Urs Schreiber

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

Equivariant super homotopy theory

talk atGeometry in Modal Homotopy Type Theory

Pittsburgh 2019

based on joint work with

H. Sati, V. Braunack-Mayer, J. Huerta, D. Fiorenza, F. Wellen

ncatlab.org/schreiber/show/Equivariant+Super+Homotopy+Theory

Motivation.

Part I.Super homotopy theory

Part II.Orbifold cohomology

Motivation

back to Contents

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory kk

set theory/type theory

foundations ofmathematics

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory kk

set theory/type theory

foundations ofmathematics

simplicialsets

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory kk

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory kk

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theory

kk

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theoryEuler-Lagrange complexes

on jet bundles

kk

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theoryEuler-Lagrange complexes

on jet bundles

BV-BRSTtheory

kk

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theoryEuler-Lagrange complexes

on jet bundles

deformationquantization

BV-BRSTtheory

kk

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theoryEuler-Lagrange complexes

on jet bundles

deformationquantization

BV-BRSTtheory

zoo of hacks fornonperturbative effectskk

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theory

for detailed exposition see:geometry+of+physics+–+perturbative+quantum+field+theory

Euler-Lagrange complexeson jet bundles

deformationquantization

BV-BRSTtheory

zoo of hacks fornonperturbative effectskk

set theory/type theory

foundations ofmathematics

for detailed exposition see:ncatlab.org/nlab/show/geometry+of+physics+–+categories+and+toposes

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theoryEuler-Lagrange complexes

on jet bundles

deformationquantization

BV-BRSTtheory

zoo of hacks fornonperturbative effectskk

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theoryEuler-Lagrange complexes

on jet bundles

deformationquantization

BV-BRSTtheory

zoo of hacks fornonperturbative effectskk

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

HoTT(!!?)

OO

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theoryEuler-Lagrange complexes

on jet bundles

deformationquantization

BV-BRSTtheory

zoo of hacks fornonperturbative effectskk

modalHoTT(??!)

OO

set theory/type theory

foundations ofmathematics

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

33

HoTT(!!?)

OO

Analytic detours and Synthetic promises

foundations ofphysics

quantumfield theory

homotopy theory

kk

modalHoTT(??!)

OO

set theory/type theory

foundations ofmathematics

33

HoTT(!!?)

OO

Part I.

Equivariant super homotopy theory

1. Super homotopy theory and the Atom of Superspace

2.

RationalSuper homotopy theory and the fundamental super p-Branes

back to Contents

Super homotopy theory

and the Atom of Superspace

back to Part I

physics mathematics

gauge principle homotopy theory

& Pauli exclusion super-geometry

= super homotopy theory

for detailed exposition see:ncatlab.org/nlab/show/geometry+of+physics+–+supergeometry

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

CartSp :=

objects: Rn-s for n ∈ N

morphisms: Rn1f

smooth//Rn2

coverings: good open covers

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

CartSp :=

objects: Rn-s for n ∈ N

morphisms: Rn1f

smooth//Rn2

coverings: good open covers

Fact (“Milnor’s exercise”):Sending Cartesian spacesto their R-algebras of smooth real-valued functionsis fully faithful:

C∞(−) : CartSp � � // CommAlgRop

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

FormalCartSp :=

objects: Rn × D-s

C∞(Rn × D

):= C∞

(Rn)⊗R

(R⊕ fin dim

nilpotent ideal

)morphisms: C∞

(Rn × D

)oo

f∗

alg.homom.C∞(Rn′ × D′

)coverings: good open covers ×idD

Fact (“Milnor’s exercise”):Sending Cartesian spacesto their R-algebras of smooth real-valued functionsis fully faithful:

C∞(−) : CartSp � � // CommAlgRop

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

SuperFormalCartSp :=

objects: Rn × D-s

C∞(Rn × D

):= C∞

(Rn)⊗R

(R⊕ fin dim

nilpotent ideal

)morphisms: C∞

(Rn × D

)oo

f∗

alg.homom.C∞(Rn′ × D′

)coverings: good open covers ×idD

Fact (“Milnor’s exercise”):Sending Cartesian spacesto their R-algebras of smooth real-valued functionsis fully faithful:

