Supertwistors, massive superparticles and k-symmetry

0.25 MB·

by  J. A. de Azcarraga

Supertwistors, massive superparticles
and κ-symmetry
J.A. de Azc´arraga,
Departamento de F´ısica Te´orica and IFIC (CSIC-UVEG),
9
46100-Burjassot (Valencia), Spain
0
0
J. M Izquierdo,
2
Departamento de F´ısica Te´orica, Universidad de Valladolid,
n
a
47011-Valladolid, Spain,
J

7 and

] J. Lukierski
h
t Institute of Theoretical Physics, Wroc law University,
-
p
50-204 Wroc law, Poland
e
h
[
August 15, 2008; revised November 28

2
v
5
5 Abstract
1
2 We consider a D = 4 two-twistor lagrangian for a massive particle
.
8 that incorporates the mass-shell condition in an algebraic way, and
0
extend it to a two-supertwistor model with N = 2 supersymmetry
8
0 and central charge identified with the mass. In the purely super-
: twistorial picture the two D = 4 supertwistors are coupled through a
v
i Wess-Zuminoterm in their fermionic sector. We demonstrate how the
X
κ-gauge symmetry appears in the purely supertwistorial formulation
r
a and reduces by half the fermionic degrees of freedom of the two super-
twistors; a formulation of the model in terms of κ-invariant degrees of
freedom is also obtained. We show that the κ-invariant supertwistor
coordinates can be obtained by dimensional (D=6 D=4) reduction
→
from a D = 6 supertwistor. We derive as well by 6 4 reduction the
→
N = 2, D=4 massive superparticle model with Wess-Zumino term
introduced in 1982. Finally, we comment on general superparticle
models constructed with more than two supertwistors.
1
1 Introduction
The conformal structure of twistor theory (see e.g. [1]-[4]) implies that rel-
ativistic particles described by single twistors are massless [1, 3, 5, 6]. To
describe massive particles at least two twistors are needed [7, 8, 3],[9]-[13]
(cf. [14]). Indeed, with two D = 4 twistors
Z = (λ , ω¯α˙) , i = 1,2 , α,α˙ = 1,2 , (1.1)
Ai αi i
the standard formula for the composite fourmomentum1
p = λ λ¯ , p = 1 p (σ )β˙α , (1.2)
αβ˙ αi β˙i µ √2 αβ˙ µ
implies the algebraic relation
p pµ = p pβ˙α = M 2 , (1.3)
µ αβ˙ | |
where the Lorentz-invariant bilinear2
M = 1 λ λαi (M¯ = 1 λ¯ λ¯α˙i) (1.4)
√2 αi √2 α˙i
gives the composite complexified mass and breaks the conformal invariance
down to the Poincar´e one. To obtain a real mass m it suffices to consider a
twistor theory invariant under the phase transformations
λ = eiϕλ , λ¯ = e iϕλ¯ ; (1.5)
′αi αi ′α˙i − α˙i
by a suitable gauge fixing, a real M M = m is obtained.
→ | |
Twistorial particle models constructed from several twistors are known
[3, 7, 8, 9]; in particular a two-twistor model was proposed recently to de-
scribe free relativistic particles with mass and spin [11]-[13]. These consider-
ations confirm that the introduction of a non-vanishing Pauli-Luban´ski vec-
tor, describing the relativistic spin fourvector in terms of twistor coordinates,
requires at least two twistors [9]. In this paper we extend the two-twistor
particle dynamics by considering two N = 2 supertwistors to describe the de-
grees of freedom of our elementary system. The supersymmetric two-twistor
geometry will be arranged in a way that leads algebraically to the condition
Ξ = M , (1.6)
where Ξ is the complex central charge of the D=4, N=2 superPoincar´e alge-
bra. Weshallusethephasesymmetry (1.5)sothatthevalueM ofthecentral
0 1
1 λαi = ǫijλα; ǫij = , ǫ ǫjk = δk. In general, a 1 σµ a , σ =
j 1 0 ij j αβ˙ ≡ √2 αβ˙ µ αβ˙
(cid:18) − (cid:19)
(σ0,σi), σβ˙α =(σ0, σi), i=1,2,3. Then, aν = 1 a σν β˙α, a aβ˙γ = 1δαaµa and
− √2 αβ˙ αβ˙ 2 γ µ
a bβ˙α =a b, a2 =(a0)2 a¯2. Complex conjugatedspinors are denoted by (λ ) λ¯ ,
αβ˙ · − αi ∗ ≡ α˙i
etc.
