String states, loops and effective actions in NC field theory and matrix models

Transcription

1 String states, loops and effective actions in NC field theory and matrix models Harold Steinacker Department of Physics, University of Vienna H.S., arxiv:

2 Motivation practical way of performing loop integrals (quantum corrections) in NC gauge theory for generic NC geometries, no symmetries understand better nature of UV/IR mixing in NC field theory derive supergravity from matrix models ( string theory) quantum corrections on fuzzy S 4 N ( emergent 4D gravity in IKKT model)

3 Outline coherent states on S 2 N and R2 θ string states some trace computations 1-loop computations on SN 2, UV/IR mixing IIB supergravity in the IKKT model

4 Fuzzy S 2 N X a = J a (N) functions on S 2 N matrix Laplacian [X a, X b ] = iε abc X c, X a X a = 1 4 (N2 1) =: R 2 N.... irrep of SU(2) on H = C N N 1... A = End(H) = l=0 (2l + 1) φ = [X a, [X a, φ]], φ End(H) quantized space, SO(3)-invariant symplectic form ω S 2 ω = 2π dim(h) = 2πN fuzzy sphere contains N quantum cells of area A N = 4πR2 N N L 2 NC

5 S 2 as group orbit: let p S 2... north pole SU(2) S 2 g g p =: x stabilizer K SU(2) S 2 = SU(2)/U(1) coherent states on SN 2 : p H N def.... highest weight vector in H N x = g x p, g x SU(2)... coherent states x a = x X a x X a S 2, x a x a = 1 4 (N 1)2 =: r 2 N x... one-to-one correspondence to points x on S 2 (up to phase)

6 relation with R 2 θ, Q.M: focus at north pole p S2 rescale R s.t. θ = 2R2 N = const [X i, X j ] = iθɛ ij + O( 1 N ) coherent state at origin = north pole: 0 p, a 0 (X 1 + ix 2 ) 0 = 0 highest weight state shifted (rotated) coherent states: x = U x 0 where U x = exp(iφ i J i ), x i = R ɛ ij φ j localization: x x = e i 2θ x i ε ij x j e x x 2 4θ covers area A N = θ

7 back to S 2 N : coherent states are optimally localized: 2 = a (X a ) 2 X a 2 = R 2 N r 2 N = N 1 2 =: L 2 NC 2 N R2 N x y 2 = ( 1+x y 2 ) N 1 exp( 1 4 φ2 (N 1)) = 1 c N δ N (x, y) char. angle φ N = π N φ = (x, y) char. angular momentum l N

8 overcompleteness: 1l H = c N S 2 dx x x, c N = dim H VolS 2 trace of any operator O End(H) tro = dim H VolS 2 dx x O x S 2 (by SU(2) invariance) generalizes to any quantized coadjoint orbit

9 operators and symbols For an operator O End(H), define symbol of O as O(x) = x O x... de-quantization of O, semi-classical limit conversely: O = c N S 2 dxõ(x) x x in particular Ŷ l m = c N S 2 dxy l m(x) x x however very delicate for large momenta, misleading in UV

10 wavefunctions and UV / IR sectors IR ( semi-classical ) sector: non-local matrix elements decay at distances x y L NC x O y x O x x y 1 c N x O x δ N (x, y) i.e. x f (X) y f (x) x y f (y) x y max. angular momentum l N, uncertainty 2 = L 2 NC optimally localized function p p =: 1 c N δ N (X; p)

11 UV sector: most O End(H) have l > N, not in semi-classical sector. best described by non-local string states ψ x,y := x y End(H) most extreme function on S 2 N : Y N 1 N 1 = p p, has l UV = N, maximally de-localized most NC functions are non-local non-local contributions in loops!

12 string states Iso Kawai Kitazawa hep-th/ ; H.S., arxiv: ) x y := ψ x,y := x y End(H) ( xy := ψ x,y := y x momentum operators expectation values ( xy P a x y ) ( xy P a P a x y ) P a O := [X a, O], O := P a P a O = tr ψ y,x [X a, ψ x,y ] = x(x) x(y) = tr ψ y,x [X a, [X a,.]]ψ x,y = E xy E xy = ( x(x) x(y)) energy of string state = length 2 + zero point energy

13 general matrix elements ( xy P a P a x y ) = x X a X a x y y + x x y X a X a y 2 x X a x y X a y nearly diagonal (2 2 + x 2 + y 2 2 x y)) x x y y E xy x x y y good localization properties in both position and momentum!!

