twistor description of superstrings d.v. uvarov nsc kharkov institute of physics and technology...
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Twistor description of superstrings
D.V. UvarovNSC Kharkov Institute of Physics and Technology
Introduction
Cartan repere variables and the string action
Twistor transform for superstrings in D=4, 6, 10 dimensions Concluding remarks
Plan of the talk:
SQS’07
Twistor theory was invented by R. Penrose as alternative approach to construction of quantum theory free of drawbacks of the traditional approach. As of today its major successes are related to the description of massless fields, whose quanta possess light-like momentum
αααααα
αα uuppp
0
The latter relation is one of the milestones of the twistor approach. 2 component spinor u
is complemented by another spinor αααα uixμ to form the twistor
4,,1, a)u,(μZ ααa
It is the spinor of SU(2,2) that is the covering group of 4-dimensional conformal group. Supersymmetry can also be incorporated into the twistor theory promoting twistor to the supertwistor (A. Ferber)
Niuθη,u)θiθi(xμ):η,u,(μZ αiαi
αiαα
iαααi
ααA ,,1,22
Realizing the fundamental of the SU(2,2|N) supergroup.
Supertwistor description of the massless superparticle provides valuable alternative to the space-time formulation as it is free of the notorious problem with κ-symmetry and makes the covariant quantization feasible (T. Shirafuji, I. Bengtsson and M. Cederwall, Y. Eisenberg and S. Solomon, M. Plyushchay, P. Howe and P. West, D.V. Volkov et.al.,…).
SQS’07
What about twistor description of (super)strings?
Not long ago in the framework of the gauge fileds/strings correspondence there were proposed several string models in supertwistor space (E. Witten, N. Berkovits, W. Siegel, I. Bars). But all of them seem to be different from Green-Schwarz superstrings.
Can GS superstrings be reformulated in terms of (super)twistors and what are the implications?
Note that the Virasoro constraints can be cast into the form
02222 m
mm
m xexexexe
reminiscent of the massless particle mass-shell condition. That observation stimulated first attempts on inclusion of twistors into the stringy mechanics (W. Shaw and L. Hughston, M. Cederwall).
The systematic approach suggests looking for the action principle formulated in terms of (super)twistors that requires an introduction of extra variables into the Polyakov or Green-Schwarz one.
One of suitable representations for the twistor transform of the d-dimensional string action was proposed by I. Bandos and A. Zheltukhin
exnenee
dS mmm
)(
)'(22222
2/12
It is classically equivalent to the Polyakov action
SQS’07
and includes the pair of light-like vectors )(2 mn and )(2
mn
from the Cartan local frame attached to the world-sheet
))(()()(22)( :),,( lkln
mnkm
Immm
km nnnnnn
IJJII nnnnnnnnnn ,0,2,0 2222222
)(2 mn )(2
mnIt follows as the equations of motion that , can be identified as the
world-sheet tangents
),(2
)'( 22222/1
mmm nenex
while other repere components are orthogonal to the world-sheet
.0 Im
mnx
Written in such form mx satisfies the Virasoro constraints by virtue of the repere
orthonormality. When D=3,4,6,10 the above action has been generalized to describe superstring:
WZm
mmss Senenee
dS
)(
)'(2222
2/12
where mmm ix is the world-sheet projection of the space-time superinvariant
1-form.
SQS’07
D=4 Cartan repere components can be realized in terms of the Newman-Penrose dyad
1v:),2()v,( uCSLu
as ununnuun v,v,vv, 22
In higher dimensions relevant spinor variables need to be identified as the Lorentz harmonics (E. Sokatchev, A. Galperin et.al, F. Delduc et.al) parametrizing the coset SO(1,D-1)/SO(1,1)xSO(D-2). For D=6 space-time we have
iba
bm
aimba
bm
amab
bm
am nnn
v~v
2
1,v~v
2
1,v~v
2
1 224,,1i
Involved D=6 spinor harmonics
)5,1()v,v(v )( Spinaa
satisfy the reality
1)(
)()()( v)*v( CC
and unimodularity conditions 1vdet )(
reducing the number of their independent components to the dimension of the Spin(1,5) group.
Since the action contains only two out of four repere vectors, dyad components are defined modulo SO(1,1)xSO(2) gauge transformations.
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The D=10 Cartan repere components admit the realization IAAAmA
ImAmAmAmAm nnn
ˆˆˆ
ˆˆˆˆˆˆ
ˆˆ2ˆˆ
ˆˆˆˆ
2ˆ v~v
8
1,v~v
8
1,v~v
8
1
in terms of D=10 spinor harmonics
satisfying 211 constraints (harmonicity conditions) that reduce the number of their independent components to the dimension of the Spin(1,9) group.
