M0brane covariant quantization and intrinsic complexity of the pure spinor approach

1
M0-brane covariant quantization and intrinsic
complexity of the pure spinor approach

• Igor A. Bandos
• Valencia 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 OUTLOOK

Relation with the Berkovits pure spinor approach
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, FedorukZima94,
  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 D11 massless superparticle or M0-brane, as
  this case is relatively less studied in
  comparison with D4 and D10

5
1.2.SummaryThe pure spinor BRST charge by
Berkovits
Fermionic constraint of the superparticle
model which obeys
Pure spinor a complex 32-component spinor
which obeys

• This conditions guaranties the nilpotency of the
  pure spinor BRST charge,

and requires the spinor ?a to be complex
6
The main result of our study

• The covariant quantization of the D11 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
the irreducible ?-symmetry generator
Complexified bosonic ghost for the ?-symmetry
restricted by
b-symmetry generator
Irreducible ?-symmetry
regularization, when we calculate cohomology of
real, ?²?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 D10, I.B. 1990 for D4 and I.B. A.
Zheltukhin 1991-92 for superstrings and
super-p-branes.
8
2. D11 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 D11, I.B.A.
  Nurmagambetov 96 for D10, I.B. 90 for D4 see
  I.B.Zheltukhin 91-94 for superstrings and
  super-p-branes, I.B. D.SorokinM.Tonin for
  super-Dp-branes

Lagrange multiplier
16 component SO(9) spinor ind
3216
32-component SO(1,10) spinor
Parametrize the coset isomorphic to
9
In principle, one can consider the action

• as constructed in terms of variables u or vaq
  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 ee(t) 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),
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
D11, but not in D10 MW cases)
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 D11
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 vvd(xv)v xdvv) the
  superparticle action can be written as

• where

momentum for the R x S? coordinate ?aq
Quantization is strightforward
Self-conjugate free fermions
Wavefunction arbitrary function of ? carrying a
representation of
the SO(16) inv. Cligfford algebra (q1,,16)
Origin of SO(16) symm.
Nicolai 86
Choise of 256 dim. SO(16) spinorial
representation
11D SUGRA (linearized)
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
16
For instance, for the fermionic constraints we
have
32
16
Remember that in the standard Brink-Schwarz
formulation the fermionic first class constriant
can be written in covariant, although
8-reducible, form
while the second class fermionic constraints
cannot be separated covariantly.

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

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
17
A practical way consists in keeping the
dependence UU(l), VV(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(1,1)
SO(9)
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 in hep-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
b-symm.
?-symm.
K9
SO(9)
SO(1,1)

• They obey the DB algebra

non-linear, W-like
the deformation of
the SO(1,1) ?SO(9)(? K9
d1, n16 SUSY
20
BRST charge for non-linear algebra of first class
constraints
sub-

• 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. ?-symm.
21
This (already reduced) BRST operator Q' can be
written as
where
and
is the BRST operator
associated to the n16, d1 SUSY algebra
generated by ?-symm. and b-sym.

• The nilpotency of already guaranties
  the consistency of the reduction

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 D4 helicity appearance in the
  quantization of D4 (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
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 real ??q this
implies
while the ?-symmetry ghost ??q enters
essentially our BRST charge
Hence a regularization is needed. This can be
done by complexifying the SO(9) spinorial
?-symmetry ghost ??
and, hence the reguilarized BRST charge is also
complex.
24
II. Complex BRST operator
Cohomologies of

• 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
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 ?-symmetry generator
which implies independence on variables
transforming nontrivially under the ?-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 ?- 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. 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 23x246 components in fundamental
pure spinor versus 16x2-29 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 D10 SUPERSTRING (22 versus
228x2-28)
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 D10 Green-Schwarz superstring case.

The Berkovits BRST charge for IIB superstring

• Our study of the M0-brane case suggests that the
  quantization of the D10 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 D10 the of degrees of freedom in
the composed and fundamental pure spinor

is the same
16x2-1022
Pure spinors
Composite pure spinors
81422
8
16-214
Hence no anomaly can be expected when replacing
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(164) and M0-brane
The set if onstraints include the above
constraints on ?s (coming from the kinematical
constraints on the harmonics) as well as