Some results for SU(N) gauge-fields on the Hypertorus |
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Authors: | Pierre van Baal |
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Institution: | 1. Institute for Theoretical Physics, Princetonplein 5, P.O. Box 80.006, NL-3508 TA, Utrecht, The Netherlands
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Abstract: | We show how to prove and to understand the formula for the “Pontryagin” indexP for SU(N) gauge fields on the HypertorusT 4, seen as a four-dimensional euclidean box with twisted boundary conditions. These twists are defined as gauge invariant integers moduloN and labelled byN μv (=?N μv ). In terms of these we can write (ν∈#x2124;) $$P = \frac{1}{{16\pi ^2 }}\int {Tr(G_{\mu v} \tilde G_{\mu v} )d_4 x = v + \left( {\frac{{N - 1}}{N}} \right) \cdot \frac{{n_{\mu v} \tilde n_{\mu v} }}{4}} $$ . Furthermore we settle the last link in the proof of the existence of zero action solutions with all possible twists satisfying \(\frac{{n_{\mu v} \tilde n_{\mu v} }}{4} = \kappa (n) = 0(\bmod N)\) for arbitraryN. |
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