Matrix model description of gauge theory (and gravity?)

Transcription

1 Matrix model description of gauge theory (and gravity?) Harold Steinacker Department of Physics, University of Vienna Humboldt Kolleg, Corfu 2015

2 Motivation aim: quantum theory of fundamental interactions incl. gravity classical geometry breaks down at Planck scale, expect quantum structure of space-time how? { quantize gravity d.o.f. quantize other, pre-geometric dof: emergent gravity (this talk)

3 emergent NC geometry admit dynamical geometry need models with well-behaved under quantization QFT on quantum geometries Matrix Models {}}{ NC gauge theory dynamical geometry simple, far-reaching, pre-geometric good properties of string theory, clear-cut definition generic feature: UV/IR mixing ONE model singled out: N = 4 SYM IKKT model

4 Matrix Models as fundamental theory 1996: BFSS model, IKTT model proposed as non-perturbative definition of M-theory / IIB string theory focus on IKKT: Ishibashi, Kawai, Kitazawa, Tsuchiya 1996 ( S[X, Ψ] = Tr [X a, X b ][X a, X b ]η aa η bb + Ψγ ) a [X a, Ψ] X a = X a Mat(N, C), a = 0,..., 9 (N ) gauge symmetry X a UX a U 1, SO(9, 1), SUSY { 1) nonpert. def. of IIB string theory (on R 10 ) (IKKT ) 2) N = 4 SUSY Yang-Mills gauge thy. on noncommutative R 4 θ governs simultaneously (quantum) space(time) & physics on it geometry, gauge theory,... emerge no need to invent new math!

5 quantization of matrix model: Z = dx a dψ e S[X,Ψ] Tr([X, X]...)Tr(...) = 1 Z dx a dψ Tr([X, X]...)Tr(...)e S[X,Ψ] gauge invariant, non-perturbative integral well-def in Euclidean case Krauth, Staudacher 1998 proposal for regularization in Minkowski case Monte-Carlo studies: Kim, Nishimura, Tsuchiya arxiv: ff evidence for expanding universe behavior, 3+1 dimensions includes integral over geometries! new techniques: eigenvalue distribution renormalization, phase trans. H.S. hep-th/ , A. Polychronakos arxiv: , Tekel arxiv: RG analysis (Grosse-Wulkenhaar), multiscale analysis (Rivasseau,... )

6 perturbative approach: choose background solution (e.g. R 4 θ ) fluctuations around R 4 θ : NC gauge theory, Filk rules, (non-)planar diagrams,... most models: strong UV/IR mixing, non-renormaliz. ONE model well-behaved (perturbatively finite?!): N = 4 NC SYM on R 4 θ (IKKT) model, in 9+1 dimensions

7 physical meaning of X a : quantized embedding function X a x a : M R 10 consistent with: spectrum of X a... possible locations in x a - directions [X a, X b ] 0 non-locality, uncertainty X a for optimally localized states = coherent states

8 quantized Poisson (symplectic) manifolds (M, θ µν (x))... 2n-dimensional manifold with Poisson structure Its quantization M θ is NC algebra such that Q : C(M) A End(H) such that Q(f ) Q(g) = Q(fg) + O(θ) [Q(f ), Q(g)] = Q(i{f, g}) + O(θ 2 ) Φ = Q(φ) End(H) quantized function φ(x) on M semi-class: in particular: (2π) n Tr Q(φ) ω n φ(x) X a x a : M R 10

9 Example: the fuzzy sphere SN 2 classical S 2 : } x a : S 2 R 3 x a x a A = C = 1 (S 2 ) fuzzy sphere S 2 N : (Hoppe, Madore) algebra A = Mat(N, C)... alg. of functions on S 2 N SO(3) action: su(2) A A (J a, φ) [π N (J a ), φ] decompose A = Mat(N, C) into irreps of SO(3): A = Mat(N, C) = (N) ( N) = (1) (3)... (2N 1) = {Ŷ 0 0} {Ŷ m} 1... {Ŷ m N 1 }.... fuzzy spherical harmonics; UV cutoff X a = π N (J a ), X a X a = R 2 1l

10 basic solutions of M.M: branes e.o.m.: δs = 0 X X b [X a, [X a, X b ]] = 0 basic solutions: (allow N ) (assume Ψ = 0) flat branes R 2n θ embedded in R10 ( ) X X a µ = c i 1l, µ = 1,..., 2n [X µ, X ν ] = iθ µν 1l Moyal-Weyl quantum plane... quantized symplectic space (R 2n, ω) ω = 1 2 θ 1 µν dx µ dx ν... Heisenberg algebra, interpreted as space of functions on R 4 θ uncertainty relations X µ X ν θ µν Weyl quantization e ikµx µ e ikµx µ