14 propagator claim: ( + µ 2 ) 1 := cn 2 M dxdy x y ) 1 E xy + µ 2 ( xy ( + µ 2 ) 1 is excellent approximation to the propagator because: ( + µ 2 ) 1 ( + µ 2 ) x y ) completely regular since E xy 2 = cn 2 ) ( dx dy x 1 x y E x y +µ 2 y ( + µ 2 ) ) x y cn 2 ) dx dy x 1 y E x y +µ 2 (Exy + µ2 ) x x y y ) x y

15 trace formula for End(H) proof: Tr End(H) O = (dim H)2 (VolM) 2 dxdy ( x y O x y ) rhs = unique functional on End(End(H)) invariant under G L G R = Tr

16 example: Tr End(H) [X a N, [X a,.]] = 2 (VolS 2 ) 2 dxdx tr( x x ) ( x a x a ) ( x x ) S 2 S 2 N = 2 (VolS 2 ) 2 dxdx ( x a x a ) S 2 S 2 N = 2 VolS 2 dx( x a (e) x a (x) ) S 2 1 N 2 (N 2 1) dx( e 4 4π 2 3 x 2 + O( 1 N )) S 2 = 1 2 N2 (N 1) 2( 1 + O( 1 N )) using rn 2 = x a x a = 1 4 (N 1)2 and 2 N 2 good agreement with exact result: N 1 Tr End(H) [X a, [X a,.]] = j(j + 1)(2j + 1) = 1 2 N2 (N 2 1). j=0

17 more generally: for any smooth function f Tr End(H) f ( ) = N 2 (VolS 2 ) 2 S 2 dx S 2 dy f (R 2 N x y ) = N2 VolS 2 S 2 dxf (r 2 N e 3 x ) π = 2π N2 VolS 2 0 dϑ sin ϑf (r N 2(1 cos θ)2 + sin 2 θ) ) 1 1 duf (2r N 2(1 u) ) = N2 2 N 0 dj 2jf (j ) N 1 j=0 (2j + 1)f ( j(j + 1) + 2 2) = Tr jmax f ( g ) = Tr jmax f ( g ) shift by 2 2 = N 1 negligible for N 1 UV dominates! works because spec( ) = spec( g ) also in UV

18 one-loop propagator for φ 4 on S 2 N S[φ] = 1 N tr ( 1 2 φ( + µ2 )φ + g 4! φ4) = S 0 [φ] + S int [φ]. 1-loop effective action (= Gaussian approx.) ( ) Γ eff [φ] = S[φ] Tr End(H) log S [φ] ( ) (ψ, S [φ]ψ) = 1 N tr ψ( + µ 2 )ψ + g 3 φ2 ψ 2 + g 6 ψφψφ expanded: Γ 1 loop [φ] = Tr log(.( + µ 2 ). + g 3.φ2. + g 6 (.φ.φ) ) = Tr log( + µ 2 1 ) + Tr. ( g +µ 2 3 φ2. + g 6 φ.φ) + O(φ 4 )

19 assume φ = φ(x) slowly varying, IR regime φψ yx φ(y)ψ yx in string basis Tr(.φ 2.) = N 2 Vol(M) 2 N = 2 Vol(M) M Similarly, planar contribution Tr(. 1 φ 2 N.) = 2 Vol(M) 2 N 2 Vol(M) 2 N = 2 Vol(M 2 ) = µ 2 N Vol(M) dxdy tr(ψ y,x φ 2 ψ x,y ) dx x φ 2 x. dxdy tr(ψ y,x ( + µ 2 ) 1 (φ 2 ψ x,y )) 1 dxdy tr(ψ rn 2 x y µ 2 y,x φ 2 ψ x,y ) 1 dxdy x φ 2 x rn 2 x y 2 + µ 2 dx φ 2 (x) M

20 assume φ = φ(x) slowly varying, IR regime φψ yx φ(y)ψ yx in string basis Tr(.φ 2.) = N 2 Vol(M) 2 N = 2 Vol(M) M Similarly, planar contribution Tr(. 1 φ 2 N.) = 2 Vol(M) 2 N 2 Vol(M) 2 N = 2 Vol(M 2 ) = µ 2 N Vol(M) dxdy tr(ψ y,x φ 2 ψ x,y ) dx x φ 2 x. dxdy tr(ψ y,x ( + µ 2 ) 1 (φ 2 ψ x,y )) 1 dxdy tr(ψ rn 2 x y µ 2 y,x φ 2 ψ x,y ) 1 dxdy x φ 2 x rn 2 x y 2 + µ 2 dx φ 2 (x) M

21 1-loop planar mass renormalization µ 2 N = N 2 Vol(S 2 ) = N2 2r 2 N 1 dy r 2 S 2 N e y 2 + µ 2 π 0 dϑ sin ϑ 1 (1 cos ϑ) 2 +sin ϑ 2 + µ2 r 2 N = du 1 2 2u+ µ2 r 2 N N j=0 2j+1 j(j+1)+µ 2