C∞(−) : CartSp � � // CommAlgRop↪→ SuperCommAlgR

op

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

e.g.super Cartesian space RD|N defined by

C∞(RD|N) := C∞

(RD)⊗R R

[{θα}Nα=1

]/(θαθβ = −θbθα

)

Fact (“Milnor’s exercise”):Sending Cartesian spacesto their R-algebras of smooth real-valued functionsis fully faithful:

C∞(−) : CartSp � � // CommAlgRop↪→ SuperCommAlgR

op

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

Include point as R0 .

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

Contract Rn to point.

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

Contract Rn × D to Rn .

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

Contract RD|N to RD .

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

Include Rn as Rn .

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

Include Rn × D as Rn × D .

The sites of super homotopy theory

1. CartSp – cartesian spaces

2. FormalCartSp – formal cartesian spaces

3. SuperFormalCartSp – super formal cartesian spaces

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

Send R0|2 to D1(1).

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

SuperFormalCartSp

even //

oo ιsup

⊥? _

Πsup

⊥//

FormalCartSp

oo ιinf ? _

Πinf

⊥//

CartSpΠ //

oo Disc

⊥? _

∗

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

H

oo ιinf ? _

Πinf

⊥//

H<

Π //

oo Disc

⊥? _

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

oo Discsup

⊥? _

Γsup

⊥//

oo coDisc

⊥? _

H

oo ιinf ? _

Πinf

⊥//

oo Discinf

⊥? _

Γinf

⊥//

H<

Π //

oo Disc

⊥? _

Γ

⊥//

H[=∞Grpd

The toposes of super homotopy theory

1. H< := Sh∞(

CartSp)

– smooth ∞-groupoids

2. H := Sh∞(

FormalCartSp)

– formal smooth ∞-groupoids

3. H := Sh∞(SuperFormalCartSp

)– super formal smooth ∞-groupoids

Adjunctions by Kan extension

H

even //

oo ιsup

⊥? _

Πsup

⊥//

oo Discsup

⊥? _

Γsup

⊥//

oo coDisc

⊥? _

H

oo ιinf ? _

Πinf

⊥//

oo Discinf

⊥? _

Γinf

⊥//

H<

Π //

oo Disc

⊥? _

Γ

⊥//

H[=∞Grpd

The Modalities of Super homotopy theory

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

supergeometric∞-groupoids︷︸︸︷

HΓ //

∗

The Modalities of Super homotopy theory

∅

nothing

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

supergeometric∞-groupoids︷︸︸︷

H oo ∆ ? _

Γ //∗

The Modalities of Super homotopy theory

∅ a ∗

pure being

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

supergeometric∞-groupoids︷︸︸︷

H oo ? _

Γ //

oo ∇ ? _

∗

The Modalities of Super homotopy theory

[∨

∅ a ∗

discrete

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

supergeometric∞-groupoids︷︸︸︷

H oo ∆ ? _

Γ //

oo ? _

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

[∨

a ]∨

∅ a ∗

continuous

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

supergeometric∞-groupoids︷︸︸︷

H oo ? _

Γ //

oo ∇ ? _

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

S a [∨

a ]∨

∅ a ∗

shaped

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

supergeometric∞-groupoids︷︸︸︷

HΠ //

oo ∆ ? _

//

oo ? _

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

=∨

S a [∨

a ]∨

∅ a ∗

infinitesimally shaped

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

supergeometric∞-groupoids︷︸︸︷

HΠ //

oo ∆ ? _

//

oo ? _

reduced differential-geometric∞-groupoids︷︸︸︷

H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

=∨

a &∨

S a [∨

a ]∨

∅ a ∗

infinitesimally discrete

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

supergeometric∞-groupoids︷︸︸︷

H//

oo ∆ ? _

Γ //

oo ? _

reduced differential-geometric∞-groupoids︷︸︸︷

H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

reduced

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

supergeometric︷︸︸︷H

oo ι ? _

Π //

oo ? _

//

oo ? _

reduced differential-geometric︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

