Algebraic study of chiral anomalies

Abstract

The algebraic structure of chiral anomalies is made globally valid on non-trivial bundles by the introduction of a fixed background connection. Some of the techniques used in the study of the anomaly are improved or generalized, including a systematic way of generating towers of "descent equations".

U(1)-extended gauge algebras in p-loop space

“…Note first that ℓ t commutes with δ ǫ since A − L and F both transform homogeneously. Then, from the definition (3.4) of C p+1 , and (3.8) it follows that (3.12) Now apply the homotopy formula [15] (3.13) this leads to the formula (3.14) where ω 1 p+1 (L, ǫ) = ω 1 p+1 (L, 0, ǫ), and we have rewritten the last term using (3.9). Finally, defining a Lie algebra-valued (p − 1)-form χ p−1 = χ a p−1 T a by 15) we obtain…”

E 7(7) symmetry in perturbatively quantised $ \mathcal{N} = 8 $ supergravity

“…with 39) and d B tanh(Φ/2) defined in terms of the covariant derivative (4.23) similarly as in (4.24). In contradistinction to the conventional Russian formula, the extended field-strength (4.38) is not only the 'horizontal' two-form curvature, but has an extra component linear in the parameter C p .…”

“…The anomalies associated to a linearly realised group are well known, and are classified by symmetric Casimirs. A nice way to derive such anomalies is by means of the 'Russian formula' [36][37][38][39] 29) to derive a (d + δ k )-cocycle from any symmetric Casimirs by use of the Cartan homotopy formula. In four dimensions, the relevant Casimir is the symmetric tensor of rank three, and the Cartan homotopy formula gives…”

Renormalization and finiteness of topological BF theories

“…Actually it is well known [31] that the ghost polynomial Tr c 5 is related via descent equations to the non-abelian gauge anomaly (or Adler-Bardeen-Bell-Jackiw anomaly)…”