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)

• Brink-Schwartz Superparticle action

• Canonical momenta

• Primary constraints

• Secondary 1st class Constraints

• No cavariant separation of 1st and 2nd class constraints in

• Universal field equation for any spin

→

• 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)

Indices : a=(1,...,D) ; α=(,) ; A=(+,-, α)

OSp(1,1|2)

Nonminimal

• Nonminimal

minimal

nonminimal extension

• Action

• Spinor

• Examples

(1) Vector

(2) Spinor

IGL(1|1)

Nonminimal

Examples

• Scalar

S=0

• Spin ½

• Vector

SuperBRST

• Solved 1st and 2nd class constraints problem

• Complete set of ghosts

• SYM Background is needed for Superstring

• Technical problem

ex)

• Something is needed to reproduce

• Two different approaches

(1) Direct Calculation to have

(2) Supersymmetrizing after finding YM b.g

(1),(2) give the same result (Constant b.g)

For arbitrary b.g

• ‘Big Picture’ like

• Extended Cohomology

Need to shrink Cohomology

ex) spin ½

Superstring

(1) should have conformal weight 1

(2) Conformal anomaly should vanish at D=10

(3) X and θ should have conformal weight 0

Classical superstring actionwith auxiliary field

• Amplitudes

Superparticle

Superstring

Ghost Pyramid Sum

• Tree amplitude

F-1 picture

satisfy the same OPE (central charge)

due to “ GP sum ”.

• Loop

IR regularization

Spinor zero mode measure

Regularized Spinor propagator

• Rules

• Contractions

• contractions

• Examples

1) Vectors only

contractions should give

4pt is the first nonvanishing amplitude

2) Super amplitude – 4pt is the first ex. again

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 (?)…