Origin of Pure Spinor Superstring

Transcript

1 Origin of Pure Spinor Superstring based on JHEP05(2005)046 (hep-th/ ) with Y. Kazama Jul. 19, Nagoya Univ. Yuri Aisaka ( ) Univ. of Tokyo (Komaba)

2 Origin of PSS in a nutshell Berkovits Pure Spinor Formalism Canonical (Dirac) quantization Derived from a classical action (essentially GS) with reparam, kappa, plus a hidden sym. 1

3 Motivation Modern string theory rounds about D-branes; But strings in (D-brane sourced) Ramond-Ramond BG s: Poorly understood Super-Poincaré covariant formulation should help: Standard RNS & GS: Either SUSY or Lorentz is opaque RNS: - Worldsheet SUSY powerful scheme of super CFT, but with spin structures, GSO, supermoduli, bosonization, picture etc. - No built-in spacetime spinor Conceptually hard to handle RR fields GS: - Fully covariant classically - Less understood worldsheet sym (kappa) - Hard to quantize except in (semi-)lc gauge PS formalism Berkovits (2000 ) A new covariant worldsheet formalism with some promising results Like GS, has built-in spacetime SUSY no conceptual difficulty with RR 2

4 With PS formalism, one hopes to understand: Strings in various BG s esp. AdS+RR String side of the AdS/CFT conjecture String spectrum in AdS 5 S 5 Quantum integrability of string side Higher genus computation Supermembrane / M(atrix) theory 3

5 Plan: 1. Introduction Motivation Review of GS; treatment of kappa & its difficulty Siegel s approach to GS 2. Review of PS Basics, Achievments & Challenges 3. Origin of PS Classical action for PS Derivation of PS via canonical quantization 4. Summary and Outlook 4

6 Classical GS action type IIB: (x m, θ Aα ) = (x m, θ α, ˆθ α ) i = 0, 1; m = 0 9; α = 1 16; A = 1, 2 where L GS = L K + L WZ { L K = 1 2 gg ij Π m i Π jm L WZ = ɛ ij Π m i (W jm Ŵ jm ) ɛ ij Wi m Ŵ jm Π m i = i x m Wi m Ŵi m, Wi m = iθγ m i θ, Ŵi m = iˆθγ m iˆθ, Γ m = ( 0 γ mαβ γ m αβ 0 ) SO(9, 1) Wess-Zumino term: Topological (no dep on g ij ) Gives rise to κ-sym (local fermionic sym; halves on-shell DOF s of θ Aα ) Due to kappa, (Semi-)LC quant: easy Covariant quant: hard Shall demonstrate for superparticle 5

7 Difficulty of Covariant Quant of GS N = 1 Brink-Schwarz Superparticle (generalisation to N = 2 is trivial) L BS = 1 2e Π2, Π m = ẋ m iθγ m θ Zero-mode part of string Kappa without WZ Symmetries: Super-Poincaré Reparam: τ τ Kappa: for a fermionic param κ α (τ) δθ α = ip m (γ m κ) α, δx m = iθγ m δθ, δe = 4e θκ 6

8 Canonical Treatment of BS I Fix reparam to conformal gauge e = 1 Canonical momenta & Poisson bracket: p m = L ẋ = Π m, p m α = L θ = ip m(γ m θ) α α {x m, p n } P = δ m n, {θ α, p β } P = δ α β Phase space constraints: D α = p α ip m (γ m θ) α 0 Gauge orbits (1st cls) Constraint surface (2nd cls) Gauge slice T = 1 2 pm p m 0 (Virasoro) Constraint algebra: {T, T } P = {T, D α } P = 0, {D α, D β } P = 2ip m γ m αβ rank 8 (p2 = 0) first class (gauge syms) 8 second class (rels between p α and θ α ) constraint surface For (Dirac) quantization, must separate explicitly; hard to maintain covariance 7

