Motivation. Super homotopy theory Orbifold …...Urs Schreiber (New York University, Abu Dhabi &...

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

simplicialsets

localizationtheory

simplicialsets

localizationtheory

zoo of modelsfor ∞-categories

quantumfield theory

homotopy theory

simplicialsets

localizationtheory

quantumfield theory

homotopy theoryEuler-Lagrange complexes

on jet bundles

simplicialsets

localizationtheory

quantumfield theory

on jet bundles

BV-BRSTtheory

simplicialsets

localizationtheory

quantumfield theory

on jet bundles

deformationquantization

BV-BRSTtheory

simplicialsets

localizationtheory

quantumfield theory

on jet bundles

BV-BRSTtheory

zoo of hacks fornonperturbative effectskk

simplicialsets

localizationtheory

quantumfield theory

homotopy theory

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

Euler-Lagrange complexeson jet bundles

BV-BRSTtheory

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

simplicialsets

localizationtheory

quantumfield theory

on jet bundles

BV-BRSTtheory

simplicialsets

localizationtheory

quantumfield theory

on jet bundles

BV-BRSTtheory

simplicialsets

localizationtheory

HoTT(!!?)

quantumfield theory

on jet bundles

BV-BRSTtheory

modalHoTT(??!)

simplicialsets

localizationtheory

HoTT(!!?)

quantumfield theory

homotopy theory

modalHoTT(??!)

HoTT(!!?)

Part I.

Equivariant super homotopy theory

1. Super homotopy theory and the Atom of Superspace

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

CartSp :=

objects: Rn-s for n ∈ N

morphisms: Rn1f

smooth//Rn2

coverings: good open covers

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

FormalCartSp :=

objects: Rn × D-s

C∞(Rn × D

):= C∞

(Rn)⊗R

(R⊕ fin dim

nilpotent ideal

)morphisms: C∞

(Rn × D

alg.homom.C∞(Rn′ × D′

)coverings: good open covers ×idD

C∞(−) : CartSp � � // CommAlgRop

SuperFormalCartSp :=

objects: Rn × D-s

C∞(Rn × D

):= C∞

(Rn)⊗R

(R⊕ fin dim

nilpotent ideal

)morphisms: C∞

(Rn × D

alg.homom.C∞(Rn′ × D′

)coverings: good open covers ×idD

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

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

C∞(RD|N) := C∞

(RD)⊗R R

[{θα}Nα=1

]/(θαθβ = −θbθα

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

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

FormalCartSp

oo ιinf ? _

CartSpΠ //

oo Disc

⊥? _

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

FormalCartSp

oo ιinf ? _

CartSpΠ //

oo Disc

⊥? _

Include point as R0 .

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

FormalCartSp

oo ιinf ? _

CartSpΠ //

oo Disc

⊥? _

Contract Rn to point.

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

FormalCartSp

oo ιinf ? _

CartSpΠ //

oo Disc

⊥? _

Contract Rn × D to Rn .

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

FormalCartSp

oo ιinf ? _

CartSpΠ //

oo Disc

⊥? _

Contract RD|N to RD .

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

FormalCartSp

oo ιinf ? _

CartSpΠ //

oo Disc

⊥? _

Include Rn as Rn .

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

FormalCartSp

oo ιinf ? _

CartSpΠ //

oo Disc

⊥? _

Include Rn × D as Rn × D .

Adjunctions

SuperFormalCartSp

even //

oo ιsup

⊥? _

FormalCartSp

oo ιinf ? _

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

⊥? _

FormalCartSp

oo ιinf ? _

CartSpΠ //

oo Disc

⊥? _

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

Adjunctions by Kan extension

SuperFormalCartSp

even //

oo ιsup

⊥? _

FormalCartSp

oo ιinf ? _

CartSpΠ //

oo Disc

⊥? _

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo ιinf ? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo Discsup

⊥? _

oo coDisc

⊥? _

oo ιinf ? _

oo Discinf

⊥? _

oo Disc

⊥? _

H[=∞Grpd

1. H< := Sh∞(

CartSp)

2. H := Sh∞(

FormalCartSp)

even //

oo ιsup

⊥? _

oo Discsup

⊥? _

oo coDisc

⊥? _

oo ιinf ? _

oo Discinf

⊥? _

oo Disc

⊥? _

H[=∞Grpd

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([?])

:= 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

// ku�B2R

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

Part II.

Orbifold cohomology

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

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

H := Sh∞(Singularities, H ⊂

)Adjunctions

Πorb//

oo Discorb

⊥? _

oo coDiscorb

⊥? _

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 ? _

H[oo ? _

oo ? _

��

� ? ��

super-geometric︷︸︸︷H ⊂

oo ? _

bosonic︷︸︸︷H ⊂

oo ? _

reduced︷︸︸︷H ⊂

oo ? _

smooth = non-singular

< a ⊂ a ≺

singular

oo ? _

H[oo ? _

oo ? _

��

� ? ��

oo ? _

< a ⊂ a ≺

singular

oo ? _

H[oo ? _

oo ? _

��

� ? ��

oo ? _

< a ⊂ a ≺

singular

oo ? _

H[oo ? _

oo ? _

��

∇� ? ��

� ? ��

oo ? _

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��

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.

≺ := ≺

(S4 � [Pin(5)

)∈ H/B[Pin(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 = 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

cocycle inequivariant Cohomotopy

≺[BPin(5)

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

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

`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

��

(G4,G7)

(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

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)��

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.

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

) such that

dG4 = 0

dG7 = −12G4 ∧G4 + 1

(NXQM5

)︸︷︷︸[Witten 96b, (5.7)]

2) Half-integrally shifted C-field flux quantization

[G4] +[

(NXQM5

)]∈ H4

)︸︷︷︸

[Witten 96a, 2.1 and 2.2]

3) M-theoretic tadpole cancellation:

dG7 = 12χ(TX) = 1

(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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