Question

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.

This post imported from StackExchange MathOverflow at 2014-08-01 07:39 (UCT), posted by SE-user Mohammad F. Tehrani

Comments

Do you know what this has to do with physics?

This post imported from StackExchange MathOverflow at 2014-08-01 07:39 (UCT), posted by SE-user J.C. Ottem

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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

This post imported from StackExchange MathOverflow at 2014-08-01 07:39 (UCT), posted by SE-user Mohammad F. Tehrani

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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.

This post imported from StackExchange MathOverflow at 2014-08-01 07:39 (UCT), posted by SE-user Jeff Harvey

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Yes, I have seen that, they are written around the same time.

This post imported from StackExchange MathOverflow at 2014-08-01 07:39 (UCT), posted by SE-user Mohammad F. Tehrani