Geometry of the nonabelian 2-index potential and twisted de Rham cohomology (Q5938853)

Mathematically, this paper deals with twisted de Rham cohomology, the de Rham cohomology of differential forms with coefficients in a flat bundle \(E\) (taking the values in the Lie algebra of the structure group \(G\) of \(E)\) with the differential \(d_A=d+A\), \(A\) is a flat connection of \(E\) and shows the Euler characteristic of this cohomology is determined by \(E\) and does not depend on the choice of \(A\) (Proposition in Section 2).NEWLINENEWLINENEWLINEPhysically, since the nonabelian 2-form field \(B_{\mu\nu}\) is invariant under the transformation \(\delta B_{\mu\nu}= d_A\Lambda\) [\textit{D. Z. Freedman} and \textit{P. K. Townsend}, Nucl. Phys. B 177, 282-296 (1981)], the authors say that the \(B_{\mu\nu}(x)\) field is not regarded as a gauge potential but as a cohomological freedom intimately related to the existence of a flat connection \(A_\mu\). The authors also remark that a nonabelian generalization of \(S\)-duality might be obtained by using Hodge star duality in the twisted de Rham cohomology.