# Yang-Mills theory in manifolds that are not simply connected

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Consider the Yang-Mills theory for the gauge field strength 2-form $F$ on a manifold $M$:

$S = \int_M( \frac{1}{2}tr (F \wedge *F) + \theta tr(F \wedge F))$.

Upon quantization there might occur additional ghost terms. The term proportional to $\theta$ I can write as a boundary term. I assume that the boundaries are only the 3-dimensional spaces at the initial time $t_i$ and the final time $t_f$. By applying Gaussian Theorem I obtain:

$\int_M tr (F \wedge F) = \int_{\partial M} tr(A \wedge dA + \frac{2}{3}A \wedge A \wedge A) = CS(\partial M, t_f)-CS(\partial M,t_i)$.

Here, $CS(\partial_M,t)$ is the Chern-Simons form for the 3-dim. space $\partial M$ evaluated at time $t$. By computing the Partition function I obtain a sum over all Chern-Simons classes. This sum goes over different topologies of the gauge bundle. If I want to compute e.g.

$<0|T A_\mu(t_f) A_\nu(t_i)|0>=D_{\mu \nu}$

I have to sum over different large gauge transformations. Let $D_{\mu \nu}^{(\Xi)}$ be the gauge boson Propagator if the gauge Transformation winds topologically in a way that is classified by $\Xi \in \pi_1(\partial M)$ with the fundamental Group $\pi$. Then I would have

$D_{\mu \nu} = \sum_{\Xi \in \pi_1(\partial M)}exp(i \theta (CS(\Xi,t_f)-CS(\Xi,t_i)))D_{\mu \nu}^{(\Xi)}$ ?

If a gauge boson winds around the manifold it propagates a longer distance than on the direct wa without winding from one Point to a neighboring Point. Therefore, gauge boson Propagators depend on how they wind around the manifold. Is my idea right? Or are there any mistakes in my Derivation? asked Nov 22, 2017

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