![Page 1: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/1.jpg)
Urs Schreiber
(New York University, Abu Dhabi & Czech Academy of Science, Prague)
Cohomotopy Theory and Branes
talk atGeometry, Topology & Physics at Abu Dhabi
NYU AD 2020
based on joint work with
H. Sati and D. Fiorenza
ncatlab.org/schreiber/show/Cohomotopy+Theory+and+Branes
![Page 2: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/2.jpg)
0) Nonabelian Differential Cohomology
Brane charge in...
1) Twisted Cohomotopy theory
2) Equivariant Cohomotopy theory
3) Differential Cohomotopy theory
implies...
4) Hanany-Witten Theory
5) Chan-Paton Data
6) BMN Matrix Model States
7) M2/M5 Brane Bound States
![Page 3: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/3.jpg)
(0)
Nonabelian Differential Cohomology
in
Cohesive Homotopy Theory
Sec. 4 of Fiorenza-Sati-Schreiber 15 [arXiv:1506.07557]
based on Sec. 6.4.14.3 of Schreiber dcct
back to top
3
![Page 4: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/4.jpg)
Geometry Topology
Physics
4
![Page 5: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/5.jpg)
Geometry Topology
Physics
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
5
![Page 6: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/6.jpg)
Geometry Topology
Physics
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
6
![Page 7: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/7.jpg)
Geometry Topology
Physics
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
7
![Page 8: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/8.jpg)
Geometry Topology
Physics
loopL∞-algebra
lXL∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
8
![Page 9: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/9.jpg)
Geometry Topology
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
9
![Page 10: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/10.jpg)
Geometry Topology
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
XRrationalized(real-ified)top. space
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
10
![Page 11: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/11.jpg)
Geometry Topology
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
Maps(Σ,XRrationalized(real-ified)top. space
)
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
11
![Page 12: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/12.jpg)
Geometry Topology
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
Maps(Σ,XRrationalized(real-ified)top. space
)
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
ratio
naliz
atio
n=
X-C
hern
char
acte
r
12
![Page 13: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/13.jpg)
Geometry Topology
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
Maps(Σ,XRrationalized(real-ified)top. space
)
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
L∞
deRham
theorem ratio
naliz
atio
n=
X-C
hern
char
acte
r
13
![Page 14: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/14.jpg)
Geometry Topology
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
Maps(Σ,XRrationalized(real-ified)top. space
)
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
L∞
deRham
theorem ratio
naliz
atio
n=
X-C
hern
char
acte
r
14
![Page 15: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/15.jpg)
Geometry Homotopy
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
Maps(Σ,XRrationalized(real-ified)top. space
)
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
L∞
deRham
theorem ratio
naliz
atio
n=
X-C
hern
char
acte
r
15
![Page 16: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/16.jpg)
Geometry Homotopy
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
Maps(Σ,XRrationalized(real-ified)top. space
)
Xconn(
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
)
L∞
deRham
theorem ratio
naliz
atio
n=
X-C
hern
char
acte
r
homotopyfiber product
16
![Page 17: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/17.jpg)
Geometry Homotopy
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
Maps(Σ,XRrationalized(real-ified)top. space
)
Xconn(
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
)
L∞
deRham
theorem ratio
naliz
atio
n=
X-C
hern
char
acte
r
curv
atur
efo
rms/
flux
dens
ity
homotopyfiber product
17
![Page 18: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/18.jpg)
Geometry Homotopy
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
Maps(Σ,XRrationalized(real-ified)top. space
)
Xconn(
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
)
L∞
deRham
theorem ratio
naliz
atio
n=
X-C
hern
char
acte
r
curv
atur
efo
rms/
flux
dens
ity
topological /instanton
sectorhomotopyfiber product
18
![Page 19: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/19.jpg)
