covariant quantum spaces, the ikkt model and gravity

Covariant Quantum Spaces, the IKKT Modeland Gravity

Harold Steinacker

Department of Physics, University of Vienna

Paris, march 2017

H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity

Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization

Motivation

need “fundamental” quantum theory of space-time and matter

issues with string theory:

compactification (why 4D? landscape?)constructive definition

proposal:

Matrix Models as fundamental theories of space-time & matter

IKKT model (IIB model) (this talk!), BFSS model (M-theory)

conjecture:4D physics arises on suitable “brane” solutionswithout target space compactification (avoid landscape!)

goal:identify promising “matrix geometries” for space-timeunderstand mechanism of 4D physics, 4D gravity



outline:

the IKKT model

covariant quantum spaces: fuzzy S4, generalizations

fluctuation modes & higher spins

geometry: metric, vielbein

realization in IKKT model:eom, (lineariz.) Einstein equations

discussion & outlook

HS, arXiv:1606.00769Marcus Sperling & HS, in preparation



The IKKT model

IKKT or IIB model Ishibashi, Kawai, Kitazawa, Tsuchiya 1996

S[Y ,Ψ] = −Tr(

[Y a,Y b][Y a′,Y b′

]ηaa′ηbb′ + Ψγa[Y a,Ψ])

Y a = Y a† ∈ Mat(N,C) , a = 0, ...,9, N large

gauge symmetry Y a → UY aU−1, SO(9,1), SUSY

proposed as non-perturbative definition of IIB string theory

origins:

quantized Schild action for IIB superstring

reduction of 10D SYM to point, N large

N = 4 SYM on noncommutative R4θ



numerical results:Kim, Nishimura, Tsuchiya arXiv:1108.1540 ff

”expanding universe“, 3+1-dim. space-time emerges

time evolution of size R(t):

3+1 large dimensions



leads to “matrix geometry”: (≈ NC geometry)

S ∼ Tr [X a,X b]2 ⇒ configurations with small [X a,X b] 6= 0should dominate

i.e. “almost-commutative” configurations, geometry

∃ basis of quasi-coherent states |x〉, (overcomplete)

minimize∑

a〈x |∆X 2a |x〉 = O([X a,X b]) � (X a)2,

X a ≈ simult. diagonal,spectrum =: M ⊂ R10

〈x |X a|x ′〉 ≈ δ(x − x ′)xa, x ∈M

embedding of branes in target space R10

X a ∼ xa : M ↪→ R10



examples of matrix geometries:

2× 2 matrices

X a =

(xa

(1)

xa(12)

xa(21)

xa(2)

)= xa

(1)|1〉〈1|+ xa(2)|2〉〈2|, a = 1, ...,D

+xa(12)|2〉〈1|+ xa

(2)|1〉〈2|

describe two points at x(1), x(2) ∈ RD

• • (“point branes”)

off-diagonal matrices ≈ strings connecting branes

spectrum of X a ↔ location in RD

choose X 3 = σ3 diagonal, & off-diagonal X 1 = σ1, X 2 = σ2

→ minimal fuzzy sphere S22 ↪→ R3



examples of matrix geometries:

2× 2 matrices

X a =

(xa

(1) xa(12)

xa(21) xa

(2)

)= xa

(1)|1〉〈1|+ xa(2)|2〉〈2|, a = 1, ...,D

+xa(12)|2〉〈1|+ xa

(2)|1〉〈2|

describe two points at x(1), x(2) ∈ RD

• 66(( • (“point branes”)

off-diagonal matrices ≈ strings connecting branes

spectrum of X a ↔ location in RD

choose X 3 = σ3 diagonal, & off-diagonal X 1 = σ1, X 2 = σ2

→ minimal fuzzy sphere S22 ↪→ R3



The fuzzy 2-sphere S2N : Hoppe, Madore

choose X a = Ja(N) ... irrep of SU(2) on H = CN

[X a,X b] = iεabcX c , X aXa =14

(N2 − 1) =: R2N .

quantized symplectic space (S2, ωN)

X a ∼ xa : S2 ↪→ R3

fully covariant under SO(3)

functions on S2N : A = End(H) =

N−1⊕l=0

(2l + 1)︸︷︷︸Y l

m



The Moyal-Weyl quantum plane R4θ:

