Anomaly structure, general gaugings, and the chiral $d=6$ descendant of the $SO(1,2)\times H$-gauged $d=8$ supergravity with vector multiplets (Salam–Sezgin, 1985)

In a 1985 letter, Salam and Sezgin coupled $d=8$, $N=1$ supergravity (sixteen supercharges) to $n$ vector multiplets whose $2n$ scalars parametrize the Kähler coset $SO(n,2)/(SO(n)\times SO(2))$, gauged an $SO(1,2)\times H$ subgroup with $H\subset SO(n-1)$, and showed that a monopole configuration embedded in $SO(1,2)$ compactifies the theory on $(\mathrm{Minkowski})_6\times S^2$ with the monopole charge fixed to $\pm1$, implying what the abstract calls a stable, chiral $N=2$ supergravity in $d=6$ — in modern notation a $(2,0)$ theory. Three interlinked issues raised by that construction still lack a dedicated treatment in the literature: the quantum consistency of the gauged eight-dimensional theory, namely the anomaly structure of its chiral fermions and of the composite chiral $U(1)$ Kähler connection together with its mixings with the Lorentz and gauge symmetries, and whether a Green–Schwarz-type cancellation is available; the construction of general gaugings and explicit models beyond the particular $SO(1,2)\times H$ gauging, which a 2023 survey by one of the original authors states remains to be worked out; and the precise identification of the chiral six-dimensional descendant together with a string/M-theory embedding of the eight-dimensional gauged theory. Modern work has built a powerful eight-dimensional anomaly machinery for the ungauged minimal theories with $l\in\{2,10,18\}$ vector multiplets, and has identified the string origin of the related six-dimensional Salam–Sezgin model, but does not settle any of the three items for the eight-dimensional gauged theory. The problem remains open.

