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 1 st 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 1 st Class and 2 nd Class constraints covariantly.  Infinitely reducible Constraints. infinite tower of ghosts Happy : 1980’~90’s : 1 st 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 1 st class Constraints No cavariant separation of 1 st and 2 nd 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 1 st class conts. Fundamental b-c ghosts Arbitrary (S)YM Background Conquest of the ghost pyramid Classical GS superstring action with auxiliary fields

8. 1 st 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. minimal nonminimal extension

11. Action Spinor

12. Examples (1)Vector (2)Spinor

13. IGL(1|1) Nonminimal

14. Examples Scalar S=0 Spin ½ Vector

15. SuperBRST Solved 1 st and 2 nd 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 (1)Direct Calculation to have (2)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 (1) should have conformal weight 1 (2)Conformal anomaly should vanish at D=10 (3)X and θ should have conformal weight 0

21. Classical superstring action with auxiliary field

22. Amplitudes Superparticle Superstring Ghost Pyramid Sum

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

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

25. Rules

26. Contractions

27. contractions

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

29. Conclusion and Future 1 st 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 (?)…