The Erwin Schrodinger International Pasteurgasse

ESI Institute for Mathematical Physics A Wien Austria

Pure in Odd Dimensions

Wo jciech Kop czynski

Vienna Preprint ESI Octob er

Supp orted by Federal Ministry of Science and Research Austria

Available via WWWESIACAT

Pure spinors in o dd dimensions

Wo jciech Kop czynski

Instytut Fizyki Teoretycznej Uniwersytet Warszawski

ul Hoza PL Warszawa Poland email kopczfuwedupl

Abstract The pap er contains description of pure simple spinors in a complex

this particularly simple approach uses extensively Fo ck bases of the under

lying vector space The pap er describ es pure spinors in a complexied vector space

using a complexied version of Fo ck bases b etter known in relativity theory as null bases

and intro ducing an imp ortant notion of real index Both these pap ers concern spinors in

an evendimensional vector space The present pap er aims at descrption of pure spinors

in o dddimensional vector spaces it is complementary to and however in one es

sential place the sucient conditions for to b e pure it go es b eyond using the

PauliKonk typ e identity found in

General spinors in o dd dimensions

Let W b e a complex vector space of dimension m m and the central

dot denotes the scalar pro duct in W Let ClW End S b e one of two irreducible

representations of the Cliord ClW in the spinor space S of complex dimension

m

In contrast to I have to make a distinction in notation b etween elements of ClW

and corresp onding element of End S since is nonfaithful Eg the letter u denotes a

vector and also its image in the Cliord algebra but not a corresp onding endomorphism of

the spinor space the latter is denoted by u The fundamental multiplication prop erty

in the Cliord algebra will b e written in the form

u v v u u v

where u v W ClW

If e e is an orthonormal basis of W

1 2m+1

e e e e

then we can intro duce an almost nul l basis n n p p e by the formulas

1 m 1 m

e ie p e ie m e e n

21 2 21 2 2m+1

The elements of this basis satisfy the relations

2

n n n n p p p p e

n p p n n e e n p e e p

Null vectors of b oth kinds span a maximal total ly nul l MTN subspaces of W N

spanfn n g P spanfp p g and we have the direct sum decomp osition

1 m 1 m

W N P spanfeg

Conversely given an MTN subspace N one can nd an MTN subspace P such that

N P fg and a unit vector e orthogonal to N and P determined up to a sign

such that the decomp osition holds One can also nd bases n n in N and

1 m

p p in P such that the orthogonality conditions hold An arbitrary vector in

1 m

W admits a unique decomp osition

u n p z e

where n N p P z C Writing

m m

X X

n x n p y p

=1 =1

we have

2 2

u u u x y x y z

1 1 m m

The orthogonal group OV also SOV acts transitively on the set of all MTN

subspaces of V this set has a natural structure of a complex manifold of dimension mm

Let us x the orientation of an orthonormal basis e The pro duct of all e

is prop ortional to the identity endomorphism of S The two inequivalent representations

of ClW in S are distinguished by the sign b elow

m

i r mid

1 2m+1 S

This formula can b e written in terms of the almost null basis as

m

n p n p e id

1 1 m m S

The transp osing mapping B S S S is the dual of the spinor space S is

dened by

T m 1

B B

It satises

T m(m+1)2

B

k

With any two spinors one bilinearly asso ciate a k vector B W dened by

k

B v hB v i B v

k

where v is an arbitrary k vector The conventions for the scalar pro duct of k vectors are

