[Paper Review] Exceptional geometry and tensor fields

Martin Cederwall, Joakim Edlund+1 more|arXiv (Cornell University)|Feb 27, 2013

Black Holes and Theoretical Physics13 references17 citations

TL;DR

This paper develops a universal, covariant tensor calculus for exceptional generalised geometry based on the exceptional groups $E_n(n)$, unifying connections, torsion, and curvature across all $n \leq 7$. It introduces tensor gauge fields transforming in $R_k$ modules, analogous to differential forms, and establishes their dynamics and self-duality conditions, providing a geometric framework for maximal supergravity and extended supermultiplets in exceptional field theory.

ABSTRACT

We present a tensor calculus for exceptional generalised geometry. Expressions for connections, torsion and curvature are given a unified formulation for different exceptional groups E_n(n). We then consider "tensor gauge fields" coupled to the exceptional generalised gravity. Many of the properties of forms on manifolds are carried over to these fields.

Motivation & Objective

• To construct a universal, covariant tensor calculus for exceptional generalised geometry valid for all $n \leq 7$, extending previous work limited to $n=4$.
• To formulate connections, torsion, and curvature in a unified, $E_n(n)$-covariant manner, ensuring consistency with generalised diffeomorphisms.
• To introduce tensor gauge fields transforming in $R_k$ modules, analogous to differential forms, and describe their dynamics and self-duality properties.
• To establish a geometric framework for coupling non-gravitational tensor fields to exceptional generalised gravity, enabling construction of supermultiplets in maximal supergravity.

Proposed method

• Derives a generalised Lie derivative acting on tensors using the invariant tensor $Y^{MN}_{PQ}$, which encodes the algebra of generalised diffeomorphisms and satisfies the section condition $Y^{MN}_{PQ} \partial_M \otimes \partial_N = 0$.
• Introduces an affine connection $\Gamma^{MN}_P$ valued in the Lie algebra $\mathfrak{e}_n(n) \oplus \mathbb{R}$, with covariant derivative $D_M = \partial_M + \Gamma_M$, ensuring tensorial transformation properties under generalised diffeomorphisms.
• Defines curvature and torsion tensors via the commutator of covariant derivatives, with expressions that are manifestly $E_n(n)$-covariant and consistent across all $n \leq 7$.
• Identifies an infinite sequence of $G$-modules $\{R_k\}$, with $R_k$ transforming in representations of $E_n(n)$, that generalize differential forms and support tensor gauge fields.
• Applies the section condition to reduce the generalised geometry to physical spacetime, selecting $n$ physical directions and projecting fields onto $R_k$ modules.
• Uses dimensional reduction and binomial counting to show that $R_k^{(n)}$ fields give rise to $R_k^{(n-1)}$ and $R_{k+1}^{(n-1)}$ fields, mimicking form field reduction.

Experimental results

Research questions

• RQ1 How can a universal, $E_n(n)$-covariant tensor calculus be constructed for exceptional generalised geometry across all $n \leq 7$?
• RQ2 What are the correct covariant expressions for connections, torsion, and curvature in exceptional field theory that unify previous $n=4$ results?
• RQ3 How do tensor gauge fields transform under generalised diffeomorphisms, and what is their geometric role in the generalised manifold?
• RQ4 What is the role of the $R_k$ modules in supporting tensor fields, and how do they relate to differential forms in ordinary geometry?
• RQ5 Can the dynamics of tensor fields be consistently coupled to exceptional generalised gravity, and how does this relate to maximal supergravity?

Key findings

• A universal, $E_n(n)$-covariant tensor calculus is constructed for exceptional generalised geometry, valid for all $n \leq 7$, extending prior work limited to $n=4$.
• The generalised Lie derivative is expressed in a form that unifies ordinary diffeomorphisms and tensor gauge transformations via the invariant tensor $Y^{MN}_{PQ}$, with explicit coefficients $\alpha_n$ and $\beta_n$ for each $n$.
• The curvature and torsion tensors are derived in a manifestly covariant way, with the nonlinear identity $Z^{MN}_{TQ}Z^{TP}_{RS} + Z^{MP}_{RQ}\delta^S_N = 0$ ensuring closure of the algebra.
• Tensor gauge fields are introduced as sections of $R_k$ modules, which behave analogously to differential forms and support self-duality conditions.
• For maximal supersymmetry, the total number of bosonic degrees of freedom from $R_k$ fields matches the 128-dimensional representation of the scalar coset, with counts of 98, 101, 100, and 94 for $n=4,5,6,7$ respectively.
• Dimensional reduction shows that $R_k^{(n)}$ fields give rise to $R_k^{(n-1)}$ and $R_{k+1}^{(n-1)}$ fields, with binomial-like counting, confirming the form-like structure of the $R_k$ sequence.