∨

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

bosonic

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

supergeometric︷︸︸︷H

oo ι ? _

Π //

oo ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

//

oo ? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

a

∨

Rh

∨

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

rheonomic

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

supergeometric︷︸︸︷H

oo ? _

Π //

oo ∆ ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

//

oo ? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

⇒ a a

∨

Rh

∨

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

even

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

super-geometric︷︸︸︷

H

ε //

oo ι ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

//

oo ? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

id

∨

a id

∨

⇒ a a

∨

Rh

∨

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

super-geometric

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

super-geometric︷︸︸︷

H

//

oo ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

//

oo ? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

id

∨

a id

∨

⇒ a a

∨

Rh

∨

< a =∨

a &∨

S a [∨

a ]∨

∅ a ∗

A1-local

The central modalities are motivic A1-localizations.

super-geometric︷︸︸︷

H

//

oo ? _

Πsup //

oo ∆sup ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

Πinf//

oo ∆inf? _

reduced︷︸︸︷H<

Π //

oo ∆ ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

id

∨

a id

∨

⇒ a a

∨

Rh

∨

< a =∨

a &∨

R1 a [∨

a ]∨

∅ a ∗

continuum-local

The central modalities are motivic A1-localizations.

super-geometric︷︸︸︷

H

//

oo ? _

Πsup //

oo ∆sup ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

Πinf//

oo ∆inf? _

reduced︷︸︸︷H<

Π //

oo ∆ ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

id

∨

a id

∨

⇒ a a

∨

Rh

∨

< a D

∨

a &∨

R1 a [∨

a ]∨

∅ a ∗

infinitum-local

The central modalities are motivic A1-localizations.

super-geometric︷︸︸︷

H

//

oo ? _

Πsup //

oo ∆sup ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

Πinf//

oo ∆inf? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

id

∨

a id

∨

⇒ a a

∨

R0|1

∨

< a D

∨

a &∨

R1 a [∨

a ]∨

∅ a ∗

superpoint-local

The central modalities are motivic A1-localizations.

super-geometric︷︸︸︷

H

//

oo ? _

Πsup //

oo ∆sup ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

//

oo ? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

The Modalities of Super homotopy theory

id

∨

a id

∨

⇒ a a

∨

R0|1

∨

< a D

∨

a &∨

R1 a [∨

a ]∨

∅ a

ii

∗⇒ emergence of Atom of Superspace from \nothing

The Modalities of Super homotopy theory

id

∨

a id

∨

⇒ a a

∨

R0|1

∨

< a D

∨

a &∨

R1 a [∨

a ]∨

∅ a

ii

∗⇒ 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([?])

:= dgcSuperAlgR,cn︸ ︷︷ ︸∞-category of

differentialgraded-commutative

superalgebras

oo CE'

(SuperL∞Algbdnil

R,cn︸ ︷︷ ︸∞-category of

nilpotentsuper L∞-algebroids

)op

Theorem. ([BM18], see Sch18)

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

rational super-∞-groupoids︷ ︸︸ ︷FDAsop

Sullivan rational homotopy theory︷ ︸︸ ︷oo O

Spec//

supergeometric∞-groupoids︷︸︸︷

H oo Disc ? _

Γ//

geometrically discrete∞-groupoids︷︸︸︷

H[

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

R0|1

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

R0|1

Type I

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

R0|1+1 oooo R0|1

Type IIB Type I

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

[HS17]

