Download - NYU AD 2020 H. Sati
Urs Schreiber
(New York University, Abu Dhabi & Czech Academy of Science, Prague)
Microscopic brane physics from Cohomotopy
talk atM-Theory and Mathematics
NYU AD 2020
based on joint work with
H. Sati
ncatlab.org/schreiber/show/Microscopic+Brane+Physics+from+Cohomotopy
0) Introduction
1) Microscopic Brane Charge
2) Orientifold Tadpole Cancellation
3) D6⊥D8D6⊥D8D6⊥D8 -Brane Intersections
4) Hanany-Witten Theory
5) Chan-Paton Data
6) BMN Matrix Model States
7) M2/M5 Brane Bound States
(0)
Introduction
Open Problem M and Hypothesis H
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Open problem QCD
Confined QCD“Millennium problem”- QCD-cosmology- nucleosynthesis- form factors- · · ·
Flavored QCD“Flavor problem”
Hig
gsse
ctor - cosm. constant
- EW hierarchy- vacuum stability- Vcb-puzzle- flavour anomalies
Leptoquark=====⇒ GUT
- · · ·
Solution
???
Open problem QCD’t Hooft doubling
Confined QCD“Millennium problem”- QCD-cosmology- nucleosynthesis- form factors- · · ·
Flavored QCD“Flavor problem”
Hig
gsse
ctor - cosm. constant
- EW hierarchy- vacuum stability- Vcb-puzzle- flavour anomalies
Leptoquark=====⇒ GUT
- · · ·
braneworld geometric engineering
AdS/QCD correspondence
Solution
??? ???
IIA super-gravitynear blackintersecting branes
flavor branes
colo
rbra
nes
. . .
Open problem QCDsmall Nc ’t Hooft doubling
Nc = 3quark colors
Confined QCD“Millennium problem”- QCD-cosmology- nucleosynthesis- form factors- · · ·
Flavored QCD“Flavor problem”
Hig
gsse
ctor - cosm. constant
- EW hierarchy- vacuum stability- Vcb-puzzle- flavour anomalies
Leptoquark=====⇒ GUT
- · · ·
braneworld geometric engineering
AdS/QCD correspondence
Solution
??? ???
IIA super-gravitynear blackintersecting branes
flavor branes
colo
rbra
nes
. . .
Open problem QCDsmall Nc ’t Hooft doubling
Nc = 3quark colors
Confined QCD“Millennium problem”- QCD-cosmology- nucleosynthesis- form factors- · · ·
Flavored QCD“Flavor problem”
Hig
gsse
ctor - cosm. constant
- EW hierarchy- vacuum stability- Vcb-puzzle- flavour anomalies
Leptoquark=====⇒ GUT
- · · ·
braneworld geometric engineering
AdS/QCD correspondence
Solution
??? ???
M-theorywith microscopicintersecting branes
flavor branes
colo
rbra
nes
. . .
Open problem QCD Open problem Msmall Nc ’t Hooft doubling
Confined QCD“Millennium problem”- QCD-cosmology- nucleosynthesis- form factors- · · ·
Flavored QCD“Flavor problem”
Hig
gsse
ctor - cosm. constant
- EW hierarchy- vacuum stability- Vcb-puzzle- flavour anomalies
Leptoquark=====⇒ GUT
- · · ·
braneworld geometric engineering
AdS/QCD correspondence
Solution
???
Hyp
othe
sis
H
M-theorywith microscopicintersecting branes
flavor branes
colo
rbra
nes
. . .
