Appendix D Homotopy
Yorikiyo Nagashima Β· 2014
Homotopy: Solutions π a (x π ) to a field equation of motion where the field possesses n degrees of internal freedom [π a (a = 1, 2, β’ β’ β’ , n)], can be considered as a projection from a four-dimensional spacetime to n-dimensional π space.If the field π is a continuous function of x π and when x π moves continuously from a point P to another point Q, the field π also moves from a point A in π space to another point B. If the path P β Q changes, the corresponding path A β B changes also (see Figure D.1a).Generally, when two different paths with the same end points can be made to overlap completely by continuous deformation of the paths, we say that they belong to the same homotopy class or that they are homotopic.The function that operates on the path and deforms (transforms) the path continuously is referred to as the homotopy.When the π space has a special structure, for instance, one like Figure D.1b, there are paths that cannot be brought to overlap completely by continuous transformations.π and π on the Figure D.1b are such examples.Then we say they belong to different homotopy classes.If they represent vacuum state of the field, they cannot be differentiated physically.If more than one degenerate vacua exist belonging to different homotopy classes, they provide conditions for the solitons to exist.We need to solve the equation of motion to prove the existence of the soliton solution.However, it helps to learn mathematics of homotopy to understand necessary conditions for existence of the solitons in general and know how to classify them.