R2,1|2

vvR0|1+1 oo

oo R0|1

Type IIB Type I

universal central extension: 3d super-Minkowski spacetime

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

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I

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

R3,1|4

ww

[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

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

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

ww

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I

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

R5,1|8

xxR3,1|4+4 oo

oo R3,1|4

ww

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I

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

R5,1|8

xx

R5,1|8+8////

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

ww

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

R9,1|16

((R5,1|8

xx

R5,1|8+8////

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

ww

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

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

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

R5,1|8

xx

R5,1|8+8////

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

ww

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

R10,1|32

��R9,1|16+16 oo

oo R9,1|16

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

R5,1|8

xx

R5,1|8+8////

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

ww

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

R10,1|32

��R9,1|16+16 oo

oo R9,1|16

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

R5,1|8

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

R10,1|32

��stringIIB

(([FSS13] stringI

��stringIIA

vvR9,1|16+16 oo

oo R9,1|16

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

R5,1|8

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

universal higher central invariant extension: stringy extended super-spacetimes

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

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

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

m2brane

��

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

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

m5brane

��

[FSS15]

m2brane

��

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

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

m5brane

��

[FSS15]

m2brane

��

d5brane

��

d3brane

��

d1brane

��

d0brane

��

d2brane

��

d4brane

��

d7brane

%%

branebouquet

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

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

m5brane

��

[FSS15]

rr M/IIADuality

$$

[FSS16a]

m2brane

��

d5brane

��

d3brane

��

d1brane

��

d0brane

(pb)

��

xx

d2brane

��

d4brane

��

d7brane

%%

branebouquet

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

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

m5brane

��

[FSS15]

rr M/IIADuality

$$

[FSS16a]

m2brane

��

d5brane

��

d3brane

��

d1brane

��

d0brane

(pb)

��

xx

d2brane

��

d4brane

��

Gaugeenhancement

{{

d7brane

%%

branebouquet

R10,1|32

��

d6brane

xxd9brane // stringIIB

(([FSS13] stringI

��stringIIA

vvd8braneoo [BSS18]

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

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

R5,1|8

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|1+1 oo

oo R0|1

Type IIB Type I Type IIA

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

m5brane

��

[FSS15]

qq M/IIADuality

$$

[FSS16a]

m2brane

��

d5brane

��

d3brane

��

d1brane

��

d0brane

(pb)

��

xx

d2brane

��

d4brane

��

Gaugeenhancement

{{

d7brane

%%

branebouquet

R10,1|32

��

d6brane

xxd9brane // stringIIB

(([FSS13] stringI

��stringIIA

vvd8braneoo [BSS18]

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

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

R5,1|8

xx

R5,1|8+8////

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

ww

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]

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

m5brane

��

[FSS15]

[FSS18] qq M/IIADuality

$$

[FSS16a]

m2brane

��

d5brane

��

d3brane

��

d1brane

��

d0brane

(pb)

��

xx

d2brane

��

d4brane

��

Gaugeenhancement

{{

d7brane

%%

branebouquet

R10,1|32

��

d6brane

xxd9brane // stringIIB

(([FSS13] stringI

��stringIIA

vvd8braneoo [BSS18]

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

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

R5,1|8

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|32

oo...

ooR0|1+1 oo

oo R0|1

Exceptional Type IIBdd

T-Duality

::Type I Type IIA [FSS16b]

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

m5brane

��

[FSS15]

[FSS18] qq M/IIADuality

$$

[FSS16a]

m2brane

��

d5brane

��

d3brane

��

d1brane

��

d0brane

(pb)

��

xx

d2brane

��

d4brane

��

Gaugeenhancement

{{

R10,1|32exc

��

d7brane

%%

branebouquet

R10,1|32

��

d6brane

xxd9brane // stringIIB

(([FSS13] stringI

��stringIIA

vvd8braneoo [BSS18]

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

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

R5,1|8

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|32

oo...

ooR0|1+1 oo

oo R0|1

Exceptional Type IIBdd

T-Duality

::Type I Type IIA [FSS16b]

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

m5brane

��

[FSS15]

[FSS18] R10,1|32exc,s

comp

++

��

��

Higher T-duality

qq M/IIA

Duality

$$

[FSS16a]

m2brane

��

d5brane

��

d3brane

��

d1brane

��

d0brane

(pb)