2In [11, 12] the variable f =λ λα = 1 M was used in place of M.
α1 2 √2
2
charge Ξ in (1.6) becomes a real parameter. An interesting outcome of our
approach will be the description of the fermionic κ-gauge transformations in
a purely twistorial formulation of the massive superparticle model. Further,
it may be shown that the presence of the mass in the twistorial framework
reduces the conformal supersymmetry realized in terms of the two N=2 su-
pertwistors to the N=2 superPoincar´e symmetry with a composite central
charge given by (1.4).
The plan of the paper is the following: In Sec. 2 we introduce a massive
two-twistorparticlemodelwithoutsupersymmetryusingahybridspinor/spacetime
formulation. Wealsopresent theretwootherequivalent spacetimeandpurely
twistorial geometry formulations and exhibit their relation with the associ-
ated six-dimensional massless particle model. In Sec. 3 we introduce the
N = 2 supersymmetric extension of the bosonic massive model, with degrees
of freedom described by two (i = 1,2) N = 2 (r = 1,2) supertwistors [15]
= (Z ,η ) = (λ ,ω¯α˙ ,η ) , (1.7)
ZRi Ai ri αi i ri
whereZ andη are,respectively, Grassmannevenandodd. Thissupersym-
Ai ri
metricmodelwillbeformulatedfirstinahybridspinor/superspace geometric
framework [5]. We discuss (cf. [16, 17]) the constraints and, in particular,
the fermionic first class constraints that generate the κ-transformations with
odd gauge parameters [18, 19].
Further, present in Sec. 3 a purely supertwistorial (two-supertwistor)
formulation of our model. Its lagrangian will be the sum of two terms,
i) a first one describing the free two-supertwistor lagrangian, and
ii) an additional bilinear term in the fermionic sector which couples the two
supertwistors; this will turn out to be a supertwistorial Wess-Zumino (WZ)
term.
The model is completed by adding the Lagrange multipliers that describe
its algebraic constraints.
Sec. 4 is devoted to quantizing the fermionic sector of the two super-
twistor model, to uncover the quaternionic structure behind it and to calcu-
lating the fermionic gauge κ-transformations. These κ-symmetries reduce by
half (from four to two) the number of complex Grassmannian supertwistorial
degrees of freedom. We find new features of the two-supertwistor formula-
tionwithrespect tosinglesupertwistor models: theappearanceofaWZterm
and the presence of the fermionic κ-gauge transformations associated with
the non-physical fermionic degrees of freedom in the multi-supertwistorial
formulation. If we impose the quaternionic SU(2) Majorana condition in the
fermionic sector, the redundant degrees of freedom of the two supertwistor
coordinates described in the fermionic sector by the κ-transformations can
be eliminated. Subsequently, we show also in Sec. 4 that it is possible to
introduce κ-invariant fermionic variables which describe the fermionic sec-
tor of our model in terms of two (rather than four) complex odd degrees of
freedom. It is then seen that these κ-invariant fermionic variables can be
3
interpreted as the dimensionally reduced (D = 6 D = 4) odd compo-
→
nents of a six-dimensional complex supertwistor, satisfying SU(2)-Majorana
conditions.
