Spontaneous supersymmetry breaking in noncritical covariant superstring theory

Transcription

1 03//0 Synthesis of integrabilities arising from gauge-string Spontaneous supersymmetry breaking in noncritical covariant superstring theory Tsunehide Kuroki (KMI, Nagoya Univ.) collaboration with M.G. Endres, H. Suzuki and F. Sugino Nucl. Phys. B 867 (03) 448 [arxiv: [hep-th]] arxiv: [hep-th] Nucl. Phys. B 876 (03) 758 [arxiv: [hep-th]].

2 Motivations LHC SUSY br. at (or just below) Planck scale? N gauge th. or matrix models: promising candidates for nonpert. def. SUSY: necessary for consistency of def. of quantum gravity : essential for superstring theory desirable scenario: SUSY for each order in /N-expansion pert. string, SUSY / spontaneously in the DSL nonpert. string but very few examples (in spite of its importance!) SUSY breaking/restoration in the large-n limit [T.K.-Sugino 08 ] SUSY DW MM

3 Review of SUSY double-well matrix model SUSY double-well matrix model: S = Ntr Properties: nilpotent SUSY: [ ] B + ib(ϕ µ ) + ψ(ϕψ + ψϕ) Qϕ = ψ, Qψ = 0, Q ψ = ib, QB = 0, Qϕ = ψ, Q ψ = 0, Qψ = ib, QB = 0, parameters: N, µ ; V (ϕ) = (ϕ µ ) finite N: SUSY br. = instanton (N = case can be checked explicitly) 3

4 In terms of eigenvalues Z = dϕdψd ψ e Ntr ( (ϕ µ ) + ψ(ϕψ+ψϕ)) = = = ( ) dλ i dψd ψ (λ) e N [ i (λ i µ ) + ψ ij (λ j +λ i )ψ ji ] i ( ) dλ i (λ i λ j ) (λ i + λ j ) e N i (λ i µ ) i i>j i,j ( ) dλ i (λ i ) (λ i λ j ) e N i (λ i µ ) i i i>j SPE : 0 = + λ j( i) i λ j j P dy ρ(y) x y + N(λ i λ i + λ µ ) λ i j P dy ρ(y) x + y = x3 µ x (ρ(x) = N ) tr δ(x ϕ) 4

5 N : two phases:. µ : two-cut phase: (ν +, ν ) (ν + + ν = ) ρ(x) = { ν+ π x (x a )(b x ) (a < x < b) ν π x (x a )(b x ) ( b < x < a) a = µ, b = µ +. µ < : one-cut phase: Order parameter: (B n = iq(b n ψ) = i Q(B n ψ)) N tr Bn =0 for n (two-cut) 0 for n = (one-cut) 5

6 µ : SUSY vacua continuously parametrized by ν + µ = : critical pt. SUSY/nonSUSY phase transition! (3rd) possible to define a superstring theory by taking a double scaling limit?: µ + 0, N with (µ )N :fixed 6

7 One-point function Nicolai mapping: [Gaiotto-Rastelli-Takayanagi 04] X = ϕ µ = Gaussian MM: c = topological gravity loop gas (O( )) model [Kostov-Staudacher 99] i N tr ϕn i : regular in µ However, this model also has N tr ϕn+, N tr ψn+, N tr ψ n+ nontrivial 7

8 One-point function (N ) N tr ϕn 0 = Ω dx x n ρ(x) = (ν + + ( ) n ν ) (µ + ) n F (Ω = [ b, a] [a, b]) ( n, 3 ), 3; 4 µ + n: even: (ν +, ν )-indep., poly. in µ n: odd: (ν + ν )-dep., logarithmic singular behavior: ω = µ : deviation from the critical pt. 4 N tr ϕk+ 0 = (ν + ν ) (const.)ω k+ ln ω + 8

9 Multi-point functions two-point functions for boson (leading part containing log) N tr ϕk N N tr ϕk+ N tr ϕl C tr ϕl tr ϕl+ N tr ϕk+ N C N tr ϕk + N tr ϕk n+ : indep. of (ν +, ν ), poly. of µ (ν + ν )ω k+ ln ω C (ν + ν ) ω k+l+ (ln ω) C,0 = (ν + ν ) n (const.)ω γ+ n i= (k i ) (ln ω) n + confirmed for general -pt funcs, first two simplest 3-pt. ones new critical behavior as power of log γ = (same as c = ) = DSL: N ω 3 /g s : fixed 9

10 two-point function for fermion N tr ψk+ tr ψ l+ N = δ kl (ν + ν ) k+ (const.)ω k+ ln ω + confirmed up to k, l = 0, 0

