Science > Physics > closed form for instanton number of quintic threefold
| Topic: |
Science > Physics |
| User: |
"galathaea" |
| Date: |
20 Jul 2007 01:58:27 AM |
| Object: |
closed form for instanton number of quintic threefold |
a (the unique) holomorphic solution to the associated picard-fuchs equation
for the canonical 3-form on a quintic threefold
is given by
oo
---
\ (5i)! i
f = / ------ (-x)
--- (i!)^5
i=0
there is also a solution with a logarithmic singularity
oo 5i
--- ---
\ (5i)! [ \ 1 ] i
f' = f(x) ln(-x) + 5 / ------ | / - | (-x)
--- (i!)^5 [ --- j ]
i=1 j=i+1
the mirror map gives q-coordinate
f'/f
q = e
and the mirror symmetry relations relating A-models to B-models gives
oo
--- s
\ 3 q 5 / q dx \3
5 + / n s ----- = ------------------ | - -- |
--- s 1-q^s (1 + 5^5 x) f(x)^2 \ x dq /
s=1
where n is the instanton number of the quintic threefold
s
are there any known closed forms available for n ?
s
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galathaea: prankster, fablist, magician, liar
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