Before Penrose, there was Dirac: for coordinate representation of D-dimensional conformal group SO(D,2) by (D+2)-vector y instead of spinor, impose 3 equations on fields:

In lightcone notation (dy

- y
^{2}= 0 constrains to cone to kill 1 coordinate- y∙∂/∂y = 0 projective to kill another
- (∂/∂y)
^{2}= 0 gives usual Klein-Gordon (massless)

y = y^{+}(1,½x^{2},x) → dy^{2}= (y^{+})^{2}dx^{2}: conformally flat

(For a particle, (y^{+})^{2} becomes the worldline metric.)

Projective lightcone is (degenerate) zero-radius limit of (A)dS

(e.g., in AdS/CFT, AdS_{5} → flat 4D space: instant holography)

→ superconformal group: SL(4|N) for N supersymmetries

(For strings, N=4 → PSL(4|4) = SL(4|4)/GL(1))

2 kinds of (super)twistors:

- Penrose (original)
- Atiyah-Drinfel'd-Hitchin-Manin (instanton)

Penrose | ADHM | |
---|---|---|

projective space | RP(3|N) → 3 bosons + N fermions on shell |
HP(1|½N) → 4 bosons + 2N fermions off shell |

group representation [ Λ, Λ } = -iI |
defining: Λ^{A} /GL(1) A = (4|N) of SL(4|N) SL(4|N): G _{A}^{B} = Λ^{B}Λ_{A} - tr GL(1): H = Λ ^{A}Λ_{A} |
"flag": Λ^{α'A} /GL(2) α' = 2 of GL(2) SL(4|N): G _{A}^{B} = Λ^{α'B}Λ_{α'A} - tr GL(2): H _{α'}^{β'} = Λ^{β'A}Λ_{α'A} |

lightcone (bosonic) |
Lorentz SO(2,2) = SL(2)^{2} 4-momentum p ^{αα̇} put on mass(less) shell: p ^{2} = det(p^{αα̇}) = 0 → p ^{αα̇} = λ^{α}λ^{α̇} → (λ ^{α}, λ^{α̇})/GL(1) |
conformal SO(3,3) = SL(4) 6-vector y ^{[ab]} put on projective lightcone: y ^{2} = Pf(y^{[ab]}) = 0 (& y∙∂/∂y = 0) → y ^{[ab]} = λ^{α'a}λ_{α'}^{b} → λ ^{α'a}/GL(2) (super: y ^{[AB}y^{CD)} = 0, etc.) |

Metric in terms of ADHM twistors (D=4):

dy^{2}= dλ^{α'a}dλ^{β'b}(λ_{α'}^{c}λ_{β'}^{d}ε_{abcd})

conformal | SO(6) | SO(5,1) | SO(4,2) | SO(3,3) |
---|---|---|---|---|

covering | SU(4) | SU*(4) | SU(2,2) | SL(4) |

reality | C | P(seudo) | C | R |

Penrose | ✓ | X | ✓ | ✓ |

ADHM | X | P: SU(2) | X | R: SL(2) |

4D sig. | ++++ | -+++ | --++ |

Penrose needs real *phase* space,
with momenta dual to coordinates

(Λ^{A}⊕Λ_{A} = R⊕R or C⊕C*);

ADHM needs real *coordinate* space

(Λ_{α'}^{A} = R⊗R or P⊗P).

So both twistors work in (--++), Penrose also in (-+++) (as invented),
ADHM also in (++++) (as invented for instantons).

Complex for either ("X") → doubling (or no reality).

φ(x) = ∫dp φ̃(p) e^{ix∙p}= ∫dλ dλ φ̂(λ,λ) e^{iλxλ}= ∫dλ φ̌(λ,λx) = ∫dλ dμ δ^{2}(μ-λx) φ̌(λ,μ)

where Λ^{A} = (λ^{α},μ^{α̇}),
Λ_{A} = (μ̄_{α},λ_{α̇})
(and similar for N>0). Thus

Λ^{α̇}= Λ^{α}x_{α}^{α̇}

On the other hand, ADHM twistors are already in coordinate space (momentum space is not so useful for nonperturbative solutions):

Λ_{β'}^{A}= Λ_{β'}^{γ}( δ_{γ}^{α}, x_{γ}^{α̇}, θ_{γ}^{a})

A = (α,α̇,a) as SL(4|N) ⊃ SL(2)^{2}SL(N) (superconformal ⊃ Lorentz ⊗ internal)

So we have in particular

Λ_{β'}^{α̇}= Λ_{β'}^{α}x_{α}^{α̇}

Similar to Penrose, but invertible for x;

to get coordinates of chiral superspace (x,θ), do one of:

Chiral superspace has no torsion.