11 Correspondence to D = IIA superstring log. singularity scaling violation in bosonic string in D = [Brezin-Kazakov-Zamolodchikov 990] [Gross-Klebanov 990][Polchinski 990] new MM interpretation: matrix=field on target space (cf. old MM) D = superstring theory with unbroken target space SUSY D = IIA superstring theory [Kutasov-Seiberg 990] [Ita-Nieder-Oz 05] action: N = Liouville theory S = π d z ( x x + φ φ + Q 4 Rφ + g ± (ψ l ± iψ x )( ψ l ± i ψ x )e (φ±ix)/q + i.e. target sp. (x, φ): D, Q =

12 target sp. SUSY: L : q + = e ϕ i H ix, R : q = e ϕ+ i H+i x ψ l ± iψ x = e ih Q + = Q = {Q +, Q } = 0 : nilpotent!, no spacetime transl. physical vertex ops.: NS: ( )-picture tachyon: T k (z) = e ϕ+ikx+p lφ (z) with p l = k R: ( /)-picture R field: V k, ϵ (z) = e ϕ+ i ϵh+ikx+p lφ (z) ϵ : chirality both WS & TS SUSY x S with R = /Q = (self-dual radius)

13 physical states: (winding background) (NS, NS) T k T k k Z + / (R+, R ) V k, + V k, k Z 0 + / (R, R+) V k, V k, + k Z 0 (NS, R ) T k V k, k Z 0 / (R+, NS) V k, + T k k Z 0 + / new matrix model interpretation natural to identify N tr ψ (NS, R) N tr ψ (R, NS) how about boson? 3

14 SUSY multiplet: identify (Q, Q) in MM (Q +, Q ) in IIA: lowest momentum (k = ±/) sector: (R+, R ) V, + V, tr ϕ Q + Q Q Q (NS, R), (R, NS) T V, V, + T tr ψ tr ψ Q Q+ Q Q (NS, NS) T T tr B i.e. N tr ψ (NS, R) T V, N tr ϕ (R+, R ) V + V, N tr ψ (R, NS) V,+ T N tr B (NS, NS) T T Z -symmetry in MM: ψ ψ, ϕ ϕ [ ] S = Ntr B + ib(ϕ µ ) + ψ(ϕψ + ψϕ) realizes ( ) F L-symm. in IIA: V, + V, + 4

15 observations: where s information of momentum? cf. Penner model [Distler-Vafa 90][Mukhi 03] Z(t, t) = dm e tr ( νm+(ν N) log M k= t k (MA ) k ), tk = νk T k T km T l T ln c=,r= = F (t, t) t k t k t kn t km power of matrices tr ϕk+ 0 (logarithmic behavior)? N 0 (R+, R ) -pt. func. 0 RR background!, resonance missing (R, R+) sector? not in MM (i.e. (asymptotic) target sp. fields), but this must be a background in IIA! In fact, (R,R+)-sector: Q +, Q -singlet. taget sp. SUSY inv.! tr A k t, t=0 5

16 Claim: SUSY DW MM = D type IIA at the level of correlation functions under: tr ϕk+ N tr ψk+ N tr ψ k+ N N tr B (R+,R ): (NS,R ): (R+,NS): (NS,NS): where the IIA correlation functions are d z i V i (z i, z i ) = i i d z V k+, + (z) V k, ( z) d z T k (z) V k, ( z) d z V k+, + (z) T k+ ( z) d z T (z) T ( z) d z i V i (z i, z i ) e (ν + ν ) k Z a kω k+ d z V k, V k, + with ω = g, g + = 0 6 N = Liouville

17 Note: RR flux term: (ν + ν ) k Z a k ω k+ d zv k, V k, + (ν + ν ): RR flux source [Takayanagi 04] k > 0: wrong branch breaking Seiberg bound: V (NL) k, = e ϕ i H ikx+p lφ with p l = + k : nonlocal disturbance on string WS 7

18 MM & IIA action: S MM = Ntr [ ] B + ib(ϕ µ ) + ψ(ϕψ + ψϕ), ( ) ω = µ 4 S IIA = π ( d z x x + φ φ + Q grφ 4 + g (ψ l iψ x )( ψ l i ψ x )eq (φ ix) }{{} T (0) T (0) + ) ω tr B d z T T g 8

19 Examples: N tr ( ib) N tr ϕk+ = (ν + ν )c k ω k+ ln ω C,0 = 4 ω N tr ϕk+ 0 d z T (z ) T ( z ) d z V k+, + (z ) V k, ( z ) = d z T (z ) T ( z ) d z V k+, + (z ) V k, ( z ) (ν + ν )a k ω k+ d z V (NL) k, (z) V (NL) k, + ( z) = (ν + ν )a k ω k+ ln g }{{} Liouville vol. (a k : finite via reguarlization) 9

