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Mathematik » Topologie » Winding number of $SO(4)$ instantons?
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Universität/Hochschule Winding number of $SO(4)$ instantons?
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Aktiv Letzter Besuch: im letzten Quartal
Dabei seit: 10.06.2018
Mitteilungen: 24
  Themenstart: 2021-08-15

I have problems figuring out how the winding number of an $SO(4)$ instanton depends on the windings of the $SU(2)$ instantons. I have $SO(4)\simeq SU(2)\times SU(2)$. The anti-instanton and the instanton shall belong to either of the $SU(2)$s. Let $a$ and $b$ be two paths on $S^3\times[t_i,t_f]$ where the latter starts on $t_f$ and the former on $t_i$. After gluing both ends together I define two connections $A$ and $B$ on the bundle over $S^3\times S^1$. Then I should have $$(CS(a_{t_f})-CS(a_{t_i}))\times ((CS(b_{t_i})-CS(b_{t_f}))=-\frac{1}2\int_{S^3\times S^1}trF_A\wedge F_A dt \times \frac{1}2\int_{S^3\times S^1}trF_B\wedge F_B dt \in 4\pi^2\mathbb{Z} \times \mathbb{Z}$$ where $CS$ is the Chern-Simons functional and $F_K$ the curvature form on the bundle over the torus. Is this the correct construction for the $SO(4)$ bundle? Antworten auch gerne auf Deutsch .


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