9 Canonical treatment of BS II Explicit splitting of D α (assume p + 0): Dȧ = pȧ ip θȧ ip i γȧb i θ b Kȧ = Dȧ 1 γ i p +pi ȧb D b : 1st cls (kappa) D a = p a ip + θ a ip i γ i aḃ θ ḃ : 2nd cls Indeed from {D α, D β } P = 2ip m γαβ m, easy to see φ I = (T, Kȧ, D a ) ( ) {D a, D b } P = 2ip + δ ab, {D a, Kḃ} P = {Kȧ, Kḃ} P = 2i p 2 δȧḃ p 0 ( Virasoro) {φ I, φ J } P 0 2p + δȧḃ + A way to see relation between kappa and conformal syms: (kappa) 2 = (Virasoro) Vectors: V ± = V 0 ± V 9, γ ± = γ 0 ± γ 9 Spinors: θ α = (θ a, θȧ) ( ) ( γ γ = 0 1, γ i i ) ( ) = aḃ 8 γȧb i, γ = γ ± : SO(8) chirality projector 8

10 Canonical treatment of BS III Quantization is most easily done by fixing kappa Semi lightcone gauge: γ + θ = 0 θȧ = 0 (Kȧ, θȧ) : 2nd class Dirac bracket on φ I (D a, Kȧ, θȧ) = 0 (Symplectic structure on φ I = 0) {A, B} D = {A, B} P {A, φ I } P {φ I, φ J } 1 P {φ J, B} P = {A, B} P 1 2ip +{A, D a} P {D a, B} P + {A, Kȧ} P {θȧ, B} P + {A, θȧ} P {Kȧ, B} P Indep variables (x m, p n ; S a = 2p + θ a ) has canonical brackets: (D a = p a ip + θ a = 0 (θ a, p a ) become self-conjugate) {x m, p n } D = η mn, {S a, S b } D = iδ ab with single 1st cls constraint T = 1 2 p2 = 0 9

11 Canonical quantization of BS Quantization is now easy: Replace {, } D by quantum commutators: {x m, p n } D = η mn [x m, p n ] = iη mn {S a, S b } D = iδ ab {S a, S b } = δ ab On quantum states, impose T = 1 2 p2 = 0 (or invoke BRST) Remarks: Usually, we further fix T imposing x + = τ etc. (full LC gauge) (x m, p n ; S a ) with canonical brackets are sometimes called free To sum up, (1) In a non-covariant gauge (θȧ = 0) BS can be systematically quantized: L BS free fields + constraint T (2) Covariant quantization is bound to be difficult: D α = 0 contains first and second cls constraint 10

12 Siegel s approach to GS (1986) Gave up classical action. Instead, asked Can one find a set of 1st cls constraints ( BRST ) s.t. Free fields + BRST = superstring? L = 1 2 x x p α θ α ˆp α ˆθ α At the time 1. Appropriate BRST was not found. 2. Also, c tot 0 = 1 2 Π Π d α θ α ˆd α ˆθ α (SUSY inv) Using pure spinor λ α, one can. Berkovits (2000) 11

13 Review of PS Superstring Pure Spinor (PS) Berkovits (2000-) Cartan (1913) Bosonic chiral spinor λ α with non-linear PS constraint: λ α γ m αβ λβ = 0 Not all 10 conditions are indep because of a Fierz In fact, only 5 constraints are indep. PS: 16 5 = 11 indep components. (λγ m λ)(λγ m λ) = 0 Under U(5) SO(9, 1): 16 = PS cond λ I = 1 8 λ 1 + ɛ IJKLMλ JK λ LM 12

14 PS formalism for superstring (Shall concentrate on the holomorphic sector) Worldsheet free CFT with vanishing center GS sector: x m, (p α, θ α ) PS sector: (ω α, λ α ) weight (1, 0) T (z) = 1 2 xm x m (c=10) p α θ α ( 32) ω α λ α (22) Free : not genuinely; But physical quantities can be computed with covariant rules BRST op: bosonic ghost against d α = p α + i(γ m θ) α p m (γm θ) α (θγ m θ), d α (z)d β (w) = 2γ m Π m /(z w) Q B = λ α d α Q 2 B = 2 λγ m λπ m = 0 Cohomology of Q B = Spacetime spectrum Passed various checks, and more! 13

15 Achievements of PS: Basics Explicit forms of Q B -inv massless vertex ops: U(z) = λ α A α (x, θ) Superfield form. of Super-Maxwell: Q B U = 0 Correct constraint: (unintegrated) γ αβ m 1 m 2...m 5 D α A β = 0, (D α = α i(γ m θ) m ) δu = Q B Ω Gauge transf. A α (x, θ) a m (x), ψ α (x): photon, photino wave func; for plane waves: U B m (λγ m θ)e ik x, k 2 = 0 (weight = 0) Also, integrated vertex ops are available U F α (λγ m θ)(γ m θ) α e ik x Similar for 1st-massive modes (alas, much much more complicated) 14