Sec. 5 exploits the above six-dimensional interpretation of the two-
supertwistor model, using a pair of N = 1, D = 4 (super)twistors to describe
a single D = 6 (super)twistor. We perform the D = 6 D = 4 dimensional
→
reduction by taking the six-momentum coordinates p , p as constants. As a
4 5
result, we recover the D=4 N=2 massive superparticle model formulated by
two of the present authors a quarter century ago, the first one with mass in
the fermionic sector introduced by means of a WZ term [18]. The last sec-
tion provides an outlook. In particular, it is argued there that using N D=4
supertwistors we can extend our construction and obtain a N-supertwistorial
model with N supersymmetries as well as with N(N−1) complex composite
2
mass-like parameters playing the role of central charges. In particular, the
D = 4,N = 4 model would be especially interesting, since its κ-invariant for-
mulation could be given by the coordinates of a single D = 10 supertwistor
with octonionic structure.
= 6
2 Massive particle model from D and its
= 4
D twistorial picture
It is known that D = 4 massive free particle models can be obtained by
dimensional reduction from massless particle ones in D > 4. Because the
twistorialmass(1.4)iscomplex, weshalltakeD = 6anddenoteM = p ip ,
4 5
M¯ = p +ip ((p ,p ,p ) are the six-momentum coordinates) and, simi−larly,
4 5 µ 4 5
z = x4 +ix5, z¯= x4 ix5. In the first order formalism, the Lagrangian of a
−
D=6 massless particle can be written as
1 e
B = p x˙µ + (Mz˙ +M¯z¯˙)+ (p2 M 2) , µ = 0,1,2,3 . (2.1)
µ
L 2 2 −| |
Using eqs. (1.2) and (1.4) the ‘mixed’ or spinor/spacetime formulation of
(2.1) is obtained, in which
1
B = λ¯ x˙β˙αλ + λ λαiz˙ +λ¯ λ¯α˙iz¯˙ , (2.2)
L β˙i αi 2√2 αi α˙i
(cid:0) (cid:1)
where the spacetime vector is expressed as a second rank spinor, x =
µ
1 (σ )β˙αx (see the first footnote) and the D=6 zero mass shell condi-
√2 µ αβ˙
tion, pkp = pµp M 2 = 0 (k = 0,1,...,5), is omitted because it is an
k µ
− | | ¯
identity in terms of the spinor variables λ ,λ . Clearly, the lagrangian (2.2)
αi α˙i
is invariant under the rigid phase transformations (1.5) if the z’s have a (cf.
(1.5)) double U(1) charge,
z = e 2iϕz , z¯ = e2iϕz¯ . (2.3)
′ − ′
4
To obtain a purely twistorial description of the lagrangian (2.2) we now
introduce a new Weyl spinor and its conjugate (ωα, ω¯α˙) and postulate for the
i i
components of the twistors Z = (λ ,ω¯α˙) the following incidence relations:
Ai αi i
ωα = i(λ¯ xβ˙α + 1 λαi z) ,
i β˙i √2
ω¯α˙ = i(xα˙βλ + 1 λ¯α˙i z¯) , (2.4)
i − βi √2
(xα˙β) = xβ˙α (xµ real). Eqs. (2.4) generalize the well known Penrose formula
†
[1]-[4]. Using them, the lagrangian (2.2) can be written as
B = i ω˙αλ +λ¯˙ ω¯α˙ + i ωα λ˙ +λ¯ ω¯˙α˙
L −2 i αi α˙i i 2 i αi α˙i i
= i Z(cid:16)¯AZ˙ Z¯˙AZ (cid:17) , (cid:16) (cid:17) (2.5)
2 i Ai − i Ai
(cid:16) (cid:17)
where the scalar product of two twistors is given by
Z¯A Z = ωα λ + λ¯ ω¯α˙ , (2.6)
i Ak i αk α˙i k
with Z¯A = (ωα ,λ¯ ).
i i α˙i
One can check, using eqs. (2.4) and the relations
λ λα = 1 ǫ M , λ¯ λ¯α˙ = 1 ǫ M¯ , (2.7)
αi j −√2 ij α˙i j −√2 ij
that
i
Z¯AZ = Mz M¯z¯ δ . (2.8)
i Aj 2 − ij
Since Z¯A Z = i Mz M¯z¯ , it fo(cid:0)llows that(cid:1)the z,z¯ coordinates may be
k Ak −
removed by the relations (2.8), which can be rewritten as (see also [6])
(cid:0) (cid:1)
1
Z¯AZ δ Z¯A Z = 0 , (2.9)
i Aj − 2 ij k Ak
an expression that does not fix the conformal norm of the twistors Z but
Ak
states that the two norms are equal.