11 embedding recovered from optimally localized states: coherent states p M = {x a = p X a p } = R 2n R 10 p ( X µ ) 2 p θ min

12 generic (curved) branes M 2n = deformed R 2n θ ( ) x X a x a µ = φ(x µ : M ) 2n R quantized embedding map (M 2n, ω)... quantized symplectic manifold embedded in R 10 ω = 1 2 θ 1 µν (x)dx µ dx ν fluctuations X a = X a + A a (X) describe NC gauge theory, dynamical eff. metric D-brane with B-field in string thy, open string metric

13 less trivial examples: squashed fuzzy CPN 2 (self-intersecting brane) X a = π (N,0) (T a ) x a : CP 2 R 8 Π R 6... SU(3) ladder op s H.S., J. Zahn arxiv: H.S, L. Schneiderbauer quantized symplectic manifold, degenerate embedding strings connecting sheets, stringy geometry stabilized in M.M. e.g. by cubic potential

14 degenerate solutions: fuzzy S 4 N X a = ˆΓ a = c α(γ a ) α β cβ on (C 4 ) SN (Γ a...so(5) Clifford) X a X a = R 2 1l Castellino, Lee, Taylor hep-th/ ; Ramgoolam,... in fact S 4 N = CP3 N /S2 Medina,O Connor hep-th/ [X a, X b ] = M ab, [M ab, X c ] = i(δ ac X b δ bc X a ), etc. fully covariant under SO(5) (cf. Snyder space) symplectic structure averaged away over fiber S 2 (cf. Doplicher Fredenhagen Roberts 1995) fluctuations X a = X a + A a (X, M) describe NC higher spin theory (?)

15 degenerate solutions: fuzzy S 4 N X a = ˆΓ a = c α(γ a ) α β cβ on (C 4 ) SN (Γ a...so(5) Clifford) X a X a = R 2 1l Castellino, Lee, Taylor hep-th/ ; Ramgoolam,... in fact S 4 N = CP3 N /S2 Medina,O Connor hep-th/ [X a, X b ] = M ab, [M ab, X c ] = i(δ ac X b δ bc X a ), etc. fully covariant under SO(5) (cf. Snyder space) symplectic structure averaged away over fiber S 2 (cf. Doplicher Fredenhagen Roberts 1995) fluctuations X a = X a + A a (X, M) describe NC higher spin theory (?)

16 classical manifolds as solution of IKKT: let (x µ, p µ = µ )... phase space ( ) X a p µ = 0... commutative R n solutions: [X a, X b ] = 0 fluctuations X a = X a + A a (x, p) describes some higher derivative / higher spin theory (?) Hanada, Kawai, Kimura hep-th/ ;... (R 2n re-appears, unlike for NC branes!)

17 stacks of branes in M.M. assume X(i) a... solutions of e.o.m. ( ) X a new solution: X a = (1) 1l n1 0 0 X(2) a 1l n 2... stacks of n 1 & n 2 coincident branes breaks U(N) to U(n 1 ) U(n 2 ) fermions may connect different branes ( ) 0 ψ(12) Ψ =, ψ (21) 0 ψ (12) transform in bifundamental (n 1 ) (n 2 ) (= strings!) interaction between R 2n θ branes consistent with IIB SUGRA (quantum effect!) (IKKT 1997, Chepelev,Makeenko, Zarembo 1997,...) can get close to particle physics Chatzistavrakidis, Zoupanos, H.S: arxiv: ; H.S.: arxiv: etc.

18 Part two: fluctuations on noncommutative branes NC gauge theory geometric fluctuations Claim A: fluctuations on branes noncommutative gauge fields Claim B: U(1) fluctuations on branes fluctuations of geometry, gravity both claims are correct 2 nd interpretation more useful, explains UV/IR mixing in M.M. consistent with string theory physical relevance not yet clear

19 claim A: fluctuations on a stack of n coincident R 4 θ branes in IKKT noncommutative U(n) N = 4 super-yang-mills on R 4 θ sketch: (Aoki, Ishibashi,Iso,Kawai,Kitazawa, Tada 1999) background solution: stack of n coinciding R 4 θ branes ( ) ( ) X X a µ X = φ i = µ 1l n µ = 0,..., 3, 0 i = 4, 5,.., 9 [ X µ, X ν ] = iθ µν... Heisenberg algebra, generate A θ End(H) add fluctuations: ( ) X X a = µ 1l n + θ µν A ν φ i A θ Mat(n, C) A µ = A µ ( X) = A µ,α ( X)λ α End(H n ) = A θ Mat(n, C)

