Kac-Moody Pre 的准备
同样的, 也一样结束了()
好热
由于老师BV是外国人, 只能用英文了()
ok
We shall talk about the affine Lie algebra of nontwisted case. Take . Consider the "base" Lie algebra with nondegenerate invariant bilinear form . Take dual basis. We have Casimir operator
We know that the Casimir operator is independent w.r.t. the choice of basis. Then, in particular,
And this proves that
Now, recall that we have defined the generalized Casimir operator acts on reduced module
of .
We shall, for the , since for the chosen dual basis, . Then
Now, we shall consider the restricted completion that is, with s.t. for all reduced module, for all but finitely many . Now, there are extend to .
Now, define the Sugawara Operators :
(need to explain, why .)
We have,
Lemma 12.8: a) For , ,
b) restricted -mod, s.t. for some , then .
Proof: (a) We proof this in the semidirect product (to in duces ) .
Then,
for .
Secondly, we could check (on Pad) that
Then, (By one Jacobi identity on Pad)
For (b), there are
And then, we shall calculate the Lie brackets of Sugawara operators. There are
replace by we obtain:
Thus, for , we have
Note that, for or , , we can still assume an perform the same argument.
It left to argue that . Here, we shall assume . Then
And then, it equals to, by replace by in the first and by , we obtain that (don't want to copy kk)
And there are , there are
Now,(cor 12.8) by the above, for restricted -mod, scalar operator with , take
a) Letting
extends to a module over ( ). In particular, extends to a module over by .
b) If is the -mod , .
: corformal anomaly of the -mod ,
vacuum anomaly of .
For nontwisted affine algebra we have the "strange" formula of Freudenthal-de Vries:
( , with the fundamental weight corresponds to , the simple imaginary root).
For the so-called modular anomaly,
Prop 12.8: restricted -mod with a non-deg contravariant Hermitian form. Then the operator and are adjoint w.r.t. the form.
recall for the compact involution, with the fixed point set of .
There are is negative def. on . Then we choose s.t. , then for , there are , , thus and thus the contravariant Hemitian form means that
Finishes the proof.
The last remark is for, f.d. v.s. over with p.d. bilinear form , complexification, extend by bilinearity. Take we have
with the level of .
where , normalized character.
做这个 pre. 主要是下个学期要学共形场论了. 也许现在应该先打打基础()
这个 Sugawara operator 挺重要的.