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I would like to know how to define spin structures combinatorially, for an oriented smooth manifold equipped with a triangulation. In the case of a 2d manifold, spin structures correspond to equivalence classes of Kasteleyn orientations (i.e. orientations of edges so that every face has an odd number of clockwise oriented edges). This fact is important in the theory of dimer models.

In general (for arbitrary dimensions) some sort of construction is proposed in http://arxiv.org/pdf/1306.4841.pdf but it is horribly complicated, I do not even see how it is related to Kasteleyn orientations in the 2d case. I would be quite happy to see a combinatorial description of spin structures for 3d and 4d triangulations.

Upd: there is an obvious generalization of the 2d definition to n dimensions. One could assign orientations to (n-1)-dimensional simplices so that for a given n-simplex an odd number of its faces have the orientation opposite to that of the n-simplex. Does this do the job, or is it wrong for some reason?

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    $\begingroup$ Though you are likely already aware of this, I include the following for context for other readers. The connection between Kasteleyn orientations and standard combinatorial spin structures as described in Ryan Budney's paper and Kevin Walker's answer is explained in Cimasoni and Reshetikhin's paper "Dimers on Surface Graphs and Spin Structures I" unige.ch/math/folks/cimasoni/spinI.pdf see in particular section 5 $\endgroup$
    – j.c.
    Commented Sep 24, 2013 at 18:21
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    $\begingroup$ It's not really clear to me what you need. When you say you need combinatorial spin structures, is your purpose the explicit computation of something? If so, what? The combinatorial spin structures in my paper are deliberately "flabby" allowing for many different ways of implementation, some perhaps more conceptually pleasant than others. $\endgroup$
    – Ryan Budney
    Commented Sep 24, 2013 at 18:21
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    $\begingroup$ If I'm understanding you correctly, you do not need combinatorial spin structures for any computation purpose. You are mainly looking to phrase things in a simple combinatorial language for aesthetic reasons? I try to avoid things like Kasteleyn orientations since they require special triangulations. My formalism is meant to work for any triangulation at all, and it's meant for direct computer implementation rather than human-readability. If you are willing to live with more restrictive triangulations, there will certainly be simplifications of the formalism. $\endgroup$
    – Ryan Budney
    Commented Sep 25, 2013 at 9:45
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    $\begingroup$ Having had a look at Cimasoni and Reshetikhin's paper, I think that the statement "In the case of a 2d manifold, spin structures correspond to equivalence classes of Kasteleyn orientations" is incorrect. The correct statement is that there is a map from pairs (Kasteleyn orientation, perfect matching) to spin structures, see Corollary 3 on page 199 of Cimasoni and Reshetikhin's paper. PS: Also, I cannot prevent myself from reminding everybody that spin structures do not form a set; they form a groupoid. $\endgroup$
    – André Henriques
    Commented Sep 25, 2013 at 20:43
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    $\begingroup$ I've done a set of revisions to the paper, hopefully you find this one easier to read. An updated version of my paper will appear on the arXiv on Tuesday (October 14th, 2014). I went over the paper in detail last week and found some sign errors and some changes of conventions that happened without explanation. I'd love to hear about simplifications of this formalism, especially ones that work for arbitrary triangulations, but ones that work for special triangulations would interest me as well. $\endgroup$
    – Ryan Budney
    Commented Oct 12, 2014 at 4:33

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Here's the standard answer, but perhaps you are looking for something different?

A spin structure on a manifold $M$ is determined by a framing of the tangent bundle $TM$, restricted to the 1-skeleton. (i.e. the framing is only defined on the 1-skeleton.) This framing is required to be the bounding framing on the boundary of each 2-cell (i.e. the framing can be extended to the 2-skeleton). Two such 1-skeleton framings determine the same spin structure if and only if they are homotopic. Such homotopies are generated by local moves which change the framing at a 0-cell and at all of the 1-cells adjacent to that 0-cell.

Kirby and Taylor have a nice paper on low-dimensional Spin and Pin structures -- http://www3.nd.edu/~taylor/papers/PSKT.pdf

[Added later]

(1) As Ryan points out in a comment, making the above idea explicit enough to be implemented on a computer is the point of his paper (which is linked to in the original question).

(2) If you are willing switch to the dual space -- functions on spin structures rather than spin structures themselves -- then there is the following relatively simple description which works in any dimension. Define a ribbon in $M$ be be a 1-dimensional submanifold $S \subset M$ equipped with a framing of $TM|_S$. Consider (finite linear combinations of) isotopy classes of ribbons in $M$ modulo the following three moves.

  • If $S$ and $S'$ differ by a framed saddle move, then $S \sim -S'$.
  • If $S$ and $S'$ differ by adding a single "kink" or "twist" to the framing, then $S \sim -S'$.
  • If $S$ and $S'$ differ by adding/removing a small unknotted circle with standard (non-bounding) framing, then $S \sim -S'$.

One can show that the above vector space is canonically isomorphic to the vector space of functions on (equivalence classes of) spin structures on $M$.

Note that if the three occurrences of $S \sim -S'$ above are changed to $S \sim +S'$, then the vector space is canonically isomorphic to finite linear combinations of elements of $H_1(M; \mathbb Z/2)$, which in turn is isomorphic to functions on $H^1(M; \mathbb Z/2)$. This is the counterpart to the fact that the set of spin structures is a torser for $H^1(M; \mathbb Z/2)$.

The above vector space naturally extends to a (fully extended) TQFT.

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    $\begingroup$ My "horribly complicated" paper is precisely what Kevin Walker is describing here, but put into the purely combinatorial language of triangulations, made suitable for computer implementation. $\endgroup$
    – Ryan Budney
    Commented Sep 24, 2013 at 18:14
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    $\begingroup$ I meant something comparable in complexity to the Kasteleyn orientation. That is, the set of spin structures is a torsor over the set of homomorphisms from the fundamental group to the cyclic group of order 2. The latter has a very simple combinatorial description. I would expect the spin structure to have a combinatorial description which involves roughly the same kind of data. $\endgroup$
    – Anton Kapustin
    Commented Sep 24, 2013 at 20:30
  • $\begingroup$ Kevin, do you have a reference that discusses what you describe as an extended TQFT? $\endgroup$
    – Ryan Thorngren
    Commented Sep 29, 2013 at 22:24
  • $\begingroup$ There are some notes on my web page which describe constructing a 3+1-dimensional TQFT from a premodular category. The ribbons-mod-relations I describe above is a very simple modular category, so the general construction for premodular categories applies. Feel free to email me if you have more questions. $\endgroup$
    – Kevin Walker
    Commented Sep 30, 2013 at 1:29

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