16 Super-Poincaré covariant rule to compute N-pt tree amplitudes A = U 1 (z 1 )U 2 (z 2 )U 3 (z 3 ) [dz 4 ]V 4 (z 4 ) [dz N ]V N (z N )... : CFT correlation func. of vertex ops. Zero-mode prescription for θ and λ: Read off the coeff. of Coincides with RNS results. (λγ m θ)(λγ n θ)(λγ r θ)(θγ mnr θ) Similar for loops finiteness proof (2004) Construction of boundary states 15

17 Achievements of PS: Applications PS analogue of Metsaev-Tseytlin GS action for AdS 5 S 5 One-loop conformal invariant Construction of infinite non-local conserved charges Supermembrane (difficult) Extended formulations with PS cond removed Genuine free (ω α, λ α ) with ghosts which effectively impose PS cond SB: covariant Komaba: minimal extension Add BRST quartet (ω I, λ I, b I, c I ) (SO(9, 1) U(5)) Quantum operator mapping to RNS / GS: Q PS Q RNS Q GS Relation to doubly supersymmetric formulation (aka superembedding) 16

18 Challenges of PS All rules postulated by hand What is Q B the BRST of? Is λ α a BRST ghost?; Origin of λγ m λ = 0? Where is Virasoro? (Q = ct vs λd?) Nature of worldsheet symmetry is obscure (Drawback of the powerful free field postulate) Desirable to have the underlying classical action for PS Geometrical understanding of Berkovits measure Application to 11D supermembranes Can we find one?...yes! 17

19 Origin of PS Superstring Classical PS Action I [hep-th/ ] (with Y. Kazama) Recall the classical GS action: L GS = L K + L WZ { L K = 1 2 gg ij Π m i Π jm L WZ = ɛ ij Π m i (W jm Ŵ jm ) ɛ ij Wi m Ŵ jm Π m i = i x m W m i Classical PS can be obtained simply by (1) gauging the origin of θ A : Ŵi m, Wi m θ A Θ A = θ A θ A (2) shifting ( to have SUSY in (x m, θ Aα ) sector) = i θγ m i θ x m y m = x m i(θγ m θ) i(ˆθγ mˆ θ) 18

20 Classical PS Action II Then, GS action turns to L PS = L K + L WZ { L K = 1 2 gg ij Π m i Π jm L WZ = ɛ ij Π m i (W jm Ŵ jm ) ɛ ij Wi m Ŵ jm Π m i = i y m W m i Θ = θ θ, Ŵi m, Wi m = iθγ m i Θ y m = x m i(θγ m θ) i(ˆθγ mˆ θ) By construction, has hidden local symmetry 2: δ θ = δθ = χ, δx m = iχγ m Θ, (δθ = δy m = 0) can be used to kill θ (equivalence to GS is manifest) or, kept along quantization Berkovits BRST sym: Q B = λ α d α 19

21 Symmetries of Classical PS Common with GS: Reparam. + Weyl Kappa among (y m, Θ α, g ij ) Super-Poincaré δ θ = 0, δθ = ɛ, δx m = iɛγ m θ, (δθ = ɛ, δy m = iɛγ m Θ) New to PS: Hidden local symmetry We now canonically quantize L PS θ Aα (GS): fix to semi-lc Physical sector (x m, θ Aα ): can maintain covariance 20

22 Canonical Quantization of PS Canonical mom. Poisson bracket: k m = L ẋ m, k α = L θ α, k α = L θ α {x m (σ), k n (σ )} P = η mn δ(σ σ ), { θ α (σ), p β (σ )} P = δ α βδ(σ σ ) {θ α (σ), p β (σ )} P = δ α βδ(σ σ ), (rest) = 0 Constraints (left/right(hatted): particle 2): T = 1 4 Πm Π m + 1 Θ α D α = (Virasoro) 0, ˆT = (Π m k m + 1 y m 2W1 m Π m 0 + Π m 1 ) (ˆΠ m ˆΠ m 0 ˆΠ m 1 ) D α = k α i(γ m θ) α k m i(γ m Θ) α (k m + Π m 1 W 1m ) 0, ˆ D α = D α = k α + i(γ m θ) α k m + i(γ m Θ) α (k m + Π m 1 W 1m) 0, ˆD α = Expect (1st cls: , 2nd cls: 8) 2 constraints. 21

