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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