研究级科学推理

Let $M_4$ be a closed four-manifold, with all gauge fields Abelian one-forms. The gauge variation of some one-loop effective action, in the background of external fields $A_L,A_R$, is $$\delta W=\kappa {\int }_{M_4}({\lambda }_L{F}_L\wedge F_L-{\lambda }_R{F}_R\wedge F_R),F_{L,R}=d{A}_{L,R}.$$ Define $$A_V=\frac{A_L+A_R}{2},A_A=\frac{A_L-A_R}{2},$$ and add the local term $$S_{\mathrm{loc}}=c{\int }_{M_4}{A}_A\wedge A_V\wedge F_V,F_V=d{A}_V.$$ Choose $c$ so that, after adding $S_{\mathrm{loc}}$, the gauge anomaly remains only where physically allowed, while the preserved charge gauge symmetry is strictly anomaly-free. Write down the expression for $c$ in terms of $\kappa$ directly.