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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
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[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].
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