the same as in The formulae and lead to the statement that the symmetry

prop erties of the blinear forms B are determined by

k

(mk )(mk +1)2

B B

k k

Therefore B for m k mo d

k

Pure spinors

Following I shall intro duce two invariantly determined subspaces M and V

asso ciated with any spinor

M fu W u g

V fB S g

1

M is always totally null TN ie M M Consider V It consists of all

u W such that

B u S

1

but

B u hB ui

1

implies u since B is an isomorphism therefore V M Since it is quite

obvious that M V we have V M It follows also that M

V V These remarks apply also in the evendimensional case For economy of

notation I shall avoid writing V in the sequel

The spinor is pure if the subspace M is maximal totally null MTN In the

evendimensional cale it simply means M M in o dd dimensions the situation

seems to b e slightly more complicated dimM M

Do there exist pure spinors in dimensions Since the representation of the

even Cl W is faithful there should exist S such that either

+

n n or n n e dep ending on the parity of m is nonzero

1 m 1 m

Then M N and is pure

The spinor similarly as in the evendimensional case can b e thought as a vacuum

m

state annihilated by the op erators n n The collection of spinors

1 m

p p

1 m

p p p p

1 2 m1 m

p p

1 m

is linearly indep endent in virtue of the anticommutation relations satised by n and p

Therefore this is a basis of S All the spinors o ccuring in are pure eg the subspace

M p p spanfn n p p g

k +1 m 1 k k +1 m

is TN and of dimension m

The direction of a pure spinor is determined by corresp onding MTN subspace Indeed

let S b e such that M M and let

p p p

0 1 1 1m 1 m

Multiplication by succesive pro ducts of n gives for any multiindex I

I

Therefore analogously like in the evendimensional case there holds

Prop osition There is a natural onetoone corresp ondence b etween the set of all

MTN subspaces of W and the set of directions of pure spinors

What can b e said ab out the spinor e which is not present in The anticommu

tation relations for e and n lead immediately to the conclusion that M e M

2

It follows from Prop osition that e and since e we have e

Prop osition Let and b e linearly indep endent pure spinors There is a basis

of S of the form such that is one of the basis vectors other than

Proof mutatis mutandis can b e transferred from

Prop osition The group Spinm acts transitively on the set of directions of

all simple spinors

Proof Following Prop osition if and are two pure spinors having dierent

directions then there holds a relation of the form

p p

k +1 m

p n p n

k +1 k +1 m m

Since all factors ab ove contain reections the group Pinm acts transitively At the

same time holds

p n p n e

k +1 k +1 m m

One of the pro ducts ab ove must represent an element of the group Spinm QED

Prop osition A necessary and sucient condition for a spinor to b e simple is

that the multivectors B for k m m m Moreover

k

the only nonvanishing multivectors of this form are

B n n

m 1 m

and

B B e

m+1 m

Proof To prove necessity consider the action of the endomorphism n n

1 m

n n on elements of a basis of S constructed out of the pure spinor It is

1 m

clear the result will b e nonvanishing only for the last element of the basis

n n p p

1 m 1 m

Therefore im n n spanf g This means that there exists a spinor such

1 m

that

n n B

1 m

But according to B B which gives

m m

B n n B n n

1 m 1 m

and this formula together with yields

B B B B

Since the spinors and ab ove are arbitrary we get thus

B n n

1 m

The tensor pro duct B can always b e expressed in terms of the basis

1 k

where m and k m of End S

1 k

m

X

m

B B

k

k =0

Applying to we get the pro of of necessity and moreover the result

To prove notice that if u is a vector and v a p vector then

p

v u u c v v u

For any mvector v we have then

m+1 m+1

B v n B n c v B n c v

m+1 m1

b ecause the rst factor ab ove is zero Similarly

B v e B v e B v B v

m+1 m

From that follows immediately the formula

The suciency pro of follows closely that of Urbantke and is based on his form of

the identity of the PauliKonk typ e

m1

X

m

B B B B B B d

1 1 k k k

k =0

where and are arbitrary spinors and co ecients d whose exact form can b e derived

k

from do not vanish for m k mo d

We notice that the nullity conditions B lead consistently with the identity

k

to

m

B B B B

1 1

Consider the map P B and the subspace f S B g of S

1

Since B is nondegenerate the subspace has co dimension Thus

dim S dim im P dim ker P dim M dim ker P

and

dim dim im P j ? dim ker P j ?