��

xx

d2brane

��

d4brane

��

Gaugeenhancement

{{

R10,1|32exc

��

d7brane

%%

branebouquet

R10,1|32

��

d6brane

xxd9brane // stringIIB

(([FSS13] stringI

��stringIIA

vvd8braneoo [BSS18]

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

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

R5,1|8

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|32

oo...

ooR0|1+1 oo

oo R0|1

Exceptional Type IIBdd

T-Duality

::Type I Type IIA [FSS16b]

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

m5brane

��

[FSS15]

[FSS18] R10,1|32exc,s

comp

++

��

��

Higher T-duality

qq M/IIA

Duality

$$

[FSS16a]

m2brane

��

d5brane

��

d3brane

��

d1brane

��

d0brane

(pb)

��

xx

d2brane

��

d4brane

��

Gaugeenhancement

{{

R10,1|32exc

��

d7brane

%%

branebouquet

R10,1|32

��

d6brane

xxd9brane // stringIIB

(([FSS13] stringI

��stringIIA

vvd8braneoo [BSS18]

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

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

R5,1|8

xx

R5,1|8+8////

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

ww

emergentspacetime

[HS17]

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

vvR0|32

oo...

ooR0|1+1 oo

oo R0|1

Exceptional Type IIBdd

T-Duality

::Type I Type IIA [FSS16b]

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

m5brane

��

[FSS15]

m2brane

��R10,1|32

��[FSS13]

R9,1|16

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

R5,1|8

xx

R5,1|8+8////

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

ww

[HS17]

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

vvR0|1+1 oo

oo R0|1

dd

emergence of fundamental M-branes from the Atom of Superspace

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

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 &&

B4R

µM2 = dLWZWM2

= i2

(ψΓa1a2ψ

)∧ ea1 ∧ ea2

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

The fundamental M2/M5-brane cocycle

m5brane

hofib(µM5)

��

[FSS15]

m2brane

hofib(µM2)

��

µM5 // S7

R

R10,1|32

µM2 &&

B4R

µM5 = dLWZWM5

= 15!

(ψΓa1···a5ψ

)∧ ea1 ∧ · · · ∧ ea5 + c3 ∧ i

2

(ψΓa1a2ψ

)∧ ea1 ∧ ea2

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

The fundamental M2/M5-brane cocycle

m5brane

hofib(µM5)

��

[FSS15]

m2brane

hofib(µM2)

��

µM5 // S7

R

hofib(c2)

��R10,1|32

µM2 &&

S4R

c2xxB4R

the quaternionic Hopf fibration (in rational homotopy theory)

The fundamental M2/M5-brane cocycle

m5brane

hofib(µM5)

��

[FSS15]

m2brane

hofib(µM2)

��

µM5 // S7

R

hofib(c2)

��R10,1|32

µM2/M5 //

µM2 &&

S4R

c2xxB4R

the unified M2/M5-cocycle

qy

The fundamental M2/M5-brane cocycle

[FSS15]

R10,1|32µM2/M5 // S4

R

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

The fundamental M2/M5-brane cocycle

[FSS15]

R10,1|32µM2/M5 // S4

R

i2

(ψΓa1a2ψ

)∧ ea1 ∧ ea2 G4

�oo

15!

(ψΓa1···a5ψ

)∧ ea1 · · · ea5 G7

�oo

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])

The fundamental M2/M5-brane cocycle

[FSS15]

R10,1|32µM2/M5 // S4

R

the unified M2/M5-cocycle

The fundamental M2/M5-brane cocycle

[FSS15]

R10,1|32µM2/M5 //

double dimensional reduction & gauge enhancement

��

S4R

R9,1|16+16µF1/D2p

// ku�B2R

D-brane charge in twisted K-theory, rationally

[BSS18]

The rational conclusion.

In

{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.

In

{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

S4

rationalization��

X

cocycle inactual Cohomotopy

99

cocycle inrational cohomotopy

//S4R

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

Part II.

