Eleven dimensional supergravity and the 
                           
                              
                                 
                                    
                                       E
                                    
                                    
                                       10
                                    
                                 
                                 /
                                 K
                                 
                                    
                                       
                                          
                                             E
                                          
                                          
                                             10
                                          
                                       
                                    
                                 
                              
                              
                           
                        
                        σ
                        -model at low 
                        
                           
                              
                                 
                                    
                                       A
                                    
                                    
                                       9
                                    
                                 
                              
                              
                           
                         levels

doi:10.1201/9781482269185-17

Chapter

Eleven dimensional supergravity and the

E

10

/
K

E

10

σ
-model at low

A

9

levels

Chapter

Eleven dimensional supergravity and the E 10 / K E 10 σ -model at low A 9 levels

ABSTRACT

Recently, the concept of a nonlinear σ-model over a coset space G/H was generalized to the case where the group G is an infinite-dimensional KacMoody group, and H its (formal) 'maximal compact subgroup'. Here, we study in detail the one-dimensional (geodesic) σ-model with https://www.w3.org/1998/Math/MathML"> G = E 10 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1076.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> H = K E 10 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1077.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We re-examine the construction of this σ-model and its relation to the bosonic sector of eleven-dimensional supergravity, up to height 30 , by using a new formulation of the equations of motion. Specifically, we make systematic use of https://www.w3.org/1998/Math/MathML"> K E 10 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1078.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -orthonormal local frames, in the sense that we decompose the 'velocity' on https://www.w3.org/1998/Math/MathML"> E 10 / K E 10 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1079.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in terms of objects which are representations of the compact subgroup https://www.w3.org/1998/Math/MathML"> K E 10 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1080.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . This new perspective may help in extending the correspondence between the https://www.w3.org/1998/Math/MathML"> E 10 / K E 10 σ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1081.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -model and supergravity beyond the level currently checked.