The spinorial geometry method of solving Killing spinor equations is reviewed as it applies to six-dimensional (1,0) supergravity. In particular, it is explained how the method is used to identify both the fractions of supersymmetry preserved by and the geometry of all supersymmetric backgrounds. Then two applications are described to systems that exhibit superconformal symmetry. The first is the proof that some six-dimensional black hole horizons are locally isometric to AdS3 × Σ3, where Σ3 is diffeomeorphic to S3. The second one is a description of all supersymmetric solutions of six-dimensional (1,0) superconformal theories and in particular of their brane solitons.

Spinorial geometry, horizons and superconformal symmetry in six dimensions

We construct Exceptional Field Theory for the group SO(5, 5) based on the extended (6+16)-dimensional spacetime, which after reduction gives the maximal D = 6 supergravity. We present both a true action and a duality-invariant pseudo-action formulations. All the fields of the theory depend on the complete extended spacetime. The U-duality group SO(5, 5) is made a geometric symmetry of the theory by virtue of introducing the generalised Lie derivative that incorporates a duality transformation. Tensor hierarchy appears as a natural consequence of the algebra of generalised Lie derivatives that are viewed as gauge transformations. Upon truncating different subsets of the extra coordinates, maximal supergravities in D = 11 and D = 10 (type IIB) can be recovered from this theory.

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Exceptional field theory: SL(5)

We construct Exceptional Field Theory for the group $SO(5,5)$ based on the extended (6+16)-dimensional spacetime, which after reduction gives the maximal $D=6$ supergravity. We present both a true action and a duality-invariant pseudo-action formulations. All the fields of the theory depend on the complete extended spacetime. The U-duality group $SO(5,5)$ is made a geometric symmetry of the theory by virtue of introducing the generalised Lie derivative that incorporates a local duality transformation. Tensor hierarchy appears as a natural consequence of the algebra of generalised Lie derivatives that are viewed as gauge transformations. Upon truncating different subsets of the extra coordinates, maximal supergravities in $D=11$ and $D=10$ (type IIB) can be recovered from this theory.

/pdf/exceptional-field-theory-so-5-5-4ing6poc05.pdf

Exceptional field theory: $SO(5,5)$

The framework of exceptional field theory is extended by introducing consistent deformations of its generalised Lie derivative. For the first time, massive type IIA super-gravity is reproduced geometrically as a solution of the section constraint. This provides a unified description of all ten- and eleven-dimensional maximal supergravities. The actionof the E7(7) deformed theory is constructed, and reduces to those of exceptional field theory and gauged maximal supergravity in respective limits. The relation of this new framework to other approaches for generating the Romans mass non-geometrically is discussed.

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The exceptional story of massive IIA supergravity

An infinite class of analytic AdS7 × S
 3 solutions has recently been found. The S
 3 is distorted into a “crescent roll” shape by the presence of D8-branes. These solutions are conjectured to be dual to a class of “linear quivers”, with a large number of gauge groups coupled to (bi-)fundamental matter and tensor fields. In this paper we perform a precise quantitative check of this correspondence, showing that the a Weyl anomalies computed in field theory and gravity agree. In the holographic limit, where the number of gauge groups is large, the field theory result is a quadratic form in the gauge group ranks involving the inverse of the A
 N
 Cartan matrix C. The agreement can be understood as a continuum limit, using the fact that C is a lattice analogue of a second derivative. The discrete data of the field theory, summarized by two partitions, become in this limit the continuous functions in the geometry. Conversely, the geometry of the internal space gets discretized at the quantum level to the discrete data of the two partitions.

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6d holographic anomaly match as a continuum limit

All nonlinear extensions of the source-free Maxwell equations preserving both $SO(2)$ electromagnetic duality invariance and conformal invariance are found, and shown to be limits of a one-parameter generalization of Born-Infeld electrodynamics. The strong-field limit is the same as that found by Bialynicki-Birula from Born-Infeld theory but the weak-field limit is a new one-parameter extension of Maxwell electrodynamics, which is interacting but admits exact light-velocity plane-wave solutions of arbitrary polarization. Small-amplitude waves on a constant uniform electromagnetic background exhibit birefringence, but one polarization mode remains lightlike.

Nonlinear duality-invariant conformal extension of Maxwell’s equations

We construct the duality-symmetric actions for a large class of six-dimensional models describing hierarchies of non-Abelian scalar, vector and tensor fields related to one another by first-order (self-)duality equations that follow from these actions. In particular, this construction provides a Lorentz invariant action for non-Abelian self-dual tensor fields. The class of models includes the bosonic sectors of the $6d$ (1,0) superconformal models of interacting non-Abelian self-dual tensor, vector, and hypermultiplets.

Igor A. Bandos

Papers

Nonlinear duality-invariant conformal extension of Maxwell’s equations

Duality-symmetric actions for non-Abelian tensor fields

Null super p-branes quantum theory in four-dimensional space-time

Superstring in doubled superspace

Superfield equations for the interacting system of D=4, N=1 supermembrane and scalar multiplet