TITLE: Complex Projective Space Spin and Dirac: Part II
QUESTION [6 upvotes]: (1) Let $M$ be a complex manifold of real dimension $2n$, and denote the line bundle of complex $(N,0)$-forms by $\Omega^{(N,0)}(M)$. When $M = CP^N$, the line bundles are indexed by the integers, and so, $\Omega^{(N,0)}(CP^N)$ must correspond to a integer. What is this integer? In the $N=1$ case, the corresponding integer is $2$. This suggests, that in general,  $\Omega^{(N,0)}(CP^N)$ corresponds to $2N$. Is this true?
(2) For the anti-canonical spin$^c$ structure of $CP^N$, the spinor bundle is isomorphic to 
$$ 
S := (\Omega^{(0,0)}(CP^N) + \cdots + 
\Omega^{(0,N)}(CP^N)) \otimes S_N,
$$
where $S_k$ is the square root of $\Omega^{(N,0)}(CP^N)$ (square root wrt tensoring as multiplication). What does the square root mean when when line bundle integer is odd? In the $N=2$ case, this is seen to reduce to $\cal{E}_{-1} \otimes \cal{E}_1$, where $\cal{E}_p$ is the line bundle corresponding to the integer $p$. Is this the $Z2$ grading on the spinor bundle? If so, what does this look like in higher dimensions?
(3) Finally, for a given spin connection $\nabla$, to define a Dirac operator we would need a Clifford representation, ie a map
$$
C:(\Omega^{(1,0)} \oplus \Omega^{(0,1)}) \otimes S \to S.
$$
For $N=2$, this should be given by a map
$$
C: (\Omega^{(1,0)} \oplus \Omega^{(0,1)}) \otimes (\cal{E}_{-1} \oplus \cal{E}_1) \to \cal{E}_1 \oplus \cal{E}_1.
$$
What is this rep? What does it look like for higher order $N$?
Note: the first subindex in the second $\cal{E} \oplus \cal{E}$ above should be $-1$, I'm having tex problems when I try to write it as such though.

REPLY [3 votes]: Here is a representation theoretical method for the computation of the (infinitesimal character of) the canonical bundle of any compact homogeneous Kahler manifold which include Grassmanians and complex projective spaces as particular cases. The computation for the complex projective space will be given explicitely. These spaces are coset spaces of a compact Lie group by an isotropy group which is a centralizer of a torus. The important step is to identify the generators of the torus in the Dynkin diagram of the compact Lie group. In the case of the $CP^N$, the torus is one dimensional and its generator corresponds to the simple root $\aplha_1$ or $\alpha_N$ of $A_N$. The other roots correspond to Grassmannians. 
A nice explanation  of this correspondence is given by John Baez here.
This method is sometimes called painted Dynkin diagrams is decribed in the following references by:  by Alexeevsky and Bordemann, Forger and Romer).
Now the infinitesimal character corresponding to the canonical bundle is the sum of all roots of $A_N$ containing the generator of the torus which will be taken here as $\alpha_1$. In $A_N$ the roots are sums of continuous strings of simple roots, thus the roots containing $\alpha_1$ are: $\alpha_1, \alpha_1+\alpha_2, \alpha_1+\alpha_2+\alpha_3,. . .$, Their sum is therefore $\lambda = N\alpha_1+(N-1)\alpha_2+(N-2)\alpha_3,. . .$. The infinitesimal characters of the basic line bundles correspond to the fundamental weights, thus we need to transform the result to the weight basis. Using:
$\alpha_1 = 2 w_1 - w_2, \alpha_i = -w_{i-1} + 2w_{i} + w_{i+1}$, for $i = 2, ..., N-1, \alpha_N = -w_{N-1}+ 2w_{N}$, we get: $\lambda = (N+1) \w_1$. The generalization to cases where the torus is multidimensional such as flag manifolds is obviousperformed by just duplication of the above procedure for evey torus generator
2) When N+1 is odd, tyen $CP^N$ does not admit a spin structure, however, one can tensor with the square root of a "line bundle" of odd infinitesimal character to obtain a $spin^c$ structure. Of course, neither the "square root of the canonical bundle" nor
the square root of the line bundle exist separately, only their product is a line bundle over $CP^N$.
3) If we choose to work with a spin connection induced from the Levi-Civita connection, 
then the Dirac oprtaor '$D: \wedge(T^{0,1}) \otimes S\rightarrow \wedge(T^{0,1}) \otimes S$ is just the sum of the twisted Dolbeault operator and its adjoint as given in page 82 of Friedrich's book.