Superconformal anomalies from superconformal Chern-Simons polynomials

Abstract

We consider the 4-dimensional $\mathcal{N}=1$ Lie superconformal algebra and search for completely "symmetric" (in the graded sense) 3-index invariant tensors. The solution we find is unique and we show that the corresponding invariant polynomial cubic in the generalized curvatures of superconformal gravity vanishes. Consequently, the associated Chern-Simons polynomial is a non-trivial anomaly cocycle. We explicitly compute this cocycle to all orders in the independent fields of superconformal gravity and establish that it is BRST equivalent to the so-called superconformal $a$-anomaly. We briefly discuss the possibility that the superconformal $c$-anomaly also admits a similar Chern-Simons formulation and the potential holographic, 5-dimensional, interpretation of our results.

50 pages; v2: Added references. Added discussion in Section 5 regarding equivalent representatives of the anomalies. Added comparison with previously published results and explicit expressions for the fermionic contributions to the anomalies; v3: typos corrected, published in JHEP