Twistor superstrings

(talk at Simons Workshop in Mathematics and Physics 2004)

ADHM supertwistors

Projective lightcone

Manifest symmetry (representation, not nonlinear realization) is important for simplification, to make things both clearer and easier to calculate.

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)

In lightcone notation (dy² = -2dy⁺dy^- + rest),

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

To avoid problems of reality, etc., spacetime signature = (--++)
→ 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: 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
lightcone (bosonic)	Lorentz SO(2,2) = SL(2)2 4-momentum pαα̇ put on mass(less) shell: p2 = det(pαα̇) = 0 → pαα̇ = λαλα̇ → (λα, λα̇)/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

Other than (--++) is more complicated:

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).

Where in twistorspace is x?

Penrose twistors defined in momentum space; get to coordinate space by Fourier → Penrose transform:

φ(x) = ∫dp φ̃(p) e^ix∙p = ∫dλ dλ φ̂(λ,λ) e^iλxλ = ∫dλ φ̌(λ,λx) = ∫dλ dμ δ²(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^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)

Chiral superspace has no torsion.

Field equations

Free field equations in terms of superconformal generators;
generalization of p² = 0 (G_α̇^β = p^β_α̇):

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

Satisfied trivially by Penrose; for ADHM, →

Λ^α'_AΛ_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:

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

Free "lightcone" solution takes ADHM → Penrose (cf. p² = 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' →

φ(Λα'A) = ∫dΛA dΛα' exp(iΛα'AΛα'ΛA) φ̂(ΛA) =	⎧ ⎨ ⎩	∫dΛA δ2(Λα'AΛA) φ̂(ΛA)
		∫dΛα' φ̌(Λα'AΛα')

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

Berkovits formulation of Nair/Witten twistor superstring, as closed string:

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'AΛ^-_α'A + g_--^ABΛ^-α'_AΛ^-_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.)

Simultaneously has usual x and GL(1) for instantons

Feynman diagrams

Second-quantization gives Feynman diagrams, graphs associated with mathematical expressions for scattering amplitudes, given in coordinate space by associating a "vertex factor" with each vertex (an interaction point in space), and a propagator (Green function for the wave equation) with each line.

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

The above 2nd-quantized Feynman diagram surface (polyhedron) can be associated with a 1st-quantized string amplitude (Nielsen, Olesen, Fairlie, Sakita, Virasoro, Douglas, Shenker, Gross, Migdal, Brézin, Kazakov), the path integral of e^-S, 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 )

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

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

For physical "partons", we want (for massless)

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?

Twistor superstrings aren't stringy (no excited states). One way to reduce strings to particles is limits on tension (inverse of slope α'):

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

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:

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

(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

L = (∇_±Λ^α'A)Λ^±_α'A + g_±α'^β'Λ^β'AΛ^±_α'A + g_±±^ABΛ^±α'_AΛ^±_α'B + L_YM

where L_YM now also has ± terms.