Mathematical Research Letters

Volume 22 (2015)

Number 3

The multiplicative anomaly of three or more commuting elliptic operators

Pages: 665 – 673

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n3.a2

Authors

Victor Castillo-Garate (Departamento de Matemática, P. Universidad Católica de Chile, Santiago, Chile)

Eduardo Friedman (Departamento de Matemática, P. Universidad Católica de Chile, Santiago, Chile)

Marius Mantoiu (Departamento de Matemática, P. Universidad Católica de Chile, Santiago, Chile)

Abstract

$\zeta$-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