20 N tr ϕk+ tr ϕl+ = (ν + ν ) c kl ω k+l+ (ln ω) N C,0 d z V k+, + (z ) V k, ( z ) d z V l+, + (z ) V l, ( z ) = d z V k+, + (z ) V k, ( z ) d z V l+, + (z ) V l, ( z ) (ν + ν )a ω + d z V, (z) V, + ( z) (ν + ν )a k+l ω k+l+ d w V (NL) k l, (w) V (NL) k+l, + ( w) = (ν + ν ) a a k+l C kl ω k+l+ ( ln g ) similar for fermion -pt. function single choice of parameters various kinds of amplitudes reproduced including coefficients strong evidence that our MM provides nonpert. def. of D = IIA superstring theory in the RR-back.! 0

21 Spontaneous SUSY breaking of superstring MM amp. in N = tree level (sphere) IIA string amp. MM in DSL = nonperturbative def. of IIA string SUSY br. order parameter in DSL order parameter: ( N tr B = i ) 4N ωf (recall iqtr ( ψ) = tr B) N tr B = i N tr (ϕ µ ) = d z T (z) T ( z) Nicolai mapping: X = ϕ µ or x i = λ i ( ) µ Z = dbdx e Ntr B +ibx = i ( µ dx i ) (x i x j ) e N i x i i>j

22 if we can ignore effect of boundary, for n N N tr Bn = db Z dx N tr Bn e Ntr ( B +ibx ) = 0 SUSY for all order of /N-expansion i.e. perturbative SUSY of IIA in the RR background (desirable) in DSL: evaluation of boundary effect orthogonal poly. Z = µ i dx i (x) e N i x i

23 orthogonal polynomial: P n (x) = x n + O(x n ), (P n, P m ) dx e µ N x P n (x)p m (x) = h n δ nm xp n (x) = P n+ (x) + s n P n (x) + r n P n (x) N tr (ϕ µ ) = N N k=0 s k without boundary, P n (x) = H n ( Nx) xh n (x) = H n+ (x) + nh n (x) s n = 0 3

24 However, taking account of the boundary, P (x) = x + c, 0 = (P 0, P ) = dx e N x (x + c) = µ = N e N µ4 + ch 0, h 0 = (P 0, P 0 ) = c = N e N µ 4 = s 0 0 h 0 dx e N x x + ch 0 µ µ dx e N x In general, s k = N h n P k ( µ ) e N µ4 nonperturbative effect: exp( NC) makes s n nonvanishing!! boundary effect nonperturbative effect [Gaiotto-Rastelli-Takayanagi 04] survives in DSL? 4

25 N tr B s k = N = i N tr (ϕ µ ) h n P k ( µ ) e N µ4 = i N N k=0 s k leading order: P k ( µ ) H k ( Nµ ) DSL: N, µ, with N((µ )/4) 3/ = /g s Nµ : turning pt. of PN P N ( ( ) Nµ 4 ) Ai N tr B leading = in 4/3 ( Ai ( 4 : proof of nonperturbative SUSY br.! g /3 s ) 4 gs /3 Ai gs /3 ( 4 g /3 s ) ) 5

26 g s : N tr B = in 4/3 3π g/3 s F = 8π g s e e 3 3 gs + O(e 64 3 gs) 3 3 gs + O(e 64 3 gs) leading part: -instanton contribution: exact via Airy func. explicit confirmation of leading part of -instanton contribution Note: zero in all orders in /N-expansion, but nonperturbatively nonzero due to boundary effect spontaneous breaking of SUSY in SUSY DW MM finite in the double scaling limit (cf. correlation functions) TS SUSY can be broken in nonpert. superstring theory (we DO NOT put a D-brane by hand!! RR flux DOES NOT break SUSY) D-brane superposition triggers SUSY / 6

27 exact result in the one-instanton sector by Ai: Ai (4/g /3 s ) 4 g /3 s Ai(4/g /3 s ) ( disk amp. with arbitrary holes and handles) 7

28 Physical interpretation MM instanton action a V (0) (0) eff (0) V eff (a) = dy (y µ ) 4 0 = ( µ µ ) µ 4 4 µ log 3 3 ω 3 (complete agreement with OP) ( ) ω = µ 4 eigenvalue tunneling, condensation of D-brane? [Hanada et. al. 04] SUSY / nonpert. effect bdy of Nicolai mapping x = µ λ = 0 the instanton is located fate of SUSY: summary finite N: SUSY / (MM inst.), N : SUSY (by exp( NC)), double scaling lim.: SUSY / (MM instanton with finite action) 8