22 nonplanar contribution Tr(.( + µ 2 ) 1 N φ.φ) = 2 Vol(M) 2 N = 2 Vol(M) 2 N = 2 Vol(M) 2 dxdy tr(ψ y,x ( + µ 2 ) 1 (φψ x,y φ)) dxdy x ( + µ 2 ) 1 φ x y φ y 1 dxdy φ(x)φ(y) rn 2 x y 2 + µ 2

23 one-loop quantum effective action: S 1 loop S 0 + g 1 dxµ 2 N 3 Vol(M) φ(x)2 + g N 2 φ(x)φ(y) 6 Vol(M) 2 r 2 dxdy N x y 2 + µ2 M r N 2 long-range non-locality from UV sector (UV/IR mixing) applies to any compact fuzzy space check for SN 2 : agrees with traditional mode expansion 1 S 1 loop = S Φ(µ2 N g h( ))Φ + o(1/n) 12π where h(l) = dt 1 t (P L(t) 1) = Chu Madore HS hep-th/ L k=1 less transparent, requires asymptotics of 6J symbols etc. 1 k + O(φ 4 )

24 Moyal-Weyl plane limit R 2 θ Γ NP = where VolM = 4πR 2 = πnθ R 2 = r 2 R 2 N = Nθ 4 gn 2 dxdy φ(x)φ(y) 6Vol(M) 2 x y 2 +µ 2 g dxdy φ(x)φ(y) 6π 2 θ 2 x y 2 +µ 2 plane wave basis φ(x) = d 2 k 2π φ k(e ixk + e ikx ). Γ NP = g 6π 2 θ d 2 kφ(k) 2 d 2 1 z 2 g 6π 2 θ d 2 kφ(k) 2 d 2 1 p 2 p i p j e ik G ij +µ 2 i θ ij p j. z 2 g+µ 2 e ik i z i replacing z i = θ ij p j, and G ij = θ ii θ jj δ i j... familiar form in NCFT, IR divergence for k 0 from UV loop, UV/IR mixing

25 significance of UV/IR mixing: UV sector in loops = virtual long strings x y lead to long-range non-locality in dxdy φ(x)φ(y) x y 2 +µ 2 interpret NCFT as (non-critical) string theory! open strings beginning and ending on D-branes universal, same on any fuzzy space M, any dimension accumulates at higher loops, inacceptable as fundamental theory except in SUSY case: cancellations!

26 higher loops t Hooft double line formalism, ribbon graphs lines labeled by positions x, preserved by propagators (!!) much simpler than in ordinary QFT, directly in position space! H.S., arxiv: (to be developed)

27 claim: analogous (quasi-) coherent states exist on generic fuzzy spaces (=quantized symplectic manifolds M R n ) L. Schneiderbauer HS arxiv: ; cf. Ishiki arxiv: defined as ground states of point branes on M X Φ = a [Xa, [X a, Φ]] = a (Xa X a Φ + ΦX a X a 2X a ΦX a ) 0 0. =.... (X a x a ) 2 φ a 0 0 φ (X a x a ) 2 0 a... shifted harmonic oscillator for φ on M ( ) X a X a 0 = 0 x a, a = 1,..., d conjecture: formula / approx. for Tr EndH holds quite generally

28 maximally SUSY case: IKKT model 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 aim: evaluate & interpret 1-loop effective action on background: R 4 θ and generic (curved) branes M R10 will see: non-local UV/IR mixing IIB supergravity (as predicted in string theory)

29 maximally SUSY case: IKKT model 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 aim: evaluate & interpret 1-loop effective action on background: R 4 θ and generic (curved) branes M R10 will see: non-local UV/IR mixing IIB supergravity (as predicted in string theory)

30 background: generic fuzzy space X a x a : M R 10 fluctuation around background X a X a + A a (X a ) one-loop effective action = Gaussian approx. around background Z [X] = dadψe S[X+A,Ψ] = e Γeff[X] Gauss where Γ eff [X] = S[X] + Γ 1loop [X] expand bare action to O(A 2 ): S[X + A] = S[X] + 2 g 2 Tr ( 2A a ( + µ 2 )X a + A a ( ( + µ 2 )δ a b + 2i[Θab,. ] [X a, [X b,.]] ) A b ) with iθ ab = [X a, X b ]