Open problem QCD Open problem Msmall Nc ’t Hooft doubling
Open problem QCD Open problem Msmall Nc ’t Hooft doubling
first consider large g2YMNc
+3 11d supergravity
Open problem QCD Open problem Msmall Nc ’t Hooft doubling
first consider large g2YMNc
+3 11d supergravity
CovariantPhaseSpace11d SuGra
=
spacetimes
formalized as: GADE
–orbi R10, 1|32
–foldssuper-orbifold
(X )
equipped with: 0) gravity 1) C-field
formalized as: Pin+-structuresuper-vielbein
(E,Ψ) differential formsflux densities
(G4, G7)
subject to: Einstein equations Page equationG = T (Ψ, G4, G7) dG7 + 1
2G4 ∧G4 = 0
Open problem QCD Open problem Msmall Nc ’t Hooft doubling
first consider large g2YMNc
+3 11d supergravity
CovariantPhaseSpace11d SuGra
=
spacetimes
formalized as: GADE
–orbi R10, 1|32
–foldssuper-orbifold
(X )
equipped with: 0) gravity 1) C-field
formalized as: Pin+-structuresuper-vielbein
(E,Ψ) differential formsflux densities
(G4, G7)
subject to: Einstein equations Page equation
equivalent to: super-torsion = 0Candiello-Lechner 93, Howe 97
flux is in rationalizedJ-twisted CohomotopySati 13, Fiorenza-Sati-S. 19a
Open problem QCD Open problem Msmall Nc ’t Hooft doubling
& large g2YMNc
+3 11d supergravitycharge-quantizedin Cohomotopy
CovariantPhaseSpace11d SuGra
=
spacetimes
formalized as: GADE
–orbi R10, 1|32
–foldssuper-orbifold
(X )
equipped with: 0) gravity 1) C-field
formalized as: Pin+-structuresuper-vielbein
(E,Ψ) differential formsflux densities
(G4, G7)
subject to: Einstein equations Page equation
equivalent to: super-torsion = 0Candiello-Lechner 93, Howe 97
flux is in rationalizedJ-twisted CohomotopySati 13, Fiorenza-Sati-S. 19a
Hypothesis HSati 13, Fiorenza-Sati-S. 19b,19c
CovariantPhaseSpaceM-Theory
=
spacetimes
formalized as: GADE
–orbi R10, 1|32
–foldssuper-orbifold
(X )
equipped with: 0) gravity 1) C-field
formalized as: Pin+-structuresuper-vielbein
(E,Ψ) differential formsflux densities
(G4, G7)
subject to: Einstein equations Page equation
formalized as: super-torsion = 0flux is in rationalizedJ-twisted CohomotopyFSS 19b,19c, SS 19a,19b,19c
Discrepancies?
Salvageable?
Looks like M-theory?
Compare implicationsof Hypothesis Hto M-folklore.
today:
Adjust fine-printin Hypothesis H(e.g. differential refinement)
Solve1) Millennium problem2) Vacuum selection problem3) Flavor problem
...
(for later)
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Implications of Hypothesis H
on on on
curved but smoothspacetimes
flat but orbi-singularspacetimes
spacetimeswith horizons
FSS 19b, FSS 19c, GS20 BSS 18, SS 19a SS 19c
topologicalanomaly cancellation:
equivariantanomaly cancellation
Dp ⊥ D(p + 2)worldvolume QFT
- shifted C-field flux quantization- C-field tadpole cancellation- M5 Hopf-WZ level quantization- DMW anomaly cancellation- C-field integral eom...
- M5/MO5 anomaly cancellation- RR-field tadpole cancellation- no irractional D-brane charge
- fuzzy funnels- BLG 3-algebras- BMN matrix model- M2/M5 bound states- AdS3-holography- Coulomb branch indices- Hanany-Witten rules- ...
H. Sati’s talkD. Fiorenza’s talk my talk
at M-Theory and Mathematics 2020
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(1)
Microscopic Brane Charge
implied by
Hypothesis H with Pontrjagin-Thom Theorem
Sati-Schreiber 19a [arXiv:1909.12277]
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(2)
Orientifold Tadpole Cancellation
implied by
Hypothesis H with Equivariant Hopf Degree Theorem
Sati-Schreiber 19a [arXiv:1909.12277]
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The following four slides showtechnical detail of therealization of this mechanism forMO5-planes atADE-singularities in heterotic M-theory
Skip over technical detailahead to section (3).
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(3)
D6⊥D8 -brane intersections
implied by
Hypothesis H with May-Segal Theorem
Sati-Schreiber 19c [arXiv:1912.10425]
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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
...
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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
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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)
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Lemma:
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=
Lemma:
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Consequence: assumingHypothesis H:
differential Cohomotopy cocycle spacereflecting D6⊥D8 -charges
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(4)
Hanany-Witten Theory
implied by
Hypothesis H with Fadell-Husseini Theorem
Sati-Schreiber 19c [arXiv:1912.10425]
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higher co-observables on D6⊥D8 -intersections
H•
topologicalphase space
t[c]
Ωc ππππ4diff
' H•
(t
Nf∈NConf1,· · ·, Nf
(R3))
(by the above)
' ⊕Nf∈NApb
NfFadell-Husseini
theorem
are algebra of horizontal chord diagrams:
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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
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Consider the subspace of skew-symmetric co-observables,
denote elements as follows:
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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
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these are the rules of Hanany-Witten theoryfor NS5 ⊥ Dp ⊥ D(p + 2)-brane intersections
if we identify horizontal chord diagrams as follows:
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(5)
Chan-Paton data
implied by
Hypothesis H with Bar-Natan’ theorem
Sati-Schreiber 19c [arXiv:1912.10425]
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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:
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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
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(6)
BMN Matrix Model States
implied by
Hypothesis H
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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
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,0) Radius fluctuation observables on N -bit fuzzy 2-spheres S2
Nare N ∈ su(2)CMetricReps weight systems on chord diagrams:
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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
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(7)
M2/M5 Brane Bound States
implied by
Hypothesis H
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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
)
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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)
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M2/M5-brane bound statesas emergent under Hypothesis H
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End.
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