23 Constraint algebra (omit δ-funcs): {T, } P 0, { D α, D β } P = 2iγαβ m Π m {D α, D β } P = { D α, D β } P = 2iγαβ m Π m { α, } P = 0 T (Virasoro): just measures weight 1st cls D α : just as GS 8 1st cls (κ) and 8 2nd cls α = D α + D α = p α + p α ip m (Θγ m ): generates the hidden sym 16 1st cls Below, (1) we fix κ by imposing semi-lc gauge: γ + θ = 0 θȧ = 0 (2) Reduce the phase space to Σ : θȧ = D α = 0 (x m, p n, S a ; p α, θ α ) (3) On Σ, we will be left with st cls constraints (T, D α ) Can invoke BRST: introduce ghosts (b, c) and (ω α, λ α ) to define Q = λ α α + ct +, Q 2 = 0 ((b, c) and S a kill a part of λ α PS constraint) 22

24 A digression on left/right splitting Similar for ( ˆT, ˆ α, ˆD α ); but at this stage, Holomorphicity Spinor species (A = 1, 2) Constraint alg splits into holomorphic/anti-hol part: {D α, ˆD β } P = 0, etc. But unlike GS, left/right variables are still entangling: More on this later Π m = k m + 1 x m i 1 (θγ m θ + ˆθγ }{{ mˆ θ) 2W } 1 m 1 y m 23

25 Explicit separation of κ Assuming Π + 0, able to separate 1st and 2nd cls. in D α : { D a { D a, D D α b } P = 2iδ ab Π + Kȧ = Dȧ 1 Π i γ i D Π + ȧb b 1st cls. = Kappa sym. Indeed { Kȧ, Kḃ} P = 8iδȧḃ T + f α D α 0, T T/Π + T = T/Π + is a good combination: {T, T } P = 0 ( Π + weight 1) T 0 T 0 (Π + 0) Can use T instead of T 24

26 1st cls constraints on reduced phase space On the constrained surface Σ : φ I (D a, θȧ, Kȧ) 0 with Dirac bracket {A, B} D = {A, B} P {A, φ I } P {φ I, φ J } 1 P {φ J, B} P T and D α = (D a, Dȧ) yield (1 + 16) 1st cls constraints with simple algebra: {Dȧ(σ), Dḃ(σ )} D = 8iδȧḃ T δ(σ σ ) {T (σ), (σ )} D = {D a (σ), (σ )} D = 0 Now Dȧ carries the info of Kappa Kȧ: (Kappa) 2 = Virasoro So far, so good. 25

27 Non-Canonical Dirac Brackets However, the phase space variables are no more canonical wrt the induced Symplectic structure (Dirac bracket) on eg. Σ : D a = Kȧ = θȧ = 0 {x m (σ), k n (σ )} D = η mn δ(σ σ ) + (i/2π + )(γ m θ) c (γ n Θ) c σ δ(σ σ ) {k m (σ), k n (σ )} D = (i/2) σ [(1/Π + )(γ m Θ) c (γ n Θ) c σ δ(σ σ )] Is not surprising; but problematic for quantization. Can one find a canonical basis? Yes, and with local combination of original fields. 26

28 Canonical Basis for {x m (σ), p n (σ )} D = η mn δ(σ σ ), {θ α (σ), p β (σ )} D = δ α β δ(σ σ ) {S a (σ), S b (σ )} D = iδ ab δ(σ σ ), (rest) = 0 p m = k m + i 1 (θγ m θ) i 1 (ˆθγ mˆ θ) p a = k a i( 1 x + iθγ + 1 θ) θ a + [2(γ i 1 θ) a θγ i θ + (γ i θ) a 1 ( θγ i θ)] [ pȧ = kȧ + i(γ m θ)ȧ 2i θγ m 1 θ + i θγ m 1 θ i 1 ( θγ m θ) ] [ i(γ i θ)ȧ 1 x i 3iθγ i 1 θ + 2iθγ i 1 θ + i 1 (ˆ θγ iˆθ) ] S a = 2Π + θ a & similar for Ŝ a, ˆp a and ˆpȧ Also, complete left/right separation takes place: eg. Π m = k m + 1 x m + i 1 (θγ m θ + ˆθγ mˆ θ) 2W m 1 Π m = p m + 1 x m + 2i 1 (θγ m θ) 2W m 1 27

