93][How97]

X is solution to 11d supergravitywith vanishing bosonic flux

G-structure is flat: ⇔ X is a “flat” super-orbifoldsolution to 11d supergravity

11d Supergravity from Super homotopy theory

Consider now V = R10,1|32 and X an orbi R10,1|32-fold.

Claim:

G := Aut Grp

(R10,1|32

)' Spin(10, 1)

G-structure on X ' super-vielbein on X' metric/field of gravity

G-structure is torsion-free: ⇔ super-torsion on X vanishes

⇔[CaLe93][How97]

X is solution to 11d supergravitywith vanishing bosonic flux

G-structure is flat: ⇔ X is a “flat” super-orbifoldsolution to 11d supergravity

⇒ all

curvature& G4-flux

hence all

higher curvature corrections

& flux quantization

crammed into orbifold singularitiesand thus taken care of by the equivarianceof charge quantization in differential equivariant Cohomotopy

Flat & fluxless except at curvature- & flux- singularities

Plausibility check:

Black M2/M5-brane solutions to SuGra interpolate ([AFFHS98]) between:

Planck-scale curved throatnear/far horizon geometry←−−−−−−−−−−−−−−→ orbifold-singularity

in flat & fluxless spaceblack M2

`P1 ,,`P1qq

AdS4 × S7/GA︸︷︷︸spherical space form

R2,1 × R8 GA︸︷︷︸orbifold du Val singularity

inconsistent :Planck-scale throat (`P 1)spurious in SuGra (`P 1)(evaded only bymacroscopic N 1)

consistent :all Planck-scale geometrycrammed into orbi-singularity(necessary formicroscopic N = 1)

Hence, indeed, a consistent & complete picture:

1. is flat & fluxless away from singularities,

2. has hidden degrees of freedom inside the singularities.

Flat & fluxless except at curvature- & flux- singularities

Plausibility check:

Black M2/M5-brane solutions to SuGra interpolate ([AFFHS98]) between:

Planck-scale curved throatnear/far horizon geometry←−−−−−−−−−−−−−−→ orbifold-singularity

in flat & fluxless spaceblack M5

`P1 ,,`P1qq

AdS7 × S4 G︸︷︷︸[MFF10]

R6,1 × R4 G︸︷︷︸orbifold du Val singularity

inconsistent :Planck-scale throat (`P 1)spurious in SuGra (`P 1)(evaded only bymacroscopic N 1)

consistent :all Planck-scale geometrycrammed into orbi-singularity(necessary formicroscopic N = 1)

Hence, indeed, a consistent & complete picture:

1. is flat & fluxless away from singularities,

2. has hidden degrees of freedom inside the singularities.

In conclusionthe following enhancement of 11d supergravitynaturally emerges out of super homotopy theory

The full covariant phase space is...

CovariantPhaseSpace :=

⊔[X ] ∈

Pin(5)[Orbi−R10,1|32Folds(H)∼

CovariantPhaseSpaceX

H(X ⊂,BSpin(10, 1)

/BAut(D10,1|32)

H(X ≺, S

)/BPin(5)[

H(X ⊂,Ω

(−, lS4

Observables := HR ≺

(CovariantPhaseSpace

)= HR ≺

(Σ∞

Pin(5)[CovariantPhaseSpace

... for each class of super-orbifolds X ...

⊔[X ] ∈

H(X ⊂,BSpin(10, 1)

/BAut(D10,1|32)

H(X ≺, S

)/BPin(5)[

H(X ⊂,Ω

(−, lS4

)= HR ≺

(Σ∞

... a super-torsion-free Spin-structureencoding the fields of supergravity...

⊔[X ] ∈

H(X ⊂,BSpin(10, 1)

/BAut(D10,1|32)

H(X ≺, S

)/BPin(5)[

H(X ⊂,Ω

(−, lS4

)= HR ≺

(Σ∞

... equipped with a compatible lift of the flux formsto a cocycle in differential equivariant Cohomotopy (charge quantization).

⊔[X ] ∈

H(X ⊂,BSpin(10, 1)

/BAut(D10,1|32)

H(X ≺, S

)/BPin(5)[

H(X ⊂,Ω

(−, lS4

)= HR ≺

(Σ∞

Part II.

Some corners of M-theory

Part II.

