M0-brane covariant quantization and intrinsic complexity of the pure spinor approach

1. M0-brane covariant quantization and intrinsic complexity of the pure spinor approach Based on I.B. arXive/0707.2336, paper in preparation and I.B., Jose A. de Azcárraga and Dmitri Sorokin, hep-th/0612252 Igor A. Bandos Valencia University and IFIC, Valencia Spain and ITP KIPT, Kharkov, Ukraine • 1- Introduction (1.1) and summary of the results (1.2)-2- M0-brane in spinor moving frame (twistor-like Lorentz harmonic) formulation: action, Hamiltonian machanics and classical BRST charge(s)-3- Covariant quantization of the M0-brane. A reduced BRST charge -4- Cohomology of , regularization and complex BRST charge-5- CONCLUSION AND OUTLOOK Relation with the Berkovits pure spinor approach Igor Bandos, M0- BRST... SQS07

2. 1.1 Introduction • Recently a significant progress in covarinat loop calculations is achieved in the frame of the Berkovits pure spinor approach: • A technique for the covariant superstring calculations was developed and first results were given • On the other hand, the pure spinor superstring was introduced as -and still remains- a set of prescriptions for quantum superstring calculations, rather than a quantization of the Green-Schwarz superstring. Igor Bandos, M0- BRST... SQS07

3. Despite a certain progress in relating the pure spinor superstring to the original Green-Schwarz formulation • and also [M. Matone, L. Mazzucato, I. Oda, D. Sorokin and M. Tonin, Nucl. Phys. B639, 182 (2002) [hep-th/0206104] to the superembedding approach • the origin and geometrical meaning of the pure spinor formalism is far from being clear. • Furthermore, a nonminimal version and other possible modifications of pure spinor formalism are under an active consideration In particular a non-minimal sector ap- peared to be needed to proceed in loops Igor Bandos, M0- BRST... SQS07

4. A deeper understanding of how the pure spinor approach appears on the way of a straightforward covariant quantization of a classical action might, in particular, provide a resource of possible non-minimal variables and give new suggestions in further development of loop calculations. • In this context, the Lorentz harmonic approach [Sokatchev 86, Nissimov, Pacheva, Solomon 87-90, Kallosh, Rahmanov 87-88, Wiegmann 89, I.B.90, Galperin, Howe, Stelle 92, Galperin, Delduc, Sokatchev 92, I.B.+ A. Zheltukhin 90-94, Galperin, Howe, Townsend 93, Fedoruk+Zima94, I.B.+ Sorokin +D.V. Volkov 95, I.B.+ D. Sorokin + +M.Tonin 97, I.B.+A. Nurmagambetov 96] looks particularly interesting: • i) In its frame a significant progress toward a covariant superstring quantization had already been made in late eighties [Nissimov, Pacheva, Solomon 87-90, Kallosh, Rahmanov 87-88]. (Although no counterpart of recent loop calculation progress was achieved) • ii) It conatains spinorial variables similar (although not identical) to the pure spinors • iii) It has clear group theoretical and geometrical meaning, is related to the super-embedding approach, and is twistor-like (in its spinor moving frame form based on Ferber-Schirafuji like action [I.B. 90, I.B.+ Zheltukhin 90-94, I.B.+ Nurmagambetov 96] • It is natural to begin the program of exploiting the spinor moving frame or twistor like Lorentz harmonic approach by studying the massless superparticle quantization. • Here we discuss the covariant quantization of the D=11 massless superparticle or M0-brane, as this case is relatively less studied in comparison with D=4 and D=10 Igor Bandos, M0- BRST... SQS07

5. 1.2.Summary:The pure spinor BRST charge by Berkovits This conditions guaranties the nilpotency of the pure spinor BRST charge, Fermionic constraint of the superparticle model which obeys Pure spinor: a complex 32-component spinor which obeys and requires the spinor Λα to be complex Igor Bandos, M0- BRST... SQS07

6. The main result of our study The covariant quantization of the D=11 massless superparticle in its spinor moving frame formulation produces a simple BRST charge which can be described as the pure spinor BRST carge by Berkovits, but with a composite pure spinor (essentially) Spinor moving frame variables: homogeneous coordinates of the coset, Igor Bandos, M0- BRST... SQS07

7. Our complex charge reads the irreducible κ-symmetry generator Complexified bosonic ghost for the κ-symmetry restricted by b-symmetry generator Irreducible κ-symmetry regularization, when we calculate cohomology of real, λ²0 Strightforward quantization of the 11D (actually, any D) superparticle in a twistor-like Lorentz harmonic formulation[I.B. + J. Lukierski 98, see I.B. + A. Nurmagambetov 1996 for D=10, I.B. 1990 for D=4 and I.B. + A. Zheltukhin 1991-92 for superstrings and super-p-branes.] Igor Bandos, M0- BRST... SQS07

