Spin chains and strings in Y p,q and L p,q r manifolds

Transcription

1 Sin chains an strings in Y,q an L,q r manifols Martin Kruczenski Princeton Universit Base on he-th/ an he-th/ w/ S. Benvenuti Scuola Normale & INFN, Pisa, Ital

2 Summar Introuction: N=4 AS/CFT: Strings Long os. Point-like strings BPS os. String ecitations insertions Long strings semiclassical states N= AS/CFT AS 5 X 5 ; X 5 is Sasaki-Einstein X 5 : T, Y,q L,q r SU SU U SU U U U U U

3 Y,q Massless geoesics an long oerators: matching of R an flavor charges. Etene strings Qualitative Eff. action aears in f.t. an is similar but not equal to string sie. L,q r Derivation of ual fiel theor Massless geoesics an long oerators: matching of R an flavor charges.

4 N=4 SYM II B on AS 5 S 5 Strings? Long oerators Strings BMN, GKP an. imension Energ Frolov-Tsetlin: Long strings Minahan-Zarembo: Sin chains Bethe Ansatz BFST X=φ i φ ;Y=φ i φ 4 ; O=TrXY Y Long chains J are classical S n eff. = λ J = J cosθ φ n sinθ cosφ, sinθ sinφ, cosθ S eff : Action of sin chain an fast strings. M.K.; Rzhov, Tsetlin, M.K.; σ

5 Strings as states of N=4 SYM on RS The oerator O= c i [ TrXY Y ] i mas to a state of the fiel th. on S State: elocalize an has a large number of articles X an Y. Q.M. : ψ = cosθ / ei φ / X sinθ / e- i φ / Y We can use v i = ψ θ i, φ i to construct: O= Tr v v v v n Coherent state

6 θ = 0 Orere set of oints looks like a string θ = π/ S Internal sace Strings are useful to escribe states of a large number of articles in the large N limit

7 N= AS 5 X 5 ; X 5 : Sasaki-Einstein s = [4] r r X 5 CY cone Put D branes at r = 0 an take near horizon limit: r [4] r r s = X 5, AS 5 X 5 T, Y,q conifol Klebanov-Witten Gauntlett, Martelli, Sarks,Walram Benvenuti, Franco, Hanan, Martelli, Sarks L,q r Cvetic, Lu, Page, Poe; Benvenuti, M.K.; Franco, Hanan, Martelli, Sarks, Vegh, Wecht; Butti, Forcella, Zaffaroni

8 [ ] [ ] = = i i sin 6 cos cos 9 s i θ i φ θ φ θ φ θ ψ [ ] cos 9 6 sin 6 - cos s φ θ ψ φ θ θ θ ψ α = q f w a = /, q q 4 4 = m T, Y,q Gauntlett, Martelli, Sarks,Walram ; < < If α β = 6 α ψ ; onl aears.

9 = b a ab g t t S τ τ τ τ τ λ - = 0 b a ab g t t τ τ τ τ λ a a L τ δ δ = b a ab g H = α P ψ J P Q P Q R R = Point-like strings massless geoesics Const.: t = κ τ ; P t = = κ ; / Q R ; Q R = P ψ - / P α

10 BPS geoesics = / Q R P = 0, J = P ψ, 0 P = α Q R Therefore, is the ratio between U an R-charges. Also: J = 0 Q R from J = P ψ

11 Dual gauge theories Benvenuti, Franco, Hanan, Martelli, Sarks 6 Eamle: Y, 5 4 Perioic re.: Hanan, Kennawa, Franco, Vegh, Wecht SU U U

12 L L Oerators of maimum an minimum sloes can be ientifie with the geoesics at = an = R-charges an flavor charges match. Also: Berenstein, Herzog, Ouang, Pinansk

13 Small Fluctuations BMN, GKP, Gomis-Ooguri, Klebanov et al. V 0 α Q R = P Quantize: δ =n, δq R =0, δp α =0, δj=0 Also J>P ψ : δ =n, δq R =0, δp α =0, δj=n Agree with quantum numbers of U an SU currents we ientif these non-bps geoesics with insertions of the currents. Also Sonnenschein et al.

14 9 6 j j i A g t s j ij = ψ 9 6 j j i j A g A t s j ij j = ψ ψ Etene Strings We consier close etene stringssuch that each oint moves aroimatel along a BPS geoesic. Effective action for such strings: We introuce a coorinate ψ = ψ - t Now we use ansatz: t = κ τ

15 j i ij i i g A S σ σ τ τψ κ = = cos 9 4 cos 4 φ θ β λ π φ θ β ψ π σ σ σ σ n Q Q S R t t t R an take the limit: τ X 0, κ, κ τ X fie. We get a reuce action: In our case:

16 Eamle of T, Angelova, Pano-Zaas, M.K. ; Strassler et al. [ ] n n λ Seff. = J cosθ φ cosθ φ σ σ J Fiel theor: TrABABAB ABAB ; SUSU... A B A B... A B A B... F B F A W= ε ab ε c Tr A a B c A b B... A B A B... A B A B..., : oublet of SU

17 For Y,q σ sloeσ; β: conjugate of sin θ, φ L L J Effective moel: ma an min sloes A C L B L P α

18 Hamiltonian H = h eff. - P i i Coherent States: O = ρ ρ i i e ip e ip α α α U i α U i θ, φ, ψ i i θ, φ, ψ L L i i i i ρ ρ = ; ρ ρ = The coherent state action is S = τ Im O O τ τ O H O which gives:

19 = cos cos 4 φ θ β π φ θ β ψ π σ σ σ σ n Q h Q S R eff t t t R where = instea of a = It is interesting that a string icture an action for the oerators emerges from the analsis.

20 Toric Varieties S z,z w z,z C * / C z, z, z w z, z, z z e i, z e i, z

21 L q r b,s s,ab s,b asbs s asb a, asbs s 0,,a r = q - s q = a s - b Also Butti, Zaffaroni, generic olgon

22 4 4 ] [ A s s R = ψ ψ α β β φ α α α 6 6 A = 4 = ψ β β φ α α ρ ρ ψ β φ α ρ ρ ] [ g g f f s L,q r Metric: Cvetic, Lu, Page, Poe = β α β α ρ < < -<<

23 To write the metric in this wa we neee to ientif the angle conjugate to the R-charge. This can be one b fining the covariantl constant holomorhic three form in the Calabi-Yau.

24 Massless geoesics

25 Fiel Theor: Benvenuti, M.K.; Franco, Hanan, Martelli, Sarks, Vegh, Wecht; Butti, Forcella, Zaffaroni Eamle: Y, = L, SU U U 5 L,

26 In general r qr gives L,q r. Using a-ma one can comute the R-chargesan the central charge. Everthing matches. The arameters can be mae from one escrition to the other.

27 Long chiral rimaries: Q V Q H can be matche to geoesics at the corners :

28 Conclusions Y,q Massless BPS geo. Chiral rimaries. Non-BPS geoesics Current insertions. Etene strings Long oerators. Effective action emerges in f.t. Similar but not equal to bulk eff. action. L,q r Foun ual gauge theories. Matche BPS geoesics with long chiral rimaries. Strings? Three imensional quiver?