11-dimensional supergravityon super-orbifold spacetimes

at low energy (`P 1)

black N = (2, 0)-M5-branesinside MK6-singularities

([Gra18, BMSS19], )

brane/anti-branepair-creation/annihilation(tachyon condensation)(Part II, Stabilization)

Green-Schwarz actionfor M-branes

at ADE-singularities[HSS18]

M-charge quantizationin differential

equivariant Cohomotopywith vanishing super-torsion

(Part I)

super-rationallocal sector

equivariantsuper-rational

local sector

sector

stablesector

stabletopological

sector ''

stableequivarianttopological

sector

equivarianttopological sector

topological sector

fractional D-brane chargeat intersecting

MK6-singularities(Part II, Comparison A)

RR-charge in twisted K-theoryupon double dimensional reduction

[BSS18]

thermal QFT n-point functionsupon KK-reduction to 4d

([BMSS19])

Fractional M5-brane charge(M-theoretic discrete torsion)in black M2-brane spacetimes

(Part II, Comparison B)

back to Contents

Part II.

([Gra18, BMSS19], )

(Part I)

local sector

sector

stablesector

stabletopological

sector ''

sector

topological sector

[BSS18]

([BMSS19])

back to Contents

Part II.

([Gra18, BMSS19], )

(Part I)

local sector

sector

stablesector

stabletopological

sector ''

sector

topological sector

[BSS18]

([BMSS19])

back to Contents

Part II.

([Gra18, BMSS19], )

(Part I)

local sector

sector

stablesector

stabletopological

sector ''

sector

topological sector

[BSS18]

([BMSS19])

back to Contents

Part II.

([Gra18, BMSS19], )

(Part I)

local sector

sector

stablesector

stabletopological

sector ''

sector

topological sector

[BSS18]

([BMSS19])

back to Contents

Part II.

([Gra18, BMSS19], )

(Part I)

local sector

sector

stablesector

stabletopological

sector ''

sector

topological sector

[BSS18]

([BMSS19])

back to Contents

Part II.

([Gra18, BMSS19], )

(Part I)

local sector

sector

stablesector

stabletopological

sector ''

sector

topological sector

[BSS18]

([BMSS19])

back to Contents

Part II.

([Gra18, BMSS19], )

(Part I)

local sector

sector

stablesector

stabletopological

sector ''

sector

topological sector

[BSS18]

([BMSS19])

back to Contents

Stable homotopy theory

and anti-branes

back to Part II

Stable Cohomotopy from observing Cohomotopy

Any space of observables is linear :Observables may be added and subtracted.

Linear + Homotopy theory = Stable homotopy theory

)︸︷︷︸

homotopy-linearspace of observables

(Σ∞G CovariantPhaseSpace︸︷︷︸free homotopy-linearized

covariant phase space

There is canonical comparison map

Σ∞H/BPin(5)[

(X , S4

≺)︸︷︷︸Cohomotopy

homotopy-linearapproximation

//Stab(H)Pin[ ≺

(Σ∞GX ,

4-shiftedequivariant

sphere spectrum︷︸︸︷S4

)︸︷︷︸

stable Cohomotopy

to fields in stable equivariant Cohomotopy

We now discuss what this means...

Brane charge – 1st order approximation

Let (C,+) bean abelian semigroupa “commutative monoid”

of charges.

charge ∈ (C,+) c1 c2 c1 + c2

singular locus −−−−

︸︷︷︸branes

︸︷︷︸coincident

branes

Fundamental example: the natural numbers

(C,+) = (N,+)

charge c = number of coincident branes ⇔ each brane carries unit charge

Brane charge – 2nd order approximation

Including anti-brane charges, hence negative brane charges,means to pass to the abelian group completion of the charge monoid:

K(C,+) :=

(c+, c−) | c± ∈ C/(

(c, c) ∼ 0)

charge ∈ K(C,+) (c1, 0) (c2, 0) (0, c2) (0, c1) (c1 + c2, c1 + c2) ∼ 0

︸︷︷︸branes

︸︷︷︸anti-branes

︸︷︷︸coincident

branes

Fundamental example: the integers: K(N,+) = (Z,+)

(c+, c−) =number of coincident branesminus number of anti-branes

⇔each brane carries unit chargeeach anti-brane carries

negative unit charge

Brane charge – 3rd order approximation

The categorification of commutative monoid(C,+)

is symmetric monoidal category(C,⊕)

(C,+) = π0 (C,⊕)

Fundamental non-linear example

Finite pointed sets with disjoint unition (C,⊕) =(

Set∗/fin,t

)this categorifies the previous example: π0

(Set∗/fin,t

)' (N,+)

Fundamental linear example for F a field

finite-dim vector spaces with direct sum (C,⊕) =(FVectfin,⊕

)this also categorifies the previous example: π0

(FVectfin,⊕

)' (N,+)