Summarizing, B in eq. (2.5) appears as the sum of two free twistor
L
lagrangians, associated with two non-null, orthogonal twistors having the
same non-vanishing length for M = 0. To specify that the purely twistorial
6
lagrangian (2.5) describes a massive particle one has to incorporate relations
(2.9) and (1.4) by means of suitable Lagrange multipliers.
= 4 = 2
3 Two-supertwistor model with D ,N
supersymmetry and mass
a) Supersymmetrization of the model (2.2) and κ-transformations.
5
The ‘hybrid’ or spinor/spacetime model (2.2) is supersymmetrized by
replacing the translation-invariant differentials dxαβ˙, dz and dz¯ by the cor-
responding supertranslation-invariant ones on D = 4, N = 2 superspace ex-
¯
tenden by a central charge and parametrized by (x ,z,z¯; θ ,θ ), r = 1,2.
αβ˙ αr α˙r
This is achieved by the replacements3
x˙β˙α ωβ˙α = x˙β˙α i√2 θ˙α θ¯β˙ θα θ¯˙β˙ , (3.1a)
→ − r r − r r
z˙ ω = z˙ +2i θ θ˙αr , z¯˙(cid:16) ω¯ = z¯˙ +2i θ¯(cid:17) θ¯˙α˙r , (3.1b)
αr α˙r
→ →
where a br a ǫrsb and z,z¯ play the rˆole of the D=4, N=2 superspace
r r s
≡
complex central charge coordinates. These replacements in (2.2) give the
supersymmetric lagrangian
1
SUSY = λ¯ ωβ˙αλ + λ λαi ω +λ¯ λ¯α˙i ω¯ . (3.2)
L β˙i αi 2√2 αi α˙i
(cid:0) (cid:1)
The action obtained from (3.2) is invariant under the supertranslations
of the N = 2 superPoincar´e group extended by a complex central charge,
xβ˙α = xβ˙α i√2(ǫα θ¯β˙ θα ǫ¯β˙) ,
′ − r r − r r
¯ ¯
θ = θ +ǫ , θ = θ +ǫ¯ ,
α′r αr αr α′˙r α˙r α˙r
z = z +2iθ ǫαr , z¯ = z¯+2i θ¯ ǫ¯α˙r ,
′ αr ′ α˙r
¯ ¯
λ = λ , λ = λ , (3.3)
′αi αi ′α˙i α˙i
which leave λ invariant. Expression (3.1a) is also invariant under the U(2)
αi
internal transformation of the odd superspace coordinates θ ,θ ; this sym-
αr α˙r
metry is broken by the ω’s of eq.(3.1b) down to the U(2) Sp(2,C) =
∩
USp(2) SU(2) internal symmetry.
≈
Thelagrangian(3.2)describesasuperparticleinamixedspinorial/superspace
configuration space (6;88) parametrized by
|
M
(6;88) = q = (xα˙β,z,z¯;λαi,λ¯α˙i θα,θ¯α˙) , (3.4)
M | { M} | r r
with4+2+8=14realbosonicand8realfermioniccoordinates. Thecanonical
momenta (π = ∂L/∂θ˙α, etc.)
αr r
∂L
= (p ,q,q¯;ρ ,ρ¯ π ,π¯ ) , = 1...22 , (3.5)
α˙β αi α˙i αr α˙r
PM ∂q˙M ≡ | M
define the following set of primary constraints:
¯
R := p λ λ = 0 ,
αβ˙ αβ˙ − αi β˙i
3The √2 in (3.1a) is needed because in this way ωµ(τ) = 1 σµ ωβ˙α(τ) = x˙µ
√2 αβ˙ −
i(θ˙ασµ θ¯β˙ θασµ θ¯˙β˙) is the pull-back to to the worldline of the particle of the super-
r αβ˙ r − r αβ˙ r
space Maurer-Cartan one-form Πµ = dxµ i(dθασµ θ¯β˙ θασµ dθ¯β˙), which is invariant
− r αβ˙ r − r αβ˙ r
under the D =4,N =2 superPoincar´etransformations in eqs. (3.3).
