Re: Complex Spin group and covering

On 11-01-2008 19:44, José Carlos Santos wrote:

To me it appears there is a contradiction between the book
'Supersymmetry for mathematicians' by Varadarajan and the book Spin
Geometry by Lawson & Michelsohn. Varadarajan claims (page 193
http://books.google.nl/books?id=sZ1-G4hQgIIC&printsec=frontcover&dq=supersymmetry+mathematicians&sig=Y1dGVGNKpZUJIjE8eSram6W_09U#PPA193,M1)

that the Spin group for complex vector space V is a double cover of
SO(V).

In the book by Lawson it is claimed (formula 2.28) that it is a 4-
sheeted cover. See http://books.google.nl/books?id=3d9JkN8w3X8C&pg=PP1&dq=spin+geometry&sig=lzJpFgpXK3fHkA2mAJyIA8SrDJQ.


My question is now, who is right? (Or, something that's perhaps more
probable, why am I wrong in seeing a contradiction?)

I hope anyone has the time to help me out with this problem. Thanks in
anticipation!

I don't see a contradiction. They are talking about different things.
Varadarajan is talking about the complex Lie group SO(n,C), which is the
group of those linear maps from C^n into C^n which preserve the bilinear
form q:C^n ---> C^n defined by

q(z_1,...z_n) = sum_k(z_k)2.

Lawson and Michelsohn are talking about the group SO(V,q), where V is a
finite-dimensional vector field over what they call a spin field. Also,
they assume that they are working with a non-degenerate quadratic form
_q_. But the specific quadratic form defined above *is* degenerate (when
the field is the complex field and dim(V) > 1), since it corresponds to
the bilinear form B defined by

B((z_1,...z_n),(w_,...,w_n)) |-> sum_k z_k*w_k

and B((1,i,0,...,0),(1,i,0,...,0)) = 0.

Forget my previous reply; it was just plain silly. I made a (huge)
mistake about the meaning of "non-degenerate". The quadratic form q
on C^n defined by

q(z_1,...z_n) = sum_k(z_k)2

*is* non-degenerate.

The problem lies elsewhere. Varadarajan defines Spin(n,C) (on page 193)
as the universal cover of SO(n,C). Then he proves (on page 198) that
Spin(n,C) is isomorphic to the group of the invertible elements _x_ of
the Clifford algebra Cl(C^n,q) such that

x.C^n.x^{-1} is a subset of C^n and x.beta(x) = 1,

where beta is the principal antiautomorphism of Cl(C^n,q).

On the other hand, Lawson and Michelsohn define Spin(n,C) (on page 18)
as the group of invertible elements Cl(C^n,q) generated by those
elements of the form

v_1 v_2 ... v_n

where each v_k belongs to C^n, _n_ is even and q(v_k) = 1 or -1 for each
_k_.

Therefore, they are not the same groups! If _v_ is an element of C^n
such that q(v) = -1, it belongs to the group that Lawson and Michelsohn
are talking about, but not the group that Varadarajan is talking about.
(However, they would be the same group over the reals; in that case, the
possibility q(v) = -1 does not occur.) The group considered by Lawson
and Michelson has, so to speak, twice as many elements as the one
considered by Varadarajan.

Best regards,

Jose Carlos Santos

.