Hierarchy of Generalised Connections

• Hierarchy of generalised connections is a structured family of geometric connections that extend the tangent bundle to include differential forms, unifying metric and gauge degrees of freedom.
• The framework employs generalised connections and the Dorfman derivative to integrate curvature, torsion, and gauge symmetries with a crucial section condition for algebra closure.
• This approach underpins advanced theories in string/M-theory and supergravity by linking exceptional symmetry groups, embedding tensor formalisms, and non-geometric fluxes.

A hierarchy of generalised connections refers to structured families of geometric connections adapted to extended tangent bundles or more abstract vector bundles that unify conventional geometric, algebraic, and gauge-theoretic aspects. Central to modern formulations in string/M-theory, higher-spin gravity, nonrelativistic geometry, and integrable systems, such a hierarchy embeds curvature, torsion, gauge symmetry, and duality in a systematic and often manifestly covariant manner. The following sections provide a technical overview of the hierarchy’s construction, algebraic and geometric principles, and significance in the context of exceptional and generalised geometry (Coimbra et al., 2011).

1. Generalised Geometry and Extended Tangent Bundles

The hierarchy arises from extending the conventional tangent bundle $TM$ of a $d$-dimensional manifold $M$ (for $d \leq 7$ in eleven-dimensional supergravity) to the so-called generalised tangent bundle, denoted $E$. The typical structure is

$$E \cong TM \oplus \Lambda^2 T^*M \oplus \Lambda^5 T^*M \oplus (T^*M \otimes \Lambda^7 T^*M)$$

with sections encoding not just vectors but also differential forms that represent gauge data associated with $p$-form potentials ($A$ fields and their duals). The bundle admits a natural action of $E_{d(d)} \times \mathbb{R}^+$, where $E_{d(d)}$ is the split real form of the exceptional Lie group, and the extra $\mathbb{R}^+$ (trombone) symmetry captures scaling degrees of freedom.

Transition functions for $E$ combine diffeomorphisms with $p$-form gauge transformations, encapsulating the unified bosonic sector—metric, three-form, its dual, and warp factor—of eleven-dimensional supergravity within a single “generalized metric” $G$, a positive-definite, $H_d$-invariant metric on $E$ constrained by the action of split frames adapted to the extended structure.

2. Generalised Connections, Symmetries, and the Dorfman Derivative

Generalised connections $D$ generalise the Levi–Civita connection to act on sections of $E$ and its tensor representations: $D_M V^A = \partial_M V^A + \Omega_M{}^A{}_B V^B$ Here, $$\Omega_M \in \mathrm{Lie}(E_{d(d)} \times \mathbb{R}^+)$$ encodes the connection coefficients in a local (generalised) frame, subject to local $H_d$ transformations.

The symmetry algebra is represented by the generalised Lie derivative (“Dorfman derivative”) capturing both diffeomorphisms and $p$-form gauge symmetries: $$\mathcal{L}_V V' = V\cdot\partial V' - (\partial \times_{\rm ad} V)\cdot V'$$ Its closure under composition is obstructed by an exact term, enforcing the so-called section condition—guaranteeing that fields depend only on a physical subset of coordinates to maintain the off-shell closure of the gauge algebra.

Generalised torsion is defined as the difference between the Dorfman derivative and its covariant version: $T(V)\cdot a = \mathcal{L}_V^D a - \mathcal{L}_V a$ where $a$ is any generalised tensor.

Unlike in conventional Riemannian geometry, there is always a family of torsion-free, $G$-compatible generalised connections, reflecting the richer structure of $E$; uniqueness occurs only for certain projections (relevant to curvature quantities) due to the enhanced local symmetry $H_d$.

3. Hierarchy of Irreducible Connections and Generalised Curvature

The extended structure typifies a “hierarchy” because different geometric data (vector, $p$-form, dual potentials) and their symmetry actions require harmonising multiple connection components, assembled according to irreducible $E_{d(d)}$-representations.

By imposing $D G = 0$ and vanishing generalised torsion, the hierarchy produces a (not unique) torsion–free, metric–compatible generalised connection. The ambiguity disappears in suitably projected components, e.g., via the $H_d$ structure, yielding uniquely defined generalised Ricci tensor $R_{MN}$ and scalar $R$.

Curvature quantities do not admit a straightforward lift from the standard definition as naive commutators fail to be tensorial. Instead, unique $H_d$-projected second-order differential operators emerge: $P_X (D^2 X) = R^\circ \cdot X$ Here, $P_X$ denotes an $H_d$-projection, and $R^\circ$ a generalised curvature acting on the projected subspace.

The entire bosonic action restricted to $d$-dimensions is given by the $E_{d(d)} \times \mathbb{R}^+$-covariant volume integrated over $R$,

$S_{\mathrm{B}} = \mathrm{vol}_G \cdot R$

and equations of motion become simply $R_{MN} = 0$.

4. Supergravity, Dualities, and Section Conditions

The framework encodes the bosonic sector of eleven-dimensional supergravity on $M^d$, with the physical fields repackaged in $G$, and all symmetry transformations realised via the generalised diffeomorphism group.

Upon dimensional reduction (e.g., to $d-1$), the formulation unifies both NSNS and RR sectors for type II supergravity democratically. Locally, it describes M-theory variants of double field theory, subject to the correct section condition ensuring the correct physical content.

The section condition is crucial: it is derived from projection conditions on pairs of derivatives (employing a canonical embedding $T^*M \rightarrow E^*$), ensuring the algebra closes and the theory remains locally geometric.

5. Embedding Tensor, Non-Geometric Fluxes, and Relations to Other Approaches

The hierarchy of generalised connections directly links to the embedding tensor formalism in flux compactifications and gauged supergravities. The generalised torsion $T$ lives in the same $E_{d(d)}$ representations as the embedding tensor and, by extension, as non-geometric fluxes, providing a systematic bridge between the geometric description and the algebraic data of gauged supergravity.

This structure also clarifies how exceptional symmetries ($E_{11}$, $E_{10}$, etc.) and duality covariant approaches (such as double field theory and exceptional field theory) can be interpreted as finite-dimensional shadows (via generalised tangent bundles) of infinite-dimensional symmetries expected in eleven-dimensional M-theory.

6. Conceptual Summary and Mathematical Significance

The hierarchy of generalised connections provides a unifying geometric framework where:

• The tangent bundle is extended to encapsulate metric and gauge degrees of freedom;
• Connections act on this extended space, governed by $E_{d(d)} \times \mathbb{R}^+$ symmetry;
• Generalised torsion and curvature are well-defined only after suitable $H_d$ projections, reflecting the enlarged local symmetry;
• The bosonic supergravity action becomes a generalised Einstein-like gravity governed by uniquely projected curvature tensors;
• Applications span the unified description of type II/M-theory, the geometrisation of fluxes and gaugings, and natural interfaces with duality-covariant (double or exceptional) field theories.

This construction illuminates the deep interplay between local and global (duality) symmetries, bundles of forms and brane charges, the role of section conditions in reduction and truncation, and the systematic emergence of physical equations from higher representation-theoretic and geometric data. It formalises and expands the classical notion of a connection into a true “hierarchy,” reflecting the compositional and highly structured nature of modern geometric and physical theories (Coimbra et al., 2011).