[Paper Review] Enhancement of hidden symmetries and Chern-Simons couplings

Marc Henneaux, Axel Kleinschmidt+1 more|arXiv (Cornell University)|May 27, 2015

Black Holes and Theoretical Physics18 references3 citations

TL;DR

This paper demonstrates that hidden symmetries of Cremmer–Julia type in three-dimensional maximal supergravity emerge only when Chern–Simons couplings take specific values, not for generic couplings. It shows that the full $ E_{8(8)} $ symmetry algebra arises only when the Chern–Simons coupling $ \kappa^2 = 1 $, with the scalar manifold metric acquiring enhanced $ SO(16) $ isotropy; for other values, the symmetry reduces to a parabolic subgroup, and the structure constants of the smaller algebra remain isomorphic to the full algebra due to rigidity theorems.

ABSTRACT

We study the role of Chern--Simons couplings for the appearance of enhanced symmetries of Cremmer--Julia type in various theories. It is shown explicitly that for generic values of the Chern--Simons coupling there is only a parabolic Lie subgroup of symmetries after reduction to three space-time dimensions but that this parabolic Lie group gets enhanced to the full and larger Cremmer--Julia Lie group of hidden symmetries if the coupling takes a specific value. This is heralded by an enhanced isotropy group of the metric on the scalar manifold. Examples of this phenomenon are discussed as well as the relation to supersymmetry. Our results are also connected with rigidity theorems of Borel-like algebras.

Motivation & Objective

• To clarify the dependence of hidden symmetries on Chern–Simons coupling coefficients in dimensional reduction.
• To demonstrate that full Cremmer–Julia symmetry $ E_{8(8)} $ only appears at critical coupling values, not generically.
• To connect the emergence of enhanced global symmetries with supersymmetry, showing that supersymmetry fixes the Chern–Simons coupling to the critical value.
• To establish that the smaller symmetry algebra for generic $ \kappa $ is isomorphic to the full hidden symmetry algebra due to rigidity theorems.
• To show that the scalar manifold metric's isotropy group enhances from $ SO(8) $ to $ SO(16) $ only when $ \kappa^2 = 1 $, signaling full $ E_{8(8)} $ symmetry.

Proposed method

• Dimensional reduction of $ D=11 $ supergravity to three spacetime dimensions to analyze the scalar manifold and global symmetries.
• Computation of the scalar metric in terms of invariant one-forms $ \tilde{\omega} $ derived from the reduction, with normalization dependent on the Chern–Simons coupling $ \kappa $.
• Reduction of the metric to canonical form $ \omega $ via redefinition of generators, revealing dependence on $ \kappa $.
• Analysis of the isotropy group of the scalar manifold metric to detect symmetry enhancement at $ \kappa^2 = 1 $.
• Application of rigidity theorems for Borel-like Lie algebras to prove that the smaller algebra for $ \kappa \neq 0 $ is isomorphic to the full $ B(E_{8(8)}) $ algebra.
• Use of dilaton vectors and kinetic term structure to identify candidate hidden symmetry groups prior to full interaction analysis.

Experimental results

Research questions

• RQ1 Does the Chern–Simons coupling value determine whether the full $ E_{8(8)} $ hidden symmetry appears in three-dimensional maximal supergravity?
• RQ2 How does the scalar manifold's metric and its isotropy group change with varying Chern–Simons coupling $ \kappa $?
• RQ3 Can the smaller symmetry algebra present for generic $ \kappa \neq 0 $ be isomorphic to the full $ B(E_{8(8)}) $ algebra despite dependence on $ \kappa $?
• RQ4 What is the role of supersymmetry in fixing the Chern–Simons coupling to the critical value $ \kappa^2 = 1 $?
• RQ5 Why does the symmetry enhance only at $ \kappa^2 = 1 $, and not at other values, despite continuous dependence of structure constants on $ \kappa $?

Key findings

• The full $ E_{8(8)} $ hidden symmetry appears only when the Chern–Simons coupling satisfies $ \kappa^2 = 1 $, corresponding to the value fixed by supersymmetry.
• For $ \kappa^2 = 1 $, the scalar manifold metric acquires an additional $ SO(16) $ isotropy symmetry acting on the 128 scalars in the spinor representation.
• For $ \kappa^2 \neq 1 $, the isotropy group reduces to $ SO(8) $, and the symmetry is limited to the parabolic subgroup $ B(E_{8(8)}) $.
• For $ \kappa = 0 $, the Borel shift symmetry is contracted, and the symmetry algebra is no longer isomorphic to $ B(E_{8(8)}) $.
• The structure constants of the smaller symmetry algebra for $ \kappa \neq 0 $ can be redefined to match those of $ B(E_{8(8)}) $, implying isomorphism due to rigidity theorems.
• The value $ \kappa^2 = 1 $ is the only one for which the full non-linear $ E_{8(8)} $ symmetry, including all negative root generators, acts consistently on the theory.