- GL(2) gauge Λ
_{β'}^{γ}= δ_{β'}^{γ}- use constrained projective lightcone coordinates y
^{AB}= Λ^{γ'A}Λ_{γ'}^{B}- use HP(1|½N) projective coordinates (x
_{β}^{α̇},θ_{β}^{a}) = Λ^{-1}_{β}^{γ'}(Λ_{γ'}^{α̇},Λ_{γ'}^{a})

generalization of p

G_{[A}^{[C}G_{B)}^{D)}- tr = 0 [ ) = graded antisymmetrize

Satisfied trivially by Penrose; for ADHM, →

Λ^{α'}_{A}Λ_{α'B}= 0

Field equations for free scalar (Λ → -i∂/∂Λ):

∂^{α'}_{A}∂_{α'B}φ = 0

After GL(2) gauge fixing, just truncate

A → (α̇,a): ∂ → (∂/∂x,∂/∂θ)

∂∂ = 0:

Free "lightcone" solution takes ADHM → Penrose (cf. p

- (∂/∂x)
^{2}= 0 (Klein-Gordon)- (∂/∂x)(∂/∂θ) = 0 (κ symmetry)
- (∂/∂θ)
^{2}= 0

Λ_{α'A}= Λ_{α'}Λ_{A}

"gauge" Λ_{α'}= δ_{α'}^{+}

This allows manifestly superconformal generalization of Penrose transform:

GL(2) says momentum-space wave functions are independent of
Λ_{α'} →

which reduces to the usual in the gauge Λ

φ(Λ _{α'}^{A}) = ∫dΛ_{A}dΛ_{α'}exp(iΛ^{α'A}Λ_{α'}Λ_{A}) φ̂(Λ_{A}) =⎧

⎨

⎩∫dΛ _{A}δ^{2}(Λ_{α'}^{A}Λ_{A}) φ̂(Λ_{A})∫dΛ _{α'}φ̌(Λ^{α'A}Λ_{α'})

Selfdual Yang-Mills in terms of ∇_{α'A} = ∂_{α'A} + iA_{α'A}:

[∇_{α'A},∇_{β'B}} = C_{α'β'}F_{AB}

where C_{α'β'} = - C_{β'α'} is the SL(2) metric.

Truncation A → (α̇,a) gives usual chiral superspace equations.

The ADHM solution writes A_{α'A} in terms of matrices that are proportional to Λ.

L = (∇_{-}Λ^{A})Λ^{-}_{A}+ L_{YM}

∇_{-} = right-handed worldsheet derivative, covariant with respect to 2D:

- coordinate
- Lorentz
- twistor GL(1)

L_{YM} is for Yang-Mills symmetry current.

3 important ingredients:

- supertwistors: take place of x
- Yang-Mills current: Yang-Mills group theory, denominators of amplitudes
- GL(1) worldsheet "instantons": perturbative helicity expansion about selfdual Y-M

This action follows as lightcone solution (see above) of ADHM twistor superstring:

L' = (∇_{-}Λ^{α'A})Λ^{-}_{α'A}+ g_{-α'}^{β'}Λ^{β'A}Λ^{-}_{α'A}+ g_{--}^{AB}Λ^{-α'}_{A}Λ^{-}_{α'B}+ L_{YM}

where now ∇ is not GL(1), which is part of GL(2) gauge field g_{-α'}^{β'}.

Two kinds of constraints imposed by gauge fields:

Simultaneously has usual x and GL(1) for instantons

- g
_{-α'}^{β'}: GL(2), kills extra coordinates- g
_{--}^{AB}: field equations (K-G, etc.)