Geometry Homotopy
Physics
Ωcl(Σ,
loopL∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(Σ,
topologicalspace
Xmapping space =
cocycle space of X-cohomology on Σ
)
Maps(Σ,XRrationalized(real-ified)top. space
)
Xconn(
smoothmanifold
Σcocycle space of
differential X-cohomology/higher gauge fields
)
L∞
deRham
theorem ratio
naliz
atio
n=
X-C
hern
char
acte
r
curv
atur
efo
rms/
flux
dens
ity
topological /instanton
sector
L∞-connection/higher gauge field
homotopyfiber product
19
![Page 20: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/20.jpg)
Geometry Homotopy
Physics
Ωcl(−,loop
L∞-algebra
lX)L∞-algebra valueddifferential forms
(flat/closed)
Maps(−,topological
space
Xmapping space =
cocycle space of X-cohomology
)
Maps(−,XRrationalized(real-ified)top. space
)
Xconn(
smoothmanifold
−cocycle space of
differential X-cohomology/higher gauge fields
)
L∞
deRham
theorem ratio
naliz
atio
n=
X-C
hern
char
acte
r
curv
atur
efo
rms/
flux
dens
ity
topological /instanton
sector
L∞-connection/higher gauge field
homotopyfiber product
all functorialin Σ
20
![Page 21: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/21.jpg)
Geometry Homotopy
Physics
Xconnmoduli stack of
differential X-cohomology/higher gauge fields
topologicalspace
Xclassifying space ofX-cohomology
21
![Page 22: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/22.jpg)
Geometry Homotopy
Physics
Xconnmoduli stack of
differential X-cohomology/higher gauge fields
topologicalspace
Xclassifying space ofX-cohomology
∞-groupoids
∈
22
![Page 23: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/23.jpg)
Geometry Homotopy
Physics
Xconnmoduli stack of
differential X-cohomology/higher gauge fields
topologicalspace
Xclassifying space ofX-cohomology
∞-groupoids
smooth
∞-groupoids
“∞-stacks on smooth manifolds”
//
oo
? _
∈
∈
23
![Page 24: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/24.jpg)
Geometry Homotopy
Physics
Xconnmoduli stack of
differential X-cohomology/higher gauge fields
topologicalspace
Xclassifying space ofX-cohomology
smooth
manifolds
∞-groupoids
smooth
∞-groupoids
“∞-stacks on smooth manifolds”
//
//
oo
? _
∈
∈
24
![Page 25: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/25.jpg)
Geometry Homotopy
Physics
Xconnmoduli stack of
differential X-cohomology/higher gauge fields
topologicalspace
Xclassifying space ofX-cohomology
smooth
manifolds
∞-groupoids
smooth
∞-groupoids
“∞-stacks on smooth manifolds”
//S
shape(“geom
. realiz.”)//
oo
⊥
? _
∈
∈
25
![Page 26: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/26.jpg)
Geometry Homotopy
Physics
Xconnmoduli stack of
differential X-cohomology/higher gauge fields
topologicalspace
Xclassifying space ofX-cohomology
smooth
manifolds
∞-groupoids
smooth
∞-groupoids
“∞-stacks on smooth manifolds”
//S
shape(“geom
. realiz.”)//
oo
⊥
? _
path ∞-groupoid//
∈
∈
26
![Page 27: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/27.jpg)
(1)
Brane Charge in Cohomotopy
implied by
Hypothesis H with Pontrjagin-Thom Theorem
Sati-Schreiber 19a [arXiv:1909.12277]
back to top
27
![Page 28: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/28.jpg)
28
![Page 29: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/29.jpg)
29
![Page 30: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/30.jpg)
30
![Page 31: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/31.jpg)
31
![Page 32: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/32.jpg)
32
![Page 33: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/33.jpg)
33
![Page 34: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/34.jpg)
34
![Page 35: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/35.jpg)
35
![Page 36: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/36.jpg)
(2)
Brane charge in Equivariant Cohomotopy
implied by
Hypothesis H with Equivariant Hopf Degree Theorem
Sati-Schreiber 19a [arXiv:1909.12277]
back to top
36
![Page 37: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/37.jpg)
37
![Page 38: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/38.jpg)
38
![Page 39: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/39.jpg)
39
![Page 40: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/40.jpg)
40
![Page 41: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/41.jpg)
41
![Page 42: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/42.jpg)
42
![Page 43: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/43.jpg)
(3)
Brane charge in Differential Cohomotopy
implied by
Hypothesis H with May-Segal Theorem
Sati-Schreiber 19c [arXiv:1912.10425]
back to top
43
![Page 44: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/44.jpg)
Cohomotopycocycle space
ππππ
boldface!