29 Quantization Quantization is trivial: just replace {A, B} D i[a, B}: [p m, x n ] = iη mn, {p α, θ β } = iδ β α, {S a, S b } = δ ab Spectrum dictated by (1 + 16) 2 1st cls constraints: T, (D a, Dȧ) 0 {Dȧ(σ), Dḃ(σ )} D = 8iδȧḃ T (w)δ(σ σ ) {T (σ), (σ )} D = {D a (σ), (σ )} D = 0 28

30 Obtained system as a free CFT Or in CFT language, we get conformal fields with free field OPE s: Constraints are where x m (z)x n (w) = η mn log(z w), S a (z)s b (w) = δ ab /(z w) p α (z)θ β (w) = δ α β /(z w) T = 1 4 Π Π Π +, D a = d a + i 2Π + S a Dȧ = dȧ + i 2 Π +Πi (γ i S)ȧ + 2 Π +(γi S)ȧ(Sγ i θ) Π m = π m 1 2 S) i 2π +(Sγm π +(Sγ θ), (Π+ = π + ) π m = i x m + θγ m θ d α = p α + i(γ m θ) α x m (γm θ) α (θγ m θ) 29

31 Separating S a from (x m, p α, θ α ), we get T = 1 π m π m 2 π π +S c S c + i π +S c θ c + i (Sγ i θ) (π + ) 3πi 1 θ) θċ θċ 2 log π + (π + ) 2(Sγi (π + ) 2 2π + D a = d a + i 2π + S a 2 Dȧ = dȧ + i (γ i π +πi S)ȧ 1 S)ȧ(Sγ i θ)+ 4 2 θȧ π +(γi π + 2 π+ θȧ (π + ) 2 some ordering ambiguities; Can fix it demanding the constraint alg close quantum mechanically: Dȧ(z)Dḃ(w) = 4δ ȧḃ T (w) z w, (rest) = 0 Those were already engineered by Berkovits-Marchioro [hep-th/ ] as a 1st cls algebra containing d α. 30

32 Derivation of PS Now, we can BRST quantize the system in a standard way: Introduce fermionic (b, c), bosonic ( ω α, λ α ) ghosts and define Q = ( λ α D α + T c λȧ λȧb) Q 2 = 0; Cohom of Q = spectrum This completes the quantization. Q must have correct cohom; a tricky way to quantize GS Furthermore, can show the equivalence with PS: cohom of Q = cohom of Q B (Q B = where λ α is constrained λγ m λ = 0 (PS) λ α d α ) Basic idea of the proof: (1) Split λ to PS direction and the rest: λ = (λ, λ ) = (2) (λ, ω ; c, b, S a ) forms 5 KO quartet and decouples cohomologically 31

33 Derivation of Q B via Similarity Transf Berkovits-Marchioro Bring a constant spinor rȧ satisfying rȧ λȧ = 1, rȧrȧ = 0 Able to define a projector to PS space: Pȧḃ = λȧrḃ 1 2 r ȧ λḃ, P ab = 1 2 γi aḃ P ḃċ λċ(γ i r) b, P ȧḃ = δ ȧḃ P ȧḃ P ab = δ ab P ab Then, one can show λ α (P λ) α, λγ m λ = 0, λ α (P λ) α where e Z e Y e X Qe X e Y e Z = δ b + δ + Q B X = c(rċdċ), Y = 1 2 (P S) c (PS) c, Z = d c(ps) c 2i + 4( θ ȧλȧ)( θḃrḃ) π + δ b = 2b( λγ + λ), δ = 2iλ c (PS) c, Q B = λ α d α {δ b, δ } = {δ b, Q B } = {δ, Q B } = 0 32

34 Summary Derived Berkovits Q B from first principles Origin of PS Classical GS action with a hidden local sym. (completely covariant) Hidden sym Q B = λ α d α (λ α : BRST ghost) Reparam (b, c)-ghost + GS S a 5 PS constraint λγ m λ = 0 33

35 Outlooks Should be able to derive Berkovits measure (Tree & Loop) Improve the last step; more clever gauge fixing? Supermembrane (or super p-branes) Just like Siegel/Berkovits, ask Can one find a clever set of 1st cls constraint s.t. Free fields + BRST = Supermembrane Appropriate constraints are not known; classical PS action might provide some hints. Application to strings in various BG s Derivation of PS in curved BG. esp. AdS 5 S 5 34