Orbifold cohomology

1.

Global equivariantSuper homotopy theory and the C-field at singularities

2. Super Cartan geometry and 11d orbifold supergravity

back to Contents

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 idea of global equivariant homotopy theory

What is an orbifold, really?

detailed exposition inncatlab.org/nlab/show/orbifold+cohomology

The idea of global equivariant homotopy theory

What is an orbifold, really?

Two opposite dual views on orbifold singularities X Gnn :

non-smooth singular points smooth but stacky quotients

orbifold singularityBG

[orb

''

Sorb

wwopposite extreme

aspects of orbifold singularitysnon-smoothsingular point∗ = ∗/G

smoothstacky quotientBG = ∗�G

The site of global equivariant homotopy theory

Singularities :=

objects: groupoids BG :={∗g��| g ∈ G

}for finite groups G

morphisms: groupoids of functorsBG −→ BG′

coverings: identity functors

Elsewhere known as the “global orbit category”

but better thought of as the ∞-category

of models for orbifold singularities.

The toposes of global equivariant homotopy theory

call the base topos

H ⊂ := Sh∞(SuperFormalCartSp

)set

H := Sh∞(Singularities, H ⊂

)Adjunctions

H

Πorb//

oo Discorb

⊥? _

Γorb

⊥//

oo coDiscorb

⊥? _

H ⊂

Cohesion of global equivariant homotopy theoryhighlighted by C. Rezk, Global homotopy theory and cohesion (2014)

The modalities of global equivariant homotopy theory

< a ⊂ a ≺

singular smoothorbi-

singular

possibly singular/orbifoldedsuper-

geometric︷︸︸︷H

//

oo ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

//

oo ? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

����

OO

� ?

OO

� ? ����

OO

� ?

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ?

super-geometric︷︸︸︷H ⊂

//

oo ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H ⊂

oo ? _

//

oo ? _

reduced︷︸︸︷H ⊂

<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H ⊂

[

oo ? _

//

oo ? _

∗

smooth = non-singular

The modalities of global equivariant homotopy theory

< a ⊂ a ≺

singular smoothorbi-

singular

possibly singular/orbifoldedsuper-

geometric︷︸︸︷H

//

oo ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

//

oo ? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

Π

��

OO

∆

� ? ����

OO

� ?

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ?

super-geometric︷︸︸︷H ⊂

//

oo ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H ⊂

oo ? _

//

oo ? _

reduced︷︸︸︷H ⊂

<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H ⊂

[

oo ? _

//

oo ? _

∗

smooth = non-singular

The modalities of global equivariant homotopy theory

< a ⊂ a ≺

singular smoothorbi-

singular

possibly singular/orbifoldedsuper-

geometric︷︸︸︷H

//

oo ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

//

oo ? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

����

OO

∆

� ?

Γ

��

OO

� ?

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ?

super-geometric︷︸︸︷H ⊂

//

oo ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H ⊂

oo ? _

//

oo ? _

reduced︷︸︸︷H ⊂

<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H ⊂

[

oo ? _

//

oo ? _

∗

smooth = non-singular

The modalities of global equivariant homotopy theory

< a ⊂ a ≺

singular smoothorbi-

singular

possibly singular/orbifoldedsuper-

geometric︷︸︸︷H

//

oo ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H

oo ? _

//

oo ? _

reduced︷︸︸︷H<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H[oo ? _

//

oo ? _

∗

����

OO

� ?

OO

� ?

Γ

��

OO

∇� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ? ��

OO

� ?

super-geometric︷︸︸︷H ⊂

//

oo ? _

//

oo ? _

//

oo ? _

bosonic︷︸︸︷H ⊂

oo ? _

//

oo ? _

reduced︷︸︸︷H ⊂

<

//

oo ? _

//

geometrically discrete∞-groupoids︷︸︸︷

H ⊂

[

oo ? _

//

oo ? _

∗

smooth = non-singular

Super-Orbifolds ([Wellen17, Sch13])

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

pV��

pX

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 := SS4 ∈ ∞Groupoids its shape.

