# Jordan-Wigner transformation on a circle and spin structures?

Is there an analog of the Jordan-Wigner transformation between fermion algebra on a circle and a Pauli algebra? For example, the continuum analog of bosonization of "compact boson $$\leftrightarrow$$ Dirac fermion" there's an analog of the bosonization map, say on a Euclidean cylinder.

As a reminder, the compact boson is a bosonic action where the target space is a circle $$S^1$$ rather than the usual $$\mathbb{R}$$. As such, when the compact boson is put on a spacetime with nontrivial topology, there can be 'winding'/'instanton' modes where there's winding of the target-space circle around a non-contractible spacetime loop.

The bosonization map in the cylindrical case is with antiperiodic (NS) boundary conditions on the Dirac fermion, there's an equivalence to the bosonic field theory. Putting periodic (R) conditions on the fermion conditions also creates a map, except that the fermionic partition function maps to the bosonic one where the modes with odd winding around the circle contribute to the partition function with an extra minus sign. This allows one to isolate the even (odd) winding boson sectors via adding (subtracting) the two partition functions.

Extra details:

• There's a more complicated map on a torus. There's four fermionic partition functions corresponding to NS/NS, NS/R, R/NS, R/R conditions around the two loops. And, there's four 'winding sectors' for the boson parition parition function, corresponding to whether there's even or odd winding around the two loops. The bosonization map says that these partition functions get rotated into each other via some 4x4 matrix (this matrix is related to quadratic forms and the so-called Arf invariant, as explained here).
• The same exact story holds for the Ising/Majorana correspondence.

I was wondering if there was such a unified description relating spin structure, a fermionic algebra on a circular spin chain of $$N$$ sites, and some bosonic algebra. And, if they could be tied together with the standard Jordan-Wigner map. (I've heard people say things along these lines but haven't found any resource that ties these all together.)