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
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
(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)
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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!)
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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.
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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 !
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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 !
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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]
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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)
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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:
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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 )
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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 )
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
.
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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)
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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)
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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)
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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)
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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.
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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.
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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{., .}
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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, ...)
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
→ 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 :
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity
Motivation IKKT model Matrix geometry Fuzzy S4N fields & kinematics Fluctuations Gravity Generalized spheres Quantization
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 )
H. Steinacker Covariant Quantum Spaces, the IKKT Model , and Gravity