Superstring in the plane-wave background with RR-flux as a conformal field theory

Transcription

1 0th December, 008 At Towards New Developments of QFT and Strings, RIKEN Superstring in the plane-wave background with RR-flux as a conformal field theory Naoto Yokoi Institute of Physics, University of Tokyo (Komaba) 1. Introduction : PP-wave as a Limit of AdS 5 S 5. Canonical Analysis and Conformal Symmetry 3. BRS Quantization and Physical Spectrum 4. Summary and Future Problems Based on the Collaboration with Yoichi Kazama, JHEP 0803 (008) 057 (arxiv: ).

2 1 Introduction : PP-Wave as a Limit of AdS 5 S 5 AdS/CFT Correspondence : One of the Profound Structures in String Theory Parameter Correspondence Type IIB Superstring Theory on AdS 5 S 5 Duality 4-Dim. N = 4 SU(N) Super Yang-Mills Theory g N = 4πg YM sn = R 4 /α Balance Between RR-Flux N and Curvature R Large RR-Flux is Crucial. RNS Formulation Has (Severe) Problem with RR-Flux = Green-Schwarz Formalism Green-Schwarz Action on AdS 5 S 5 with RR-Flux Has Been Constructed. E.g. Bosonic Part is a Non-Linear Sigma Model on SO(,4) SO(1,4) SO(6) SO(5). Interacting and Massive Theory = Left and Right Moving Sectors Couple.

3 PP-wave as a Limit of AdS 5 S 5 = Zoom-In of a Null Geodesics 5 AdS 5 S t X c ZOOM IN PP-wave Geometry ds = dx + dx µ x I dx+ + dx I (I = 1 8), RR-flux F +134 = F = µ Non-Trivial Curvature and RR-flux is Still There! The Green-Schwarz Action in Light-Cone Gauge = Massive Free Field Theory. (Metsaev) But, ANY String Theory Has Also Massless CFT Description. (Cf. Friedan-Martinec-Shenker) How Can We Reconcile the Massive Picture with Powerful CFT Description? = We Try to Formulate and Quantize the String Theory as EXACT CFT.

4 Canonical Analysis and Conformal Symmetry GS Action in a Conformally Inv. Gauge g ij = η ij, γ + θ A = 0 (A = 1, ) 1 : L GS = L Kin + L WZ, L Kin = T ) ηij( i X + j X + i X I j X I µ X I i X + j X + }{{} Coupling to Curvature +i T η ij( i X + ( θ 1 j θ 1 + θ j θ ) ) +µ i X + j X + θ 1 θ, }{{} Coupling to RR-Flux L WZ = i T ɛ ij i X + ( θ 1 j θ 1 θ j θ ), Soln. of EoM + X + = 0 = X + = X + L (σ +) + X + R (σ ), X I ( = Σ n a I n u n + ã I ) ( nũn, θ A = Σ n b A n u n + b ) A n ũn. u n (ũ n ) = e i λ ± n X + R +λ n X + L, λ ± n = ω n ± n α p, ω + n = M + n. 1 To Fix κ-symmetry. This Becomes Free Massive Theory in the LC Gauge 0 X + p +.

5 The Solutions are Inseparable Functions of σ + and σ. Can we Construct PURELY Left (or Right) Moving Virasoro Generator T + (σ + )? ( ) f+ Virasoro Generators T ± in terms of the Soln. = T + = T + X + L (σ + ). Depends on ONLY σ + = However, Completely Different from Flat (µ = 0) Case. Hamiltonian Analysis Brackets for Modes (a n, b n ) from the ETC for Fields. = We Do NOT Know the Completeness for u n and Can NOT Obtain the Brackets. Soln. of EoM + Brackets for t-indep. Modes }{{} Not Obtained Here = Correlators at Unequal-Times. However, String Theory Has Conformal Symmetry including the HAMILTONIAN. Virasoro Alg. Based on Fields at t = 0 Gives Dynamical Information! Physical Spectrum = (Gauge) Constraints Dynamics = Construction of Physical Primary Fields For Free Boson, T + 1 ` + φ L. f+ is an Arbitrary Fn. of σ +

6 With Dimensionless Fields (A µ (σ) X µ, B µ (σ) P µ, S A (σ) θ A ) at t = 0, π T + = Π + Π + 1 Π I + i S σ S + ˆµ Π + Π + A I iˆµ Π + Π + S 1 S, where Π = 1 (B + σ A) X, Π = 1 (B + σ A) X. Quantum Op. Requires Ordering = Phase-Space Normal Ordering for Fourier Modes A n (n 1), B n (n 0), S A n (n 1) as Annihilation Operators. Except for Central Charge Terms, Quantum Operator Anomalies Appear from C B = C F C B = C F = 1 (π) 1 (π) = iˆµ π ([ 1 Π I (σ), ˆµ ] ) Π + Π + A I (σ ) (σ σ ), [ iˆµ Π + Π + S 1 S iˆµ ] (σ), Π + Π + S 1 S (σ ) ( ) Π + Π + δ (σ σ ) + σ ( Π + Π + )δ(σ σ ). These Two Operator Anomalies Exactly Cancel Out!,

7 3 BRS Quantization and Physical Spectrum BRS Quantization Requires NILPOTENT BRS Charge Q B Virasoro Generator with Central Charge 6 is Needed. = Quantum Correction Term T + = 1 From the Virasoro Generator, Q B = n Physical States as Q B -Cohomology π σ ln Π + should be Introduced. ( c n L + n 1 m (m n) c m c n b m+n ). The Decomposition Q B = Q 1 + Q 0 + Q n 1 by Light-Cone No.: Π ± n ±1. Isomorphism Q B -Cohomology Q 1 -Cohomology H T with L + 0 Ψ = 0 L + 0 = L 0 = 0 in H T = On-Shell Condition and Level-Matching. On-Shell Condition H = 0 Gives Light-Cone Hamiltonian with Massive Oscillators α I n = 1 (B I n ω na I n ), [ α I n, αj m] = ωn δ IJ δ m+n,0. Correctly Reproduce the Light-Cone Gauge Spectrum as BRS-Cohomology!

8 4 Summary and Future Problems Summary We Have Investigated Both the Classical and Quantum Aspects of Superstring Theory in the PP-Wave Background with a Conformally Invariant Gauge as an Exact CFT. Quantum Virasoro Generators 3 are Constructed and the Light-Cone Gauge Spectrum is Correctly Reproduced as the BRS-Cohomology. I Hope (Interesting) Details will Be Reported Somewhere! (Here?) Future Problems (Now in Progress) Analysis of Global Symmetries Realization of the PP-Wave Superalgebra. (1, 1) Primary Fields and Correlation Fn. = Flat GS-String as a CFT. Application to the BMN and AdS/CFT Correspondence, 3 Purely Bosonic Part and Other Orderings Suffer from Operator Anomalies.