[X a,X b] = iθab 1l ... Heisenberg algebra

quantized symplectic space (R4, ω)

admits translations, breaks rotations

functions on R4θ: A = End(H) 3 φ =

∫d4k eikX φ(k)

any quantized symplectic spaceM⊂ RD

matrices X a = quantized embedding maps

X a ∼ xa : M ↪→ RD



(quasi)coherent states & string states

given matrix geometry X a and point xa ∈ RD

quasi-coherent state = ground state |x〉 of∑a

(X a − xa)2|x〉 = E(x)|x〉

Berenstein - Dzienkowski arXiv:1204.2788, Ishiki arXiv:1503.01230,Schneiderbauer - HS arXiv:1606.00646

string state: |x〉〈y | ∈ End(H)

strings connecting different branes ↔ block-diagonal matrices

Xa =

(X a |x〉〈y ||y〉〈x | Y a

)string states dominate quantum effects! (UV/IR mixing)



can measure matrix geometries {X a}:

measure energy E(x) of string connectingM with point x ∈ RD

location ofM⊂ RD ↔ minima of E(x),

Mathematica package “Bprobe” DOI 10.5281/zenodo.45045Schneiderbauer - HS arXiv:1606.00646

examples:

squashed fuzzy CP2N ⊂ R6 fuzzy torus T 2

N



in IKKT model:IIB supergravity interactions between branesarise upon integrating out off-diagonal “strings”

X a =

(∗

∗

)→ correct ∼ 1

(x−y)8 propagators (gravitons, ...) in R10

IKKT, Kabat-Taylor, van Raamsdonk, Chepelev-Tseytlin,...

H.S. arXiv:1606.00646



back to IKKT model: perturbative approach:

background = set of matrices (2 + µ2

2 )X a = 0, 2 = [X a, [Xa, .]]

X a ∼ xa : M ↪→ R10

add fluctuations Y a = X a +Aa

expand action to second oder in Aa

S[Y ] = S[X ] +2g2 TrAa

((2 +

12µ2)δa

b + 2[[X a,X b], . ]− [X a, [X b, .]])Ab

fluctuations A describe

gauge theory (NCFT) onM (”open strings“ ending onM)

effective metric Gµν(x) ∼ θµµ′(x)θνν

′(x)gµ′ν′(x) , dynamical

⇒ dynamical geometry, ”emergent gravity“ onM(6= 10D gravity!!) (review: H.S. arXiv:1003.4134 )

cf. Rivelles, H-S. YangH. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity


how to choose the background braneM⊂ R10?

quantized 4D symplecticM4 ⊂ R10 :

θµν breaks local Lorentz-invariance”invisible“ semi-classically, significant in quantum corrections

look for Lorentz / SO(4) covariant 4D quantum space:

∃ fully covariant fuzzy four-sphere S4N

Grosse-Klimcik-Presnajder; Castelino-Lee-Taylor; Ramgoolam; Kimura;

Hasebe; Azuma-Bal-Nagao-Nisimura; Karabail-Nair; ...

price to pay: “internal structure” → higher spin theory

here:work out lowest spin modes on S4

Λ in IKKT model→ (linearized) Einstein equations HS arXiv:1606.00769



covariant fuzzy four-spheres

5 hermitian matrices Xa, a = 1, ...,5 acting on HN∑a

X 2a = R2

covariance: Xa ∈ End(HN) transform as vectors of SO(5)

[Mab,Xc ] = i(δacXb − δbcXa),

[Mab,Mcd ] = i(δacMbd − δadMbc − δbcMad + δbdMac) .

Mab ... so(5) generators acting on HN

denote[X a,X b] =: iΘab

particular realization so(6) ∼= su(4) generatorsMab:

X a = rMa6, a = 1, ...,5 , Θab = r2Mab

(cf. Snyder, Yang 1947)



covariant fuzzy four-spheres

5 hermitian matrices Xa, a = 1, ...,5 acting on HN∑a

X 2a = R2

covariance: Xa ∈ End(HN) transform as vectors of SO(5)

[Mab,Xc ] = i(δacXb − δbcXa),

[Mab,Mcd ] = i(δacMbd − δadMbc − δbcMad + δbdMac) .

Mab ... so(5) generators acting on HN

denote[X a,X b] =: iΘab

particular realization so(6) ∼= su(4) generatorsMab:

X a = rMa6, a = 1, ...,5 , Θab = r2Mab

(cf. Snyder, Yang 1947)H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity


basic fuzzy 4-sphere S4N :

Grosse-Klimcik-Presnajder 1996; Castelino-Lee-Taylor

Ramgoolam; Medina-o’Connor, Dolan, ...

choose HN = (0,0,N)so(6)∼= (C4)⊗SN

satisfiesXaXa = R21l, R2 ∼ 1

4 r2N2

εabcdeXaXbXcXdXe = (N + 2)R2r3 (volume quantiz.)

