Subject: Re: two postings
From: Gerd Laures
Date: Fri, 01 Jun 2007 15:19:10 +0200
On Fri, 2007-06-01 at 08:28 -0400, Don Davis wrote:
>> Two postings: A question and a conference update..........DMD
>> _______________________________________________
>>
>> Subject: possible question for the list
>> From: Jack Morava
>> Date: Fri, 1 Jun 2007 08:02:30 -0400 (EDT)
>>
>> I suspect the answer to this question is known (but not by me!):
>>
>> What is the (Ochanine-Landweber-Ravenel-Stong) elliptic genus of
>> the quaternionic projective space HP_n? [Its A^-genus HP_n vanishes
>> (if n > 0), for example because it admits a nontrivial circle action.]
>>
>> Help, go-to's, or references would be very much appreciated.
>>
>> Best,
>> Jack
It is the appropriate power of epsilon if n is even and 0 else. You can
find it in Hirzebruch, Berger, Jung: manifolds and modular forms on page
29.
Regards,
Gerd
____________________________________________________________________
Subject: RE: two postings
From: "John Greenlees"
Date: Fri, 1 Jun 2007 17:17:07 +0100
Reference for Jack:
MR1990835 (2004k:58035)
Gálvez, Imma(4-SHEF-SM); Tonks, Andrew(4-LNDMU-CTM)
Differential operators and the Witten genus for projective spaces and
Milnor manifolds.
Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 1, 123--131.
58J26 (55P60 57R20 57R75 58J20)
____________________________________________________________________
Subject: Re: two postings
From: Peter Landweber
Date: Fri, 1 Jun 2007 14:45:36 -0400
On Jun 1, 2007, at 2:32 PM, Peter Landweber wrote:
> Dear Jack,
>
> Here's the answer to your question about the original elliptic genus of
HP^n.
>
> Of course, one knows that for HP2 one gets epsilon (where delta and
epsilon are the usual parameters; epsilon is also viewed as a modular form
of weight 4). It says so on pages 4 and 56 of Springer LNM 1326.
>
> Now continue to page 57, to learn that the elliptic genus vanishes on
HP^n when n is odd, while if n=2m its value is epsilon^m. I could dig
this out from an unpublished paper, but I believe Hirzebruch found a nicer
way to prove it:
>
> Hirzebruch, F. and Slodowy, P., Elliptic genera, involutions, and
homogeneous spin manifolds, Geom. Dedicata 35 (1990), 309--343
>
> Best wishes,
> Peter
>
________________________________________________________________________
Subject: Great!
From: Jack Morava
Date: Fri, 1 Jun 2007 14:50:29 -0400 (EDT)
To: Peter Landweber
Dear Peter,
It's great to hear from you!
Gerd Laures also wrote me about this; he points out that it's in
Hirzebruch's book (p. 29). In fact I'm not very familiar with that
-- I'd only seen it in German before a couple of weeks ago, and I
think some vague memory of that must have prompted my question.
Anyway here's a note I sent around after encountering the book:
Hope things go well with you, thanks again & all best,
(:+{)} Jack
---------- Forwarded message ----------
Date: Mon, 30 Apr 2007 19:20:10 -0400 (EDT)
From: Jack Morava
To: Michael J. Hopkins , Mark Mahowald
Cc: Nora Ganter , MCKAY john
,
Haynes Miller
Subject: cube root of Hirzebruch's problem?
Hi folks:
I'm using this bulk e-mail to record some folklore I learned last weekend
at John McKay's birthday conference in Montreal, which might deserve wider
dissemination:
In {\sc Manifolds and modular forms} (which I'd only seen in German before
the meeting) Hirzebruch et al construct an eight-dimensional (string)
manifold $M_8$ with Witten genus $j(q3)^{1/3}$ (I think!). I believe this
is the first in the sequence $M_{8k}$ of $8k$-dimensional manifolds you
construct in \S 2 of your paper.
It seems to be common knowledge in the Moonshine community that
$j(q3)^{1/3}$ is the McKay-Thompson series associated to an element of
order three in the Monster. The centralizer of such an element is [I'm
told] Thompson's simple group, which constitutes a big chunk of
$E_8({\mathbb F}_3)$.
The existence of an interesting action of the Thompson group on (some
manifold string cobordant to ?) $M_8$ would apparently be a corollary to
Hirzebruch's prize problem. It occurred to me to wonder if $M_8$ might
have a model which is an algebraic variety, maybe defined over
$\mathbb{Z}$, whose mod three reduction might admit an action of
$E_8(\mathbb{F}_3)$. [I guess it would have to be Calabi-Yau.]
I don't have any feel for these things, this is really just a shot in the
dark; but this corollary to H's question looks like it might be more
accessible, so I thought you might be interested.
The conference was great fun. All best,
(:+{)} Jack