Then

S4

≺ := ≺

(S4 � [Pin(5)

)∈ H/B[Pin(5)

is a [Pin(5)-orbi R4-fold

S4

≺ := S ≺

(S4 � Pin(5)[

)is its shape orbi-space

Equivariant Cohomotpy of Super-orbifolds

Let

R10,1|32 ∈ Grp(H) D = 11, 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

≺

)=

X

%%

cocycle inequivariant Cohomotopy

//S4

≺

yy

≺[BPin(5)

s{

and so the cohomology set is

H(X , S4

≺

):= π0

(H/ ≺[BPin(5)

) (X , S4

≺

)

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

X ⊂

frame−→ BAut(DV)

Def.:for G

homom.−→ Aut(DV)

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

X ⊂

frame &&

//BG

ww

BAut(DV)Erz

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

V

frame &&

//BG

ww

BAut(DV)Eli

rz

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

`P�1 ,,`P�1qq

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

`P�1 ,,`P�1qq

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 conclusion, in super homotopy theory emerges:

CovariantPhaseSpace :=

⊔[X ] ∈

[Pin(3)× Pin(5)Orbi−R10,1|32Folds/∼

S(S7�[

(Spin(3)×Spin(5)

))

��

// S(S4�[Spin(5)

)

��

gauge field

X(NQM5

QM2, NXQM5

)//

g

��

FrX

��

(G4,G7)

22

(TQM2, NXQM2

)((

B[(Spin(3)× Spin(5)

)//

��

B[Spin(5)

gravity

B[(Spin(2, 1)× Spin(8)

)//

��

B[Spin(8)

B[AutGrp

(D10,1|32)

��

B[Spin(10, 1)

BAut(D10,1|32)

�Aut(X )

MotiveOfObservables := Σ∞[Pin(3)×[Pin(5)(CovariantPhaseSpace

)

In conclusion, in super homotopy theory emerges:

This is the core ingredients of what is known as

M-theory on Spin(7)-manifolds with space-filling M2-branes

R10,1 ' R2,1

Pin(2,1)��

⊕ R3 ⊕

Pin(3)��

R5

Pin(5)��

M5 × × −M2 × − −

T-dual to

F-theory on elliptically fibered 8-manifolds with space-filling D3-branes(whose near-horizon geometry is AdS5 × S5)

This happens to bethe phenomenologically relevant sector of type II string theory.

In conclusion, in super homotopy theory emerges:

Theorem (Fiorenza-Sati-S.):

The C-field charge quantized in twisted Cohomotopy this way,implies cancellation of M-theory anomalies:

1) M5-brane anomaly counterterm:

G4 ∈ Z4(X,Q

)G7 ∈ C7

(X,Q

) such that

dG4 = 0

dG7 = −12G4 ∧G4 + 1

8p2

(NXQM5

)︸ ︷︷ ︸[Witten 96b, (5.7)]

2) Half-integrally shifted C-field flux quantization

[G4] +[

14p1

(NXQM5

)]∈ H4

(X,Z

)︸ ︷︷ ︸

[Witten 96a, 2.1 and 2.2]

3) M-theoretic tadpole cancellation:

dG7 = 12χ(TX) = 1

4

(p2(TX)− 1

4p1(TX)2)︸ ︷︷ ︸

[Sethi-Vafa-Witten 96] [Witten 96a, 3]

References

[BMSS19]V. Braunack-Mayer, H. Sati, U. Schreiber,Towards microscopic M-Theory:Differential equivariant Cohomotopyin preparation

[SSS09]H. Sati, U. Schreiber, J. Stasheff,Twisted differential string and fivebrane structuresCommunications in Mathematical Physics, 315 (1), 169-213 (2012)[arXiv:0910.4001]

[FSS13]D. Fiorenza, H. Sati, U. Schreiber,Super Lie n-algebras, higher WZW models, and super p-branes,Intern. J. Geom. Meth. Mod. Phys. 12 (2015) 1550018 ,[arXiv:1308.5264]

[FSS15]D. Fiorenza, H. Sati, U. Schreiber,The WZW term of the M5-brane and differential cohomotopy,J. Math. Phys. 56, 102301 (2015)[arXiv:1506.07557]

[FSS16a]D. Fiorenza, H. Sati, U. Schreiber,Sphere valued supercocycles in M-theory and type IIA string theory,J. Geom. Phys. 114 (2017) 91-108,[arXiv:1606.03206].

