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They will play an important role in what follows and we shall show later on how they relate to chiral (gauge) anomalies. 36 A. Cardona, C. Ducourtioux, S. Paycha In order to describe weighted trace anomalies, it is useful to recall properties of logarithms of admissible elliptic operators. O. A ∈ ∗adm (M, E) is defined by log A = d z Ellord>0 dz |z=0 A , and depends on the spectral cut one chooses to define the complex power Az . O. s. A first weighted trace anomaly: The coboundary. It is by now a well known fact that, despite their name, weighted traces are not traces; given A, B ∈ Cl(M, E) we have [M, MN, CDMP]: ∂tr Q (A, B) = tr Q ([A, B]) = − 1 res (A[log Q, B]) , ordQ (4) where ∂tr Q denotes the coboundary of the linear functional tr Q on the Lie algebra Cl(M, E) in the Hochschild cohomology.

A++ A−+ B+− , one easily sees it is sufB−− 0 A+− 0 B+− ficient to check the formula for the odd operators and , A−+ 0 B−+ 0 since the result for the even part follows from (4). Proof. Writing A := A+− , B := A−− B++ B−+ From Tracial Anomalies to Anomalies in Quantum Field Theory 37 0 A− 0 and B = A+ 0 B+ acting on sections of some super-vector bundle E := E + ⊕ E − . We have: Let us therefore consider two odd operators A = B− 0 str Q ({A, B}) = = = = tr Q ( {A, B}) tr Q ( AB + BA) tr Q (−A+ B − + B − A+ − B + A− + A− B + ) tr Q ([B − , A+ ]) + tr Q ([A− , B + ]) 1 1 = res(A+ [log Q, B − ]) − res(A− [log Q, B + ]) ordQ ordQ where we have used (4) 1 1 = − res( A+ {log Q, B − }) − res( A− {log Q, B + }) ordQ ordQ where we have used the fact that Q (and hence log Q) is even 1 = − sres(A{log Q, B}).

Specializing to ∇ E = d in the case of a trivial bundle E and differentiating this expression yields: DetA+ −1 Det 1 d str Q A−1 dA 2 1 1 = [d, str Q ] A−1 dA − str Q A−1 dAA−1 dA . 2 2 DetA+ = Formula (38) then follows applying this formula to the vectors U and V . 5. The Chern Simons Term as an Integrated Trace Anomaly In this section and the next one, we specialize to the case of a trivial fibration M → X, with constant fibre given by a closed spin manifold M. Let W → M be a vector bundle with constant fibre above (x, m) ∈ M given by Wx,m := M ×W , where W is an exterior vector bundle on M and let E → M be a Hermitian Clifford vector bundle with constant fibre given by a Hermitian Clifford vector bundle Ex,m := M × E, where E = S × W , S being the spin bundle on M.

### Communications in Mathematical Physics - Volume 242 by M. Aizenman (Chief Editor)

by Paul

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