$Spin(n,n)\times\mathbb{R}^+$ Generalised Geometry and Consistent Truncations on Branes

Abstract: In this note we show how the consistent truncations on half-supersymmetric branes of Leung and Stelle and Lin, Skrzypek and Stelle fit into the general exceptional generalised geometry analysis of Cassani \emph{et al.}. Each solution defines a torsion-free $Spin(n)$ structure in the $Spin(n,n)\times \mathbb{R}^+$ generalised geometry introduced by Strickland--Constable, where $n$ is the dimension of the space transverse to the brane. Embedding this into the appropriate exceptional generalised geometry then defines the truncation. As a by-product we derive a new consistent truncation on the IIA NS5-brane to six-dimensional $\mathcal{N}=(2,0)$ supergravity coupled to a tensor mutliplet, and new consistent truncations on the D6- and D7-branes to seven- and eight-dimensional pure half-maximal supergravity respectively.

• The paper introduces a unified approach by embedding torsion-free Spin(n) structures within Spin(n,n)×R+ generalised geometry to construct consistent truncations on branes.
• It develops explicit truncation ansätze for various branes, including IIA NS5, D6, and D7 reductions, ensuring compatibility with lower-dimensional supergravity equations.
• The work unifies previous ad-hoc methods and provides systematic tools for flux compactifications, intersecting brane analyses, and potential holographic applications.

$Spin(n,n)\times\mathbb{R}^+$ Generalised Geometry and Consistent Truncations on Branes

Introduction and Theoretical Framework

This work provides a unifying perspective on the construction of consistent truncations in supergravity theories via the machinery of generalised geometry, specifically the $Spin(n,n)\times\mathbb{R}^+$ extension, and its embedding into exceptional generalised geometry (EGG). The results consolidate and extend prior analyses of half-supersymmetric brane backgrounds, showing that all associated consistent truncations can be understood as arising from torsion-free $Spin(n)$ structures in $Spin(n,n)\times\mathbb{R}^+$ generalised geometry, following the approach outlined by Cassani et al. [Cassani:2019vcl].

A consistent truncation, in this context, is a dimensional reduction where the truncated sector is guaranteed to yield solutions of the original higher-dimensional theory for arbitrary lower-dimensional solutions. Such truncations are not common and typically require an underlying symmetry or geometric structure, often linked with the preservation of supersymmetry and encoded in the structure of the EGG.

A pivotal assertion of this work is the systematic embedding of the $Spin(n)$-structure (for $n$ the dimension transverse to the brane) into $E_{n(n)}$ (or $E_{n+1(n+1)}$ for type II), where the relevant singlet structures are both topological (arising from group theory decompositions inside the exceptional group) and geometrical (through vanishing, or more generally, singlet, intrinsic torsion). The mechanism ensures the truncated lower-dimensional theory remains consistent and, in many cases, ungauged (corresponding to vanishing intrinsic torsion).

$Spin(n,n)\times\mathbb{R}^+$ Generalised Geometry: Core Structures

The $Spin(n,n)\times\mathbb{R}^+$ framework is a generalisation of conventional $G$-structure geometry. Here, the generalised tangent bundle incorporates not only vector fields but also higher $p$-form gauge symmetries inherent to supergravity. The split structure (upon which the consistent truncations are based) is encoded in the chiral spinor module and its dual, with a natural pairing, Dorfman derivative, and compatible notion of torsion, paralleling standard Riemannian geometry. The maximally compact subgroup is $Spin(n)\times Spin(n)$, with the generalised metric parametrising the coset $$Spin(n,n)/(Spin(n)\times Spin(n))\times\mathbb{R}^+$$.

Explicitly, for a $p$-brane background, the brane solution is always associated to a metric of the form

$$ds^2 = H^{2\alpha}(y)\eta_{\mu\nu}dx^\mu dx^\nu + H^{2\beta}(y)\delta_{mn}dy^m dy^n, \qquad F = - *_\delta dH(y)$$

with $H(y)$ harmonic in the transverse $\mathbb{R}^n$. The generalised structure is determined by constructing invariant generalised vector fields $\Xi^m$ that satisfy a torsion-free condition under the twisted exterior differential $d_F$, ensuring their compatibility with the generalised connection. The singlet property is realised through the identification of $Spin(n)$ as a subgroup of $Spin(n,n)$ under which these objects are invariant.

