Chern-Simons Theory and Its Applications
Malik Matwi · viXra · 2020
Chern-Simons theory is a gauge theory in $2+1$ dimensional spaceime. This theory does not depend on additional structures, like a metric structure, thus it is a topological quantum theory that measures topological invariants like linking numbers, Jones polynomial, and other quantum invariants for knots and 3-manifolds. The equations of motion of Chern-Simons action is vanishing of the curvature $F = 0$. No metric is used in forming the action principle. One might expect the path integral to be a topological invariant of $3$ manifolds. The difference for the equation of motion with the Maxwell theory is that the Maxwell theory has non-trivial solution of curvature $F e0$ in absence of matter, while the Chern-Simons theory has solution only with $F=0$. The Chern-Simons theory has non-trivial solution with $F e0$ only when the gauge field couples with matter. Since the action functional of the Chern-Simons theory is first order in space-time derivatives, its Legendre transform gives the trivial Hamiltonian $H=0$. So there is no dynamics, and the only dynamics would be inherited from coupling to dynamical matter fields