8. 2. D=11 massless superparticle (M0-brane) in spinormoving frame (twistor-like Lorentz harmonic) formulation • The action of massless superparticle in spinor moving frame (Lorentz harmonic) formulation is [see I.B.+J. Lukierski 98 for D=11, I.B.+A. Nurmagambetov 96 for D=10, I.B. 90 for D=4; see I.B.+Zheltukhin 91-94 for superstrings and super-p-branes, I.B. +D.Sorokin+M.Tonin for super-Dp-branes] Lagrange multiplier 16 component SO(9) spinor ind 32×16 32-component SO(1,10) spinor Parametrize the coset isomorphic to Igor Bandos, M0- BRST... SQS07

9. In principle, one can consider the action as constructed in terms of variables uˉ ˉ or vαqˉ constrained by and However, it is more convenient to treat them as parts of the moving frame and spinor moving frame matrices Igor Bandos, M0- BRST... SQS07

10. How to arrive at the spinor moving frame action • Let us start from the first order form of the Brink Schwarz action, where e=e(τ) is the Lagrange multiplier which produces the mass shell constrint • A simple observation: if we have a solution of this constraint (in terms of some new variables) we can substitute it into the action and arrive at a classically equivalent, but different formulation of the model, schematically • One easily finds a non-covariant solution, The general solution, in an arbitrary frame is related to this by Lorentz ratation Igor Bandos, M0- BRST... SQS07

11. Now, to every element of the SO(1,10), one always can associate an element V (actually, two elements, ±V) of Spin(1,10), To this end, one writes the conditions of the Gamma matrix conservation and the concervation of C (when exists, i.e. in D=11, but not in D=10 MW cases) and use them as defining constraints for the new spinor moving frame variables (spinorial Lorentz harmonics) (a)=-- Igor Bandos, M0- BRST... SQS07

12. In a theory with certain gauge symmetry (including SO(1,1) acting on sign indices) the constrained set of 16 11D Majorana spinors which obey parametrize the following coset isomorphic to the celestial sphere, S9 in D=11 This is the case for our superparticle model Igor Bandos, M0- BRST... SQS07

13. Quantization of physical degrees of freedom – supertwistor quantization [I.B. + J.A. de Azcárraga + D. Sorokin, hep-th/0612252] • Using the Leibnitz rules (dx vv=d(xv)v –xdvv) the superparticle action can be written as where momentum for the R+ x S⁹ coordinate λαq Quantization is strightforward: Self-conjugate free fermions Wavefunction = arbitrary function of λ carrying a representation of Origin of SO(16) symm. [Nicolai 86] the SO(16) inv. Cligfford algebra (q=1,…,16) Choise of 256 dim. SO(16) spinorial representation 11D SUGRA (linearized) Igor Bandos, M0- BRST... SQS07

14. Hamiltonian mechnaics and momenta • Phase space contains coordinates which are subject to the set of primary contraints including • The defining constraints for the harmonics (the second class) and • The primary constraints following from the def. of the canonical momentum which are the mixture of the first and second class constraints Igor Bandos, M0- BRST... SQS07

15. The presence ofharmonics allows toseparate the first and the second class constraints covariantly 16 For instance, for the fermionic constraints we have • The relation between the standard and irreducible form of the κ-symmetry is due to the bosonic constraint generalizing the Cartan-Penrose relation 32 16 Remember that in the standard Brink-Schwarz formulation the fermionic first class constriant can be written in covariant, although ∞-reducible, form while the second class fermionic constraints cannot be separated covariantly. Igor Bandos, M0- BRST... SQS07

16. Remark on vector and spinor harmonics and their defining constraints • In principle, the defining constraints can be solved explicitly in terms of the SO(1,10) parameter The identification of the harmonics with the coordinates of SO(1,10)/H corresponds to setting to zero the H coordinates, in our case In distinction to the general expression the above eqs are not Lorentz covariant. Although the use of the explicit parametrization is not practical, it is useful to keep in mind the mere fact of their existence which, in particular, exhibits that U and V carry the same degrees of freedom This allows us to switch from U- to V-language, and back, when convenient Igor Bandos, M0- BRST... SQS07

17. A practical way consists in keeping the dependence U=U(l), V=V(l) on the SO(1,10) group parameter l =l() implicit For the vector harmonics U  SO(1,10) these are Namely, relaized by means of the second class constraints. Following Dirac, one can introduce Dirac brackets allowing to treat these second class constraints as strong equality. They would be equivalent to the Poisson brackets formulated in terms of the uncontrained parameter of SO(1,10), The (non-comm.) translations on the SO(1,10) group manifold are described by which obey and can be split as SO(1,1) SO(9) SO(1,10)/[SO(1,1)x SO(9)] Igor Bandos, M0- BRST... SQS07

18. The other second class constraints are • One can introduce Dirac brackets allowing to treat them as strong equality Altogether, on the final Dirac brackets (the form of which can be found in hep-th/0707.2336) all the 2-nd class constraints are implicitly resolved = are treated as strong equalities Igor Bandos, M0- BRST... SQS07

19. The first class constriants of the M0-brane model are b-symm. κ-symm. • They obey the DB algebra K9 SO(9) SO(1,1) non-linear, W-like the deformation of the SO(1,1) SO(9)( K9 d=1, n=16 SUSY Igor Bandos, M0- BRST... SQS07

