# [time 900] Re: [time 899] Virasoro conditions,renormalization groupinvariance,unitarity,cohomology

Fri, 8 Oct 1999 00:55:16 +0900

Dear Matti,

Thanks for posting your paper. I read it but before going to physical
justification part I again stumbled on mathematical part: the proof that (19)
vanishes. As I reread your [time 894], I found it is interesting idea but does
not seem to work. I calculated like a blind mathematician:

m_1=Ym_0, Y=\sum_{k>0}(-X)^k =X(1+Y),

(BTW note (1+r)^{-1}=\sum_{k=or>0} (-r)^k not r^k !)

= Xm_0 + Xm_1 (1)

= Xm_0 + XPm_1 + X(1-P)m_1

XPm_1 = 0 yields

m_1 = Xm_0 + X(1-P)m_1

= Xm_0 + Zm_1, Z=X(1-P),

= Xm_0 + ZXm_0 + ZXm_1 (by (1))

= (1+Z)Xm_0 + ZXPm_1 + ZX(1-P)m_1

=(1+Z)Xm_0 + ZX(1-P)m_1 (by XPm_1=0)

= ....

= (1+Z+Z^2+Z^3+...)Xm_0 +lim_{k->infty}Z^kXm_1

=(1-Z)^{-1}Xm_0 + lim_{k->infty}Z^kXm_1.

This does not seem to vanish in general.

If this does vanish, your T is 0, not only that sum T+T^dagger=0: One would
need to use cancellation.

Maybe you want to find another proof by yourself, but [time 892} will be a
hint.

Best wishes,
Hitoshi

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