m0-brane covariant quantization and intrinsic complexity of the pure spinor approach

Igor Bandos, M0- BRST... SQS07

1

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

Igor A. BandosValencia University and IFIC, Valencia Spain

and ITP KIPT, Kharkov, Ukraine

Based on I.B. arXive/0707.2336, paper in preparation and I.B., Jose A. de Azcárraga and Dmitri Sorokin, hep-th/0612252

-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 OUTLOOKRelation with the

Berkovits pure spinorapproach


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.


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


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)

• 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

• ii) It conatains spinorial variables similar (although not identical) to the pure spinors


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 modelwhich obeys

Pure spinor: a complex32-component spinor

which obeys

and requires the spinor Λα to be complex


6

The main result of our studyThe 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,


7

Our complex charge reads

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.]

the irreducible κ-symmetrygenerator

Complexified bosonic ghost for the κ-symmetry restricted by

b-symmetry generator

Irreducible κ-symmetry

regularization, when we calculate cohomology ofreal, λ²0


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]

16 component

SO(9) spinor ind

32-component

SO(1,10) spinor

32×16

Lagrangemultiplier

Parametrize the coset isomorphic to


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


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


11

Now, to every element of the SO(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)

one always can associate an element V (actually, two elements, ±V) of Spin(1,10),

and use them as defining constraints for the new spinor moving frame variables

(spinorial Lorentz harmonics)

(a)=--


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


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 S coordinate ⁹ λαq

Self-conjugatefree fermions

Quantization is strightforward:

Wavefunction = arbitrary function of λ carrying a representation of

the SO(16) inv. Cligfford algebra (q=1,…,16)

Choise of 256 dim. SO(16) spinorial representation 11D SUGRA (linearized)

Origin of SO(16) symm. [Nicolai 86]


14

Hamiltonian mechnaics• Phase space contains coordinates

and momenta

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


15

The presence of harmonics allows to separate the first and the second class constraints covariantly

• The relation between the standard and irreducible form of the κ-symmetry is due to the bosonic constraint generalizing the Cartan-Penrose relation

For instance, for the fermionic constraints we have

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.

32

16

16


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 Vcarry the same degrees of freedom

This allows us to switch from U- to V-language, and back,

when convenient


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

Namely, relaized by means of the second class constraints. For the vector harmonics U SO(1,10) these are

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(9)SO(1,1)SO(1,10)/[SO(1,1)x SO(9)]


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 inhep-th/0707.2336) all the 2-nd class constraints are implicitly resolved = are treated as strong equalities


19

The first class constriants of the M0-brane model are

• They obey the DB algebra

d=1, n=16 SUSY

the SO(1,1) SO(9)( K9

the deformation of

non-linear, W-like

SO(9) SO(1,1)

K99

κ-symm. b-symm.


20

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,

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

`taking care of them in a different manner’ in the pragmatic spirit of the Berkovits approach,

characterized by the BRST charge

sub-

Bosonic ghostFor irred. κ-symm.

(already this is unpractical-too long)


21

This (already reduced) BRST operator Q' can be written as

• The nilpotency of already guaranties the consistency of the reduction

where

and

is the BRST operator

associated to the n=16, d=1 SUSY algebra generated by κ-symm. and b-sym.


22

We will use here this reduction

• 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

as it is very much in the pragmatic spirit of the pure spinor approach

• 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


23

Cohomologies of I. They are located at

This is a problem because, for real λ⁺q this 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.

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


24

II. Complex BRST operator

• Action of the regularized BRST on the wavefunction

Cohomologies of

can be written in terms of simpler BRST operator

where or, more explicitly

The cohomology of = cohomology of at


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.


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. generatorhas 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

i) 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)


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.

The Berkovits BRST charge for IIB superstring

• This makes interesting the similar investigation of the D=10 Green-Schwarz superstring case.

• 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


28

Important: in D=10 the # of degrees of freedom in the composed and ‘fundamental’ pure spinor is the same

16x2-10=22

16-2=14

8 8+14=22

Hence no anomalycan be expected when replacing

Pure spinors

Composite pure spinors


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!


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):


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

m0-brane covariant quantization and intrinsic complexity of the pure spinor approach

Documents

pure spinor superstring

m0 brst

berkovits pure spinor

pure spinor approachigor

frame twistor

greenschwarz superstring

classical brst charges

significant progress