http://www.neverendingbooks.org/index.php/superpotentials-and-calabi-yaus.html lieven le bruyn's blog Fri, 20 Jan 2012 16:50:41 +0100 hourly 1 http://wordpress.org/?v=3.3.1 http://www.neverendingbooks.org/index.php/superpotentials-and-calabi-yaus.html/comment-page-1#comment-4275 now what? | neverendingbooks Tue, 15 Jan 2008 06:20:02 +0000 http://www.neverendingbooks.org/?p=61#comment-4275 <p>[...] of you may have noticed that I’ve closed the open series on tori-cryptography and on superpotentials in a rather abrupt manner. It took me that long to realize that none of you is waiting for this [...]</p> [...] of you may have noticed that I’ve closed the open series on tori-cryptography and on superpotentials in a rather abrupt manner. It took me that long to realize that none of you is waiting for this [...]

]]> http://www.neverendingbooks.org/index.php/superpotentials-and-calabi-yaus.html/comment-page-1#comment-3630 the modular group and superpotentials (1) at neverendingbooks Wed, 26 Dec 2007 13:36:32 +0000 http://www.neverendingbooks.org/?p=61#comment-3630 <p>[...] I will go over the last post at a more leisurely pace, focussing on a couple of far more trivial examples. Here’s the goal [...]</p> [...] I will go over the last post at a more leisurely pace, focussing on a couple of far more trivial examples. Here’s the goal [...]

]]> http://www.neverendingbooks.org/index.php/superpotentials-and-calabi-yaus.html/comment-page-1#comment-3607 Geert Mon, 24 Dec 2007 09:18:51 +0000 http://www.neverendingbooks.org/?p=61#comment-3607 <p>@lieven: yes, I realize you want to know what the representation theory of these algebras tells us about the representation theory of the modular group and I am looking forward to a follow-up post in which you explain this in more detail!</p> @lieven: yes, I realize you want to know what the representation theory of these algebras tells us about the representation theory of the modular group and I am looking forward to a follow-up post in which you explain this in more detail!

]]> http://www.neverendingbooks.org/index.php/superpotentials-and-calabi-yaus.html/comment-page-1#comment-3606 lieven Mon, 24 Dec 2007 09:04:39 +0000 http://www.neverendingbooks.org/?p=61#comment-3606 <p>Yep, Raf already told me... but there seems to be some confusion about what a CY-algebra should be. A much easier example (corresponding to the trivial representation of the modular group) gives as quiver a triangle with cyclic orientation and the defining equations from the necklace are that all paths of length 2 vanish. This algebra is not CY in Raf's sense because it is not a finite global dimension (all projectives have even dimension, so simples cant have a finite resolution) but seems to be CY in other people's sense (Keller-Reiten call it 3-CY as does Ginzburg). Anyway, fitting one definition or another is not what im after, i would like to know what the representation theory of these algebras have to do with the monodromy representation of the modular group and with representation spaces of the modular group.</p> Yep, Raf already told me… but there seems to be some confusion about what a CY-algebra should be. A much easier example (corresponding to the trivial representation of the modular group) gives as quiver a triangle with cyclic orientation and the defining equations from the necklace are that all paths of length 2 vanish. This algebra is not CY in Raf’s sense because it is not a finite global dimension (all projectives have even dimension, so simples cant have a finite resolution) but seems to be CY in other people’s sense (Keller-Reiten call it 3-CY as does Ginzburg). Anyway, fitting one definition or another is not what im after, i would like to know what the representation theory of these algebras have to do with the monodromy representation of the modular group and with representation spaces of the modular group.

]]> http://www.neverendingbooks.org/index.php/superpotentials-and-calabi-yaus.html/comment-page-1#comment-3605 Geert Mon, 24 Dec 2007 08:47:47 +0000 http://www.neverendingbooks.org/?p=61#comment-3605 <p>I briefly checked Raf's paper on graded Calabi-Yau algebras of dimension three (<a href="http://arxiv.org/abs/math/0603558" rel="nofollow">arXiv:math/0603558</a>) and Theorem 3.1 states that for a quiver and a necklace to define a Calabi-Yau algebra each vertex in the quiver must be the source and target of two arrows. This is not the case in the quiver you wrote down here, so it cannot be graded Calabi-Yau in Raf's definition. Maybe some other algebraic CY-definition would do the trick? I'm far from an expert in these matters, so that's for someone else to answer.</p> I briefly checked Raf’s paper on graded Calabi-Yau algebras of dimension three (arXiv:math/0603558) and Theorem 3.1 states that for a quiver and a necklace to define a Calabi-Yau algebra each vertex in the quiver must be the source and target of two arrows. This is not the case in the quiver you wrote down here, so it cannot be graded Calabi-Yau in Raf’s definition.
Maybe some other algebraic CY-definition would do the trick? I’m far from an expert in these matters, so that’s for someone else to answer.

]]> http://www.neverendingbooks.org/index.php/superpotentials-and-calabi-yaus.html/comment-page-1#comment-3590 Kea Sun, 23 Dec 2007 18:42:19 +0000 http://www.neverendingbooks.org/?p=61#comment-3590 <p>Unbelievably FANTASTIC! Thanks. I wish I had more time right now to try and understand all this.</p> Unbelievably FANTASTIC! Thanks. I wish I had more time right now to try and understand all this.

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