The multiplicative anomaly of three or more commuting elliptic operators

Abstract

zeta-regularized determinants are well-known to fail to be multiplicative, so that in general det(zeta)(AB) not equal det(zeta)(A) det(zeta)(B). Hence one is lead to study the n-fold multiplicative anomaly

M-n(A(1),..., A(n)) := det(zeta)(Pi(n)(i=1) A(i))/Pi(n)(i=1)det(zeta()A(i))

attached to n (suitable) operators A1,..., An. 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))(m1+...+mn) = Pi(1 <= i<j <= n) M-2(A(i), A(j))(mi+mj),

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.