29 -pt. func. HnormalizedL.08 exact: N=0 p exact: N= -inst. HfullL -inst. HleadingL -inst. HfullL + -inst. HleadingL p= p= p=3 0.5 p=4.0 t.5.0

30 pt. func. normalized exact: N inst. full inst. full inst. leading t

31 FH,0L exact: N=0 p exact: N= -inst. HfullL -inst. HleadingL -inst. HfullL + -inst. HleadingL p= t p=.5.0

32 Conclusions & Discussions at last we would get nonperturbative formulation of covariant superstring theory with (perturbatively) unbroken target space SUSY! (target sp. interpretation) agreement in fundamental correlation functions cf. KK not in D-brane decay rate [Takayanagi 04] But nonperturbatively, target space SUSY is broken spontaneously without introducing source for it by hand. Even quite difficult in field theory case noncritical (restricted to R = ), nilpotent SUSY SUSY version of Penner model matrix reloaded interpretation: origin of MM: effective aciton on IIA D-particle? (power=winding or momentum large-n reduced model?) origin of breakdown of Seiberg bound? (D-brane?) 9

33 identification of missing states (positive winding tachyon, discrete states, ), more general correlation functions, s = correlation functions, usefulness of orthogonal polynomial with boundary, or Nicolai mapping application of Yang-Mills type? (essentially Gaussian, but taking account of boundaries) SUSY is not for using it, but (may be) for breaking it! 30

34 A Sectors for finite N define the sector with (ν +, ν ) for finite N: decomposition of integration region of eigenvalues: divide the integration region for each λ i : dλ i = 0 dλ i + (ν +, ν )-sector: ν + N eigenvalues integrated over R 0, N Z = NC ν+ NZ (ν+,ν ) ν + N=0 Z (ν+,ν ) = i>j ν + N i= 0 (dλ i λ i ) N j=ν + N+ (λ i λ j ) e N i (λ i µ ) 0 dλ i ν N ones over R 0 0 (dλ j λ j ) flipping sign: λ j λ j Z (ν+,ν ) = ( ) ν N Z (,0) 3

35 Note: Z = 0: corresponding to Witten index (SUSY breaking case) this argument can be applied to correlation func. (confirmed up to 3-pt.): N tr ϕn (ν + + ν ) =, N tr ϕn+ (ν + ν ) simple (ν +, ν )-dep. calculations can be reduced to (, 0)-sector 3

36 B N = Liouville theory N = (, ) WS SUSY, flat Euclidean: S = d zdθ + θ d θ + d θ Φ Φ 8π + g d zdθ + d θ + e Q Φ + ḡ d zdθ d θ e Q Φ π π Φ: chiral ( s.f.: ) ( ) θ iθ+ Φ = θ i θ + Φ = 0, ( ) ( ) θ + iθ Φ = θ i θ Φ = 0 + Φ = ϕ + i θ + ψ + + i θ + ψ+ + θ + θ+ F +, Φ = ϕ + i θ ψ + i θ ψ + θ θ F + S = ) d z ( x x + φ φ + ψ + ψ + ψ + ψ π + ig d zψ πq + ψ+ e Q ϕ + iḡ d zψ πq ψ e Q ϕ 33

37 ϕ = φ + ix, rescaled ψ ± = ψ l iψ x, F = F = 0, curved sp. linear dilation (N = WS superconf. alg.): S = ( d z x x + φ φ + Q grφ + g± (ψ l ± iψ x )( ψ l ± i ψ x )eq (φ±ix) π 4 ) + fermion kin. terms 34

38 C Instanton action in SUSY double-well matrix model ( ) Z = dλ i λ i (λ i λ j ) e i N (λ i µ ) i i>j ( ) N = dx x dλ i λ i (x λ i ) = i i= N i>j (λ i λ j ) e N i= N (λ i µ ) e N (x µ ) (x = λ N ) dx x det(x ϕ ) (N ) e N (x µ ) dx x e NV eff(x) V eff (x) = (x µ ) N log det(x ϕ ) = (x µ ) N log e Re tr log(x ϕ ) = (x µ ) N log e Re tr log(x ϕ ) + ( Re tr log(x ϕ )) c + 35

39 V (0) eff (x) = (x µ ) Re N tr log(x ϕ ) 0 = x (x µ ) Re dy N tr y ϕ = Re x dy (y µ ) 4 0 a V (0) (0) eff (0) V eff (a) = 0 dy (y µ ) 4 = µ µ log(µ µ 4 4) log 3 3 ω 3 (complete agreement with OP) ( ) ω = µ 4 36