Vtyjkyuk

Following this, we consider a more advanced question, below we take $N=3$ for all $N$.

1. Consider the group:

$$mathcalG=fracSU(N)_A times SU(N)_B_1 times SU(N)_B_2 times U(1)(mathbbZ_N)^2,$$

this group can be understood as a 4-multiplet
$$(g_A, g_B,1,g_B,2, e^i theta) in SU(N)_A times SU(N)_B_1 times SU(N)_B_2times U(1),$$
such that
$(e^i frac2piN, 1, 1, e^-i frac2piN)$ is the first $mathbbZ_N$ generator mod out in $G$, while $(1,e^i frac2piN,e^i frac2piN, e^-i frac2piN)$ is the second $mathbbZ_N$ generator mod out in $G$. Namely (1) the center of $SU(N)_A$, (2) the center of $SU(N)_B,1$ together with the center of $SU(N)_B,2$, and (3) the $e^i frac2piNin U(1)$ overlap, thus we only mod out the twice redundant $(mathbbZ_N)^2$.

Now consider the subgroup $H$ of the $G$ as:

$$H=fracSU(N)_A,BmathbbZ_Ntimes mathbbZ_2,$$

where
this group can be understood as a doublet
$(g_A,B, g_1) in SU(N)_A,Btimes mathbbZ_2= (SU(N)_A,B, pm 1 ) =H subset G,$
where there is a one-to-one correspondence between the doublet
$$(g_A,B, g_1) in H$$ and the 4-multiplet
$$(g_A,B,g_A,B^*,g_A,B^*, g_1) in G$$ with $g_A,B=g_A=g_B,1^*=g_B,2^*$, where $g^*$ means the complex conjugation (without the transpose $T$) of $g$.

Finally, we need $fracSU(N)_A,BmathbbZ_N$ to mod out $mathbbZ_N$, where we consider
$(g_A,B',g_A,B'^*)=(e^i frac2piN,e^-i frac2piN)mathbbI in mathbbZ_N$; this particular rank-$N$ diagonal matrix is the $mathbbZ_N$ we modded out .

We have that
$$fracSU(N)_A,BmathbbZ_N subset SU(N)_A times SU(N)_B_1 times SU(N)_B_2 , $$
$$ mathbbZ_2 = pm 1 subset U(1). $$

So this explains how $H$ is embedded as a subgroup in $G$.

Question:

• What is the precise space of the set of left coests
  $$
  G/H=?
  $$
  Is this certain smooth homogeneous space like a sphere or a complex/real projective space?




• What is the homotopy group?
  $$
  pi_j(G/H)=?
  $$
  for $j=1,2,3,4,5$.

Some background info that I prepared for you:

(1). $pi_i(U(N))=pi_i(fracSU(N)times U(1)mathbbZ_N)$:
$$pi_m(U(N))=pi_m(SU(N)), text for m geq 2$$
$$pi_1(U(N))=mathbbZ, ;;pi_1(SU(N))=0,$$

(2). $pi_1(mathbbZ_N)=mathbbZ_N$ and $pi_i(mathbbZ_N)=0$ for $igeq 2$.