6
R := q 1 λ λαi = q M = 0 ,
− 2√2 αi − 2
R¯ := q¯ 1 λ¯ λ¯αi = q M¯ = 0 ,
− 2√2 αi − 2
R := ρ = 0 , R := ρ¯ = 0 , (3.6)
αi αi α˙i α˙i
and (G¯ = (G )+, π¯ = (π )+)
α˙r αr α˙r αr
− −
G := π +i√2p θ¯β˙ iM ǫ θ = 0 , (3.7a)
αr αr αβ˙ r − rs αs
G¯ := π¯ +i√2θβ p iM¯ ǫ θ¯ = 0 . (3.7b)
α˙r α˙r r βα˙ − rs α˙s
Let us restrict ourselves to the set (3.7a), (3.7b) of fermionic constraints,
which determine the elements of the Poisson brackets (PB) matrix
G ,G , G ,G¯
= { αr βs} { αr β˙s} . (3.8)
CAB (cid:18) {G¯α˙r,Gβs}, {G¯α˙r,G¯β˙s} (cid:19)
Using the canonical PB
¯
θ ,π = ǫ δ , θ , π¯ = ǫ δ , (3.9)
{ αr βs} αβ rs { α˙r β˙s} ǫ˙β˙ rs
it follows that the four 4 4 blocks of the matrix are given by (p =
× CAB β˙α
(p )T)
αβ˙
ǫ ǫ M √2δ p
= 2i − αβ rs rs αβ˙ . (3.10)
CAB √2δ p ǫ ǫ M¯
(cid:18) rs βα˙ − α˙β˙ rs (cid:19)
Using the formula for the determinant of a 2 2 blocks matrix,
×
A B
det = detA det(D CA 1B) , = , (3.11)
−
C · − C C D
(cid:18) (cid:19)
one finds that
det = 28(p2 M 2)4 . (3.12)
C −| |
The first constraint in (3.6) reproduces eq. (1.2), implying
p2 M 2 = 0 (3.13)
−| |
(eq. (1.3)), and thus we conclude from (3.12) that the 8 8 PB constraints
×
matrix (3.10) is of rank four.
We may now derive four first class fermionic constraints by multiplying
respectively (3.7a) by pαγ˙ and (3.7b) by ǫ M. This gives
rs
M i
C¯ = παp + ǫ π¯ θ¯ (p2 M 2) = 0. (3.14)
β˙r r αβ˙ √2 rs β˙s − √2 β˙r −| |
7
Equation (3.14) determines in principle four complex constraints, but their
complex-conjugate ones are equivalent to them. Indeed, if we multiply the
constraints (C+ = (C¯ ))
βr − β˙r
M¯ i
C = p π¯α˙ + ǫ π θ (p2 M 2) = 0 , (3.15)
βr βα˙ r √2 rs βs − √2 βr −| |
by pγ˙β we get back the constraints (3.14), plus terms that contain the factor
(p2 M 2). Therefore our model has effectively only four real first class con-
−| |
straints, which in the 8-dimensional real Grassmann odd sector of the config-
uration space (3.4) generate four real odd gauge transformations. These are
the κ-symmetries of the model (3.2) that eliminate the unphysical fermionic
gauge degrees of freedom i.e., half of the odd N=2 superspace coordinates.
