Timeline for mirror symmetry with algebraic geometry?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Dec 12, 2010 at 20:01 | comment | added | Chris Brav | Thank you, Laie. I had not known that there were no known examples of non-toric-type Hodge numbers for CY3s. This is indeed intriguing since, of course, there are a great many examples of CY3s that are not obviously (to me) of toric-type. | |
| Dec 12, 2010 at 19:47 | comment | added | Laie | As I mentioned, the idea was to very briefly explain very roughly in two different ways how it can be understood that polynomial structures appear in the context of mirror symmetry. For this purpose I focused on certain relevant classes of CYs that have been important in the past. It is at this point in time not possible for many reasons to make universally valid statements about the relation between CYs and CFTs. It is intriguing though that the limits for the Hodge numbers of weighted CY hypersurfaces obtained in one of the 1990 mirror papers are still valid limits for all known CYs. | |
| Dec 12, 2010 at 18:07 | comment | added | Chris Brav | Is it really true that all or even most of the conformal field theories on the string worldsheet have a limit giving an LG potential that defines a Calabi-Yau hypersurface in a toric variety? This seems to me like it should put serious restrictions on the possible Hodge numbers for such Calabi-Yaus that would not be apparent (to me) from starting out with a type II sigma model. Or perhaps there are just a lot more toric ambient spaces than I assume. | |
| Dec 11, 2010 at 19:18 | comment | added | Spiro Karigiannis | @Kevin Lin: Thanks for pointing out that other MO question. Indeed, I always assume that Calabi-Yau means holonomy exactly SU(n), because otherwise it really reduces to a "simpler" situation. It's holonomy SU(n) that yields exactly 2 parallel spinors. See hal.archives-ouvertes.fr/docs/00/12/60/70/PDF/2000jmp.pdf for example | |
| Dec 11, 2010 at 19:09 | comment | added | Spiro Karigiannis | It's true that in complex dimension greater than or equal to 3, all Calabi-Yau manifolds are projective. (It's not true for K3 surfaces.) However, I didn't think this was crucial. From what I've heard, it's the existence of a parallel spinor (for supersymmetry) that forces one to use Calabi-Yau manifolds. In fact, I believe currently many string theorists are considering non-Kahler complex manifolds with trivial canonical bundle, and these are probably not all algebraic (although I am not sure...) | |
| Dec 11, 2010 at 18:57 | comment | added | Kevin H. Lin | "Calabi-Yau manifolds are described by polynomials" --- Aaron Bergman's answer here mathoverflow.net/questions/30629/… suggests that this issue is a bit subtle... | |
| Dec 11, 2010 at 18:46 | history | answered | Laie | CC BY-SA 2.5 |