Inner structure of Spinc(4)Spinc(4) gauge potential on 4-dimensional manifolds

The decomposition of Spinc(4)Spinc(4) gauge potential in terms of the Dirac 4-spinor is investigated, where an important characterizing equation ΔAμ=-λAμΔAμ=-λAμ has been discovered. Here, λλ is the vacuum expectation value of the spinor field, λ=∥Φ∥2λ=∥Φ∥2, and AμAμ the twisting U(1)U(1) potential. It is found that when λλ takes constant values, the characterizing equation becomes an eigenvalue problem of the Laplacian operator. It provides a revenue to determine the modulus of the spinor field by using the Laplacian spectral theory. The above study could be useful in determining the spinor field and twisting potential in the Seiberg–Witten equations. Moreover, topological characteristic numbers of instantons in the self-dual sub-space are also discussed.