Exceptional Generalised Geometry

• Exceptional Generalised Geometry is a unified geometric framework that extends conventional and Courant geometry by incorporating exceptional Lie groups as U-duality symmetries in string/M-theory.
• It encodes flux compactifications, consistent truncations, and the interplay of NSNS, RR, and higher-degree fields via generalized tangent bundles and patching with U-duality groups.
• Its algebraic structures, including the exceptional Dorfman bracket and derived L∞-algebras, provide a systematic method to unify gauge symmetries and analyze lower-dimensional effective theories.

Exceptional Generalised Geometry (EGG) is a geometric framework that extends conventional and Courant generalised geometry by incorporating the exceptional Lie groups $E_{d(d)}$ ($d=3,\ldots,8$), which act as U-duality groups in string/M-theory compactifications. EGG is engineered to make the non-perturbative symmetries of M-theory and type II supergravity manifest, unifying diffeomorphisms and p-form gauge symmetries, and geometrising not only the $B$-field but also the full spectrum of RR and higher-degree potentials, their duals, and the dual graviton. EGG provides a systematic language for flux compactifications, consistent truncations, and duality-covariant constructions across supergravity and string theory.

1. Generalised Tangent Bundle and Structure Group

The key mathematical object in EGG is the exceptional generalised tangent bundle $E$, constructed as the extension of the standard tangent bundle $TM$ plus a tower of differential-form bundles, chosen so that its fibres transform in the fundamental representation $R_1$ of $E_{d(d)}\times\mathbb{R}^+$, the continuous U-duality group for $d$-dimensional internal spaces. For example:

• $E_{7(7)}$: $$E \simeq TM \oplus \Lambda^2 T^*M \oplus \Lambda^5 T^*M \oplus (T^*M \otimes \Lambda^7 T^*M)$$
• $E_{6(6)}$: $$E \simeq TM \oplus \Lambda^2 T^*M \oplus \Lambda^5 T^*M$$
• $E_{5(5)}\cong SO(5,5)$: $$E \simeq TM \oplus \Lambda^2 T^*M \oplus \Lambda^5 T^*M$$
• $E_{4(4)}\cong SL(5)$: $E \simeq TM \oplus \Lambda^2 T^*M$

On overlaps of local patches, generalised vectors are glued using $E_{d(d)}\times\mathbb{R}^+$ transition functions, which combine diffeomorphisms and gauge transformations for all NSNS and RR fields, as well as non-trivial p-form gerbe data, thereby encoding fluxes geometrically (Josse et al., 2 Dec 2025, Larfors, 2015, Cederwall, 2013, Baraglia, 2011, Cassani et al., 2016).

2. Exceptional Dorfman Bracket and Generalised Lie Derivative

The algebra of generalised diffeomorphisms is governed by the exceptional Dorfman bracket (also called the generalised Lie derivative). For generalised vectors $U^M$, $V^M$ in $R_1$, the bracket is

$$(L_U V)^M = U^N\partial_N V^M - V^N \partial_N U^M + Y^{MN}{}_{PQ}\,\partial_N U^P\,V^Q$$

where $Y^{MN}{}_{PQ}$ is an $E_{d(d)}$-invariant tensor and the strong section condition

$Y^{MN}{}_{PQ}\,\partial_M\otimes\partial_N=0$

ensures closure and encodes the physical requirement that all fields depend on an $n$-dimensional physical slice of the extended coordinates (Palmkvist, 2015, Cederwall, 2013, Bugden et al., 2021).

The bracket, and its antisymmetrisation (the so-called C-bracket), unify diffeomorphisms and all p-form gauge symmetries, and encode the flux backgrounds via their structure constants. On the level of derived geometry, these brackets can be seen as derived brackets from a QP-manifold, leading to an underlying $L_\infty$-algebra encoding the tensor hierarchy (Arvanitakis, 2021, Arvanitakis, 2018, Baraglia, 2011, Lupercio et al., 2012).

3. Generalised Connections, Curvature, and Torsion Hierarchy

EGG admits a universal, $E_{d(d)}$-covariant calculus of generalised connections, metric, torsion, and curvature. The generalised (affine) connection $\Gamma_{MN}{}^P$ acts on generalised tensors with appropriate weights. The torsion tensor is defined as

$$T_{MN}{}^P = \Gamma_{MN}{}^P + Z^{PQ}{}_{RN}\,\Gamma_{QM}{}^R$$

where $Z=Y-\delta\otimes\delta$, and the torsion-free, metric-compatibility conditions are imposed for integrability (Cederwall et al., 2013, Godazgar et al., 2014, Hassler et al., 2023).

A remarkable recent advance is the identification of a full hierarchy of connections and curvatures mirroring the tensor hierarchy of gauged supergravity. One obtains, at each level, higher-rank generalised curvatures (Riemann, Ricci, etc.) built as quadratic expressions in the intrinsic torsion and embedding tensor: $R_{ABC}{}^D = X^\beta_{AB} X_{\beta C}{}^D$ These generalized Riemann tensors are fully covariant under exceptional generalised diffeomorphisms and reduce to their standard geometrical analogues when fluxes and non-metric data are switched off (Hassler et al., 2023, Godazgar et al., 2014, Cederwall et al., 2013).

