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I'm a mathematician interested in mathematical physics, but, again, penetrating the physics literature is not so easy for me.

In the mathematical literature, a lot has been written about closed GW-invariants, opposed to open ones, at least as far as I understand. For example, the closed GW-invariants of the point resp. the projective space $\mathbb P ^1$ are well-known due to Witten-Kontsevich resp. Okounkov-Pandharipande (and many others). Recently, Pandharipande, Solomon and Tessler have initiated a mathematical (sound) framework for the study of open GW-invariants of the point (roughly, by constructing an intersection theory on moduli spaces of bordered Riemann surfaces).

I'd like to know any references concerning open analogues of the work of Okounkov and Pandharipande, i.e., open GW-invariants for $\mathbb P ^1$. This should be well known to physicists at least, but I couldn't find any literature (except for open GW invariants for 3 CYs).

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The open Gromov-Witten invariants for $\mathbb{P}^1$ were computed by Hori in 'Linear Models of Supersymmetric D-branes' (http://arxiv.org/abs/hep-th/0012179), using both direct computation via the nonlinear sigma model (NLSM) with target space $X=\mathbb{P}^1$, as well as computation via the mirror Landau-Ginzburg model which is dual to this NLSM. His mirror computation can also be generalized to the case where $X$ is a toric manifold with semipositive-definite first Chern class.

Additional details are contained in Chapter 39 of the 'Mirror Symmetry' monograph by Hori et al.

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