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:

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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.,…).

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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

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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.

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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

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.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.

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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

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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

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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

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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

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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

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.

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

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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

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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

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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

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.

.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

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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.