(IHP-December 2006) 1 / 34 Geometric configurations and E 10 - PowerPoint PPT Presentation

(IHP-December 2006) 1 / 34

Geometric configurations and E 10 subalgebras of cosmological inspiration M. Henneaux, M. Leston, D. Persson, Ph. S. High Energy, Cosmology and Strings Paris, December 12 (IHP-December 2006) 2 / 34

Summary: We re-examine previously found cosmological solutions to eleven-dimensional supergravity in the light of the E 10 -approach to M-theory. We focus on the solutions with non zero electric field determined by geometric configurations ( n m , g 3 ) , n ≤ 10 . We show that these solutions are associated with rank g regular subalgebras of E 10 , the Dynkin diagrams of which are the (line) incidence diagrams of the geometric configurations. Our analysis provides as a byproduct an interesting class of rank-10 Coxeter subgroups of the Weyl group of E 10 . (IHP-December 2006) 3 / 34

Talk based on : J. Demaret, J.-L. Hanquin, M. Henneaux, Ph. S. Cosmological models in Eleven-dimensional Supergravity Nucl. Phus. B 252 , 538 (1985) M. Henneaux, M. Leston, D. Persson, Ph. S. Geometric Configurations, Regular Subalgebras of E 10 and M-Theory Cosmology JHEP 0610 (2006) 021 (hep-th/0606123) M. Henneaux, M. Leston, D. Persson, Ph. S. A special Class of Rank 10 and 11 of Coxeter groups (hep-th/0610278) (IHP-December 2006) 4 / 34

11 - D, Binachi I supergravity solutions Field configurations − N 2 [ t ] dt 2 + g ij [ t ] dx i dx j ds 2 = F αβγδ = F αβγδ [ t ] Field equations • dynamical equations √ g K a � � d − N √ gF aρστ F bρστ + N √ gF λρστ F λρστ δ a b = b dt 2 144 F 0 abc N √ g � � d 1 144 η 0 abcd 1 d 2 d 3 e 1 e 2 e 3 e 4 F 0 d 1 d 2 d 3 F e 1 e 2 e 3 e 4 = dt dF a 1 a 2 a 3 a 4 = 0 dt (IHP-December 2006) 5 / 34

11 - D, Binachi I supergravity solutions Field configurations − N 2 [ t ] dt 2 + g ij [ t ] dx i dx j ds 2 = F αβγδ = F αβγδ [ t ] Field equations • dynamical equations √ g K a � � d − N √ gF aρστ F bρστ + N √ gF λρστ F λρστ δ a b = b dt 2 144 F 0 abc N √ g � � d 1 144 η 0 abcd 1 d 2 d 3 e 1 e 2 e 3 e 4 F 0 d 1 d 2 d 3 F e 1 e 2 e 3 e 4 = dt dF a 1 a 2 a 3 a 4 = 0 dt (IHP-December 2006) 5 / 34

• Constraint equations a − K 2 + 1 + 1 48 F abcd F abcd = 0 K a b K b 12 F ⊥ abc F abc Hamiltonian C. ⊥ 1 6 NF 0 bcd F abcd = 0 Momentum C. ε 0 abc 1 c 2 c 3 c 4 d 1 d 2 d 3 d 4 F c 1 c 2 c 3 c 4 F d 1 d 2 d 3 d 4 = 0 Gauss law where K ab = ( − 1 / 2 N )˙ g ab and F ⊥ abc = (1 /N ) F 0 abc . (IHP-December 2006) 6 / 34

Bianchi I configurations Diagonal field configurations Diagonal metric implies diagonal extrinsic curvature K ab Evolution and constraint equations imply diagonal energy-momentum tensor: F aρστ F bρστ ∝ δ a b • Freund-Rubin ansatz: 10=3+7 11 = − N 2 dt 2 + ds 2 ds 2 3 + ds 2 7 F 0 abc ∝ 1 √ gN ε 0 abc ( a, b, c = 1 , 2 , 3) [ P.G.O. Freund, M.A. Rubin, Phys. Lett. 97B (1980) 233 ] • Different splittings: 10 = n + (10 − n ) , n ≥ 0 11 = − N 2 dt 2 + R 2 [ t ] � a ≤ n ( dx a ) 2 + S 2 [ t ] � ds 2 a ≥ n ( dx a ) 2 Only F 0 abc � = 0 (IHP-December 2006) 7 / 34

Einstein-Maxwell equations imply: 1 F 0 abc = E apq E bpq = f 2 δ a N √ gE abc , b • n=1, 2 No non-trivial three-index tensor • n=3 E abc = f ε abc : solution proportional to the Levi-Civita tensor • n=4 Let A a = ε abcd E bcd : A a A b ∝ δ a b i.e. A a = 0 • n=5 Let B ab = ε abcde E cde , B ac B cb ∝ δ a b i.e. B 2 = µ 2 Id in matrix notations, but B is antisymmetric and the dimension odd: B = 0 (IHP-December 2006) 8 / 34

Einstein-Maxwell equations imply: 1 F 0 abc = E apq E bpq = f 2 δ a N √ gE abc , b • n=1, 2 No non-trivial three-index tensor • n=3 E abc = f ε abc : solution proportional to the Levi-Civita tensor • n=4 Let A a = ε abcd E bcd : A a A b ∝ δ a b i.e. A a = 0 • n=5 Let B ab = ε abcde E cde , B ac B cb ∝ δ a b i.e. B 2 = µ 2 Id in matrix notations, but B is antisymmetric and the dimension odd: B = 0 (IHP-December 2006) 8 / 34

