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B-Model Gromov-Witten Theory pp 499–538Cite as

Birkhäuser

Eynard-Orantin B-Model and Its Application in Mirror Symmetry

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Abstract

We describe the Eynard-Orantin recursive algorithm on a spectral curve, and give a biased survey on its roles as B-models which predict various higher genus A-model invariants via mirror symmetry.

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Notes

  1. 1.

    We reserve the variable X = e −x for other purposes. In many but not all examples, it will be the first coordinate.

  2. 2.

    We allow such cycles to be non-geometric, i.e. elements in \(H_1({\overline {\Sigma }};\mathbb {C})\).

  3. 3.

    In [19], Eynard-Orantin recursions on such formal spectral curves are called local topological recursions.

  4. 4.

    It should converge near “large radius limit” τ 0. We decompose τ = τ′ + τ″, \(\tau '\in H^2_{\mathbb {T}}(X;\mathcal {Q})\) and \(\tau ''\in H^{\neq 2}_{\mathbb {T}}(X;\mathcal {Q})\). Here \(\tau _0^{\prime }=-\infty \) and \(\tau _0^{\prime \prime }=0\). This fact allows us to avoid Novikov variables. It is a highly non-trivial statement (see [45]). A common practice is to utilize Novikov variables first, and the convergence follows from the B-model after establishing a mirror symmetry statement.

  5. 5.

    If one insists on algebraic geometry, we can use relative Gromov-Witten invariants as the definition. This involves partially compactifying \(\mathbb {C}^3\) into the total space of \(\mathcal {O}_{\mathbb {P}^1}(-1-f)\oplus \mathcal {O}_{\mathbb {P}^1}(f)\), and define \(N^{\mathbb {C}^3,L,f}_{g,n,\mu }\) as the relative Gromov-Witten invariants on this space relative to the fiber divisor at the infinity in \(\mathbb {P}^1\). The tangency condition at the divisor is given by μ. (See [48, 53, 54].)

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Acknowledgements

The author would like to thank Chiu-Chu Melissa Liu and Zhengyu Zong for the exciting and pleasant collaboration—this survey is based on the joint works with them. He also thanks Yongbin Ruan for helpful discussion. BF is partially supported by a start-up grant at Peking University and the Recruitment Program for Global Experts in China.

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  1. Beijing International Center for Mathematical Research, Peking University, Beijing, China

    Bohan Fang

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  1. Bohan Fang
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Editors and Affiliations

  1. San Francisco State University, San Francisco, CA, USA

    Emily Clader

  2. University of Michigan, Ann Arbor, MI, USA

    Yongbin Ruan

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Fang, B. (2018). Eynard-Orantin B-Model and Its Application in Mirror Symmetry. In: Clader, E., Ruan, Y. (eds) B-Model Gromov-Witten Theory. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-94220-9_5

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