Group theory question about the N large limit

[llorgos]

Hi!

I keep hearing that in the large N limit (so I am talking in specific AdS/CFT but more general too I guess) U(N) and SU(N) are isomorphic. So if I construct, say, the $\mathcal{N}=1$ SYM Lagrangian in the large N limit, I can take as gauge group both of the ones mentioned above.

Why is this true?

 

[LBloom]

It's not, U(N) and SU(N) are never isomorphic. Maybe in the large N limit all the relevant physics is encoded in the SU(N) and any extra group factors from U(N) correspond to center of mass motion of the branes, but that concerns physics not group theory.

 

[tom.stoer]

Many simplifications in the large-N limit are due to algebraic identities arising in contractions of su(N) matrices.

One example for fund-rep. generators T with adjoint-rep. index a=1..N²-1 is

$2\,T^a_{ij}\,T^a_{kl} = \delta_{il}\,\delta_{jk} - \frac{1}{N}\delta_{ij}\,\delta_{kl}$

and setting 1/N = 0 in the large-N limit. There are other identified for the anti-symm. structure constants f and the less well-known symm. structure constants d derived from this identity. In addition one can derive similar approximations for terms with three or more generators T.

Another approx. in 1+1 dim. QCD is valid for "mesonic" operators. If there are fields q carrying a fund-rep. index i you define

$Q(x,y) = \frac{1}{N}q_i(x)\,q_k(y)$

You may use a an approximation of Q in terms of the vev + mesonic fluctuations like

$Q(x,y) = \langle 0|Q(x-y)|0\rangle + \frac{1}{\sqrt{N}}\tilde{Q}(x,y) + \ldots$

which means that you have an 1/N expansion in terms of vev + free mesons + interaction terms.