)9,1()v,v(v ˆˆ)ˆ(
ˆ SpinAA
Having introduced appropriate formulation of the superstring action and relevant spinor variables, consider its twistor transform starting with the D=4 N=1 space-time case. The superstring action in terms of Ferber N=1 supertwistors
and their conjugate acquires the form
)(214WZkin
NDtw LLdS
eeeie
L WZkin
)()'(4
222/1
)('8
WZZWWZWZs
L
uxuiuZ A 2,:),,(
v2,v:),v,( xiW A
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It depends on the world-sheet projections of the SU(2,2|1) invariant 1-forms dWWWWddZZZZd WZ ,
*)(, WZZWWZ dWZWZd as well as the projections of 1-forms constructed out of the covariant differentials of Grassmann-odd supertwistor components
,, DDDD )*,(, DD
where the covariant differentials dDdD ,~
include derivation coefficients
,vv2
1
2
1
ududdWIZdZIW
.vv,~
ddWIWududZIZ
It should be noted that supertwistors are constrained by 4 algebraic relations ,0,0 ZWWZWWZZ
ensuring reality of the superspace bosonic body.
The twistor transformed action functional is invariant under the κ-symmetry transformations in their irreducible realization that can be seen e.g. by inspecting fermionic equations of motion
SQS’071
.0)1(,0)1( 22
DesDes
Definite choice of the value of 1s turns one of the equations into identity.
Among the bosonic equations of motion there are the twistor counterparts
0,)'(2,)'(2 22/122/1 ZWWZWZ eiei
of the nondynamical equations of space-time formulation
)(2
)'( 22222/1
mmm nene
that resolve the Virasoro constraints.
Substituting 0 ZWWZ back into the action it can be cast into the following
more simple κ-symmetry gauged fixed form
)()()'(4
222/1
24.. WWWWZZZZ
ee
iedS D
fg ed 2
where Z and W stand either for twistor or N=1 supertwistor. So above action
corresponds to κ-symmetry gauge fixed D=4 N=1 superstring: Z is supertwistor and
W is twistor or vice versa depending on the sign of the WZ term,
and also D=4 bosonic string: both and Z W are twistors,
and D=4 N=2 superstring: both Z and W are supertwistors.
SQS’07
Generalization to higher dimensions requires properly generalizing (super)twistors.
In 6 dimensions N=1 superconformal group is isomorphic to OSp(8*|2) supergroup (P. Claus et.al.) so we consider the supertwistor to realize its fundamental representation
2,1,2,141,),,v,( a,aaa iZ i a
where primary spinor a and projectional av parts are presented by D=6 symplectic
MW spinors of opposite chiralities ba
v)*v(,)*( 1ab
bab
a
CC
Supertwistor components are assumed to be incident
to D=6 N=1 superspace coordinates mmxx ~ and i being also the symplectic
MW spinor. To twistor transform D=6 superstring, similarly to 4-dimensional case, we need the pair of supertwistors
),v,(),,v,( - aiaaaaiaaa ZZ
whose projectional parts form the spinor harmonic matrix
)5,1()v,v(v )( Spinaa
Introduced supertwistors are subject to 10 constraints
0
aababa ZGZZGZZGZ
where
iiiiix
aa v2),2(v aa
SQS’07
1
jiijij
CCi
CG
,,0
0ˆˆˆˆ
ˆˆ
is the OSp(8*|2) metric. Their solution can be cast into the form of the above adduced incidence relations to D=6 N=1 superspace coordinates.
D=6 N=1 superstring in the first-order form involving Lorentz harmonics
,2216 WZkin
NDLH LdLdS
,)()(2
22222/1
enenee
L mmmkin
i
mim
WZis
L
'
imi
mm iddx
acquires the form in terms of the supertwistors
))'(2()'(2
1 2/1222222/1
16
eeedS NDtw
iid
is
22222
2
1
2
1
'
where 1-forms constructed from supertwistor variables have been introduced
,2
1,
2
1 22 baba
baab ZGdZZGdZ
iaa
aaaai ZGdZdZGZ )(
4
1
SQS’07
and
iaa
aiai
aiai
i DD )(
4
1
that include SO(1,5)-covariant differentials
,2
1~2
1
4
1 222 bi
ab
ijijai
aaiiai
ai
ai dD
.~2
1~2
1
4
1 222 bib
aijijia
aaiiai
ai
ai dD
Corresponding derivation coefficients are defined by spinor harmonics
,vv,vvvv 222 iaa
aaiaa
aa ddd
.vv~vv,vv2 ab
ijbab
aijba
ijiaa
aai ddd
Taking into account constraints imposed on supertwistors one derives the following equations of motion
,0,)'( 22/12 ie
,0)( 2/1
2222 iaa
aiai
ii DDis
ee
.0)1()1( 22 ia
ia DesDes
By choosing definite value of s half of the fermionic equations turn into identities manifesting κ-invariance of the supertwistor action.
