# Open Gromov-Witten invariants via lozalization with $\mathbb{C}^{*}$ (not $S^1$) action

Amplitudes of open A-model on a Calabi-Yau 3-fold $X$ with branes are given by the open Gromov-Witten invariants of $X$. It is known how to compute them if there is a toric action on a manifold, which preserves all the branes and the image of the curve (see Katz, Liu, Enumerative Geometry of Stable Maps with Lagrangian Boundary Conditions and Multiple Covers of the Disc and Graber, Zaslow, Open-String Gromov-Witten Invariants: Calculations and a Mirror "Theorem" for examples). It is done via equivariant localization on the fixed points of the toric action.

All the references I read, consider the toric action with integer weights (thus $S^1$), while the brane configuration I have is preserved only by certain $\mathbb{C}^{*}$ action, which rotates some of the coordinates by $\mathrm{e}^{\mathrm{i} b}$, while the others by $\mathrm{e}^{\mathrm{i/b}}$, for irrational $b$. If I try to follow the usual procedure used for the cases of $S^1$ action, I get the result which contradicts my physical expectations. I suspect that the reason is that this procedure must be modified appropriately in order to deal with irrational weights.

Could anybody recommend any reference in which such a problem is discussed?

• Unless $b^2$ is rational, the set $(e^{ ibn}, e^{-in/b})$ is dense in $S^1 \times S^1$. So if your manifold is closed in the coordinates you're using it should extend to a $S^1 \times S^1$ action. – Will Sawin Apr 21 '18 at 13:56
• The manifold is not closed. It is an $S^2 \times S^3$ conifold, which is also an $\cal{O}(-1) \oplus \cal{O}(-1)$ bundle over $\mathbb{P}^1$, so the orbits of the $\mathbb{C}^{*}$ action are infinite helices, and just two closed 1-cycles. – Andrey Feldman Apr 21 '18 at 14:17