generalized fuzzy 4-spheres S4Λ:

H.S, arXiv:1606.00769, M. Sperling - H.S in preparation

choose e.g. HΛ = (n,0,N)so(6)

... thick sphere; R2 := XaXa not sharp

bundle over S4N with fiber CP2

n (→ fuzzy extra dim’s!)



local description: pick north pole p ∈ S4

→ tangential & radial generators

X a =

(Xµ

X 5

), xµ ∼ Xµ, µ = 1, ...,4...tangential coords at p

separate SO(5) into SO(4) & translations

Mab =

(Mµν Pµ−Pµ 0

)where Pµ =Mµ5

rescalePµ =

1R

gµνPν (cf. Wigner contraction)

algebra[Pµ,X ν ] ' iδνµ,

[Pµ,Pν ] = iR2Mµν → 0

[Xµ,X ν ] =: iθµν = ir2Mµν ≈ 0



start with basic S4N : HN = (0,0,N)so(6)

∼= (0,N)so(5)∼= (C4)⊗SN

geometry from coherent states |p〉:

{pa = 〈p|Xa|p〉} = S4

minimal uncertainty

〈p|∑

(∆Xa)2|p〉 ≈ 4R2

N=: ∆2

closer inspection:

∃ degenerate space of coherent states at p ∈ S4

→ “internal” fuzzy S2N+1 fiber



hidden bundle structure:

CP3 3 ψ↓ ↓S4 3 ψγ iψ = x i

Ho-Ramgoolam, Medina-O’Connor, Abe, ...

fuzzy S4N is really fuzzy CP3

N , hidden extra dimensions S2 !

local Poisson tensor (↔ commutation relations)

−i[Xµ,X ν ] ∼ θµν(x , ξ)

rotates along fiber ξ ∈ S2 !

is averaged [θµν(x , ξ)]0 = 0 over fiber → local SO(4) preserved,

4D “covariant” quantum space



fields and harmonics on S4N

”functions“ on S4N :

φ ∈ End(HN) ∼=⊕

m≤n≤N

(n −m,2m)so(5)

(n,0) modes = scalar functions on S4:

φ(X ) = φa1...an X a1 ...X an

(n,2) modes = selfdual 2-forms on S4

φbc(X )Mbc = φa1...an;bcX a1 ...X anMbc ( ∼= φbc(x)dxb ∧ dxc )

etc.tower of higher spin modes

from ”twisted“ would-be KK modes on S2

(local SO(4) acts non-trivially on S2 fiber)H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity


Fluctuation modes on S4Λ

organize tangential fluctuations at p ∈ S4 as

Aµ = θµνAν

where

Aν(x) = Aν(x) +Aνρ(x)Pρ + Aνρσ(x)Mρσ︸︷︷︸AνabMab ...SO(5) connection

+...

rank 2 tensor field

Aνρ(x) =12

(hνρ + aνρ) hνρ = hρν ... metric fluctuation

rank 3 tensor field

Aνρσ(x)Mρσ ... so(4) connection

rank 1 field Aν(x) ... gauge fieldH. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity


gauge transformations:

Y a → UY aU−1 = U(X a +Aa)U−1 leads to (U = eiΛ)

δAa = i[Λ,X a] + i[Λ,Aa]

expand

Λ = Λ0 +12

ΛabMab + ...

... U(1)× SO(5)× ... - valued gauge trafosdiffeos from δv := i[vρPρ, .]

δhµν = (∂µvν + ∂νvµ)− vρ∂ρhµν + (Λ · h)µν

δAµρσ = 12∂µΛσρ(x)− vρ∂ρAµρσ + (Λ · A)µρσ

etc.



metric and vielbein

consider scalar field φ = φ(X ) (= transversal fluctuation Aa(X ))

kinetic term

−[Xα, φ][Xα, φ] ∼ eαφeαφ = γµν∂µφ∂νφ,

vielbeineα := {Xα, .} = eαµ∂µ

eαµ = θαµ

note: Poisson structure → frame bundle!metric (open string)

γµν = gαβeαµ eβν = 14 ∆4 gµν

averaging over internal S2:

[eαν ]0 = 0, [γµν ]0 = γµν =∆4

4gµν ... SO(5) invariant !



perturbed vielbein:

Y a = X a +Aa

ea := {Y a, .} ∼ eaµ[A]∂µ ... vielbein

eαµ[A] ∼ θαβ(δµβ + Aβρgρµ) + 1r2 θ

ανθρσ{Aνρσ, xµ}

using {Pρ,Xµ} ∼ gρµ (!)

effective (open string!) metric (drop radial vielbein)

−[Y a, φ][Ya, φ] ∼ eaφeaφ = γµν∂µφ∂νφ+ ...

