6D (2,0)-superconformal field theory

The existence of the 6D (2,0)-superconformal field theory is a prediction, actually, of string theory. It is actually poorly understood. This is because, there is no understanding of the theory in terms of an action functional. This theory is still a source of physical and mathematical insights, despite its difficulties.

The 6D (2,0)-superconformal field theory has proven effective in studying the general properties of some quantum field theories. The 6D (2,0)-superconformal field theory subsumes a large amount of effective quantum field theories. It also points to new dualities relating these theories. 

Luis Alday, David Gaiotto and Yujji Tachikawa showed that by compactifying the 6D (2,0)-superconformal field theory on a surface, allows one to obtain a 4-dimensional quantum field theory.

There is also a duality known as the AGT correspondence. The AGT correspondence relates the 6D (2,0)-superconformal field theory to certain physical concepts related to the surface itself. Theorists have recently extended these ideas by studying theories obtained by compactifying down to 3 dimensions. 

Besides quantum field theory, the 6D (2,0)-superconformal field theory has had application in pure mathematics. For example, Edward Witten used the 6D (2,0)-superconformal field theory to give a physical explanation for a conjectural relationship in mathematics known as the geometric Langlands correspondence and later he showed that the 6D (2,0)-superconformal field theory could be used to understand Khovanov homology. Khovanov homology, is given to us by Mikhail Khovanov around the year 2000. Khovanov homnology provides a tool in knot theory. Knot theory is the branch of mathematics that seeks to understand the different kinds of possible knots. There have also been the attempts to relate the 6D (2,0)-superconformal field theory to hyperkahler geometry by Davide Gaiotto, Greg Moore, and Andrew Neitzke.