4(X) :=
pointedmapping space
Maps∗/(X,S4
)
π0
(ππππ4(X)
)=
Cohomotopycohomology
classes
= πnot
boldface
4(X) Cohomotopyset
π1
(ππππ4(X)
)=
Cohomotopygauge
transformations
π2
(ππππ4(X)
)=
Cohomotopygauge of gaugetransformations
...
44
![Page 45: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/45.jpg)
Cohomotopy cocycle spacevanishing at ∞ on Euclidean 3-space
ππππ4((R3)cpt
) hmtpy'←−−−−assign unit charge
in Cohomotopyto each point
Conf(R3,D1
)May-Segal theorem
configuration space of pointsin R3 × R1
which are:1) unordered
2) distinct after projection to R3
3) allowed to vanish to ∞ along R1
=
R3×0 R3×∞R3×∞
projection to R3point
in R3×R1point
disappearedto infinity
along R1
45
![Page 46: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/46.jpg)
ππππ4((Rd)cpt ∧ (R4−d)+
)oo hmtpy'
hence: a form of differential Cohomotopy
ππππ4diff
((Rd)cpt ∧ (R4−d)+
):=
assigns configuration spaces:
Conf(Rd,D4−d)
46
![Page 47: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/47.jpg)
Lemma:
47
![Page 48: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/48.jpg)
=
Lemma:
48
![Page 49: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/49.jpg)
Consequence: assumingHypothesis H:
differential Cohomotopy cocycle spacereflecting D6⊥D8 -charges
49
![Page 50: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/50.jpg)
0) Nonabelian Differential Cohomology
Brane charge in...
1) Twisted Cohomotopy theory
2) Equivariant Cohomotopy theory
3) Differential Cohomotopy theory
implies...
4) Hanany-Witten Theory
5) Chan-Paton Data
6) BMN Matrix Model States
7) M2/M5 Brane Bound States
![Page 51: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/51.jpg)
(4)
Hanany-Witten Theory
implied by
Hypothesis H with Fadell-Husseini Theorem
Sati-Schreiber 19c [arXiv:1912.10425]
back to top
51
![Page 52: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/52.jpg)
higher co-observables on D6⊥D8 -intersections
H•
topologicalphase space
t[c]
Ωc ππππ4diff
' H•
(t
Nf∈NΩConf1,· · ·, Nf
(R3))
(by the above)
' ⊕Nf∈NApb
NfFadell-Husseini
theorem
are algebra of horizontal chord diagrams:
52
![Page 53: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/53.jpg)
Horizontal chord diagrams form algebra under concatenation of strands.
tik tij = tiktij
This is universal enveloping algebra of the infinitesimal braid Lie algebra (Kohno):
[tij, tkl] = 0
[tij, tik + tjk] = 0
53
![Page 54: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/54.jpg)
Consider the subspace of skew-symmetric co-observables,
denote elements as follows:
54
![Page 55: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/55.jpg)
In the subspace of skew-symmetric co-observables we find:
the 2T relationsbecome theordering constraint
=form of anynon-vanishing element
skew-symmetrybecomes thes-rule
the 4T relationsbecome thebreaking rule
55
![Page 56: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/56.jpg)
these are the rules of Hanany-Witten theoryfor NS5 ⊥ Dp ⊥ D(p + 2)-brane intersections
if we identify horizontal chord diagrams as follows:
56
![Page 57: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/57.jpg)
(5)
Chan-Paton data
implied by
Hypothesis H with Bar-Natan theorem
Sati-Schreiber 19c [arXiv:1912.10425]
back to top
57
![Page 58: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/58.jpg)
higher co-states on D6⊥D8 -intersections
H•
topologicalphase space
t[c]
Ωc ππππ4diff
' H•(t
Nf∈NConf1,· · ·, Nf
(R3))
(by the above)
' ⊕Nf∈NWpb
NfKohno & Cohen-Gitler
theorem
are horizontal weight systems:
58