The explicit expression of these κ-transformations, parametrized by a pair of
anticommuting Weyl spinors κ and their complex conjugates κ¯ , is given
αr α˙r
by the graded Poisson brackets
M¯
δ θα := κβC ,θα = κβ C ,θα = ǫ κα ,
κ r { s βs r} s{ βs r} − rs√2 s
δ θ := κ¯β˙C¯ ,θ = κ¯β˙ C¯ ,θ = p κ¯β˙ ,
κ¯ αr { s β˙s αr} s{ β˙s αr} − αβ˙ r
δ θ¯α˙ := κβC ,θ¯α˙ = κβ C ,θ¯α˙ = pα˙β κ ,
κ r { s βs r} s{ βs r} βr
M
δ θ¯α˙ := κ¯β˙C¯ ,θ¯α˙ = κ¯β˙ C¯ ,θ¯α˙ = ǫ κ¯α˙ . (3.16)
κ¯ r { s β˙s r} s{ β˙s r} − rs√2 s
If δ now denotes the variation under both κ and κ¯, the variation of the
κ
four-dimensional spinor (θ ,θ¯α˙) is written as
αr r
δ θαr = −ǫrsδαβ√M¯2 −δrspαβ˙ κβs . (3.17)
κ θ¯α˙ δ pα˙β ǫ δα˙ M κ¯β˙
(cid:18) r (cid:19) rs − rs β˙ √2 !(cid:18) s (cid:19)
The above matrix can be rewritten as the product
ǫ δγ M¯ 0 δ δβ ǫ √2p
− rt α√2 ts γ − ts M¯ γβ˙ . (3.18)
0 ǫ δα˙ M ǫ √2pγ˙β δ δγ˙
− rt γ˙ √2 ! ts M ts β˙ !
The first matrix just produces a scaling of the κ-transformations, and the
second matrix is the sum of the four-dimensional unit matrix plus one with
only non-zero 2 2 antidiagonal boxes that squares to one i.e., it is a projec-
×
tion operator. Thus, δ θ in eq. (3.17) has the standard projector structure
κ r
effectively halving the parameters of the κ-transformations.
The behaviour of the remaining configuration space variables (3.4) under
δ is given by
κ
δ xβ˙α = i√2(δ θα θ¯β˙ θα δ θ¯β˙ ) ,
κ κ r r − r κ r
8
δ z = 2iθ δ θαr ,
κ αr κ
δ z¯ = −2iθ¯ δ θ¯α˙r ,
κ α˙r κ
− ¯
δ λ = δ λ = 0 . (3.19)
κ αi κ α˙i
These relations differ from eqs. (3.3) by the replacement ǫ δ θ; the rela-
κ
→ −
tive minus sign characterizes κ-symmetry as a ‘right’ (local) supersymmetry
(see e.g. [20]). The κ-transformations (3.16), (3.19), may be used to check
explicitly the κ-invariance of the action based on (3.2).
b) From the hybrid (spinor/spacetime) formulation to the purely super twisto-
rial one.
To introduce a purely supertwistorial formulation of the model (3.2), eqs.
(2.4) are further extended in the two supertwistors case by4
i
ωα = iλ¯ (xβ˙α i√2θα θ¯β˙ )+ ǫij(λα z +2iλ θαrθβ) ,
i β˙i − r r √2 j βj r
i
ω¯α˙ = i(xα˙β +i√2θβ θ¯α˙ )λ ǫij(λ¯α˙ z¯ 2iλ¯ θ¯α˙ θ¯β˙r),(3.20)
i − r r βi − √2 j − β˙j r
which are the generalized incidence relations for the bosonic components of
the Z part of (eq. (1.7)). These relations, which involve the fermionic
Ai Ri
Z
superspace coordinates besides the real spacetime xα˙β and complex z vari-
ables (cf. eq. (2.4)), have to be complemented by those affecting the odd
composite variables η , η¯ that make up [15] the coordinates triple of the
ri ri
two D=4 N = 2 supertwistors,
¯R = (ωα, λ¯ ,η¯ ) , = (λ , ω¯α˙ , η ) , i = 1,2 , (3.21)
Zri i α˙i ri ZRri αi i ri
where r = 1,2 is the N=2 supertwistor index. Eqs. (3.20) are accordingly
supplemented by [15]
η = √2θα λ , η¯ = √2θ¯α˙ λ¯ . (3.22)
ri r αi ri r α˙i
Using (1.4), the above expressions can be inverted with the result
λαj η λ¯α˙j η¯
θα = rj , θ¯α˙ = rj . (3.23)
r M r M¯
It follows from the definition (3.22) and from eqs. (3.3) that under super-
symmetry the odd supertwistor variables transform as
δ η = √2ǫαλ , δ η¯ = √2ǫ¯α˙λ¯ . (3.24)
ǫ ri r αi ǫ¯ ri r α˙i
Finally we note that, in terms of the fermionic composite coordinates, the
ω,ω¯ components (3.20) of the two N = 2 supertwistors can be rewritten as
1
ωα = i(λ¯ xβ˙α + λαiz)+(θα η¯ θαrηi) ,
i β˙i √2 r ri − r
4See [21] for similar relations in the framework of D=6 Lorentz harmonics.
