Supersymmetry generator(s) ?

[ygor.geurts@gmail.com]

Hi,

I'm confused about the following three questinos.

Suppose we have local N=(1,0) supersymmetry in 10-dimensional
minkowski spacetime. In 10-dimensions we have two real irreducible
unique 16-dimensional majorana-weyl representations of Spin(9,1)
denoted by S^+ and S^- respectively. For N=(1,0) local supersymmetry,
the super poincare algebra is now given by (V \oplus o(V)) \oplus \Pi
(S^+)*. Then by choosing a basis {P_\mu} for V and {Q_a} for S^* the
odd part of the supersymmetry algebra is given by {Q_a,Q_a} = -2
\Gamma^\mu_{ab} P_\mu. Now it is clear that the supersymmetry
generators are given by the 16 basis vectors Q_a. So the a-index
denotes different vectors. In many physics books however they write
down the same supersymmetry algebra bracket and claim that the Q_a is
just a single 16-dimensional spinor, where a denotes the component-
index. So my question is, what is going on ?

Then, another question that is bugging me is this. This Q_a is a lie
algebra basis vector of the super Poincare group, which is the
isometry group of the super Minkwowski spacetime. The problem is that
I have read at some places that the Q_a is a section of a spinor
bundle over Minkowski spacetime, but I'm not entirely sure that I
understand how a basis vector of a lie algebra of the Poincare group
can be interpreted as a section of a spinor bundle over Minkowski
spacetime. Could anyone perhaps clarify this for me?

I hope anyone can help me out on the above questions,

Thanks in advance!

Ygor

[Rock Brentwood]

On Jun 21, 4:59 am, ygor.geu...@gmail.com wrote:
> The problem is that
> I have read at some places that the Q_a is a section of a spinor
> bundle over Minkowski spacetime, but I'm not entirely sure that I
> understand how a basis vector of a lie algebra of the Poincare group
> can be interpreted as a section of a spinor bundle over Minkowski
> spacetime.

The momentum generator P_{mu} is directly related to the partial
differential operator, which I'll denote here @_{mu}. This factors.

For Minkowski space, take l = (@_t + @_z)/root(2), m = (@_x + i @_y)/
root(2), m* = (@_x - i @_y)/root(2), n = (@_t - @_z)/root(2), in units
where c = 1.

Factor the basis 4 = (l,m,m*,n) into 4 = 2_R x 2_L where 2_R: (o, i)
and 2_L: (o*, i*) are the right-helicity and left-helicity spinor
bases. The factoring is as
l = o x o*, m = o x i*, m* = i x o*, n = i x i*,
where x denotes tensor product. The elements of the respective bases
are "dyads". Impose the convention
l = o* x o, m = i* x o, m* = o* x i, n = i* x i
(this is essentially equivalent to requiring that the "sigma"
coefficients form Hermitian matrices).

The dyads (i,o) = (Q_0, Q_1) and (i*, o*) = (Q_0', Q_1') are,
theselves, the generators. They're spinor fields, so they're sections
over the left and right helicity spinor bundles. They're generators,
with the relation
P_l = l = 1/2 (o x o* + o* x o) = 1/2 (Q_0 Q_0* + Q_0* Q_0) = 1/2
{Q_0, Q_0*}.
By convention, a different scaling is adopted for the Q's so that the
1/2 factor is normalized to 1. The spinors, themselves, in each
subspace 2_R and 2_L are treated as Grassmann numbers, so one also has
{Q_a, Q_b} = 0 = {Q_a', Q_b'}. The coefficients with respect to the
bases 4, 2_R and 2_L are sigma^l_{00'} = sigma^m_{01'} =
sigma^m*_{10'} = sigma^n_{11'} = 1, all other sigma^{mu}_{AB'} being
0.

I don't think you'll see anywhere in the literature the explicit
connection of the spinor dyads and Q operators. But it's implied and
is probably what is indirectly being referred to by the passage you're
asking the question of.