But if follows from that im P j ? M and also that ker P j ? ker P Sub

tracting from we get

dim M dim M

which means that M is MTN QED

Prop osition If and are pure spinors then the dimension of the intersection

M M is the least integer k such that B The multivector B is

k k

prop ortional to the exterior pro duct of the vectors of a basis of this intersection

Proof We can nd an almost null basis such that

M spanfn n g

1 m

M spanfn n p p g

1 k k +1 m

p p

k +1 m

and k dim M M Since

m1

p n p n n n n

m m m 1 m 1 m1

the recursive pro cedure gives

mk k (2m+1k )2

p p n n n n

k +1 m 1 m 1 k

k vector m k vector

On the other hand

m

X

m

B p p B

l k +1 m

l=0

p p n n

k +1 m 1 m

The representation when applied to the tail of the formula gives images of multi

vectors of degrees higher then k eg m k vector is a image of a k vector

This observation proves the prop osition QED

In and dimensions the nullity conditions are empty ie each spinor is pure

In dimensions there app ears the rst constraint B in dimensions we have

0

two constraints B and B and so on

0 1

Real index of a pure spinor

Let now W b e a complexication of a real vector space V W C V and let the

scalar pro duct in W b e inherited from V where it has the signature k l k l m

If N is a TN subspace of W it can b e decomp osed as

N C K M M

0

where K consists of all real vectors b elonging to N C K N N M is a subspace

0

in N fu N u v v N g complementary to C K and the Hermitian

scalar pro duct u v u v in M is nondegenerate Although the decomp osition

is nonunique the dimensions r n of N N and M resp ectively and the signature

0 0

n n of the Hermitian scalar pro duct in M are determined uniquely by N Following

+

the argumentation of we get the inequalities

r n n k r n n l

0 + 0

If they are satised for some nonnegative integers r n n and n one can construct a

0 +

suitable TN subspace N of W

If N is MTN then

r n n n m

0 +

Taking into account the fact that k l m we get from and

n

0

and

n

k r

l r

n and resp ectively n

+

k r

l r

For a given value of r there is only one integer solution n n of Therefore the

+

real index r determines uniquely all interesting integer parameters characterizing an MTN

subspace N and is restricted by

r mink l

In contrast to the evendimensional case there are no empty places in the sequence

Prop osition Let N and P b e two MTN subspaces of W C V such that

dim N P m Then the following four conditions are equivalent

N P I N P N P

I I N and P have the same real index

2

I I I There exists a real vector u N P such that u

1

IV There exists a real reection mapping N onto P P u N u

Proof is identical to that of Theorem in

Prop osition The groups Ok l and SOk l act transitively on each set of all

MTN subspaces of W C V with a given real index

Proof for the group Ok l is identical to that of Theorem in

If N denotes a subspace of N consisting of real parts of all vectors b elonging to the

MTN subspace N of W then

N K M

where the signature of the scalar pro duct in M is n n The scalar pro duct in

+

M has signature k n l n which is either r r or r r Therefore

+

there exists a real MTN subspace L of M and a vector e M orthogonal to K

2

and L and normalized e or such that

M K L span feg

R

The vector e is determined up to a sign We have also the decomp ositions

V K L span feg M

R

M W C K C L spanfeg M

1

It is clear that e N e N thus the vector e provides us with a real reection

b elonging to the stability subgroup of N in the orthogonal group Ok l Since there

exists an element of Ok l mapping N onto P there exists also an element of SOk l

p erforming this op eration QED

Remark For o dd dimensions only if mink l the Euclidean case the action

of Ok l on the set of all MTN subspaces is transitive For even dimensions the fact

o ccured also in the case mink l

Prop osition The connected comp onent of unity SO k l of the group Ok l

o

acts transitively on each set of all MTN subspaces of W C V with a given real index

r

Proof Mutatis mutandis that of Theorem of Ref

of W adapted to the decomp osition Let us intro duce a basis k l e m m

_

i j A

B

Here and b elow i j r A B AB m r The only nonvanishing

scalar pro ducts of the vectors of this basis are

k l m m e e

_ _

i j ij A

B AB

where

diag

_

AB

z z

n n

+

Any linear map A V V which leaves N invariant AN N can b e describ ed in

terms of this basis as cf

j

k Ak a

i j i

j B

Am b k c m

A A j A B

_

B j j B

m f l g e Al d k e m e

_

i i j i i i j i B

B

_

B j j B

m j l e Ae h k i m

_

j j B

B

where a d f g h j are real and det a The map A b elongs to Ok l if and only

if

1

y T y

c c f a b ae c

T T T y T

d f f d e e e e g g

t

j i f h g k

T

Instead of d we can use s d f then shows that its symmetric part is determined

T

by e and g whereas its skewsymmetric part t s s is arbitrary Therefore an element