,2
1,
2
1 22 biaiba
biaiab DD
SQS’07
In the proposed formulation κ-symmetry can be gauged fixed without violation of the
Lorentz invariance by substituting nondynamical equation 0i back into the action.
Explicit form of the gauge-fixed action depends on s. When s=1 we have
,))'(4()'(4
1 2/12ˆ
ˆ222/1
61.,.
aa
aa
Dsfg ZZezzeedS
and accordingly when s=-1
,))'(4()'(4
1 2/1ˆ
ˆ2222/1
61.,.
aa
aa
Dsfg zzeZZeedS
where az and az are bosonic D=6 twistors that can be identified as Spin(6,2) symplectic MW spinors
.)*(,)*(ˆ
ˆˆˆˆˆˆˆ b
baab
aba zBzzBz
Similarly it is possible to formulate the κ-symmetry gauge-fixed action for D=6 N=(2,0) superstring in terms of OSp(8*|2) supertwistors
as well as, for the bosonic string
,))'(4()'(4
1 2/12222/1
)0,2(,6..
a
aa
aND
fg ZZeZZeedS
.))'(4()'(4
1 2/1ˆ
ˆ2ˆ
ˆ222/1
6
aa
aa
D zzezzeedS
SQS’07
Twistor transform for the D=10 superstring assumes elaborating appropriate supertwistor variables. Minimal superconformal group in 10 dimensions, that contains conformal group generators, is isomorphic to OSp(32|1) (J. van Holten and A. van Proeyen). So 10-dimensional supertwistor is required to realize its fundamental representation (I. Bandos and J. Lukierski, I. Bandos, J. Lukierski and D. Sorokin)
16,1ˆ),,v,( ˆˆ ΛZ
with its primary spinor ˆ and projectional v parts given by Spin(1,9) MW spinors of opposite chiralities. Application to the twistor description of superstring suggests introductionof two sets of 8 supertwistors
),v,(),,v,( ˆˆ
ˆˆ AAAAAAAA ZZ
ΛΛ
discussed in I. Bandos, J. de Azcarraga, C. Miquel-Espanya. Note that Av and
Av
constitute spinor Lorentz-harmonic matrix ).9,1(v )ˆ(ˆ Spin Imposition of constraints
,0 ΣΛΣ
ΛΣΛΣ
ΛΣΛΣ
ΛBABABA ZGZZGZZGZ
where
i
G
00
00
00ˆˆ
ˆ
ˆ
ΛΣ
is the OSp(32|1) metric, and
SQS’07
.
0)vv(ˆˆˆˆ
ˆˆ5ˆ1ˆ
AAAAmm
allows to bring incidence relations to D=10 N=1 superspace coordinates ,~ ˆˆˆ
ˆˆˆ mmxx
ˆ to the form ;v4,v)8( ˆ
ˆˆˆˆˆˆˆ
AAAA ix
ˆˆˆ
ˆˆˆˆˆ v4,v)8( AAAAix
generalizing Penrose-Ferber relations. The first order D=10 superstring action that includes Lorentz-harmonic variables (I. Bandos and A. Zheltukhin)
,22110 WZkin
NDLH LdLdS
,)()(2
ˆ2ˆ
22ˆ
22/1
enenee
L mmmkin
,'
ˆˆˆˆ
ˆˆ
mm
WZis
L
where
ˆˆ
ˆˆˆˆˆ mmm iddx is D=10 N=1 supersymmetric 1-form,
after the twistor transform reads
))'(2()'(2
1 2/1222222/1
10
eeedS Dtw
.2
1
2
1
'22222
IId
is
SQS’07
It comprises world-sheet projections of OSp(32|1) invariant 1-forms ,
8
1,
8
1 22 -ΣΛΣ
ΛΣΛΣ
ΛAAAA ZGdZZGdZ
)(16
1
ΣΛΣΛAAAA
IAA
I ZGdZZGdZ
and those constructed from the fermionic components of supertwistors
),(16
1,
8
1,
8
1 22 AAAAIAA
IAAAA DDDD
where SO(1,9)-covariant differentials
,ˆ4
1ˆ2
1ˆ4
1 222 BIJAB
IJA
IAA
IAAA dD
BIJBA
IJA
IAA
IAAA dD ~ˆ
4
1~ˆ2
1ˆ4
1 222
include components of Cartan 1-form constructed from the spinor harmonics
,vv4
1ˆ),vvvv(4
1ˆ ˆˆ
2ˆˆ
ˆˆ
22
AIAAA
IAAAA ddd
When deriving superstring equations of motion, above adduced constraints imposed on supertwistors have to be taken into account. As the result, similarly to lower dimensional cases, one obtains the set of nondynamical equations
,0,)'( 22/12 Ie
).v~vvv(8
1ˆ,v~v4
1ˆ ˆˆ
ˆˆ
ˆˆ
2
BIJBAAB
IJABA
IJA
IAAA
I ddd
SQS’07
and
,0)'(4
ˆˆ2/1
2222 A
IAAA
II Dis
ee
.0)1()1( 22 AA DesDes
The latter equations imply that twistor transformed action is κ-invariant. κ-Symmetry gauge
fixed action can be obtained by substituting back nondynamical equation .0IExplicit form of the gauge fixed action depends on the value of s
2/12222/1
101.,. )'(16
)'(16
1
AAAA
Dsfg ZZezzeedS Λ
Λ-A
-Α
or
.')(16')(16
1 2/12222/1
101.,.