4. Intrinsic Torsion, $G_S$-Structures, and Consistent Truncations

The consistent truncation of higher-dimensional supergravity to lower-dimensional gauged supergravity is systematically encoded by a reduction of the structure group of $E$ to a subgroup $G_S$ (a generalised $G_S$-structure), defined by invariant tensors (e.g., generalised vectors $K_I$, adjoint elements $J_A$) such that the generalised connection's intrinsic torsion is a singlet of $G_S$ and is constant. The intrinsic torsion is equated with the embedding tensor of the lower-dimensional gauged supergravity, dictating both the gauge algebra and the spectrum (Josse et al., 2 Dec 2025, Cassani et al., 2019, Felice, 2018).

The classification of all $G_S$-structures with constant singlet intrinsic torsion for a given $E_{d(d)}$ yields all possible consistent truncations to lower-dimensional maximal and half-maximal gauged supergravities:

• $D=4$: $E_{7(7)}$, fiberwise $\mathbf{56}_1$ (see Table 1 in (Josse et al., 2 Dec 2025))
• $D=5$: $E_{6(6)}$, $\mathbf{27}_1$
• $D=6$: $SO(5,5)$, $\mathbf{16}^c_{-1}$
• $D=7$: $SL(5)$, $\mathbf{10}_1$

For each, the possible $G_S$ are correlated with the amount of supersymmetry preserved in the lower-dimensional theory, the number of vector multiplets, R-symmetry, scalar cosets, and the allowed gaugings (via the decomposition of the $\mathbf{912}$ embedding tensor for $E_{7(7)}$) (Josse et al., 2 Dec 2025).

Consistent truncations are constructed by expanding all generalised tensors in a basis of globally-defined $G_S$-singlets, ensuring the equations of motion reduce consistently to lower dimensions (Josse et al., 2 Dec 2025, Cassani et al., 2016, Cassani et al., 2019, Felice, 2018).

5. Applications: Flux Compactifications, Brane Dynamics, and Dualities

EGG provides a manifestly U-duality-covariant framework for flux compactifications, exceptional calibrations, and brane current algebras:

• Flux backgrounds are encoded as global sections of the generalised bundle patched with U-duality data, including both NSNS/RR fluxes and dual graviton/dual forms. The nontrivial patching is described by twisting forms $H$ satisfying Maurer–Cartan equations, whose moduli classify distinct flux backgrounds (Baraglia, 2011, Larfors, 2015).
• Supersymmetric AdS vacua correspond to torsion-free or integrable generalised $G_S$-structures. The moduli space of exceptional Sasaki–Einstein structures gives the space of marginal/ exactly marginal deformations of dual SCFTs (Ashmore et al., 2016, Ashmore et al., 2016).
• Brane worldvolume dynamics, including Wess-Zumino terms and anomaly-free current algebras, are captured through the $L_\infty$ tensor hierarchy structure arising from graded symplectic (QP) manifolds underlying EGG (Arvanitakis, 2021, Arvanitakis, 2018).
• Non-abelian and Poisson-Lie T/U-dualities are algebraically unified via Exceptional Drinfeld Algebras (EDAs), whose constant “structure constants” provide generalised parallelisations realising all uplifts to maximal gauged supergravities and duality webs (Blair et al., 2020).

6. Classification and Examples of Generalised Homogeneous Spaces

Exceptional generalised homogeneous spaces—the analogues of group cosets $H\backslash G$—are precisely those spaces admitting an equivariant generalised frame with constant intrinsic torsion and constant higher curvatures. On these spaces, all generalised connection and curvature data are $H$-singlets and constant. Examples include maximally symmetric spheres $S^d\subset E_{d(d)}$, hyperboloids, Drinfeld doubles, and “three-algebra” geometries uplifting CSO-type gaugings (Hassler et al., 2023, Josse et al., 2 Dec 2025, Blair et al., 2020, Cassani et al., 2016).

A selection of $S^d$ consistent truncation examples is summarised below:

Dim. $d$	Group	Tangent Bundle (fundamental)	Maximal $G_S$	Embedding Tensor Representation	Example
7	$E_{7(7)}$	$\mathbf{56}_1$	$1$	$\mathbf{912}$	$S^7$, SO(8)
6	$E_{6(6)}$	$\mathbf{27}_1$	$1$	$\mathbf{351}'$	$S^6$, ISO(7)
5	$SO(5,5)$	$\mathbf{16}^c_{-1}$	$1$	$\mathbf{144}$	$S^5$, SO(6)
4	$SL(5)$	$\mathbf{10}_1$	$1$	$\mathbf{40}$	$S^4$, SO(5)

The selection of the $G_S$-structure determines the allowed multiplet content and scalar coset in the truncated theory (Josse et al., 2 Dec 2025).

7. Algebraic Structures and Derived Brackets

The algebraic underpinnings of EGG can be formalised in terms of Leibniz algebroids, derived brackets, and $L_\infty$-algebroids constructed from graded Lie/DGLA data. The Dorfman bracket and higher tensor hierarchy are realised as derived and higher derived brackets of these structures, incorporating twisting by fluxes (encoded in the Maurer–Cartan equation), and producing moduli spaces governed by algebraic varieties (Kuranishi spaces) (Baraglia, 2011, Lupercio et al., 2012, Arvanitakis, 2021).

The relation to Borcherds superalgebras and the universal construction via Chevalley-Serre data not only recovers the structure of the generalised Lie derivative and the section condition, but also reveals the underlying universality of exceptional geometry across all $E_{d(d)}$ ($d<8$) (Palmkvist, 2015).

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The modern theory of exceptional generalised geometry interweaves deep algebraic, geometric, and representation-theoretic structures to deliver a unified, duality-covariant framework for supergravity, flux compactifications, duality webs, and consistent truncations, with a systematic dictionary linking group-theoretic data to physical spectra, gauge algebras, and scalar topologies in lower-dimensional effective field theories.