Tag: Spin

I am interested if it is possible to represent a spin structure and the invariant Arf associated with it in terms of some kind of local fields.

For example, the first Chern class of a group of complex lines in a variety is an element of $ H ^ 2 (X, mathbb {Z}) $, that in a variety 2d equivalent to the specification of a whole number, the number of Chern $ c_1 in mathbb {Z} $.

All these concepts have local interpretations after putting a $ U (1) $ connection in the package, in terms of the 1-way connection $ A_ dx ^ mu $ and the curvature field of 2 shapes in the collector, $ F = ( partial_ mu A_ nu) dx ^ mu wedge dx ^ nu $. In this case, Chern's class can be considered as $ frac {1} {2 pi} F in H ^ 2 (X, mathbb {Z}) subset H ^ 2_ text {(from Rham)} (X, mathbb {R}) $. And Chern's number is given as the integral of this local amount, $ c_1 = frac {1} {2 pi} int_X F $.

I was curious to know whether or not there was an analogous interpretation of a spin structure and the invariant Arf.

From my point of view (pedestrian, physical) (as explained in this document), a turn structure is an element $ rho in H_1 (X, mathbb {Z} _2) $ that specifies periodic or antiperiodic boundary conditions in each non-trivial cycle, and the invariant Arf is $ (- 1) ^ {ind (D_ rho)} $, where $ D_ rho $ is the Dirac operator associated with the spin structure and $ ind (D_ rho) $ It is the index of the operator.

My question could also be formulated as if the index of a Dirac operator is an integral part of some local quantity that is "canonically" associated with the spin structure.