31 one-loop effective action in IKKT model Γ 1loop [X] = = SUSY = = ( 1 2 Tr log( + µ 2 M (A) ab [Θab,.]) 1 ( Tr n>0 1 n ( 1 2 Tr ) log( M(ψ) ab [Θab,.]) 2 log( ) ( ( 1( M (A) ab [Θab,.] + µ 2 ) ) n 1 2 ( 1 M (ψ) ab [Θab,.]) n ) 1 4 ( 1 (M (A) ab [Θab,.]) ( 1 M (ψ) ab [Θab,.]) 4 + O( 1 [Θ ab,.]) µ2 Tr 1 + O(µ 4 ) ) 1 ( 4 Tr 1 [Θ a 1b 1,... 1 [Θ a 4b 4,.]]]] ( 4gb1 a 2 g b2 a 3 g b3 a 4 g b4 a 1 4g b1 a 2 g b2 a 4 g b4 a 3 g b3 a 1 4g b1 a 3 g b3 a 2 g b2 a 4 ) g b +g b1 a 2 g b2 a 1 g b3 a 4 g b4 a 3 + g b1 a 3 g b3 a 1 g b2 a 4 g b4 a 2 + g b1 a 4 g b4 a 1 g b2 a 3 g b3 a µ2 Tr ( 1) ) ) where (M (ψ) ab )α β = 1 4i [γ a, γ b ] α β (M (A) ab )c d = i(δb cδ ad δaδ c bd )

32 use string states to evaluate this!! 1 1 ( x y ) x y x y [Θ ab 1,.]( x y ) δθ ab (x, y) x y x y on slowly-varying background, where evaluate trace as Γ 1loop;4 [X] 1 (dim H) 2 4 (VolM) 2 Γ 1loop;µ 2[X] 5 2 where = 3 4 δθ ab (x, y) := Θ ab (x) Θ ab (y) Ω x Ω y δθ a 1 b 1 (x,y)δθ a 2 b 2 (x,y)δθ a 3 b 3 (x,y)δθ a 4 b 4 (x,y) ( x y ) 4 3 ( ) 4g b1 a 2 g b2 a 3 g b3 a 4 g b4 a 1 + g b1 a 2 g b2 a 1 g b3 a 4 g b4 a 3 dxdyρ(x)ρ(y) S 4[δΘ(x,y)] ( x y ) 4 dxdyρ(x)ρ(y) µ 2 x y S 4 [δθ] = 4trδΘ 4 + (trδθ 2 ) 2

33 induced IIB supergravity physics of effective action: conjecture (IKKT): gives IIB supergravity on R 10 (relation with string theory!) in particular: D-branes = (soliton) solutions of IIB sugra (open) strings end on D-branes, lead to NC gauge theory direct verification (starting from M.M.) so far only for special geometries: interaction between flat branes long-distance interaction between D-blobs... IKKT, Kabat-Taylor, van Raamsdonk, Chepelev-Tseytlin,...

34 rewrite as Γ 1loop;4 [X] π5 2 Here ( dxdyρ(x)ρ(y) 2S 4 (Θ(x)) D(x y) +16 ( Θ ae (x)θ ef (x)θ fb (x) Θab (x)θ ef (x)θ ef (x) ) D (AS) ab;cd (x, y) Θdc (y) 8T ab (x)d (S) ab;cd (x, y) T cd (y) ) +4Θ aab (x)θ ef (x)d (AS) abef ;cdgh (x, y) Θcd (y)θ gh (y). D (S) ab;cd (x, y) = ( g acg bd + g ad g bc 1 4 g abg cd ) D(x y)... graviton propagator in 10D D (AS) abef ;cdgh (x, y) = ( g acg bd g egg fh + g acg bh g ed g fg g acg bg g ed g fh ) D(x y)... prop. for rank 4 AS tensor D (AS) ab;cd (x, y) = ( g acg bd g ad g bc ) D(x y)... prop. for rank 2 AS tensor D(x y) = 3 2π 5 1 ( x y ) 4 3 2π 5 1 x y D propagator T ab = Θ ac Θ cb... closed string 10D e-m tensor... consistent with interpretation in terms of IIB sugra

35 similarly for fields on fuzzy branes: expand Γ 1loop;4 [X] in F Θ ij = Θ ij + F ij leading interaction between separated blobs F(x) = F A (x) + F B (x): Γ 1loop;4 [F A, F B ] π 5 gravitational interaction of gauge fields etc. ( ) dxdyρ(x)ρ(y) 8TA ab (x)d(s) cd ab;cd (x, y) TB (y)+... however, is not physical gravity, decays like r 8, short range! way out: compactify extra dimensions orthodox string theory conjecture: 4D gravity emerges on suitable 4-dim. NC branes works indeed e.g. on fuzzy S 4 N! H.S., arxiv:

36 summary & outlook coherent states x, string states x y useful on fuzzy spaces greatly simplify loop computations on NC spaces, position space works for generic (curved) geometries explains UV/IR mixing long-range non-locality supersymm. IKKT model mild non-locality, IIB supergravity candidate for theory of fundamental interactions including gravity