What Is Topological Charge in Quantum Chromodynamics?

[Bobhawke]

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

 

[humanino]

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.

 

[sir-pinski]

for your question! Topological charge is a concept that arises in the study of topological field theories, specifically in the field of quantum chromodynamics (QCD). In simple terms, topological charge is a numerical quantity that characterizes the topological structure of a field configuration in QCD.

To understand topological charge, we first need to understand what is meant by "topology". In mathematics, topology is the study of the properties of geometric objects that are preserved under continuous deformations. In physics, topology plays a crucial role in understanding the behavior of physical systems, particularly in the field of quantum field theory.

In the context of QCD, topological charge is a measure of the winding or twisting of the gluon field, which is responsible for the strong force between quarks. This winding or twisting is related to the topological structure of the vacuum state of QCD, specifically the non-trivial topology of the gauge fields. Topological charge is a conserved quantity, meaning it remains constant even as the field configuration changes.

One way to visualize topological charge is to imagine a rubber band wrapped around a sphere. If we deform the rubber band, it will still have one loop, or winding, around the sphere. This winding is analogous to the topological charge in QCD - it remains unchanged even as the field configuration changes. In QCD, topological charge can take on integer values, with each value representing a different topological sector.

Topological charge has important implications in the study of QCD, as it affects the behavior of the theory at high energies and temperatures. It also plays a key role in the understanding of certain phenomena, such as the formation of hadrons (particles made of quarks) and the breaking of chiral symmetry.

I hope this explanation helps clarify the concept of topological charge. If you would like to delve deeper into the topic, I recommend checking out resources on topological field theories and their applications in QCD.