9
1
ω¯α˙ = i(xα˙β λ + λ¯α˙iz¯) (θ¯α˙ η θ¯α˙r η¯i) , (3.25)
i − βi √2 − r ri − r
which are the supersymmetric generalizations of eqs. (2.4).
The supersymmetric extension (3.2) of the bosonic model (2.2) can be
written in the form
SUSY = B i λ θ˙α η¯ +λ¯ θ¯˙α˙ η
L L − αi r ri α˙i r ri
+i Mθ(cid:16) θ˙βr +M¯ θ¯ θ¯˙βr . (cid:17) (3.26)
βr β˙r
(cid:16) (cid:17)
A calculation now shows (modulo a total time derivative) that, after intro-
ducing (3.23), the purely supertwistorial lagrangian for our model reads
SUSY = SUSY + SUSY ,
L L1 L2
SUSY i(Z¯A Z˙ Z¯˙A Z ) ,
L1 ≡ 2 i Ai − i Ai
i i
SUSY (η η¯˙ η˙ η¯ ) η˙ ηri +η¯˙ η¯ri , (3.27)
L2 ≡ √2 ri ri − ri ri − √2 ri ri
(cid:0) (cid:1)
wherethescalarproductoftheZ¯A = (ωα, λ¯ )andZ = (λ , ω¯α˙)twistors,
i i α˙i Ai αi i
in which ωα, ω¯α˙ are those in (3.20), is given by eq. (2.6). Using eq. (3.23) it
i i
is seen that the SUSY part of SUSY depends only on η,η¯,λ,λ¯ and on the
L1 L
time derivatives of the λ’s; all the dependence of SUSY on the derivatives
L
of the η,η¯ variables is contained in SUSY above.
L2
The SU(2,2 2)-invariant product of N=2 supertwistors (3.21) is defined
|
by5
¯R = Z¯A Z +√2 η¯ η . (3.28)
Z iZRj i Aj ri rj
The bosonic subgroup of SU(2,2 2) is SU(2,2) U(2) SO(2,4) U(2),
| × ≈ ×
of which SU(2,2) acts on the A indices and U(2) on the index r; each factor
preserves the two terms in (3.28) independently. Using (3.28)f, the lagrangian
(3.27) takes the form
i i
SUSY = ¯R ˙ ¯˙ R η˙ ηri +η¯˙ η¯ri . (3.29)
L 2 Z i ZRi −Zi ZRi − √2 ri ri
(cid:16) (cid:17)
(cid:0) (cid:1)
The first term in (3.29) is the free lagrangian for two supertwistors, which
are coupled only through the second term. This last one is the pull-back
to the worldline of the supertwistor space one-form dη η +dη¯ η¯, a potential
one-formof the closed, supersymmetry-invariant two-form dη dη+dη¯dη¯and,
thus, SUSY is the supertwistorial WZ part of SUSY (for the geometry of
L2 L
WZ rerms, see [22]). In fact, using δ η, δ η¯in (3.24), it is seen that dηη+dη¯η¯
ǫ ǫ¯
is invariant modulo an exact term.
5The different components of the (super)twistors are not dimensionally homogeneous.
In natural units, [λ] = L−21, [ω¯] = L12, [η] = L0; the scalar products of (super)twistors
(eqs. (2.6), (3.28)) are, of course, dimensionless. Note also that (1.7) implies that the
components λ, ω¯ and η of the (super)twistors transform in the same manner under a
symmetry group acting on the index i that labels the two (super)twistors.
10