Descent Equations From The Third Chern Class And Its Solutions

Chern-Simons forms were named after S.S.Chern and J.H.Simons for their work[1].They are also important in gauge theory.In mathematics,the Chern-Simons forms are certain secondary characteristic classes.Descent equations play an significant role in the theory of characteristics and have certain applications in theoretical physics,such as in the Chern-Simons field theory and the theory of anomalies.There is a strong connec-tion between the secondary characteristic classes and the Chern-Simons characteristic polynomial.In some documents,Chern-Simons characteristic polynomial is also named invariant polynomial.The second Chern class is defined as P = tr(F ∧ F)where F is the curvature 2-form,and P is the invariant polynomials.Thus,the descent equations starting from the second Chern class is defined as △Q2(k-1)(A0…,Ak;(?)△k)= dQ2(k)(A0,…,Ak;△k),k = 1,2[2]where A is the connection 1-form.A is coboundary operator,d is the exterior differential operator.That means,the coboundary of the k-Chern-Simons cochain exactly equals to the exterior differential of k + 1-Chern-Simons cochain.They play important roles in the applications of the issues about nomalies.Because of the limitations of the original definition,the descent equations terminate when k equals 2.The differential solutions can only be 4-forms,3-forms and 2-forms without solutions of 1-form and 0-form.A.Alekseev,F.Naef,X.M.Xu and C.C.Zhu gave a degree completion of descent equations starting from the second Chern class,and get a universal solutions[3]without discussing the descent equations starting from higher rank.P=tr(F∧F∧F)is the third Chern class,and the descent equations starting from the third Chern class is △Q3(k-1)(A0,…,Ak;(?)△k)=dQ3(k)(A0,…Ak;△k),k = 1,2,3.Because of the limitations of the original definition,the descent equations starting from the third Chern class will terminate when k=3,and have no expression of 2-form,1-form and 0-form.In this paper,we give a degree completion of descent equations starting from the Chern-Simons starting from the third Chern class,to obtain the universal solutions differ-ent from the solutions of ω1,and ω0 in[3],then get all the solutions of the descent equa-tions.Our main tool is the Kashiwara-Vergne theory.We give a definition of the universal enveloping algebra of free Lie algebra,then define a new complex space,Ω<x1,…,xn>with its differential operator d and coboundary operator δ,to complete the expansion of descent equations starting from the Chern-Simons of the third Chern class,and to obtain the expressions of the solutions,especially the constructions of 1-form ω1 and 1-form ω0.It can be verified that this kind of constructions of ω0 and ω1 can be applied to the descent equations for the cases of any degree.