ADHM supertwistors

Projective lightcone

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:

• 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)

Penrose vs. ADHM

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 [ Λ̄, Λ } = -iI	defining: ΛA /GL(1) A = (4|N) of SL(4|N) SL(4|N): GAB = ΛBΛ̄A - tr GL(1): H = ΛAΛ̄A	"flag": Λα'A /GL(2) α' = 2 of GL(2) SL(4|N): GAB = Λα'BΛ̄α'A - tr GL(2): Hα'β' = Λβ'AΛ̄α'A
lightcone (bosonic)	Lorentz SO(2,2) = SL(2)2 4-momentum pαα̇ put on mass(less) shell: p2 = det(pαα̇) = 0 → pαα̇ = λαλ̄α̇ → (λα, λ̄α̇)/GL(1)	conformal SO(3,3) = SL(4) 6-vector y[ab] put on projective lightcone: y2 = Pf(y[ab]) = 0 (& y∙∂/∂y = 0) → y[ab] = λα'aλα'b → λα'a/GL(2) (super: y[AByCD) = 0, etc.)

(Can generalize GL(2) → GL(2|n), but not useful for selfduality)

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

dy² = dλ^α'adλ^β'b(λ_α'^cλ_β'^dε_abcd)

Signature

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⊕Λ̄_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).

Where in twistorspace is x?

φ(x) = ∫dp φ̃(p) e^ix∙p = ∫dλ dλ̄ φ̂(λ,λ̄) e^iλxλ̄ = ∫dλ φ̌(λ,λx) = ∫dλ dμ δ²(μ-λx) φ̌(λ,μ)

where Λ^A = (λ^α,μ^α̇), Λ̄_A = (μ̄_α,λ̄_α̇) (and similar for N>0). Thus

Λ^α̇ = Λ^αx_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)²SL(N) (superconformal ⊃ Lorentz ⊗ internal)

So we have in particular

Λ_β'^α̇ = Λ_β'^αx_α^α̇

Similar to Penrose, but invertible for x;
to get coordinates of chiral superspace (x,θ), do one of:

• GL(2) gauge Λ_β'^γ = δ_β'^γ
• use constrained projective lightcone coordinates y^AB = Λ^γ'AΛ_γ'^B
• use HP(1|½N) projective coordinates (x_β^α̇,θ_β^a) = Λ^-1_β^γ'(Λ_γ'^α̇,Λ_γ'^a)

Field equations

G_[A^[CG_B)^D) - tr = 0                 [   ) = graded antisymmetrize

Satisfied trivially by Penrose; for ADHM, →

Λ̄^α'_AΛ̄_α'B = 0

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

∂^α'_A∂_α'B φ = 0

After GL(2) gauge fixing, just truncate

A → (α̇,a):     ∂ → (∂/∂x,∂/∂θ)

∂∂ = 0:

• (∂/∂x)² = 0         (Klein-Gordon)
• (∂/∂x)(∂/∂θ) = 0     (κ symmetry)
• (∂/∂θ)² = 0

Λ̄_α'A = Λ̄_α'Λ̄_A
"gauge" Λ̄_α' = δ_α'⁺

This allows manifestly superconformal generalization of Penrose transform:
GL(2) says momentum-space wave functions are independent of Λ̄_α' →

φ(Λα'A) = ∫dΛ̄A dΛ̄α' exp(iΛα'AΛ̄α'Λ̄A) φ̂(Λ̄A) =	{	∫dΛ̄A δ2(Λα'AΛ̄A) φ̂(Λ̄A)
		∫dΛ̄α' φ̌(Λα'AΛ̄α')

which reduces to the usual in the gauge Λ_β'^γ = δ_β'^γ

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.

ADHM twistor superstrings

Topological

L = (∇_-Λ^A)Λ̄^-_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)Λ̄^-_α'A + g_-α'^β'Λ^β'AΛ̄^-_α'A + g_--^ABΛ̄^-α'_AΛ̄^-_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:

• g_-α'^β':     GL(2), kills extra coordinates
• g_--^AB:     field equations (K-G, etc.)

Feynman diagrams

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-1N^F = (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).

Random lattice

• 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> (x_i - x_j)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).

Wrong-sign φ⁴

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> 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 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.

QCD superstring?

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

• spacelike:   (∂x)² → ẋ²   (usual)
• lightlike:   (∂x)² → 0,   but   τ^±±(∇_±x)² → τ^--(∇_-x)²   (chiral)

L = (∇_±Λ^α'A)Λ̄^±_α'A + g_±α'^β'Λ^β'AΛ̄^±_α'A + g_±±^ABΛ̄^±α'_AΛ̄^±_α'B + L_YM

where L_YM now also has ± terms.