Unified on a deeper level:pointed sets may be regarded as the vector spacesover the “absolute ground-field with one element” F1(

Set∗/fin,t

)'(F1Vectfin,⊕

)Spec(F)

algebraic base

Spec(Z)

Spec(F1)combinatorialabsolute base

Brane charge in generalized cohomology

Brane/anti-brane annihilation may be varying over spacetime X

enhance discrete abelian group of charges to a space of charges

The homotopification of abelian group(A,+)

is ∞-loop space / spectrumA

(A,+) = π0 (A)

brane chargelocally constant locally varyingX −→ (A,+)︸︷︷︸

discreteabelian group

X −→ A︸︷︷︸∞-loop spaceor spectrum

Hence brane charge group on spacetime X is generalized cohomology group:

A(X) := π0Maps ( X , A )

Example: D-brane/anti-D-brane bound statesopen string tachyon condensation profile:

X −→ KU

K-theory spectrum

(conjecturally, or similar)

Algebraic K-Theory – locally varying brane/anti-brane annihilation

in conclusion:

brane/anti-brane annihilation

abelian group completionof charge monoid

locally varying brane charge spectral enhancement

of charge group

combine:

Thecategorification /homotopification

of abelian group completion

(C,⊕

)) is algebraic K-theory spectrum

K(C,⊕

):= ΩB⊕C

(C,⊕

))= π0

(K(C,⊕

Algebraic K-Theory – Examples

algebraic K-theory spectrum of a field F

KF ' K(FVectfin

) complex algebraic K-theory KC

is flat K-theory:differentialK-theory︷︸︸︷

kuproject out

topological class

**KC︸︷︷︸

algebraicK-theory

includeflat cocycles 33

absolute algebraic K-theory KF1 := K(F1Vectfin

(Set∗/fin

)is stable Cohomotopy theory (Barrat-Priddy-Quillen theorem):

KF1︸︷︷︸absolute algebraicK-theory spectrum

' S︸︷︷︸sphere

spectrum

Brane charge on Orbifolds – Equivariant generalized cohomology

A representation sphere SV

:= one-point compactification

of linear representation V

A G-equivariant spectrum A isa spectrum of G-spaces indexed by representation spheres, hence

1. a system of pointed G-spaces

a linear G-representation

2. with equivariant suspension morphisms SV ∧ AW

σV,W−→ AV⊕W

Examples

• The equivariant suspension spectrum of a G-space SV

Σ∞GX : V 7→ SV ∧X .

• The equivariant K-theory of a contractible space is the representation ring

KUG( Rd,1

) ' KUG( ∗

) ' RC(G) ' Z

[ irreps︷︸︸︷ρ1, · · · , ρn

]“fractional D-branes”

In conclusion, from Part I:A compelling candidate for M-brane charge cohomology theory is...

by discussionin Part I

non-linearbrane/anti-brane

bound states

fundamental& black

M-branesby [HSS18]

absoluteK-theory

ADE-equivariantCohomotopy

stableCohomotopy

equivariantize

stabilize

in RO-degreeH

quaternions

SHGADE' Σ

GADESH

a ADE-equivariantstable

Cohomotopyin RO-degree H

Hypothesis H:

Thegeneralized cohomology theory

forM-brane charge

ADE-equivariantstable

Cohomotopyin RO-degree H

Hypothesis H predicts M-brane charge groups: SHGADE

︸︷︷︸11d spacetime

orbifold

How does this compare to / clarify folklore of perturbative string theory:

intersecting MK6-branes fractional D-branes ?

M-theoretic “discrete torsion” of fractional M5-branes ?

GUT at E-type singularities ?

· · ·

This we discuss now −→

Strategy for testing Hypothesis H

1. Identify suitable comparison homomorphism

putativeM-brane charge

inequivariant stable Cohomotopy

“β α”

putativeperturbative

approximation

putativeD-brane charge

inequivariant K-theory

2. Compute:

the co-kernel of β α; reflectsD-brane configurations

that do not liftto M-theory

the kernel of β α; reflectsM-brane degrees of freedom

invisible inperturbative string theory

Hypothesis H finds support if the cokernel of β α is

1. small⇔ putative M-brane charge mostly reproduces string theory folklore,

2. plausible ⇔ the putative D-brane states in the co-kernel are dubious.

If so, Hypothsis H predicts the kernel of β α as hidden M-theoretic DOFs.