γµν ∼ eαµ[A]e να [A]



linearize & average over fiber →

γµν = γµν + [δγµν ]0

metric fluctuation:

[δγµν ]0 = ∆4

4

(hµν + kµν

)=: ∆4

4 hµν

kµν ∼ ∂ρAµρν + ∂ρAνρµ

note:

metric hµν = combination of

momentum modes hµν

kµν ∼ ∂ρAµρν + (µ↔ ν)



effective metric & conformal factor

kinetic term for scalar fields:

S[φ] ∼∫M

d4x γµν∂µφ∂νφ ∼∫M

d4x√|Gµν |Gµν∂µϕ∂νϕ

effective metric

Gµν = c∆4 γ

µν , c =√

∆4|γ−1µν | = 1− 1

2 h + ...

metric fluctuation

Gµν = gµν + Hµν , Hµν = hµν − 12

gµν h

de Donder gauge

∂µHµν − 12∂νH = 0



action for gravitational modes:

quadratic action

S2[A] ∼∫M

d4x(

L2R

(Aµν(2 + 8r2 + 1

2µ2)Aµν

+R4Aνσρ(2 + 8r2 + r2P0 + 1

2µ2)Aνσρ + 2αR4 r2 Aνσρ∂σAνρ︸︷︷︸

mixing

)

where α = 1 and

L2R = ∆4[pµpµ]0 ... thickness of S4

Λ

Λ = (n1,n2,N)

need L2R > 0, generalized fuzzy sphere S4

Λ with Λ = (n,0,N), n > 0



coupling to matter:

δAS[matter] ∼∫M

d4x hµνTµν

equations of motion

(2 + 12µ

2)Aµν =g2

YM∆4

4L2R

Tµν + 4αR2

L2R∂σAµσν(

2 + µ2

2

)Aνσρ = (PSD)

(4αR2 ∂σAνρ +

4g2YM∆4

R2 ∂σTνρ).

PSD ... projector on SD antisymm.

set aµν = 0 (decouples)

mixing α → two scaling regimes:



regime G: p2 � α2

L2R

= m2 : neglect mass terms,

... ”gravity” regime (requires “thick” fuzzy sphere S4Λ !)

e.o.m.2 hµν ∝ Tµν

→ linearized Einstein equations

regime C: extreme IR: p2 � α2

L2R

= m2, (“cosmological”)

kµν ∝ Tµν2hµν ≈ 0

mass terms dominate, no propagation, no gravity

or: α = 0, no mixingarises for momentum embedding (below), or self-dual action(add FµνFρσεµνρσ )



linearized curvature and Einstein equations

assume regime G

lin. Einstein tensor

Gµν [g + H] ≈ − 3R2 gµν +

12∂ · ∂hµν

drop background curvature ∼ 1R2 (& local effects)

Gµν ≈ 8πGNTµν (linearized)

Newton constant

GN =(

(α +32

)(α

3+ 1)− 4L2

Rµ2)g2

YM∆4

48πL2R

.



gravity mechanism requires generalized fuzzy sphere S4Λ with

a) thick fuzzy sphere with√

N � n� N

b) or decoupling of ∂σAµσν via α = 0

thick S4Λ:

fuzzy extra dimensions (squashed CP2)H.S. arXiv:1504.05703 , H.S. & J. Zahn, arXiv:1409.1440

interesting for particle physics (chiral gauge thy)

dim. red. to 4D (mass gap?!)

IR cutoff for gravity: only wavelengths ≤ LRα gravitate

details (IR cutoff) depend on representation Λ

(in progress)



alternative: momentum embedding in M.M.:

Yµ = Pµ, µ = 1, ...,4

fluctuations

Yµ = Pµ + δYµ =(δµν + hµν(x)

)Pν + ...

... avoids mixing term α, no mass term for graviton (Goldstone boson)

(work in progress w/ M. Sperling)



generalized spheres from coadjoint SO(6) orbits

(co)adjoint orbits

O[Λ] = {g · HΛ · g−1; g ∈ SO(6)} ⊂ so(6) ∼= R15

embedding functions mab : O[Λ] ↪→ R15 defined by

mab = tr(X Σab), a,b = 1, ...,6, X ∈ O[Λ]

Σab ... generators of so(6)

project to R5 via the projection

Π : O[Λ] ⊂ R15 → R5

only xa = ma6 describe embedding in R5

xa = tr(XΣa6) = −12

tr(Xγa)



basic sphere S4N : Λ = (0,0,N)so(6), HΛ = N|ψ0〉〈ψ0|

O[Λ] = SU(4)/SU(3)× U(1) ∼= CP3

projection to 4-sphere (=Hopf map):

CP3 3 ψ↓ ↓S4 3 ψγaψ = xa

xaxa = R2N

mµν selfdual, describes S2

CP3 is a S2 bundle over S4.



generalized sphere S4Λ: Λ = (n,0,N)so(6)

HΛ = N|ψ0〉〈ψ0|+ n|ψ1〉〈ψ1|

Let P ... spectral projection n→ 0

O[Λ]

P ↓O[NΛ1] ∼= CP3 xa

→ S4 ↪→ R5 .

