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Covariant quantization of the Superstring with fundamental b-c ghosts.

Covariant quantization of the Superstring with fundamental b-c ghosts. Kiyoung Lee (Stony Brook) 2006. 5. 4. UNC. Outline. 1. Brief History 2. Review of 1 st quantized BRST formalism 3. Superparticle BRST 4. Superparticle BRST in SYM background 5. Superstring BRST 6. Amplitudes

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Covariant quantization of the Superstring with fundamental b-c ghosts.

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  1. Covariant quantization of the Superstring with fundamental b-c ghosts. Kiyoung Lee (Stony Brook) 2006. 5. 4. UNC

  2. Outline 1. Brief History 2. Review of 1st quantized BRST formalism 3. Superparticle BRST 4. Superparticle BRST in SYM background 5. Superstring BRST 6. Amplitudes 7. Conclusion and future research

  3. Brief History • Sad : 1989~90 : Superparticle and Superstring (first-)quantization was attemped.(BV approach) • Separation of 1st Class and 2nd Class constraints covariantly. • Infinitely reducible Constraints. infinite tower of ghosts • Happy : 1980’~90’s : 1st quantized BRST formalism was established. • Universal field equation for any spin. • Universal free action for any spin. • SuperBRST with complete infinite tower of ghosts solved “sad” problem.(still reducible)

  4. Brink-Schwartz Superparticle action • Canonical momenta • Primary constraints

  5. Secondary 1st class Constraints • No cavariant separation of 1st and 2nd class constraints in

  6. Universal field equation for any spin →

  7. Detouring : 2000 : Pure Spinor formalism Termination in ghost pyramid Complicating composite b ghost Picture changing again • Fundamental : 2005 : Direct attack on infinitely reducible 1st class conts. Fundamental b-c ghosts Arbitrary (S)YM Background Conquest of the ghost pyramid Classical GS superstring action with auxiliary fields

  8. 1st quantized BRST • Adding 2+2 extra unphysical dimensions 2+2 SO(D-1,1) SO(D,2|2) L.C L.C 2+2 SO(D-2) SO(D-1,1|2) Indices : i=(a,α) ; a=(1,...,D) ; α=(,) ; A=(+,-, α)

  9. Indices : a=(1,...,D) ; α=(,) ; A=(+,-, α) OSp(1,1|2) Nonminimal

  10. Nonminimal minimal nonminimal extension

  11. Action • Spinor

  12. Examples • Vector • Spinor

  13. IGL(1|1) Nonminimal

  14. Examples • Scalar S=0 • Spin ½ • Vector

  15. SuperBRST • Solved1st and 2nd class constraints problem • Complete set of ghosts • SYM Background is needed for Superstring

  16. Technical problem ex) • Something is needed to reproduce

  17. Two different approaches • Direct Calculation to have • Supersymmetrizing after finding YM b.g (1),(2) give the same result (Constant b.g)

  18. For arbitrary b.g • ‘Big Picture’ like • Extended Cohomology Need to shrink Cohomology ex) spin ½

  19. Superstring • should have conformal weight 1 • Conformal anomaly should vanish at D=10 • X and θ should have conformal weight 0

  20. Classical superstring actionwith auxiliary field

  21. Amplitudes Superparticle Superstring Ghost Pyramid Sum

  22. Tree amplitude F-1 picture satisfy the same OPE (central charge) due to “ GP sum ”.

  23. Loop IR regularization Spinor zero mode measure Regularized Spinor propagator

  24. Rules

  25. Contractions

  26. contractions

  27. Examples 1) Vectors only contractions should give 4pt is the first nonvanishing amplitude 2) Super amplitude – 4pt is the first ex. again

  28. Conclusion and Future • 1st quantized BRST operator for GS superstring with fundamental b-c ghosts was constructed. • Tree and 1 loop amplitudes can be calculated in a manifestly supersymmetric and Lorentz covariant manner. • Multiloop amplitude will be calculated. → Geometry is crucial (?)…

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