Twistor description of superstrings D.V. Uvarov NSC 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 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.,…).

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 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

SQS’07 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 formsatisfies the Virasoro constraints by virtue of the repere orthonormality. When D=3,4,6,10 the above action has been generalized to describe superstring: whereis 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 as 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 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. Since the action contains only two out of four repere vectors, dyad components are defined modulo SO(1,1)xSO(2) gauge transformations.

SQS’07 The D=10 Cartan repere components admit the realization 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. 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

SQS’07 It depends on the world-sheet projections of the SU(2,2|1) 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 κ-symmetry transformations in their irreducible realization that can be seen e.g. by inspecting fermionic equations of motion

SQS’07 1 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. Substitutingback into the action it can be cast into the following more simple κ-symmetry gauged fixed form whereandstand 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: both andare twistors, and D=4 N=2 superstring: bothandare 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 where primary spinorand projectionalparts are presented by D=6 symplectic MW spinors of opposite chiralities Supertwistor components are assumed to be incident to D=6 N=1 superspace coordinatesandbeing 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 where

SQS’07 1 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

SQS’07 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 κ-invariance of the supertwistor action.

SQS’07 In the proposed formulation κ-symmetry can be gauged fixed without violation of the Lorentz invariance by substituting nondynamical equationback into the action. Explicit form of the gauge-fixed action depends on s. When s=1 we have and accordingly when s=-1 whereandare 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

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) with its primary spinorand projectionalparts 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 thatand constitute spinor Lorentz-harmonicmatrixImposition of constraints where is the OSp(32|1) metric, and

SQS’07. 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) whereis D=10 N=1 supersymmetric 1-form, after the twistor transform reads

SQS’07 It comprises world-sheet projections of OSp(32|1) 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

SQS’07 and 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 or whereandare 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

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 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 supertwistorsandover harmonic basis and whereThen the quadratic in supertwistors 1-forms that enter the action become

SQS’07. Noting that harmonic variables parametrize the coset SO(1,9)/SO(1,1)xSO(8) and hence depend on the pair of 8-vectorsallows to expand Cartan 1-form components in the power series where … stand for higher order terms inAdduced expressions satisfy Maurer-Cartan equations up to the second order. As the result the superstring action acquires the form Soadmit interpretation of the generalized light-cone momenta. Integrating them out gives Type IIB superstring action in the light-cone gauge

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.