Systematics of Consistent Truncations via Exceptional Generalised Geometry

Using this generalised geometry, the paper constructs consistent truncation ansätze for branes in type IIA, type IIB, and eleven-dimensional supergravity (for $3 \le p \le 7$, both D$p$ and NS5/M5 branes). For each brane, the embedding of $Spin(n)$ into the appropriate exceptional group is detailed, including the determination of scalar, vector, and tensor fields of the truncated theory from group-theoretic branching rules and the generalised geometric singlet condition.

Numerous truncations are constructed explicitly, and the structure of the resulting supergravities is catalogued:

• D$p$-branes ($3 \le p \le 7$): The truncation yields lower-dimensional pure ungauged half-maximal supergravity or related couplings, with scalars typically associated to the axion-dilaton sector or volume/lapse deformations.
• NS5-brane (IIA/IIB cases): The IIA case produces an $\mathcal{N}=(2,0)$ six-dimensional theory coupled to a tensor multiplet—a previously unconstructed truncation—while the IIB case gives pure $\mathcal{N}=(1,1)$ supergravity. For the IIA NS5, detailed checks of the consistency were given using conventional supergravity equations in addition to the generalised geometric argument.
• D6/D7 and M5 branes: New truncations to seven- and eight-dimensional half-maximal supergravities and pure six-dimensional $(2,0)$ supergravity are constructed.

A consistent truncation ansatz is derived for each, ensuring that field strengths and Bianchi identities are compatible with the higher-dimensional equations, including appropriate dualisation constraints (e.g., self-duality in six-dimensional tensor fields).

New Results, Numerical Claims, and Contradictions

• New Consistent Truncations: The consistent truncations on the type IIA NS5-brane giving $\mathcal{N}=(2,0)$ supergravity coupled to a tensor multiplet, and those for the D6, D7 branes, are claimed as new results. These expand the dictionary of known consistent truncations, especially in cases not predicted by conventional group manifold or sphere reductions.
• Strong Structural Claim: All known brane truncations, including various intersecting cases, are systematically captured by the $Spin(n,n)\times \mathbb{R}^+$ framework as special cases of the EGG approach, with the torsion-free $Spin(n)$ structure as the unifying ingredient.

Practical and Theoretical Implications

Practically, these results provide robust, systematic algorithms for constructing consistent truncation ansätze on brane backgrounds, advancing beyond ad-hoc or case-by-case methods and giving a general toolset for the construction of lower-dimensional gauged supergravities from string/M-theory. Theoretically, the embedding of $Spin(n,n)$ into $E_{n(n)}$, and the resulting hierarchy of $G$-structures, reveals new links between exceptional geometry, flux backgrounds, and the global characteristics of supersymmetric compactifications.

A particularly profound implication is that some of these truncations cannot be explained solely by conventional parallelisability; i.e., the intrinsic torsion associated with the identity structure is non-singlet, and only in the generalised (often fluxed) extension does a singlet torsion condition emerge.

Furthermore, the results suggest an extension: smearing over $k$ directions generically enlarges the torsion-free structure from $Spin(n)$ to $Spin(n-k)$, resulting in additional vector or tensor multiplets in the truncated theory, as seen in the IIA NS5 analysis.

Outlook and Future Developments

The identification of intrinsically generalised geometric truncations beyond the sphere and group manifold reductions opens a question regarding the full classification of half-maximal consistent truncations—a challenging and open problem. The methods of this paper provide a promising framework for further extending systematic reductions to nontrivial flux and intersecting brane backgrounds, potentially enabling new holographic dual descriptions and systematic searches for gapped string/M-theory sectors.

The generalised geometric methods highlighted also seem readily adaptable to applications in flux compactifications, consistent AdS/CFT reductions, and studies of moduli stabilization, as well as the systematic coupling of complete matter sectors in lower-dimensional gauged supergravities.

Conclusion

By leveraging the $Spin(n,n)\times\mathbb{R}^+$ generalised geometry and its embedding in exceptional generalised geometry, this work presents a comprehensive and unified approach to the construction of consistent truncations for half-supersymmetric brane backgrounds. The explicit identification of torsion-free $Spin(n)$ structures provides a group-theoretically and geometrically robust origin for these truncations, reconciling and extending previous results and demonstrating consistency for new cases, notably the IIA NS5, D6, and D7 reductions. This formalism strengthens the bridge between geometric structure in higher-dimensional (super)gravity and the properties of the associated lower-dimensional effective theories.