The multiplicative anomaly of three or more commuting elliptic operators

$\zeta$-regularized determinants are well-known to fail to be multiplicative, so that in general $\mathrm{det}_\zeta (AB) \neq \mathrm{det}_\zeta (A) \mathrm{det}_\zeta (B)$. Hence one is lead to study the $n$-fold multiplicative anomaly\[M_n(A_1,...,A_n) :=\frac{\det_\zeta\!\!\Big(\!\prod_{i=1}^n A_i\Big)}{\prod_{i=1}^n \det_\zeta(A_i)}\]attached to $n$ (suitable) operators $A_1, \dotsc, A_n$. We show that if the $A_i$ are commuting pseudo-differential elliptic operators, then their joint multiplicative anomaly can be expressed in terms of the pairwise multiplicative anomalies. Namely\[M_n(A_1,...,A_n)^{m_1+\cdots+m_n} =\prod_{1\le i<j\le n}M_2(A_i,A_j)^{m_i+m_j},\]where $m_j$ is the order of $A_j$ . The proof relies on Wodzicki’s 1987 formula for the pairwise multiplicative anomaly $M_2(A,B)$ of two commuting elliptic operators.

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Published 20 May 2015