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
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 et.al.,…).
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,
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
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
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 et.al, F. Delduc et.al) 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:
and D=4 N=2 superstring: both
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
Generalization to higher dimensions requires properly generalizing (super)twistors.
where primary spinor
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
To twistor transform D=6 superstring, similarly to 4-dimensional case, we need the pair
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
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
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.
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
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.