Timeline for Zero and Negative Gromov-Witten invariants in genus 0
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 31, 2013 at 18:43 | vote | accept | HNuer | ||
| Jan 30, 2013 at 17:51 | comment | added | Jim Bryan | This example, often called "local $\mathbf{P}^2$, is extensively studied in both the math and the physics literature. Here is one paper which analyses this example from the mirror symmetry / Picard-Fuchs point of view: arxiv.org/pdf/0710.0049v2.pdf Note that even though local $\mathbb{P}^2$ is non-compact, any compact CY3 $X$ which contains a $\mathbb{P}^2$ will have part of its GW theory identical to that of local $\mathbb{P}^2$. | |
| Jan 30, 2013 at 17:51 | comment | added | Jim Bryan | The total space of a locally free sheaf $F$ on $X$ is given by $Spec(Sym(F^\vee))$ --- you are missing the dual in your comment above. The invariants $n^0_d$ are only expected to be positive integers when the number of genus 0 curves in the class $d$ is finite. | |
| Jan 30, 2013 at 14:27 | comment | added | HNuer | Two questions: 1) is your $K_{\mathbb P^2}$ Calabi-Yau? If you mean by the total space of the canonical bundle the scheme $Spec_{\mathbb P^2} Sym(\omgea_{\mathbb P^2})$ then I believe this has canonical bundle $\pi^*\mathcal O_{\mathbb P^2}(-6)$. So did you mean the total space of the line bundle which has sheaf of sections $\omega_{\mathbb P^2}$, which I agree would be Calabi-Yau. 2)Also, aren't the GV invariants conjectured to be nonnegative (for example see p. 19 of arxiv.org/abs/hep-th/9111025), so is that simply not believed anymore, or are they talking about different things? | |
| Jan 30, 2013 at 0:27 | comment | added | Jim Bryan | the reference for this description of genus zero BPS invariants is the paper by Sheldon Katz: arxiv.org/pdf/math/0601193.pdf | |
| Jan 30, 2013 at 0:26 | history | answered | Jim Bryan | CC BY-SA 3.0 |