Projective Spectrum and Cyclic Cohomolgy

For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital algebra ${\mathcal B}$ over $\mathbb{C}$, its {\em projective spectrum} $P(A)$ or $p(A)$ is the collection of $z\in \mathbb{C}^n$, or respectively $z\in \mathbb{P}^{n-1}$ such that the multi-parameter pencil $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible in ${\mathcal B}$. ${\mathcal B}$-valued $1$-form $A^{-1}(z)dA(z)$ contains much topological information about $P^c(A):=\mathbb{C}^n\setminus P(A)$. In commutative cases, invariant multi-linear functionals are effective tools to extract that information. This paper shows that in non-commutative cases, the cyclic cohomology of ${\mathcal B}$ does a similar job. In fact, a Chen-Weil type map $κ$ from the cyclic cohomology of ${\mathcal B}$ to the de Rham cohomology $H^*_d(P^c(A),\ \mathbb{C})$ is established. As an example, we prove a closed high-order form of the classical Jacobi's formula.

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