Background

Supergravity couples general relativity to supersymmetry, so that every bosonic field has a fermionic partner. In $d=8$ spacetime dimensions the minimal ($N=1$) supergravity carries sixteen real supercharges and contains the graviton, a chiral gravitino (the spin-3/2 fermionic superpartner of the graviton), two vector fields (the graviphotons), a chiral spin-1/2 fermion (the dilatino), a scalar (the dilaton), and a two-form antisymmetric tensor potential $B_{\mu\nu}$. Coupling $n$ vector multiplets — each supplying a vector, a chiral gaugino and two real scalars — adds $2n$ real scalars which parametrize the coset manifold $SO(n,2)/(SO(n)\times SO(2))$, the dilaton remaining a separate real scalar of the gravity multiplet; a coset $G/H$ is the space of ways of embedding the subgroup $H$ in the group $G$, here a Kähler manifold (a complex manifold with a compatible notion of integration), because the $2n$ real scalars combine into $n$ complex scalars. Gauging means promoting a subgroup of the global symmetry $SO(n,2)$ to a local symmetry, using the $n+2$ available vector fields as gauge bosons. Salam and Sezgin gauged a subgroup $SO(1,2)\times H$ with $H\subset SO(n-1)$ and $\dim H=n-1$; the non-compact $SO(1,2)$ generators act nonlinearly, giving combinations of the vectors a mass and generating a scalar potential of the form $-e^{-\sigma}C^aC^a$. The denominator factor $SO(2)\simeq U(1)$ of the coset acts on the chiral fermions through a composite connection $Q_\mu$ built from the scalar fields (the Kähler connection): the fermions are inert under the global $SO(n,2)$ but charged under this composite chiral $U(1)$, so the theory contains chiral fermions coupled to both genuine and composite gauge fields. Chiral fermions generically produce anomalies: a symmetry that holds classically can fail under quantization because the fermion path-integral measure is not invariant; in $D$ dimensions the anomaly is encoded in a $(D+2)$-form anomaly polynomial built from the curvatures of all connections that couple to the fermions. The Green–Schwarz (GS) mechanism cancels such anomalies through a form field whose own gauge variation absorbs the anomalous variation; for the composite $U(1)$ of minimal $d=8$ supergravity the relevant polynomial is $I_{10}=(F^Q/2\pi)\wedge X_8(R,F)$, where $F^Q$ is the curvature of the composite connection and $X_8$ is built from the Riemann curvature $R$ and the gauge field strengths $F$. Salam and Sezgin then embedded a Dirac monopole in the compact $U(1)$ inside the gauged $SO(1,2)$ and compactified on a two-sphere: the field equations fix the monopole charge to $\pm1$, and the Killing-spinor projection $(\nu+\gamma_7)\epsilon=0$ leaves exactly two Majorana–Weyl spinors (spinors that are simultaneously real and of definite chirality) in six dimensions, i.e. sixteen supercharges of a single chirality. The compactification is thus to $(\mathrm{Minkowski})_6\times S^2$ preserving $N=2$ chiral supersymmetry in $d=6$ — in modern notation the $(2,0)$ theory, whose gravity multiplet contains a chiral gravitino and a self-dual two-form. Two adjacent works must be distinguished from this 1985 letter: the 1984 six-dimensional Salam–Sezgin model (a gauged Einstein–Maxwell supergravity in $d=6$ compactified on $\mathrm{Minkowski}_4\times S^2$, which is the object called the 'Salam–Sezgin model' in the modern brane-world literature), and the maximal $D=8$ supergravity with thirty-two supercharges, constructed by the same authors (Salam and Sezgin, Nucl. Phys. B 258 (1985) 284). The string and M-theory origin of the six-dimensional model was identified in 2003–04, but no analogous construction is known for the eight-dimensional gauged theory considered here. The modern context sharpens the question: the ungauged minimal $d=8$ supergravities are by now strongly constrained — string constructions give $l=18$ (heterotic or type I on $T^2$), $l=10$ (the CHL string, an asymmetric $\mathbb{Z}_2$ orbifold of the heterotic string), and $l=2$ (certain orbifolds) vector multiplets, and anomaly and rank arguments, together with cobordism constraints (conditions that the spacetime manifolds on which the theory can be defined be mutually boundable, as expected of a theory embeddable in quantum gravity), indicate that a consistent minimal eight-dimensional supergravity must have an even number $l$ of vector multiplets, with $l\in\{2,10,18\}$ under the standard rank constraints. The composite-$U(1)$ anomaly of these ungauged theories is cancelled by modular-invariant higher-derivative counterterms built from modular forms (automorphic functions transforming covariantly under the discrete duality group) of Borcherds-product type. This machinery covers the same scalar coset as the Salam–Sezgin letter but not its gauging, and the confrontation of the two has not been carried out.

Problem Statement

Settle the quantum consistency and completion of the $SO(1,2)\times H$-gauged $d=8$, $N=1$ supergravity coupled to $n$ vector multiplets constructed by Salam and Sezgin (1985), and determine the fate of its monopole compactification. The problem comprises three jointly supporting parts, all raised by the 1985 construction. (1) Anomaly analysis: compute the anomaly structure of the gauged eight-dimensional theory — the local and global anomalies of its chiral fermion spectrum (gravitino, dilatino, gaugini) under Lorentz transformations, the gauged $SO(1,2)\times H$, and the composite chiral $U(1)$ Kähler connection of the coset $SO(n,2)/(SO(n)\times SO(2))$, including all mixed anomalies among these factors — and establish whether the theory is anomaly-free, and if not whether any cancellation mechanism is available, together with any resulting consistency restrictions (such as allowed values of $n$ or constraints on the gauge group $H$). (2) General gaugings and models: construct the general gaugings of $d=8$, $N=1$ supergravity with vector multiplets, determining which gauge groups, vector mass spectra, and scalar potentials are consistent, and building explicit models, including the Salam–Sezgin gauging as a special case. (3) Descendant and embedding: identify precisely the chiral $d=6$, $N=2$ (modern $(2,0)$) supergravity produced by the $\mathrm{Minkowski}_6\times S^2$ monopole compactification — its field content, interactions, and anomaly cancellation — and establish whether the eight-dimensional gauged theory admits a string/M-theory embedding, relating it where possible to the known eight-dimensional string landscape with $l\in\{2,10,18\}$ vector multiplets.

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