Outline

Since the sphere spectrum Sis the initial commutative ring spectrum,

there is a unique multiplicative comparison morphismfrom stable cohomotopy

to every other multiplicative cohomology theory A,called the

equivariant generalized Boardman homomorphism

SαG(X) G //Aα

Here we present two cases:

1. Comparison map A to

K-theory and RR-charge of fractional D-branes

2. Comparison map B to

ordinary cohomology and “discrete torsion” of fractional M5-branes

Comparison A to

K-theoryand

fractional RR-charge of D-branes

back to Part II

Finite subgroups GADE ⊂ SU(2) – Classification

DynkinLabel

Finitesubgroupof SU(2)

Name ofgroup

An≥1 Zn+1 CyclicDn≥4 2D2(n−2) Binary dihedralE6 2T Binary tetrahedralE7 2O Binary octahedralE8 2I Binary icosahedral

Assumption: In the following, consider finite groupsG = GDE ⊂ E ⊂ SU(2)in the D- or E-seriesandin the exceptional subgroup lattice. next slide //

Implies in particular:G-orbi-folds are G-orienti-folds,the relevant K-theory for fractional D-brane chargeat G-fixed points is is KO-theory

Finite subgroups GADE ⊂ SU(2) – Exceptional subgroup lattice

The comparison homomorphism A

charge groupof M-branes

at two intersectingMK6 ADE-singularities

visible inM-theory

charge latticeof fractional M-branesat an MK6-singularity

visible inM-theory

charge latticeof fractional D-branes

at orientifold singularity

expected inperturbativestring theory

equivariant stable cohomotopy in RO(G×G′)-degree H equivariant KO-theoryin degree 0

SH(G×G′)(R10,1)

zoom in onto one MK6// S(HG′)

perturbative approximation//KO0

G(R6,1)

SH(G×G′)(∗)α

form geometric G′-fixed spectrum

// S0G(∗)Boardman homomorphism

//' KO0

G(∗)'

A (G)β

linearize G-actions

//RO (G)

module over(G×G′)-Burnside ring

Thm. 1 G-Burnside ring Thm. 2 G-Representation ring

Theorem 1

visisible inM-theory

Proof.Use Prop. II 9.13 in[LewisMaySteinberger86].2

equivariant stable cohomotopy in RO(G×G′)-degree H

SH(G×G′)(R10,1)

SH(G×G′)(∗)α

// S0G(∗)'

Is surjective.

G-Burnside ring hence: coker(β α) ' coker(β)

The comparison homomorphism A

equivariant stable cohomotopy in RO(G×G′)-degree H equivariant KO-theoryin degree 0

SH(G×G′)(R10,1)

G(R6,1)

SH(G×G′)(∗)α

// S0G(∗)Bordman homomorphism

//' KO0

G(∗)'

A (G)β

linearize G-actions

//RO (G)

Thm. 1 G-Burnside ring Thm. 2 G-Representation ring

Theorem 2

irrationalcharacters︷︸︸︷

ROirrational(G)'

RO(G) / ROint(G)︸︷︷︸integral

characters

equivariant stable cohomotopyin degree 0

equivariant KO-theoryin degree 0

S(HG′)

G(R6,1)

S0G(∗)Bordman homomorphism

//' KO0

G(∗)'

A (G)β

linearize G-actions

//RO (G)

G-Burnside ringcoker(β) '

ROirrational(G)

G-Representation ring

Theorem 2 – Ingredients

G-actionson finite sets(not-linear)

(GSetfin,t

linearizeF[−]

//(GRepF,⊕

G-representationson F-vector spaces

(linear)

Burnside ring ofvirtual G-sets

A(G)β

//RF(G)Representation ring

of virtual representations

G-set S

//F[S]

G “permutation

representation”

the cokernel of β reflectslinear algebra

invisible topure combinatorics

the kernel of β reflectspure combinatorics

invisible tolinear algebras

Theorem 2 – Proof

Compute:

1. set of conjugacy classes

of subgroups H ⊂ G

2. the Burnside product [G/Hi]× [G/Hj] =⊔

n`ij︸︷︷︸structureconstants

· [G/H`]

3. its matrix of total multiplicities multij :=∑n`ij

4. its integral row reduction H︸︷︷︸upper

triangular

:= U︸︷︷︸∈ GL(N,Z)

·mult

Lemma. The rows of H span im(β) ⊂ RF(G).

This yields an effective algorithm computing coker(β) = RF(G)/im(β)

Simon Burton has implemented this algorithm in Python.

