Twistor description of superstrings

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Twistor description of superstrings

• D.V. Uvarov
• NSC Kharkov Institute of Physics and Technology

Plan of the talk
Introduction Cartan repere variables and the
string action Twistor transform for
superstrings in D4, 6, 10 dimensions
Concluding remarks
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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

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(2,2N)
supergroup.
Supertwistor description of the massless
superparticle provides valuable alternative to
the space-time formulation as it is free of the
notorious problem with ?-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
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
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• and includes the pair of light-like vectors

and
from the Cartan local frame attached to the
world-sheet
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 constraints by virtue of
the repere
orthonormality.
When D3,4,6,10 the above action has been
generalized to describe superstring
where
is the world-sheet projection of the space-time
superinvariant
1-form.
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• D4 Cartan repere components can be realized in
  terms of the Newman-Penrose dyad

as
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 SO(1,D-1)/SO(1,1)x
SO(D-2).
For D6 space-time we have
Involved D6 spinor harmonics
satisfy the reality
and unimodularity conditions
reducing the number of their independent
components to the dimension of the Spin(1,5)
group.
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• The D10 Cartan repere components admit the
  realization

in terms of D10 spinor harmonics
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 D4 N1 space-time case. The
superstring action in terms of Ferber N1
supertwistors
and their conjugate acquires the form
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• It depends on the world-sheet projections of the
  SU(2,21) invariant 1-forms

as well as the projections of 1-forms constructed
out of the covariant differentials of
Grassmann-odd supertwistor components
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 ?-symmetry transformations
in their irreducible realization that can be
seen e.g. by inspecting fermionic equations of
motion
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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.
Substituting
back into the action it can be cast into the
following
more simple ?-symmetry gauged fixed form
where
and
stand either for twistor or N1 supertwistor. So
above action
corresponds to ?-symmetry gauge fixed D4 N1
superstring
is supertwistor and
is twistor or vice versa depending on the sign of
the WZ term,
and also D4 bosonic string
both
and
are twistors,
and D4 N2 superstring both
and
are supertwistors.
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• Generalization to higher dimensions requires
  properly generalizing (super)twistors.

In 6 dimensions N1 superconformal group is
isomorphic to OSp(82) supergroup (P. Claus
et.al.) so we consider the supertwistor to
realize its fundamental representation
where primary spinor
and projectional
parts are presented by D6 symplectic
MW spinors of opposite chiralities
Supertwistor components are assumed to be
incident
to D6 N1 superspace coordinates
and
being also the symplectic
MW spinor.
To twistor transform D6 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
where
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is the OSp(82) metric. Their solution can be
cast into the form of the above adduced
incidence relations to D6 N1 superspace
coordinates.
D6 N1 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
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• and

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 ?-invariance of the supertwistor
action.
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• In the proposed formulation ?-symmetry can be
  gauged fixed without violation of the

Lorentz invariance by substituting nondynamical
equation
back into the action.
Explicit form of the gauge-fixed action depends
on s. When s1 we have
and accordingly when s-1
where
and
are bosonic D6 twistors that can be identified
as Spin(6,2)
symplectic MW spinors
Similarly it is possible to formulate the
?-symmetry gauge-fixed action for D6 N(2,0)
superstring in terms of OSp(82) supertwistors
as well as, for the bosonic string
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• Twistor transform for the D10 superstring
  assumes elaborating appropriate supertwistor

variables. Minimal superconformal group in 10
dimensions, that contains conformal group
generators, is isomorphic to OSp(321) (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)
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
and
constitute spinor Lorentz-harmonic
matrix
Imposition of constraints
where
is the OSp(321) metric, and
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• .

allows to bring incidence relations to D10 N1
superspace coordinates
to the form
generalizing Penrose-Ferber relations.
The first order D10 superstring action that
includes Lorentz-harmonic variables (I. Bandos
and A. Zheltukhin)
where
is D10 N1 supersymmetric 1-form,
after the twistor transform reads
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• It comprises world-sheet projections of OSp(321)
  invariant 1-forms

and those constructed from the fermionic
components of supertwistors
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
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• and

The latter equations imply that twistor
transformed action is ?-invariant. ?-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
or
where
and
are bosonic Sp(32) twistors subject to the same
as supertwistors
algebraic constraints to satisfy Penrose-type
incidence relations. Note that D10 bosonic
string and ?-symmetry gauge fixed Type IIB
superstring actions can be brought to the
similar form
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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
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
and
over harmonic basis
and
where
Then the quadratic in supertwistors 1-forms
that enter the action become
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• .

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
So
admit interpretation of the generalized
light-cone momenta.
Integrating them out gives Type IIB superstring
action in the light-cone gauge
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• Concluding remarks

The advantage of the Lorentz-harmonic formulation
is the irreducible realization of the
?-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
?-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.