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  Yang-Mills theory in manifolds that are not simply connected

+ 4 like - 0 dislike

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 in Theoretical Physics by anonymous [ no revision ]

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