⇒ Proof of Theorem 2: By brute force automatized computation. 2

Theorem 2 – Proof

Theorem 2 – Physics interpretation

Let V ∈ KG(∗) ' R(G) a fractional D-brane ↔ G-representation,

RR-charge in the g-twisted closed string sectoris the value of its character at g:

(g) =1

|G|χV

([DouglasGreeneMorrisson97, (3.8)], [DiGo00, (2.4)], [BCR00, (4.65) with (4.41)], [EGJ05, (4.5)], [ReSc13, 4.102])

Theorem 2 – Physics reformulation:Hypothesis H impliesthat fractional D-branes with irrational RR-charge are spurious.

Physically plausible?

Some V ∈ KG(∗) must be spurious [BDHKMMS02, 4.5.2].

Irrational RR-charge called a paradox in [BachasDouglasSchweigert00, (2.8)],also [Taylor00, Zho01, Rajan02], apparently unresolved.

If this is indeed a paradox,then hypothesis H exactly resolves it.

Theorem 2 – Physics interpretation

Regard coker(β) under McKay correspondence :irreps ρ ∈ RC (GADE)

vertices of corresponding

ADE-type Dynkin diagram

Most exceptional Example: G = 2I :

integral/non-irrational

characters

Dynkindiagram

hence:im(β)|irred ⊂ RO(2I)|irred ↔ SU(5)︸︷︷︸

actualGUT group

⊂ E8︸︷︷︸stringy

GUT group

Comparison B to

ordinary Cohomologyand

“discrete torsion” of fractional M5-branes

back to Part II

The comparison homomorphism B

Away from the singular locusof a black M2-brane

· · · `P1

++`P1ss

AdS4 ×S7/GA︸︷︷︸spherical space form

R2,1 ×R8 GA︸︷︷︸orbifold du Val singularity

the orbifold is smooth and, for A-type singularities, so is the RO-degree:

coefficientbundle

((R10,1 − R2,1

(R10,1 − R2,1

)/G × R4

spacetimeorbifold

(R10,1 − R2,1

⇒ relevant comparison morphism is equivariant Boardman homomorphism

(S7)︸︷︷︸

equivariantstable Cohomotopy

βGA // HZ4GA

(S7)︸︷︷︸

equivariantordinary cohomology

' H4(S7/GA,Z

)︸︷︷︸Borel

equivariance

Theorem 3 i): factors through plain Boardman homomorphism:

putativeC-field flux /

M5-brane chargevisible in M-theory

“β α”

approximation

putativeM-theoretic discrete torsion

of fractional M5-brane charge[AharonyBergmanJafferis08, Sharpe03]

equivariant Boardmamn homomorphism

βGA //H4GA

(S7/GA,Z

Carlsson-Segal completion S4(S7/GA

plain Boardman homomorphism

Proof. Use [Schwede18, Example 4.5.19]. 2.

Theorem 3 ii): 4 coker(β) = 0

Proof. By [Arlettaz04, Theorem 1.2 b)]. 2

Theorem 3 iii): factors isomorphically through ko4:

putativeC-field flux /

M5-brane chargevisible in M-theory

“β α”

approximation

putativeM-theoretic discrete torsion

of fractional M5-brane charge[AharonyBergmanJafferis08, Sharpe03]

equivariant Boardmamn homomorphism

βGA //H4(S7/GA,Z

Carlsson-Segal completion S4(S7/GA

'Boardman homomorphism

for ko

//ko4(S7/GA)

p1first

Pontryagin class

Proof. By the AHSS and using π•≤2(S) = π•≤2(ko) 2.

Physically reasonable?

This coker(β) is KO-version of what was argued for KU in [DiaconescuMooreWitten00].

Conclusion1. Part I – Motivation of differential equivariant Cohomotopy:

(a) Derivation of equivariant cohomotopy/Q from first principlesvia rational super homotopy theory [FSS13, FSS16a, FSS16b, BSS18, HSS18]

(b) actual Cohomotopy is the minimal non-rational lift –differential equivariant Cohomotopy of super-orbifoldsexists in global equivariant super homotopy theory

(c) Hypothesis H:the observables of M-theory are the differential equivariant real coho-mology of the moduli stack of supertorsion-free differential equivariantCohomotopy of spacetime Pin(5)[-orbi R10,1|32-folds

2. Part II – Consistency checks of Hypothesis H:

(a) reproduces fractional D-brane charge in equivariant K-theory

i. excluding exactly the spurious irrational RR-charges,

ii. which may correspond, via McKay, to breaking E8 to SU(5) GUT

(b) reproduces discrete torsion of fractional M5-branes with DMW-correction.

In particular, equivariant stable cohomotopy somehow

unifies ordinary cohomology of the C-field with K-theory of D-branes.

back to Contents

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