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Below is a brief introduction to spin$^{\mathbf{C}}$ structure that I took from Wikipedia. For more information, one should refer to https://en.wikipedia.org/wiki/Spin_structure#SpinC_structures.

A spin$^{\mathbf{C}}$ structure is analogous to a spin structure for orientable Riemannian manifold, but uses the spin$^{\mathbf{C}}$ group, which is defined by the exact sequence \begin{equation} 1 \rightarrow \mathbf{Z}_2 \rightarrow \text{spin}^{\mathbf{C}}(n) \rightarrow SO(n)\times U(1) \rightarrow 1. \end{equation}

A spin$^{\mathbf{C}}$ structure exists iff the third integral Stiefel-Whitney class of the manifold vanishes. Moreover, the set of spin$^{\mathbf{C}}$ structures has a free transitive action of $H^2(M, \mathbf{Z})$. Thus, spin$^{\mathbf{C}}$ correspond to elements of $H^2(M, \mathbf{Z})$ although not in a natural way.

For spin structure on an orientable two dimensional manifold equipped with a triangulation, there is a nice combinatorial representation in terms of the Kasteleyn orientation (i.e. orientations of edges so that every face has an odd number of clockwise oriented edges). This correspondence is detailed in https://arxiv.org/abs/math-ph/0608070. See also this post Combinatorial spin structures.

I would like to know if there is a similar combinatorial representation of spin$^{\mathbf{C}}$ structures on orientable manifolds in two dimension.

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  • $\begingroup$ Is $2D$ two or even? As far as I remember, $\operatorname{Spin}^{\mathbb{C}}$-structures are the integral lifts of $w_2$. If $2D=2$ and, hence, $w_2=0$, there is a distinguished lift, do the set is naturally isomorphic to $\mathbb{Z}$. $\endgroup$
    – Alex Degtyarev
    Commented Nov 7, 2017 at 19:26
  • $\begingroup$ 2D means two dimension. Yes, the spin$^{\mathbf{C}}$ structures are integral lifts of $w_2$. But I'm not sure how to translate this into the orientations of the edges of the graph on the manifold. i.e., something analogous to the Kasteleyn orientation. $\endgroup$
    – Zitao Wang
    Commented Nov 7, 2017 at 20:00
  • $\begingroup$ In general $\text{spin}^{\mathbf{C}}$-structures correspond to homotopy classes of complex structures over the 2-skeleton which can be extended to the 3-skeleton. So for surfaces, $\text{spin}^{\mathbf{C}}$-structures correspond to homotopy classes of complex structures on the triangulation. $\endgroup$
    – HYL
    Commented Nov 8, 2017 at 2:00
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    $\begingroup$ Yes, there are similar combinatorial representations. The 2nd link you give cites my preprint. The last section of my preprint shows how you adapt the first part of the paper to describe $Spin^c$ structures in the same language. I'll update my preprint soon as I think (sigh!) the version on the arXiv is still difficult to read in spots. $\endgroup$
    – Ryan Budney
    Commented Nov 8, 2017 at 3:35
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    $\begingroup$ The revised version (v3) should appear on the arXiv tomorrow. Hopefully it's far less painful to read. $\endgroup$
    – Ryan Budney
    Commented Nov 8, 2017 at 3:45

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