# Physicists Euler number conjecture

Physicist's Euler number conjecture says:

If $G \subset SL(n,\mathbb{C})$ is a finite group, $X=\mathbb{C}^n/G$ is the quotient space and $f:Y \rightarrow X$ a crepant resolution (always exists for $n\leq 3$). Then there exists a basis of $H^*(Y,\mathbb{Q})$ consisting of algebraic cycles in one-to-one correspondence with conjugacy classes of $G$.

I have seen some works (by Reid,...) which date back to 2000. What are the recent results around this conjecture?

See : The McKay correspondence for finite sungroups of SL(3,C), by Miles Reid and Yukari Ito.

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Do you know what this has to do with physics? –  J.C. Ottem Feb 27 '12 at 1:19
Physicists interested in String theory came up with this, while studying strings on the resolved Calabi-Yau, See: L. Dixon, J. Harvey, C. Vafa and E. Witten, Strings on orbifolds –  Mohammad F. Tehrani Feb 27 '12 at 1:41
You might take a look at Miles Reid's Bourbaki seminar, warwick.ac.uk/~masda/McKay/Bour/Bour.pdf the book "Orbifolds in Mathematics and Physics" from 2001 and descriptions of results from the Newton Institute workshop "Higher Dimensional Complex Geometry" in 2002 which are available on their website. There must be more recent summaries of results than these, but I don't know where. –  Jeff Harvey Feb 27 '12 at 13:38
Yes, I have seen that, they are written around the same time. –  Mohammad F. Tehrani Feb 27 '12 at 15:43