S4Λ = CP2 bundle over CP3 ∼= S4 × S2

R2 ... non-trivial spectrum, ”thick“ 4-sphere embedded in R5

[R2,X b] 6= 0.



quantized (“fuzzy”) coadjoint orbits S4Λ

replace functions mab on O[Λ] by generatorsMab acting on HΛ,

Λ ... (dominant) integral weight.

End(HΛ) ... quantized algebra of functions

cf. E. Hawkins, q-alg/9708030

same geometry in target space as O[Λ]

semi-classical limit: [., .]→ i{., .}



Quantization

defined by

e−Γeff[X ] =

∫dAdΨe−S[X+A,Ψ]

well-behaved due to max. SUSY

one-loop results: using string states H.S. arXiv:1606.00646

TrEnd(H)O =(dimH)2

(VolM)2

∫M×M

dxdy(|x〉〈y |)O(|y〉〈x |) .

stabilization of S4N via vacuum energy, bare mass µ2 > 0

(cf. H.S. : arXiv:1510.05779)

tangential fluctuations:

Γ1−loop[F2] = −8π2

3(dimH)2

(VolM4)2

∫M4

dx Fµν− (ξ)F−µν(ξ)

... anti-selfdual, absorbed via αH. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity


summary:

4D covariant quantum spaces in IKKT model→ tower of higher spin modesPoisson structure → frame bundlethick fuzzy sphere S4

Λ → ≈ (lin.) 4-D Einstein equations( provided dim. red. )

classical mechanism, protected by max. SUSY

IR modifications (“cutoff”), additional modes

many open issues(mass gap, Minkowski, non-lin. regime, conf. factor, ...)



outlook

more general embeddings of S4Λ: (ongoing w/ M. Sperling)

fuzzy extra dim’s→ mass gap, 4D ?!momentum embeddings Pa

chiral gauge theory expected

new view on string theory: no target space compactification !4D gravity on brane independent of bulk gravity

issues, open questions:

nonlinear case, higher spin modes, fermions, ...more complete mode expansion

Lorentzian case



→ equations of motion

L2R(2 + 8r2 + 1

2µ2)Aµν = − g2

YM∆8

16 Tµν + αR2∆4∂σAµσν

R2(2 + 8r2 + µ2

2

)Aνσρ = (PSD)σ

′ρ′

σρ

(− α∆4∂σ′Aνρ′ + g2

YM∆8∂σ′Tνρ′)

(2 + 4r2 + 12µ

2)κ = − g2YM∆8

8R T .

P0SD = 1

4 (δδ − δδ + ε) ... SD antisymmetric projector

neglect radial fluctuations κ, set aµν = 0

use 2 = [X a, [Xa, .]] ∼ ∆4

4 ∂ · ∂

→ equations for kµν ∼ ∂ρAµρν and hµν :



separate kµν into a “local” and a propagating ”gravitational“ part,

kµν = k (loc)µν + k (grav)

µν , k (loc)µν ∝ −Tµν .

eom

(∂ · ∂ −m2k ) k (grav)

µν =g2

YM∆4

3

(α

2L2R−m2

k

)Tµν + 4α

3 m2 hµν ,

(∂ · ∂ − 4m2)hµν =g2

YM∆4

2L2R

Tµν − αL2

Rkµν

where m2k := 4m2 − α2

3L2R≥ 0

formal solutions:

k (grav)µν =

g2YM∆4

3L2R

(α3

(α + 3

2

)− 4L2

Rm2)

12g−m2

kTµν

hµν =g2

YM∆4

3L2R

( (α+ 3

2

)2g−4m2 Tµν −

α

L2R

(α3

(α+ 3

2

)−4L2

Rm2

)(2g−m2

k )(2g−4m2)Tµν

).

effective gravitational metric

h(grav)µν := hµν + k (grav)

µν

(drop ”contact term“ k (loc)µν )


covariant quantum spaces, the ikkt model and gravity

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