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 (dy2 = -2dy+dy- + rest),
- y2 = 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+(1,½x2,x) → dy2 = (y+)2dx2: 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, AdS5 → flat 4D space: instant holography)
2 kinds of (super)twistors:
- Penrose (original)
- Atiyah-Drinfel'd-Hitchin-Manin (instanton)
→ 3 bosons + N fermions
→ 4 bosons + 2N fermions
[ Λ, Λ } = -iI
|defining: ΛA /GL(1)
A = (4|N) of SL(4|N)
SL(4|N): GAB = ΛBΛA - tr
GL(1): H = ΛAΛA
|"flag": Λα'A /GL(2)
α' = 2 of GL(2)
SL(4|N): GAB = Λα'BΛα'A - tr
GL(2): Hα'β' = Λβ'AΛα'A
|Lorentz SO(2,2) = SL(2)2
put on mass(less) shell:
p2 = det(pαα̇) = 0
→ pαα̇ = λαλα̇
→ (λα, λα̇)/GL(1)
|conformal SO(3,3) = SL(4)
put on projective lightcone:
y2 = Pf(y[ab]) = 0 (& y∙∂/∂y = 0)
→ y[ab] = λα'aλα'b
(super: y[AByCD) = 0, etc.)
Metric in terms of ADHM twistors (D=4):
dy2 = dλα'adλβ'b(λα'cλβ'dεabcd)
|ADHM||X||P: SU(2)||X||R: SL(2)|
Penrose needs real phase space,
with momenta dual to coordinates
(ΛA⊕ΛA = R⊕R or C⊕C*);
ADHM needs real coordinate space
(Λα'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) eix∙p = ∫dλ dλ φ̂(λ,λ) eiλxλ = ∫dλ φ̌(λ,λx) = ∫dλ dμ δ2(μ-λx) φ̌(λ,μ)
where ΛA = (λα,μα̇), ΛA = (μ̄α,λα̇) (and similar for N>0). Thus
Λα̇ = Λαxαα̇
On the other hand, ADHM twistors are already in coordinate space (momentum space is not so useful for nonperturbative solutions):
Λβ'A = Λβ'γ ( δγα, xγα̇, θγa)
A = (α,α̇,a) as SL(4|N) ⊃ SL(2)2SL(N) (superconformal ⊃ Lorentz ⊗ internal)
So we have in particular
Λβ'α̇ = Λβ'αxαα̇
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 Λβ'γ = δβ'γ
- use constrained projective lightcone coordinates yAB = Λγ'AΛγ'B
- use HP(1|½N) projective coordinates (xβα̇,θβa) = Λ-1βγ'(Λγ'α̇,Λγ'a)
G[A[CGB)D) - tr = 0 [ ) = graded antisymmetrize
Satisfied trivially by Penrose; for ADHM, →
Λα'AΛα'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. p2 = 0):
- (∂/∂x)2 = 0 (Klein-Gordon)
- (∂/∂x)(∂/∂θ) = 0 (κ symmetry)
- (∂/∂θ)2 = 0
Λα'A = Λα'ΛA
"gauge" Λα' = δα'+
This allows manifestly superconformal generalization of Penrose transform:
GL(2) says momentum-space wave functions are independent of Λα' →
which reduces to the usual in the gauge Λβ'γ = δβ'γ
φ(Λα'A) = ∫dΛA dΛα' exp(iΛα'AΛα'ΛA) φ̂(ΛA) = ⎧
∫dΛA δ2(Λα'AΛA) φ̂(ΛA) ∫dΛα' φ̌(Λα'AΛα')
Selfdual Yang-Mills in terms of ∇α'A = ∂α'A + iAα'A:
[∇α'A,∇β'B} = Cα'β'FAB
where Cα'β' = - Cβ'α' is the SL(2) metric.
Truncation A → (α̇,a) gives usual chiral superspace equations.
The ADHM solution writes Aα'A in terms of matrices that are proportional to Λ.
L = (∇-ΛA)Λ-A + LYM
∇- = right-handed worldsheet derivative, covariant with respect to 2D:
- twistor GL(1)
LYM 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' = (∇-Λα'A)Λ-α'A + g-α'β'Λβ'AΛ-α'A + g--ABΛ-α'AΛ-α'B + LYM
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
(g2)ℓ-1NF = (Ng2)ℓ-1N-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/g2 (= 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
- Ng2 that counts loops = - exp( - coefficient of worldsheet cosmological term )
- Feynman propagator = exp( - worldsheet action for x )
½∫(∂x)2/α' → ½∑<ij> (xi - xj)2/α'
for string "slope" α' and links <ij> (lines), which gives the unusual propagator
Δ = exp(-x2/2α')
which accounts for several of the unusual features of known string theories (those that don't seem to apply to hadrons).
Δ = 1/x2
in D=4, or other powers in D≠4 from Fourier transforming 1/p2. 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/x2 = ∫0∞ dτ exp( -τx2 )
The discretized x action is then
½∑<ij> τij(xi - xj)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 φ4 theory.
In field theory, a coupling G in the action gives a factor -G at a vertex, because of exp(-S2) for 2nd-quantized S2 when perturbatively expanding in the interaction part of -S2. But the strings 1st-quantized action S1 always gives positive amplitudes because exp(-S1) is always positive. Thus, G is always negative in this correspondence. In this case, we then have "wrong-sign" Gφ4=g2φ4 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 σ0 = ±∞.) Thus, the chiral action of the twistor superstring can come as a tensionless limit only from a string with ordinary parton propagators: Since both relate to QCD, this is not totally unexpected. Working backwards, we then find the stringy twistor superstring action
- spacelike: (∂x)2 → ẋ2 (usual)
- lightlike: (∂x)2 → 0, but τ±±(∇±x)2 → τ--(∇-x)2 (chiral)
L = (∇±Λα'A)Λ±α'A + g±α'β'Λβ'AΛ±α'A + g±±ABΛ±α'AΛ±α'B + LYM
where LYM now also has ± terms.