A of the stability subgroup in Ok l of the subspace N with the real index equal to r is

r n +n r r

+

parametrized by a c e t g GLr R U n n C C R R

+

r

R Z The dimension of the stability subgroup is

2

2 2 2

r r r m r r r m r r m r

The dimension of the orbit is

mm r r

Notice that according to only these MTN subspaces and pure spinors for which

r are generic ie they form op en sets in the top ological spaces of all MTN subspaces

and of all directions of pure spinors resp ectively This situation is distinct from that in

the evendimensional case when also MTN subspaces and pure spinors with r were

generic

Let us intro duce the charge conjugation C S S ny the formula

(k l)(k l1)2 1

C C

The charge conjugate of the spinor is then C S The same argumentation as

c

that in leads to

M Prop osition There always holds M

c

2

For any MTN subspace N we can dene its Kahler bivector j W provided we

distinguished in advance a subspace M N

iu v for any u v M

j u v

otherwise

The Kahler bivector uniquely determines a complex structure in the space

N N N N

by mappings of the form N u n c j N

Prop osition The algebraic constraints for a pure spinor to have the real index

equal to r are

B B

r c r 1 c

The only nonvanishing multivectors of the form B are

q c

B k k

r c 1 r

p p

B i p B j

r +2p c r c

for p m r and

B B

r +2p+1 c r +2p c

for p m r

Proof The formulas and follow from Prop Prop and its pro of

To prove for p notice that

B l l e B l l e

r +1 c 1 r c 1 r

B l l B l l

c 1 r r c 1 r

and from it follows that

B B e

r +1 c r c

To prove notice that the statements concerning the expression

k k m m m m l B l

_ _

j j B B i r +q c i

1 1 1

A A

1

in the pro of of Theorem of Ref remain valid In particular the expression do es

not vanish only if r and q an integer Considering a similar

expression

k k m e m m m l B l

_ _

j j B B i r +q c i

1 1 1

A A

1

we get mo dulo a sign

c c k ck c m B m

j j B r +q 22 c B

1 1

e m l l m

_ _

i i

1

A A

1

If q then in we get B moreover the interior pro ducts

r +q 22 c

vanish if or Since we have B B and since

c c p c c c p c

M the remarks concerning vanishing of hold to o if the M C K

c

roles of and are reversed To have nonvanishing expression we need and

since now r q we obtain r These conditions lead to the

following three solutions

r q

r q

r q

The degree r q of the B multivector in is r in the cases and

and r in the case Therefore in virtue of the expression vanishes in the

cases and Since in the case q is o dd nonvanishing of has no meaning

for the value of the multivector on the left hand side of It follows that the formula

can b e proven using the same argumentation as in The simplest way of proving

for p is to rep eat the argumentation for the case p using instead of

as the starting p oint QED

Acknowledgments

I thank Professor Helmut Urbantke for interesting discussions The pap er was written

during the stay of the author in the Erwin Schrodinger Institute in Vienna the hospitality

extended to him is gratefull acknowledged The rep orted research was partially supp orted

by a grant from the Polish Commitee of Scientic Research KBN

References

P Budinich and A Trautman J Math Phys

W Kop czynski and A Trautman J Math Phys

H Urbantke in Spinors Twistors Cliord and Quantum Deformations

Sobotka eds Z Oziewicz et al Kluwer Academic Publishers

W Kop czynski in Proc of First PolishGerman Symposium on Mathematical Physics

Rydzyna eds H Do ebner and R Raczka World Scientic Singap ore