Composite anomalies in supergravity

Abstract

It is argued that the vectors of N-extended supergravity theories should be thought of as self-dual tensor fields, and that they contribute to the anomalies of the (S)U(N) currents. Some implications of these anomalies are discussed. The total anomaly is found to vanish for N > 4, so the composite gauging scenario is consistent in these cases.

Quantized gauged massless Rarita-Schwinger fields

“…Thus our conclusion in (I) that the classical theory of gauged Rarita-Schwinger fields is consistent extends to the quantized theory of gauged Rarita-Schwinger fields as well. As noted in (I), this means that in constructing grand unified theories, one can contemplate an anomaly cancellation mechanism in which the gauge anomalies of Rarita-Schwinger fields cancel against those of spin-1 2 fields, as first suggested in [12] and as used in the SU (8) family unification model of [13]. Some final remarks:…”

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

“…On the other hand, as shown by Marcus [17], the rigid SU(8) ⊂ E 7(7) left after gauge-fixing is non-anomalous, implying the absence of anomalies for the rigid su(8) current Ward identities in the gauge-fixed formulation of the theory. This is because the rigid su(8) symmetry acts linearly on the vector fields, whose chiral nature under SU (8) implies that there is an extra contribution to the anomaly from the vector fields which precisely compensates the contribution from the fermion fields.…”

“…It has been shown in [25] that self-dual form fields contribute to (gravitational) anomalies, just like chiral fermion fields, by means of a formal Fujikawa-like path integral derivation. This result can be understood geometrically from the family's index theorem [26], and it has been used in [17] to establish the absence of anomalies for the su(8) current Ward identities in N = 8 supergravity. Here we will exploit the duality invariant formulation to provide a full fledged Feynman diagram computation of the vector field contribution which confirms the expected result, and therefore the absence of anomalies in the theory.…”

“…It follows from this result that, although the non-linear character of the e 7 (7) symmetry is such that the associated anomalies involve infinitely many correlation functions with arbitrarily many scalar field insertions, the WessZumino consistency condition implies that the corresponding coefficients are all determined in function of the linear su(8) anomaly coefficient -thereby saving us the labour of having to determine an infinitude of anomalous diagrams! Now, thanks to a crucial insight of [17], it is known that for N = 8 supergravity, the anomalous contributions to the current (rigid) su(8) Ward identities from the fermions cancel against the contributions from the vector fields because the latter are also chiral under SU (8). Therefore the non-linear e 7 (7) Ward identities are likewise free of anomalies.…”

Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

“…Perturbative finiteness of d = 4 N = 8 SUGRA is possible only if its E 7(7) symmetry is anomalyfree [26,27,28,29]. Even when the symmetry is anomalous, as in pure N = 4 SUGRA, its Ward identities can restrict counterterms [30,31,32].…”