Twistor description of superstrings
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Twistor description of superstrings. Plan of the talk:. Introduction Cartan repere variables and the string action Twistor transform for superstrings in D=4, 6, 10 dimensions Concluding remarks. D.V. Uvarov NSC Kharkov Institute of Physics and Technology. SQS’07.

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Twistor description of superstrings

Plan of the talk:


Cartan repere variables and the string action

Twistor transform for superstrings in D=4, 6, 10 dimensions

Concluding remarks

D.V. Uvarov

NSC Kharkov Institute of Physics and Technology


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

The latter relation is one of the milestones of the twistor approach. 2 component spinor

is complemented by another spinor

to form the twistor

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)

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,…).


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.

What about twistor description of (super)strings?

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

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

It is classically equivalent to the Polyakov action



from the Cartan local frame attached to the world-sheet

and includes the pair of light-like vectors

It follows as the equations of motion that


can be identified as the

world-sheet tangents

while other repere components are orthogonal to the world-sheet

Written in such form

satisfies the Virasoro constraintsby virtue of the repere


When D=3,4,6,10 the above action has been generalized to describe superstring:


is the world-sheet projection of the space-timesuperinvariant




D=4 Cartan repere components can be realized in terms of the Newman-Penrose dyad

Since the action contains only two out of four repere vectors, dyad components are defined

modulo SO(1,1)xSO(2) gauge transformations.

In higher dimensions relevant spinor variables need to be identified as the Lorentz harmonics

(E. Sokatchev, A. Galperin, F. Delduc parametrizing the coset


For D=6 space-time we have

Involved D=6 spinor harmonics

satisfy the reality

and unimodularity conditions

reducing the number of their independent components to the dimension of the Spin(1,5) group.


in terms of D=10 spinor harmonics

The D=10 Cartan repere components admit the realization

satisfying 211 constraints (harmonicity conditions) that reduce the number of their

independent components to the dimension of the Spin(1,9) group.

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


as well as the projections of 1-forms constructed out of the covariant differentials of

Grassmann-odd supertwistor components

It depends on the world-sheet projections of the SU(2,2|1) invariant 1-forms

where the covariant differentials

include derivation coefficients

It should be noted that supertwistors are constrained by 4 algebraic relations

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


Definite choice of the value of

turns one of the equations into identity.

Among the bosonic equations of motion there are the twistor counterparts


of the nondynamical equations of space-time formulation

that resolve the Virasoro constraints.


back into the action it can be cast intothe following

more simple κ-symmetry gauged fixed form



stand either for twistor or N=1 supertwistor. So above action

corresponds to κ-symmetry gauge fixed D=4 N=1 superstring:

is supertwistor and

is twistor or vice versa depending on the sign of the WZ term,

and also D=4 bosonic string:


are twistors,


and D=4 N=2 superstring: both


are supertwistors.


In 6 dimensions N=1 superconformal group is isomorphic to OSp(8*|2) supergroup

(P. Claus so we consider the supertwistor to realize its fundamental representation

Generalization to higher dimensions requires properly generalizing (super)twistors.

where primary spinor

and projectional

parts are presented by D=6 symplectic

MW spinors of opposite chiralities

Supertwistor components are assumed to be incident

to D=6 N=1 superspace coordinates


being also the symplectic

MW spinor.

To twistor transform D=6 superstring, similarly to 4-dimensional case, we need the pair

of supertwistors

whose projectional parts form the spinor harmonic matrix

Introduced supertwistors are subject to 10 constraints



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

acquires the form in terms of the supertwistors

where 1-forms constructed from supertwistor variables have been introduced


that include SO(1,5)-covariant differentials


Corresponding derivation coefficients are defined by spinor harmonics

Taking into account constraints imposed on supertwistors one derives the following equations

of motion

By choosing definite value of s half of the fermionic equations turn into identities

manifesting κ-invariance of the supertwistor action.


Lorentz invariance by substituting nondynamical equation

back into the action.

Explicit form of the gauge-fixed action depends on s. When s=1 we have

In the proposed formulation κ-symmetry can be gauged fixed without violation of the

and accordingly when s=-1



are bosonic D=6 twistors that can be identified as Spin(6,2)

symplectic MW spinors

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


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)

Twistor transform for the D=10 superstring assumes elaborating appropriate supertwistor

with its primary spinor

and projectional

parts given by Spin(1,9) MW spinors of

opposite chiralities. Application to the twistor description of superstring suggests introduction

of two sets of 8 supertwistors

discussed in I. Bandos, J. de Azcarraga, C. Miquel-Espanya.

Note that


constitute spinor Lorentz-harmonic


Imposition of constraints


is the OSp(32|1) metric, and


allows to bring incidence relations to D=10 N=1 superspace coordinates

to the form


generalizing Penrose-Ferber relations.

The first order D=10 superstring action that includes Lorentz-harmonic variables

(I. Bandos and A. Zheltukhin)


is D=10 N=1 supersymmetric 1-form,

after the twistor transform reads


and those constructed from the fermionic components of supertwistors

It comprises world-sheet projections of OSp(32|1) invariant 1-forms

where SO(1,9)-covariant differentials

include components of Cartan 1-form constructed from the spinor harmonics

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



The latter equations imply that twistor transformed action is κ-invariant. κ-Symmetry gauge

fixed action can be obtained by substituting back nondynamical equation

Explicit form of the gauge fixed action depends on the value of s




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


the Green-Schwarz superstring. To this end it is convenient to consider Lorentz-harmonic

variables normalized up to the scale

Let us consider how the above action can be matched to light-cone gauge formulation of

This affects only the cosmological term in the first-order superstring action

and allows to gauge out all zweibein components.

Further expand primary spinor parts of supertwistors


over harmonic basis



Then the quadratic in supertwistors 1-forms

that enter the action become



Noting that harmonic variables parametrize the coset SO(1,9)/SO(1,1)xSO(8) and hence

depend on the pair of 8-vectors

allows to expand Cartan 1-form components in

the power series

where … stand for higher order terms in

Adduced expressions satisfy

Maurer-Cartan equations up to the second order.

As the result the superstring action acquires the form


admit interpretation of the generalized light-cone momenta.

Integrating them out gives Type IIB superstring action in the light-cone gauge


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.

Concluding remarks

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