20 define derivatives as inner derivations: [ X µ, φ(x)] =: iθ µν ν φ(x), [ µ, ν ] = 0 thus [X µ, φ(x)] = iθ µν D ν φ(x), D µ = µ + i[a µ,.] [X µ, X ν ] = iθ µν + iθ µµ θ νν ( µ A ν ν A µ + [A µ, A ν ]) = iθ µν + iθ µµ θ νν F µ ν S = Tr([X a, X b ][X a, X b ]) F µ ν... Yang-Mills field strength is gauge-invariant: X a U 1 X a U tangential fluctuations X µ = X µ + θ µν A ν transform as A µ U 1 A µ U + iu 1 µ U...u(n) gauge fields! transversal fluctuations φ i U 1 φ i U...u(n) scalar fields!

21 insert in IKKT action: S = Λ 4 0 ([X Tr a, X b ][X a, X b ] + ΨΓ ) a [X a, Ψ] = d 4 x ( 1 G tr n (FF) 4g 2 G Gµν D µ Φ i D ν Φ i 1 4 g2 [Φ i, Φ j ][Φ i, Φ j ] + ψ γ µ (i µ + [A µ,.])ψ + g ψγ ) i [Φ i, ψ] + ρθ ab θ ab where G µν = ρθ µν θ νν η µ ν, ρ = θ 1 γ µ = ρ 1/2 θ νµ γ ν, 1 4g 2 = Λ4 0 (2π) 2 ρ 1 IKKT on stack of n branes U(n) N = 4 SYM coupled to G µν (cf. large N reduction! )

22 very simple & compelling origin of gauge theory however, misleading for U(1) sector: deformations of branes are obviously geometrical d.o.f. cannot disentangle U(1) from SU(n) UV/IR mixing different physics because U(1) is gravity sector!

23 claim B: fluctuations on a stack of n coincident R 4 θ branes in IKKT U(1) tr dynamical G µν (x), SU(n) SYM coupled to G µν (x) H.S., JHEP 0712:049 (2007) (review: JHEP 0902:044,(2009), Class.Quant.Grav. 27 (2010) ) analogous for finite matrix geometries, A = Mat(N, C) explains UV/IR mixing (quantitatively!)

24 metric structure on branes: fluctuations governed by matrix Laplacian S[ϕ] = Tr [X a, ϕ][x b, ϕ] η ab = Tr ϕ ϕ encodes metric! ϕ η ab [X a, [X b, ϕ]] e.g. on S 2 N : φ = 1 C N J a J a φ SO(3) invariant Ŷ m l = 1 C N l(l + 1)Ŷ m l spectrum identical with classical case g φ = 1 µ ( g g µν ν φ) g effective metric = round metric on S 2

25 geometry of generic NC branes: X a x a : M R 10 Lemma: assume dim M > 2. Then f (X) η ab {x a, {x b, f (x)}} = e σ G f (x)... Matrix Laplace- operator, effective metric G µν (x) = e σ θ µµ (x)θ νν (x) g µ ν (x) effective metric (cf. open string m.) g µν (x) = µ x a ν x b η ab induced metric on M 4 θ (cf. closed string m.) e 2σ = θ 1 µν g µν (H.S. Nucl.Phys. B810 (2009) ) follows by coupling to scalar field ϕ: S[ϕ] = Tr [X a, ϕ][x b, ϕ] g ab d 2n x G G µν (x) µ ϕ ν ϕ = dϕ G dϕ

26 stack of coincident curved branes su(n) gauge thy generic background branes X a = general CR [ X µ, X ν ] = iθ µν ( X) ( ) X µ 1l n φ i 1l n fluctuations: X a = ( ) X µ 1l N + A µ φ i 1l N + Φ i A µ, Φ i 1l n d.o.f. change background X a, geometrical d.o.f. θ µν, g µν write A µ = θ µν A ν, note [ X µ, f ] iθ µν ν f [X µ, X ν ] = iθ µν + iθ µµ θ νν ( µ A ν ν A µ + [A µ, A ν ]) = iθ µν + iθ µµ θ νν F µ ν field strength