[FSS16b]D. Fiorenza, H. Sati, U. Schreiber,T-Duality from super Lie n-algebra cocycles for super p-branes,ATMP Volume 22 (2018) Number 5,[arXiv:1611.06536].

[HS17]J. Huerta and U. Schreiber,M-Theory from the superpoint,Lett. in Math. Phys. 2018,[arXiv:1702.01774] [hep-th]

[FSS18]D. Fiorenza, H. Sati, U. Schreiber,Higher T-duality of M-branes,[arXiv:1803.05634].andH. Sati, U. Schreiber, Higher T-duality in M-theory via local supersym-metry,Physics Letters B, Volume 781, 10 June 2018, Pages 694-698[arXiv:1805.00233]

[HSS18]J. Huerta, H. Sati, U. Schreiber,Real ADE-equivariant (co)homotpy of super M-branes,Communications in Mathematical Physics 2019[arXiv:1805.05987].

[BSS18]V. Braunack-Mayer, H. Sati, U. SchreiberGauge enhancement via Parameterized stable homotopy theory,Communications in Mathematical Physics 2019[arXiv:1806.01115]

[Sati13]H. Sati,Framed M-branes, corners, and topological invariants,J. Math. Phys. 59 (2018), 062304,[arXiv:1310.1060]

[Sch13]U. Schreiber,Differential cohomology in a cohesive ∞-toposlecture notes:geometry of physics – categories and toposesgeometry of physics – supergeometry

[Rezk14]C. Rezk,Global Homotopy Theory and Cohesion,2014[faculty.math.illinois.edu/rezk/global-cohesion.pdf]

[Wellen17]F. Wellen,Cartan Geometry in Modal Homotopy Type TheoryPhD Thesis, KIT 2017[arXiv:1806.05966]

[BM18]V. Braunack-Mayer,Rational parametrised stable homotopy theory,PhD thesis, Zurich 2018[ncatlab.org/schreiber/files/VBM RPSHT.pdf]

[CaLe93] A. Candiello, K. Lechner, Duality in Supergravity Theories,Nucl.Phys. B412 (1994) 479-501 [arXiv:hep-th/9309143]

[Sethi-Vafa-Witten 96] S. Sethi, C. Vafa, E. Witten, Constraints on Low-Dimensional String Compactifications, Nucl. Phys. B480:213-224, 1996[arXiv:hep-th/9606122]

[Witten 96a] E. Witten, On Flux Quantization In M-Theory And The Ef-fective Action, J. Geom. Phys.22:1-13, 1997 [arXiv:hep-th/9609122]

[Witten 96b] E. Witten, Five-Brane Effective Action In M-Theory, J. Geom.Phys.22:103-133, 1997 [arXiv:hep-th/9610234]

[How97] P. Howe, Weyl Superspace, Physics Letters B Volume 415, Issue 2, 11December 1997, Pages 149155 [arXiv:hep-th/9707184]

[AFFHS98] B. Acharya, J. Figueroa-O’Farrill, C. Hull, B. Spence, Branesat conical singularities and holography, Adv. Theor.Math. Phys.2:1249-1286, 1999 [arXiv:hep-th/9808014]

[MFFGME09] P. de Medeiros, J. Figueroa-O’Farrill, S. Gadhia, E. Mendez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP0910:038,2009 [arXiv:0909.0163]

[MFF10] P. de Medeiros, J. Figueroa-O’Farrill, Half-BPS M2-brane orb-ifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408.[arXiv:1007.4761]

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