Let $$E\rightarrow B$$ E → B be a smooth vector bundle of rank n , and let $$P \in I^p(GL(n,{\mathbb {R}}))$$ P ∈ I p ( G L ( n , R ) ) be a $$GL(n,{\mathbb {R}})$$ G L ( n , R ) -invariant polynomial of degree p compatible with a universal integral characteristic class $$ u \in H^{2p}(BGL(n,{\mathbb {R}}),{\mathbb {Z}})$$ u ∈ H 2 p ( B G L ( n , R ) , Z ) . Cheeger–Simons theory associates a rigid invariant in $$H^{2p-1}(B,{\mathbb {R}}/{\mathbb {Z}})$$ H 2 p - 1 ( B , R / Z ) to any flat connection on this bundle. Generalizing this result, Jaya Iyer (Lett Math Phys 106 (1):131–146, 2016) constructed maps $$H_r({\mathcal {D}}(E)) \rightarrow H^{2p-r-1}(B,{\mathbb {R}}/{\mathbb {Z}})$$ H r ( D ( E ) ) → H 2 p - r - 1 ( B , R / Z ) for $$p>r+1$$ p > r + 1 . Here, $${\mathcal {D}}(E)$$ D ( E ) is the simplicial abelian group whose group of r -simplices is freely generated by $$(r+1)$$ ( r + 1 ) -tuples of relatively flat connections. In this article, we construct such maps for the cases $$pr+1$$ p > r + 1 using fiber integration of differential characters. We find that for $$p>r+1$$ p > r + 1 case, the invariants constructed here coincide with those obtained by Jaya Iyer, and that in the $$p

中文翻译：

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使用微分特征的纤维积分的平面连接族的不变量

令 $$E\rightarrow B$$ E → B 是秩为 n 的光滑向量丛，令 $$P \in I^p(GL(n,{\mathbb {R}}))$$ P ∈ I p ( GL ( n , R ) ) 是一个 $$GL(n,{\mathbb {R}})$$ GL ( n , R ) - 与通用积分特征类兼容的 p 次不变多项式 $$ u \在 H^{2p}(BGL(n,{\mathbb {R}}),{\mathbb {Z}})$$ u ∈ H 2 p ( BGL ( n , R ) , Z ) 。Cheeger–Simons 理论将 $$H^{2p-1}(B,{\mathbb {R}}/{\mathbb {Z}})$$ H 2 p - 1 ( B , R / Z ) 连接到此包上的任何平面连接。概括这个结果，Jaya Iyer (Lett Math Phys 106 (1):131–146, 2016) 构建了地图 $$H_r({\mathcal {D}}(E)) \rightarrow H^{2p-r-1}( B,{\mathbb {R}}/{\mathbb {Z}})$$ H r ( D ( E ) ) → H 2 p - r - 1 ( B , R / Z ) $$p>r+ 1$$ p > r + 1 。这里，$${\mathcal {D}}(E)$$ D ( E ) 是单纯阿贝尔群，其 r -单纯形群由 $$(r+1)$$ ( r + 1 ) - 元组自由生成相对平坦的连接。在本文中，我们使用微分字符的纤维积分为 $$pr+1$$ p > r + 1 的情况构建这样的映射。我们发现对于 $$p>r+1$$ p > r + 1 的情况，这里构造的不变量与 Jaya Iyer 得到的不变量一致，而在 $$p