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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 1st quantized BRST formalism
3. Superparticle BRST
4. Superparticle BRST in SYM background
5. Superstring BRST
6. Amplitudes
7. Conclusion and future research
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)
• 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
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)
SuperBRST
• Solved 1st and 2nd class constraints problem
• Complete set of ghosts
• SYM Background is needed for Superstring
• Two different approaches
(1) Direct Calculation to have
(2) Supersymmetrizing after finding YM b.g
(1),(2) give the same result (Constant b.g)
Superstring
(1) should have conformal weight 1
(2) Conformal anomaly should vanish at D=10
(3) X and θ should have conformal weight 0
• Examples
1) Vectors only
contractions should give
4pt is the first nonvanishing amplitude
2) Super amplitude – 4pt is the first ex. again