To relate to first-quantization, we'll consider the fields to be scalars, carrying no Lorentz indices (which may be hidden if the coordinates include fermions). However, they will carry internal symmetry indices, by making the scalars N×N matrices. (From now on, N ≠ # of supersymmetries.) In 't Hooft's notation (inspired by string theory), this is indicated by replacing lines with double lines, which are continuous (no branches, and ending only on asymptotic states), reflecting the N-fold symmetry of the action, which has an overall trace.

This notation gives a 2-dimensionality to Feynman graphs, by filling in the space between lines, and also that inside closed loops of such lines. Such closed loops get factors of N from summing over the internal symmetry index, and by Euler's theorem any graph then gets a factor of

(g^{2})^{ℓ-1}N^{F}= (Ng^{2})^{ℓ-1}N^{-2(h-1)}[P-V-F = 2(h-1), P-V = ℓ-1 → F = ℓ-1 -2(h-1)]

for ℓ loops of the original lines, F faces (P propagators & V vertices), and h handles (genus), where g is the coupling, which appears in the action as an overall 1/g^{2} (= 1/ħ).

For the usual string, the x part of the action is latticized as

- polyhedron = "random" lattice (discretization) of worldsheet
- sum over Feynman diagrams = integral over worldsheet metrics
- 1/N that counts genus = string coupling
- Ng
^{2}that counts loops = - exp( - coefficient of worldsheet cosmological term )- Feynman propagator = exp( - worldsheet action for x )

½∫(∂x)^{2}/α' → ½∑_{<ij>}(x_{i}- x_{j})^{2}/α'

for string "slope" α' and links <ij> (lines), which gives the unusual propagator

Δ = exp(-x^{2}/2α')

which accounts for several of the unusual features of known string theories (those that don't seem to apply to hadrons).

Δ = 1/x^{2}

in D=4, or other powers in D≠4 from Fourier transforming 1/p^{2}. However, note that "T-duality" (symmetry under Fourier transformation) requires D=4 for this propagator (but no restriction for the Gaussian one).

On the other hand, we need an exponential propagator for e^{-S}, so use

1/x^{2}= ∫_{0}^{∞}dτ exp( -τx^{2})

The discretized x action is then

½∑_{<ij>}τ_{ij}(x_{i}- x_{j})^{2}

Note, e.g., this means 2 components for τ at each vertex on a regular (flat) square lattice. We can enforce this in general by making τ a traceless tensor: In the continuum limit,

S → ½∫τ^{±±}(∇_{±}x)^{2}

Sometimes we prefer a 1st-order formalism, where

L = (∇_{±}x)∙p^{±}-½(τ^{±±})^{-1}(p^{±})^{2}

Since τ is associated with lightlike directions, the lattice is thus a lightlike lattice, with any vertex having 4 lightlike directions, corresponding to φ^{4} theory.

In field theory, a coupling G in the action gives a factor -G at a vertex, because of exp(-S_{2}) for 2nd-quantized S_{2} when perturbatively expanding in the interaction part of -S_{2}. But the strings 1st-quantized action S_{1} always gives positive amplitudes because exp(-S_{1}) is always positive. Thus, G is always negative in this correspondence. In this case, we then have "wrong-sign" Gφ^{4}=g^{2}φ^{4} theory (potential unbounded below). However, in D=4 this theory (because of the wrong sign) is "asymptotically free": better behaved perturbatively.

They're taken after rescaling 1 worldsheet coordinate by α', so α' → 0 collapses the string in that direction, while α' → ∞ expands it to ∞. The latter limit can be taken explicitly on a sting action. It leads to a degenerate worldsheet metric: The worldsheet breaks up into worldlines. There are 2 possible degenerate directions:

- α' → 0: truncation to massless states
- α' → ∞: no tension; string "falls apart" into constituent partons

(Not timelike, since then the partons would not propagate to σ

- spacelike: (∂x)
^{2}→ ẋ^{2}(usual)- lightlike: (∂x)
^{2}→ 0, but τ^{±±}(∇_{±}x)^{2}→ τ^{--}(∇_{-}x)^{2}(chiral)

L = (∇_{±}Λ^{α'A})Λ^{±}_{α'A}+ g_{±α'}^{β'}Λ^{β'A}Λ^{±}_{α'A}+ g_{±±}^{AB}Λ^{±α'}_{A}Λ^{±}_{α'B}+ L_{YM}

where L_{YM} now also has ± terms.