![Page 59: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/59.jpg)
All horizontal weight systems w : Apb → C come from Chan-Paton data:1) metric Lie representations ρ 2) stacks of coincident strands 3) winding monodromies:
w
Bar-Natantheorem
59
![Page 61: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/61.jpg)
(6)
BMN Matrix Model States
implied by
Hypothesis H
back to top
61
![Page 62: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/62.jpg)
ρ ∈ su(2)C MetricReps equivalently identified with:
0) configuration of concentric fuzzy 2-spheres
1) fuzzy funnel state in DBI model for Dp⊥D(p + 2)
2) susy state in BMN matrix model for M2/M5
corresponding weight systems w(ρ,σ)
: Apb → C are:
0) radius fluctuation amplitudes of fuzzy 2-spheres
1)
2)invariant multi-trace observables in
DBI modelBMN model
62
![Page 63: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/63.jpg)
,0) Radius fluctuation observables on N -bit fuzzy 2-spheres S2
Nare N ∈ su(2)CMetricReps weight systems on chord diagrams:
63
![Page 64: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/64.jpg)
1,2) weight system wρ is the observable aspect of matrix model state ρ:
linear combinations offinite-dim suC-representations
Span(su(2)CMetricReps
)naive funnel- / susy-states of
DBI model / BMN matrix model
ρ 7→wρ//
weight systems onhorizontal chord diagrams
Wpb
states of DBI model / BMN matrix modeas observed by invariant multi-trace observables
64
![Page 65: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/65.jpg)
(7)
M2/M5 Brane Bound States
implied by
Hypothesis H
back to top
65
![Page 66: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/66.jpg)
Given a sequence of susy states in the BMN matrix model
M2/M5-brane state(finite-dim su(2)C-rep)︷ ︸︸ ︷
(V, ρ) := ⊕i︸︷︷︸
stacks of coincident branes(direct sum over irreps)
(M2/M5-brane charge in ith stack
(ith irrep with multiplicity)︷ ︸︸ ︷N
(M2)
i ·N(M5)
i
)∈ su(2)CMetricReps/∼
this is argued to converge to macroscopic M2- or M5-branesdepending on how the sequence behaves in the large N limit:
Stacks of macroscopic...
M2-branes M5-branes
If for all i N(M5)
i →∞ N(M2)
i →∞(
the relevantlarge N limit)
)
with fixed N(M2)
i N(M5)
i
(the number of coincident branes
in the ith stack
)
and fixed N(M2)i /N N
(M5)i /N
(the charge/light-cone momentum
carried by the ith stack
)
66
![Page 67: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/67.jpg)
Given a sequence of susy states in the BMN matrix model
M2/M5-brane state(finite-dim su(2)C-rep)︷ ︸︸ ︷
(V, ρ) := ⊕i︸︷︷︸
stacks of coincident branes(direct sum over irreps)
(M2/M5-brane charge in ith stack
(ith irrep with multiplicity)︷ ︸︸ ︷N
(M2)
i ·N(M5)
i
)∈ su(2)CMetricReps/∼
the large Nlimit does not exist
here:
Span(su(2)CMetricReps
) ρ 7→wρ//
butdoes exist in weight systems
Wpb
if we normalize by the scale of the fuzzy 2-sphere geometry:
4π 22n((N
(M5))2−1)1/2+n wN
(M5)
︸ ︷︷ ︸Single M2-brane state in BMN model
(multiple of suC-weight system)
∈ Wpb
states as seen by multi-trace observables(weight systems on chord diagrams)
67
![Page 68: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/68.jpg)
M2/M5-brane bound statesas emergent under Hypothesis H
68
![Page 69: Cohomotopy Theory and Branessmooth manifold cocycle space of di erential X-cohomology/ higher gauge elds 11. Geometry Topology Physics cl( ; loop L 1-algebra lX) L 1-algebra valued](https://reader035.vdocuments.mx/reader035/viewer/2022071011/5fc99c193f3db94e3d282f1c/html5/thumbnails/69.jpg)
End.
back to top
69