AAAAD
sfg zzeZZeedS AA
ΛΛ
where -AAz and
AzA are bosonic Sp(32) twistors subject to the same as supertwistors algebraic constraints to satisfy Penrose-type incidence relations. Note that D=10 bosonic string and κ-symmetry gauge fixed Type IIB superstring actions can be brought to the similar form
,)'(16)'(16
1 2/12222/1
10
AAAAD zzezzeedS A
A-A
-Α
.')(16')(16
1 2/12222/1
,10..
AAAA
IIBDfg ZZeZZeedS Λ
ΛΛ
Λ
SQS’07
Let us consider how the above action can be matched to light-cone gauge formulation of the Green-Schwarz superstring. To this end it is convenient to consider Lorentz-harmonic variables normalized up to the scale
)ˆ()ˆ(
2ˆ)ˆ(
)ˆ(ˆ
)ˆ()ˆ(
)ˆ(ˆ
ˆ)ˆ( ,vv n
kmk
nm nnnn
This affects only the cosmological term in the first-order superstring action
)()()( 222 neded
and allows to gauge out all zweibein components.
Further expand primary spinor parts of supertwistors AZ Λ and
AZΛ over harmonic basis
,v)4
(v)4
(1 ˆˆ2ˆ
BBAIBA
IBBAABA
ix
ix
n
and
,v)4
~(v)4
(1 ˆˆ2ˆ
BBA
IBA
IBBABAA
ix
ix
n
where ,2ˆˆ2 mmnxx .ˆ
ˆ Im
mI nxx Then the quadratic in supertwistors 1-forms
that enter the action become
,~ˆ6416
)~32
(ˆ)ˆ2
1(
8
1 222222
IJIJAA
IIIAA
id
i
ixdnxdxZGdZ
Σ
ΛΣΛ
SQS’07
.
.ˆ6416
)32
(ˆ)ˆ2
1(
8
1 222222
IJIJAA
IIIAA
id
i
ixdnxdxZGdZ Σ
ΛΣΛ
Noting that harmonic variables parametrize the coset SO(1,9)/SO(1,1)xSO(8) and hence
depend on the pair of 8-vectors Ip 2 allows to expand Cartan 1-form components in
the power series
,2ˆ,)(2ˆ 22222222 IIIIII dpdpppdp
JIJIJIJIIJ dpppdppdpdpp 22222222ˆ
where … stand for higher order terms in .2Ip Adduced expressions satisfy
Maurer-Cartan equations up to the second order.
As the result the superstring action acquires the form
.)('16
)2('
2' 22
2222
22
22..
AAAA
IIIIIIIIBcl d
ippxpxpdS
So Ip 2 admit interpretation of the generalized light-cone momenta.
Integrating them out gives Type IIB superstring action in the light-cone gauge
).1616
('
12222
2..
AAAA
IIIIBcl
iixxdS
SQS’07
Concluding remarks
The advantage of the Lorentz-harmonic formulation is the irreducible realization
of the κ-symmetry and the possibility of fixing the gauge in the manifestly Lorentz-
covariant way, in contrast to the original Green-Schwarz formulation. In the supertwistor
formulation κ-symmery gauge fixed action acquires very simple form – it is quadratic in
supertwistors. But they appear to be constrained variables. Hence one can try to solve
those constraints at the cost of giving up manifest Lorentz-covariance or treat them as
they stand using elaborated Dirac or conversion prescriptions.
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