What Is Topological Charge in Quantum Chromodynamics?

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Topological charge in Quantum Chromodynamics (QCD) is an integer that characterizes the topology of a manifold, exemplified by the number of holes in surfaces like spheres or tori. In QCD, multiple equivalent vacua exist, connected by global transformations, with instantons representing spontaneous tunneling between these states. The discussion illustrates this concept using an analogy of a rope with attached masses, demonstrating how a "knot" can form, resulting in a non-zero topological charge, specifically a winding number. Instantons and solitons are closely related concepts within this framework.

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I have been going through some papers on lattice QCD lately, and many of them mention "topological charge". I was wondering if someone could either explain what is meant by this term, or point me to a resource that has an explanation.

Thanks
 
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An integer (or set of them) which characterizes topologically a given manifold. The most given example is the number of holes in a 2-dimensional closed surface, like a sphere or a torus. In QCD there are several equivalent vacua for perturbation theory which are related by global transformations, and instantons correpond to spontaneous tunneling between them.

Think of an infinitely long rope to which masses are attached every inch. Each mass is also connected to its two neighbours. In the vacuum state, all masses are just down. If you kick one mass, an oscillation will propagate. Now imagine that you keep two masses separated by 10 yards fixed in there down position, and you make a complete turn with one mass in between. Release now only one of the two masses, while keeping the other still fixed. Wait long enough and forget about what happened. Far away on the right, and far away on the left, everything points down : vacuum. But in between, there is a "knot". The field of masses has acquired a non-zero topological charge, namely a winding number. Truth is, in this case we have a soliton. But instantons and soliton are as closely related as euclidean and mikowskian geometries.
 

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