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--- notes:2022-12-29-teleman-s-paper-on-2d-3d-tft [2022/12/30 00:10] – created pzhou
+++ notes:2022-12-29-teleman-s-paper-on-2d-3d-tft [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 20: / Line 20: @@
 The first surprising thing is that, Teleman tells me, this is related to my well-known and loved, toric mirror symmetry for $\C$ (but with $\C^*$ acting on it). The mirror is a space with superpotential, $\varphi: \C^* \to \C$, $\varphi_V(z) = z$. I am going to look for critical point, but I am not going to find it.
-If we take the universal cover of $\C^*$, we get $\varphi: \C \to \C$, $\varphi(u) = e^u$, and we can run the game of the Legendre transformation. $\psi(w) = crit-val(-e^u + w u)$, something like that, and we get, $w = e^{u}$, so $\psi(w) = w (\log w - 1)$, which is not very much well-defined! However, $d \psi(w) = \log w dw $, is somehow well-defined. Indeed, $\Gamma_{d e^u}$ is multivalued as one project to the $w$ direction. So, what am I doing here?
+If we take the universal cover of $\C^*_z$, we get $\varphi: \C_u \to \C$, $\varphi(u) = e^u$, and we can run the game of the Legendre transformation. $\psi(w) = crit-val(-e^u + w u)$, something like that, and we get, $w = e^{u}$, so $\psi(w) = w (\log w - 1)$, which is not very much well-defined! However, $d \psi(w) = \log w dw $, is somehow well-defined. Indeed, the graph Lagrangian $\Gamma_{d e^u}$ is multivalued as one projects to the $w$ direction. So, what am I doing here?
+So, forget about Legendre transformation, what do I want to do with this Lagrangian? It is a meromorphic section over $\C_\tau$, with $z = \tau$ (meromorphic because when $\tau=0$, we get stuck, since $z$ cannot be $0$).
+So, you say. Let's glue in another copy of pure Coulomb branch, after identify fiber by fiber, this Lagrangian become $1$ in the second copy. (but why should I do this? Why not glue in more copies? Why identify this particular Lagrangian? This toric mirror symmetry seems to come from NOWHERE.)