The multiplicative anomaly of three or more commuting elliptic operators

Abstract

Abstract. ζ-regularized determinants are well-known to fail to be multiplicative, so that in general det ζ (AB) = det ζ (A) det ζ (B). Hence one is lead to study the n-fold multiplicative anomaly.., 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. Namelywhere 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.

Calculation of the multiplicative anomaly

“…where g is the total degeneracy, g = i g i . Similar formulae can be found relating M k to M n for any n, 1 < n < k. Equation ( 12) is a particular example of a general result of Castillo-Garate, Friedman and Mǎntoiu, [7], having the same basic definitions but with D i quite general operators. The proof is an inductive one, and uses general properties of the Wodzicki expression for the defect.…”

“…N and R will depend on the parameters in the eigenvalues, λ m . I note the important fact that there are no cubic, or higher powers of α 2 in (7). Expression ( 7) can be implemented easily on a machine, but it does not directly yield the answer in its simplest factorised form without further manipulation.…”

“…Expression ( 7) can be implemented easily on a machine, but it does not directly yield the answer in its simplest factorised form without further manipulation. The symmetrised sums appearing in (7),…”

“…The formula (7) has been evaluated by brute force for various d and k. I will not write out the expressions but rather will appeal to the sum structure (11) which organises them in a more symmetrical way. Then I need give only M (i, j) for each dimension.…”

Effective action of conformal spins on spheres with multiplicative and conformal anomalies

“…In the present approach, I deal with spheres ab initio along the lines of Tseytlin in [9]. 3 My treatment differs from his only in a more commodious computation of the ζ-functions, I believe. In this regard, the method used in [1] extends the early work of Allen, [7].…”