27 effective action on M 4 θ (semi-classical): S = Λ 4 0 ([X Tr a, X b ][X a, X b ] + ΨΓ ) a [X a, Ψ] d 4 x ( 1 G tr n (FF) 4g 2 G (DΦi DΦ i G 1 4 g2 [Φ i, Φ j ][Φ i, Φ j ] + ψ γ µ (i µ + [A µ,.])ψ + g ψγ ) i [Φ i, ψ] + 2η(θ θ + tr n F F) where G µν (x) = ρθ µν (x)θ νν (x)g µ ν (x), ρ = θ 1 γ µ (x) = ρ 1/2 θ νµ (x)γ ν, η = Gg 1 4g 2 = Λ4 0 (2π) 2 ρ 1 IKKT on stack of branes SU(n) N = 4 SYM coupled to G µν dynamical G µν (x)! ( gravity?!) H.S., JHEP 0712:049 (2007), JHEP 0902:044,(2009), Class.Quant.Grav. 27 (2010)

28 fermions Ψ... A - valued Majorana-Weyl spinor of SO(9, 1) S[Ψ] = Tr ΨΓ a [X a, Ψ] Tr Ψ /DΨ d 4 x θ 1 Ψi γ µ ( µ + [A µ,.])ψ, with γ µ = ρ 1/2 Γ a θ νµ ν x a { γ µ, γ ν } = 2 G µν (x) Ψ decomposes into 4 Weyl fermions N = 4 SYM

29 result: trace-u(1) sector defines geometry M 2n R 10 SU(n) fluctuations of matrices X a, Ψ gauge fields, scalar fields, fermions on M 2n (NOT 10 dim!) all fields couple to metric G µν (x) determined by θ µν (x), embedding dynamical ( emergent ) gravity matrix e.o.m [X a, [X a, X b ]]η aa = 0 G x a = 0, minimal surface µ (e σ θµν 1 ) = e σ G ρν θ ρµ µ η η G µν g µν covariant formulation in semi-classical limit (H.S. Nucl.Phys. B810 (2009) )

30 2 interpretations for quantization: Z = dx a dψ e S[X] S[Ψ] 1 on R 4 θ : X µ = X µ + θ µν A ν, X µ...moyal-weyl NC gauge theory on R 4 θ, UV/IR mixing in U(1) sector IKKT model: N = 4 SYM, perturb. finite!(?) 2 on M 4 R 10 : U(1) absorbed in θ µν (x), g µν quantized gravity, induced E-H. action S eff d 4 x G ( Λ 4 + cλ 2 4 R[G] +... ) explanation for UV/IR mixing & U(1) entanglement good quantization for theory with dynamical geometry

31 2 interpretations for quantization: Z = dx a dψ e S[X] S[Ψ] 1 on R 4 θ : X µ = X µ + θ µν A ν, X µ...moyal-weyl NC gauge theory on R 4 θ, UV/IR mixing in U(1) sector IKKT model: N = 4 SYM, perturb. finite!(?) 2 on M 4 R 10 : U(1) absorbed in θ µν (x), g µν quantized gravity, induced E-H. action S eff d 4 x G ( Λ 4 + cλ 2 4 R[G] +... ) explanation for UV/IR mixing & U(1) entanglement good quantization for theory with dynamical geometry

32 semi-classical limit of UV/IR mixing: interaction of two scalar field components S int Tr([φ 1, φ 2 ][φ 1, φ 2 ]) = 2Tr(φ 1 φ 2 φ 1 φ 2 φ 2 1φ 2 2) integrate out A φ 2 eff. action for φ φ 1 phase factors for non-planar diagrams, e ikx e ilx = e ikθl e ilx e ikx (planar non-planar diagram) Λ 2 (1 1 1 p2 Λ 2 Λ 4 NC ) usual treatment: high UV cutoff Λ Λ NC IR divergence 1, accumulates p 2

33 p 2 Λ 2 Λ 4 NC different limit: low UV cutoff 1 (max. SUSY!) ) Λ (1 2 1 = p2 Λ 4 + O(p 4 Λ 6 ) 1 p2 Λ 2 Λ 4 Λ 4 NC NC phase factors [e ikx, e ilx ] = 2i sin( kθl 2 )ei(l+k)x can be understood semi-classically: [φ, A] {φ, A} = θ µν µ φ ν A integrate out A [φ, A][φ, A] A θ µµ θ νν ( µ A ν A) µ φ ν φ }{{} Λ 4 G µν 1-loop correction to kinetic term (metric!) of φ: δs kin [φ] [A, φ][a, φ] A Λ 4 θ µµ θ νν G µ ν µφ ν φ Λ 4 g µν µ φ ν φ