20. sub- BRST charge for non-linear algebra of first class constraints • One can guess that the complete BRST charge associated with the above first class constraint algebra is not too practical. Indeed, even omitting the SO(1,1) and SO(9) symmetry generators, `taking care of them in a different manner’ in the pragmatic spirit of the Berkovits approach, e.g. by imposing them on the wave functions were the cohomologies of the BRST operators are calculated, we arrive at the following nonlinear algebra of characterized by the BRST charge (already this is unpractical-too long) Bosonic ghost For irred. κ-symm. Igor Bandos, M0- BRST... SQS07

21. This (already reduced) BRST operator Q'can be written as where • The nilpotency of already guaranties the consistency of the reduction and is the BRST operator associated to the n=16, d=1 SUSY algebra generated by κ-symm. and b-sym. Igor Bandos, M0- BRST... SQS07

22. We will use here this reduction as it is very much in the pragmatic spirit of the pure spinor approach • It can be achieved by setting K9 ghost to zero, • In the classical theory such a reduction can appear as a result of the gauge fixing, e.g., one may keep in mind the explicit parametrization with • Although the question of how to realize a counterpart of such a classical gauge fixing in quantum description looks quite interesting, and its study might bring light on a counterpart of the effect of the D=4 helicity appearance in the quantization of D=4 (super)particle it is out of the score of the present discussion. Thus, we are going to study the cohomology of Igor Bandos, M0- BRST... SQS07

23. Cohomologies of I. They are located at A way to see that: assumiing that one finds that the cohomology is trivial. Hence nontrivial cohomology, if exists, can be described by the wavefunctions This is a problem because, for realλ⁺qthis implies while the κ-symmetry ghost λ⁺q enters essentially our BRST charge Hence a regularization is needed. This can be done by complexifying the SO(9) spinorial κ-symmetry ghost λ⁺ and, hence the reguilarized BRST charge is also complex. Igor Bandos, M0- BRST... SQS07

24. Cohomologies of II. Complex BRST operator • Action of the regularized BRST on the wavefunction can be written in terms of simpler BRST operator where or, more explicitly The cohomology of = cohomology of at Igor Bandos, M0- BRST... SQS07

25. The further study shows that the cohomology is nontrivial only in the sector with ghost number -2 for the cohomology of This cohomology is described by the kernel of the quantum κ-symmetry generator which implies independence on variables transforming nontrivially under the κ-symmetry and b-symmetry I.e., the nontrivial cohomology is described by the wavefunctions which depend on the `physical variables' only (on the variables invariant under the κ- and b-symmetry). This brings us to the starting point of the quantization in terms of physical degrees of freedom – the supertwistor quantization of I.B.+J.de A.+ D.S. 2006. Igor Bandos, M0- BRST... SQS07

26. But the most important point is that our is closely related with the Berkovits pure spinor BRST charge Our study shows that the b-symm. generator has no influence on cohomologies And this can be written as the Berkovits BRST charge, but with the composite pure spinor, Some mismatch of degrees of feedom can be observed: 23x2=46 components in ‘fundamental’ pure spinor versus 16x2-2+9= 39 for the composed one. However • It is not clear that all degrees of freedom in pure spinor are important when • superparticle is considered ii) NO MISMATCH FOR D=10 SUPERSTRING (22 versus 22=8x2-2+8) Igor Bandos, M0- BRST... SQS07

27. Conclusion and outlook • The main conclusion of our study is that the twistor-like Lorentz harmonic approach (spinor moving frame approach), is able to produce a simple and practical BRST charge. • This makes interesting the similar investigation of the D=10 Green-Schwarz superstring case. The Berkovits BRST charge for IIB superstring • Our study of the M0-brane case suggests that the quantization of the D=10 Green-Schwarz superstring in its spinor moving frame formulation [I.B.+A. Zheltukhin 91-92]  basically the same BRST charge, but with composite Goldstone fields for the Lorentz symmetry breaking by superstring worldsheet Igor Bandos, M0- BRST... SQS07

28. Important: in D=10 the # of degrees of freedom in the composed and ‘fundamental’ pure spinor is the same 16x2-10=22 Pure spinors Composite pure spinors 8+14=22 8 16-2=14 Hence no anomaly can be expected when replacing Igor Bandos, M0- BRST... SQS07

29. The quantization of Green Schwarz superstring in the spinor moving frame formulation of [I.B.+ A. Zheltukhin 90-92] is under investigation now. Thank you for your attention! Igor Bandos, M0- BRST... SQS07

30. Appendices • Spinor moving frame action for superstring: Auxiliary worldsheet vielbein SO(1,9)/[SO(1,1)xSO(8)] Lorentz harmonics WZ term (standard) • Action with S9xS9 harmonics (two sets of particle-like harmonics): Igor Bandos, M0- BRST... SQS07

31. OSp(1|64) and M0-brane The set if onstraints include the above constraints on λ’s (coming from the kinematical constraints on the harmonics) as well as Igor Bandos, M0- BRST... SQS07