Einstein-Maxwell equations imply: 1 F 0 abc = E apq E bpq = f 2 δ a N √ gE abc , b • n=1, 2 No non-trivial three-index tensor • n=3 E abc = f ε abc : solution proportional to the Levi-Civita tensor • n=4 Let A a = ε abcd E bcd : A a A b ∝ δ a b i.e. A a = 0 • n=5 Let B ab = ε abcde E cde , B ac B cb ∝ δ a b i.e. B 2 = µ 2 Id in matrix notations, but B is antisymmetric and the dimension odd: B = 0 (IHP-December 2006) 8 / 34

Einstein-Maxwell equations imply: 1 F 0 abc = E apq E bpq = f 2 δ a N √ gE abc , b • n=1, 2 No non-trivial three-index tensor • n=3 E abc = f ε abc : solution proportional to the Levi-Civita tensor • n=4 Let A a = ε abcd E bcd : A a A b ∝ δ a b i.e. A a = 0 • n=5 Let B ab = ε abcde E cde , B ac B cb ∝ δ a b i.e. B 2 = µ 2 Id in matrix notations, but B is antisymmetric and the dimension odd: B = 0 (IHP-December 2006) 8 / 34

Einstein-Maxwell equations imply: 1 F 0 abc = E apq E bpq = f 2 δ a N √ gE abc , b • n=1, 2 No non-trivial three-index tensor • n=3 E abc = f ε abc : solution proportional to the Levi-Civita tensor • n=4 Let A a = ε abcd E bcd : A a A b ∝ δ a b i.e. A a = 0 • n=5 Let B ab = ε abcde E cde , B ac B cb ∝ δ a b i.e. B 2 = µ 2 Id in matrix notations, but B is antisymmetric and the dimension odd: B = 0 (IHP-December 2006) 8 / 34

In dimensions greater than five Special solutions are obtained by imposing the following conditions: 1 given a pair of indices ( a, b ) , there is at most one c such that E abc � = 0 2 for each index a there are exactly m pairs ( b, c ) such that E abc � = 0 , 3 all non-vanishing E abc are equal up to sign : E abc = ± h Condition 1 implies E apq E bpq = 0 when a � = b ; conditions 2 and 3 imply E apq E bpq = mh 2 δ a b (IHP-December 2006) 9 / 34

Geometric configurations Incidence rules The first two conditions can be reformulated in terms of geometric configurations ( n m , g 3 ) i.e. set of n points with g distinguished subsets, called lines, such that 0 Each line contains exactly three points and defines an E abc component 1 Two points determine at most one line (condition 1) 2 Each point belongs to m lines (condition 2) [ S. Kantor, “Die configurationen (3 , 3) 10 ”, K. Academie der Wissenschaften, Vienna, Sitzungsbereichte der matematisch naturewissenshaftlichen classe, 84 II, 1291-1314 (1881). D. Hilbert and S. Cohn-Vossen, “Geometry and the Imagination”,(Chelsea, New York, 1952) W. Page and H. L. Dorwart, “Numerical Patterns and Geometrical Configurations”, Mathematics Magazine 57 , No. 2, 82-92 (1984). ] (IHP-December 2006) 10 / 34

Geometric configurations Incidence rules The first two conditions can be reformulated in terms of geometric configurations ( n m , g 3 ) i.e. set of n points with g distinguished subsets, called lines, such that 0 Each line contains exactly three points and defines an E abc component 1 Two points determine at most one line (condition 1) 2 Each point belongs to m lines (condition 2) [ S. Kantor, “Die configurationen (3 , 3) 10 ”, K. Academie der Wissenschaften, Vienna, Sitzungsbereichte der matematisch naturewissenshaftlichen classe, 84 II, 1291-1314 (1881). D. Hilbert and S. Cohn-Vossen, “Geometry and the Imagination”,(Chelsea, New York, 1952) W. Page and H. L. Dorwart, “Numerical Patterns and Geometrical Configurations”, Mathematics Magazine 57 , No. 2, 82-92 (1984). ] (IHP-December 2006) 10 / 34

Geometric configurations Some examples 3 1 2 4 2 7 3 4 1 6 5 5 Figure: (7 3 , 7 3 ) : The 6 Fano plane; the Figure: (6 2 , 4 3 ) : The first multiplication table of configuration with intersecting lines. the octonions. (IHP-December 2006) 11 / 34

Geometric configurations Two other examples 1 2 3 7 (9) 1 9 (5) 3 2 (1) (7) 5 4 6 10 (3) (4) 6 (2) (10) 4 (6) 5 (8) 7 8 9 8 Figure: (9 3 , 9 3 ) 1 : The so-called Figure: (10 3 , 10 3 ) 3 : The Pappus configuration . Desargues configuration, dual to the Petersen graph. (IHP-December 2006) 12 / 34

The “symmetric space” E 10 / K ( E 10 ) Definitions • The Kac-Moody algebra : E 10 10 6 9 8 7 5 4 3 2 1 Figure: The Dynkin diagram of E 10 . Labels i = 1 , . . . , 9 enumerate the nodes corresponding to simple roots, α i , of the A 9 subalgebra and the exceptional node, labeled “ 10 ”, is associated to the root α 10 that defines the level decomposition. [ h i , h j ] = 0 , [ h i , e j ] = A ij e j , [ h i , f j ] = − A ij f j , [ e i , f j ] = δ ij h j ( ad e i ) (1 − A ij ) e j = 0 ( ad f i ) (1 − A ij ) f j = 0 , . [ V. Kac, “Infinite dimensional Lie algebras”, 3rd Ed., Cambridge University Press (1990). ] (IHP-December 2006) 13 / 34