34 2 θ νµ F µα g αβ ν ϕ i β ϕ i ( θ µν F µν )g αβ β ϕ i α ϕ i + h.o. careful treatment: 1-loop eff. action due to fermion loops: (all terms of dim 6): Γ eff = Λ4 Λ 4 NC 1 2 Λ4 NC d 4 x (g αβ D (2π) 2 α ϕ i D β ϕ i ( θµν ) F θρσ νµ F σρ + ( θ σσ F σσ )(F θf θ) ( + Λ2 d 4 x 11 Λ 4 (2π) NC 2 2 F ρη g F στ Ḡ ρσḡητ 12 g ϕ i ϕ i ) Λ4 NC ( θ µν F µν ) G ( θ ρσ F ρσ ) Λ6 d 4 x (...) +... Λ 8 (2π) NC 2 (all of this is due to UV/IR mixing, low cutoff, U(1) only ) (D. Blaschke, H.S., M. Wohlgenannt JHEP 1103 (2011) ) )

35 summarized in effective generalized matrix model: ( ) X re-assemble effective action: X a µ = + 0 Γ L [X] = Tr L 4 1 2H 2 H ab H ab + 1 L L 2 10,curv [X]+... ( ) θ µν A ν φ i d 4 x Λ 4 (x) g(x) L 10,curv [X] = c 1 [X c, H ab ][X c, H ab ] + c 2 H cd [X c, [X a, X b ]][X d, [X a, X b ]] +... H ab = [X a, X c ][X b, X c ] + (a b), H = H ab η ab (D. Blaschke, H.S. M. Wohlgenannt arxiv: ) SO(D) manifest, broken by background (e.g. R 4 θ ) highly non-trivial predictions for (NC) gauge theory expect generalization to nonabelian N = 4 SYM: full SO(9, 1)! effective generalized matrix model = powerful new tool for (NC) gauge theory and (emergent) gravity

36 summarized in effective generalized matrix model: ( ) X re-assemble effective action: X a µ = + 0 Γ L [X] = Tr L 4 1 2H 2 H ab H ab + 1 L L 2 10,curv [X]+... ( ) θ µν A ν φ i L 10,curv [X] = c 1 [X c, H ab ][X c, H ab ] + c 2 H cd [X c, [X a, X b ]][X d, [X a, X b ]] +... H ab = [X a, X c ][X b, X c ] + (a b), H = H ab η ab (D. Blaschke, H.S. M. Wohlgenannt arxiv: ) SO(D) manifest, broken by background (e.g. R 4 θ ) highly non-trivial predictions for (NC) gauge theory expect generalization to nonabelian N = 4 SYM: full SO(9, 1)! effective generalized matrix model = powerful new tool for (NC) gauge theory and (emergent) gravity

37 higher-order terms, curvature result: H ab := 1 2 [[X a, X c ], [X b, X c ]] + T ab := H ab 1 4 ηab H, H := H ab η ab = [X c, X d ][X c, X d ], X := [X b, [X b, X]] for 4-dim. M R D with g µν = G µν : Tr ( ) 2T ab X a X b T ab H ab 2 (2π) d 4 x g e 2σ R 2 Tr([[X a, X c ], [X c, X b ]][X a, X b ] 2 X a X a ) 1 (2π) d 4 x g e ( σ e σ θ µη θ ρα R µηρα 2R + µ σ µ σ ) (Blaschke, H.S. arxiv: ) (cf. Arnlind, Hoppe, Huisken arxiv: ) contains Einstein-Hilbert- type action from matrix model pre-geometric origin, background indep.

38 gravity from U(1) on branes? + good quantum theory of geometry + E-H action induced + θ µν invisible to scalar fields, gauge fields θ µν couples to U(1) d.o.f., Lorentz-breaking effects R abcd θ ab θ cd 1-loop induced action (D. Klammer H.S. 2009) + NC gauge field F µν 2 propagating Ricci-flat dof (Rivelles 2003; cf. Yang 2004) grav. field on R 4 couples to derivative of T µν (class.) + non-deriv. coupling to T µν in presence of extrinsic curvature (H.S ff) maybe better: covariant quantum spaces e.g. SN 4, manifest Lorentz/ Euclidean inv.

39 summary, conclusion matrix-models Tr[X a, X b ][X a, X b ] η aa η bb + fermions dynamical NC branes emergent geometry, gravity? fluctuations of matrices gauge theory on brane all ingredients for physics rich solutions of IKKT model with M 4 K building blocks for intersecting branes ( standard model?) need better understanding of quantum effects perturbative: effective quantum matrix model action new, adapted methods: eigenvalue distribution, localization,... identify appropriate background... very rich model, more to be discovered

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