The Superspace Geometry of Gravitational Chern–Simons Forms and Their Couplings to Linear Multiplets: A Self-Contained Presentation

Abstract

The superspace geometry of Chern-Simons forms is shown to be closely related to that of the 3-form multiplet. This observation allows to simplify considerably the geometric structure of supersymmetric Chern-Simons forms and their coupling to linear multiplets. The analysis is carried through in U K (1) superspace, relevant at the same time for supergravity-matter couplings and for chirally extended supergravity.

More modular invariant anomalous U(1) breaking

“…The latter point was demonstrated in detail, at both the superfield and the component field levels, for the toy model studied in [1]. L Q is the quantum correction [11,12,13] that contains the field theory anomalies canceled by the GS terms: 10) where P χ is the chiral projection operator [14]: P χ W α = W α , that reduces in the flat space limit to (16✷) −1D2 D 2 , and the L-dependent piece f a (L) is the "2-loop" contribution [11]. The string-loop contribution is [15] …”

“…Suppose a modification of the GS counterterm Lagrangian such that This is a straightforward generalization of the coupling of a single two-form field strength to Chern-Simons three-forms, as has been described for instance in [22,10,9]. It then follows from this definition that The anomaly cancellation coefficientsĉ ab can then be matched to those obtained from the underlying theory; e.g., any of the matrices enumerated in [25].…”

“…Except where noted above, we use the U(1) K superspace formalism [8,7,9]. (For a review of the U(1) K superspace formalism see [9]; for a review of the linear multiplet formulation see [10]. )…”

Anomaly Structure of Regularized Supergravity

“…In Appendix B we calculate the chiral anomaly for a general supergravity theory. As has been recently emphasized [19], in supergravity the fermion connections and corresponding field strengths contain many more operators than the Yang-Mills [8]- [13] and space-time curvature [20]- [22] terms that have been studied previously in the context of anomaly cancellation. These additional operators include [21,16,17] the Kähler U (1) K connection for all fermions, the reparameterization connection for chiral fermions, an axion coupling in the gaugino connection and a (matrix-valued) connection [17] linear in the Yang-Mills field strength in the gaugino-gravitino sector.…”

Modular-invariant anomalous U(1) breaking

“…Except where noted below, we use the U (1) K superspace formalism [5,6]. (For a review of the U (1) K superspace formalism see [6]; for a review of the linear multiplet formulation see [7]. )…”