Gauged supergravity and E 10
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1 Gauged supergravity and E 10 Jakob Palmkvist Albert-Einstein-Institut in collaboration with Eric Bergshoeff, Olaf Hohm, Axel Kleinschmidt, Hermann Nicolai and Teake Nutma arxiv: JHEP01(2009)020
2 Three-dimensional maximal supergravity is globally invariant under E 8 (Julia 1983)
3 Three-dimensional maximal supergravity is globally invariant under E 8 and SL(2, R). (Julia 1983)
4 Three-dimensional maximal supergravity is globally invariant under E 8 and SL(2, R). Can we unify these symmetries into E 10? (Julia 1983) (Damour, Henneaux, Nicolai 2002)
5 Three-dimensional maximal supergravity is globally invariant under E 8 and SL(2, R). Can we unify these symmetries into E 10? What if we promote a subgroup of E 8 to a local symmetry? (Julia 1983) (Damour, Henneaux, Nicolai 2002)
6 Outline
7 Outline Gauged maximal supergravity in three dimensions
8 Outline Gauged maximal supergravity in three dimensions The E 10 /K(E 10 ) coset model
9 Outline Gauged maximal supergravity in three dimensions The E 10 /K(E 10 ) coset model Comparison between the two theories
10 Consider the bosonic sector of maximal (N = 16) supergravity in three dimensions:
11 Consider the bosonic sector of maximal (N = 16) supergravity in three dimensions: Pure gravity does not propagate
12 Consider the bosonic sector of maximal (N = 16) supergravity in three dimensions: Pure gravity does not propagate Vectors are dual to scalars
13 Consider the bosonic sector of maximal (N = 16) supergravity in three dimensions: Pure gravity does not propagate Vectors are dual to scalars All propagating degrees of freedom of supergravity are scalars.
14 Consider the bosonic sector of maximal (N = 16) supergravity in three dimensions: L = 1 4 er ηαβ η AB P α A P β B α, β = 0, 1, 2 A, B = 1, 2,..., 248
15 The Maurer-Cartan form of the gravity sector decomposes into its symmetric and antisymmetric parts: (e 1 µ e) αβ = P µ(αβ) + Q µ[αβ]
16 The Maurer-Cartan form of the gravity sector decomposes into its symmetric and antisymmetric parts: (e 1 µ e) αβ = P µ(αβ) + Q µ[αβ] The Maurer-Cartan form of the scalar sector decomposes into a spinor and the adjoint of SO(16): (E 1 µ E) A P µ A, Q µ IJ
17 The Maurer-Cartan form of the scalar sector decomposes into a spinor and the adjoint of SO(16): (E 1 µ E) A P µ A, Q µ IJ
18 The Maurer-Cartan form of the scalar sector decomposes into a spinor and the adjoint of SO(16): (E 1 µ E) A P µ A, Q µ IJ
19 The Maurer-Cartan form of the scalar sector decomposes into a spinor and the adjoint of SO(16): (E 1 µ E) A P µ A, Q µ IJ A, B,... = 1, 2,..., 248 A, B,... = 1, 2,..., 128 I, J,... = 1, 2,..., 16
20 The scalar sector is described by an element E of the coset E 8 /SO(16), and is invariant under the transformations E(x) ge(x)h(x) g E 8 h(x) SO(16)
21 Gauging the theory: Promote a subgroup G E 8 to a local symmetry E(x) g(x)e(x)h(x) g(x) G E 8 h(x) SO(16)
22 Gauging the theory: Promote a subgroup G E 8 to a local symmetry E(x) g(x)e(x)h(x) g(x) G E 8 h(x) SO(16) This is done by dualizing the scalars to vectors A µ A.
23 Gauging the theory: Replace partial derivatives with gauge covariant ones P µ A = (E 1 µ E) A (E 1 D µ E) A
24 Gauging the theory: Replace partial derivatives with gauge covariant ones P µ A = (E 1 µ E) A (E 1 D µ E) A Add a potential term L V
25 Gauging the theory: Replace partial derivatives with gauge covariant ones P µ A = (E 1 µ E) A (E 1 D µ E) A Add a potential term L V Add a Chern-Simons term L CS
26 We can keep an E 8 covariant notation by using the embedding tensor Θ MN, the projection of the Killing form η MN onto the gauge group G E 8.
27 We can keep an E 8 covariant notation by using the embedding tensor Θ MN, the projection of the Killing form η MN onto the gauge group G E 8. Thus it has two symmetric E 8 indices.
28 We can keep an E 8 covariant notation by using the embedding tensor Θ MN, the projection of the Killing form η MN onto the gauge group G E 8. Thus it has two symmetric E 8 indices. The symmetric product of two adjoint E 8 representations:
29 We can keep an E 8 covariant notation by using the embedding tensor Θ MN, the projection of the Killing form η MN onto the gauge group G E 8. Thus it has two symmetric E 8 indices. The symmetric product of two adjoint E 8 representations: ( ) + =
30 We can keep an E 8 covariant notation by using the embedding tensor Θ MN, the projection of the Killing form η MN onto the gauge group G E 8. Thus it has two symmetric E 8 indices. The condition for supersymmetry: Θ MN (Nicolai, Samtleben 2001)
31 Θ MN
32 We divide the embedding tensor into its irreducible parts: Θ MN Θ MN = η MN θ + Θ MN
33 We divide the embedding tensor into its irreducible parts: θ Θ MN
34 We divide the embedding tensor into its irreducible parts: θ 1 Θ MN 3875
35 We divide the embedding tensor into its irreducible parts: θ 1 Θ MN 3875
36 We divide the embedding tensor into its irreducible parts: θ 1 Θ MN 3875 The embedding tensor in flat indices: T AB (x) = E A M (x)e B N (x) Θ MN
37 L V = eg2 112 (3 T AB TAB + T A IJ TA IJ T IJ KL TIJ KL ) + 2eg 2 θ 2
38 L V = eg2 112 (3 T AB TAB + T A IJ TA IJ T IJ KL TIJ KL ) + 2eg 2 θ 2 L CS = g 4 εµνρ Θ MN A µ M ν A ρ N g2 12 εµνρ Θ MN Θ PQ f MP RA µ N A ν Q A ρ R
39 L V = eg2 112 (3 T AB TAB + T A IJ TA IJ T IJ KL TIJ KL ) + 2eg 2 θ 2 L CS = g 4 εµνρ Θ MN A µ M ν A ρ N g2 12 εµνρ Θ MN Θ PQ f MP RA µ N A ν Q A ρ R E 1 D µ E = E 1 µ E + ga µ M Θ MN (E 1 t N E)
40 To compare with the E 10 model, we must choose...
41 To compare with the E 10 model, we must choose... An ADM-like split of the dreibein: N 0 0 e µ α = 0 0 e m a
42 To compare with the E 10 model, we must choose... An ADM-like split of the dreibein: N 0 0 e µ α = 0 0 e m a A temporal gauge for the vector fields: A t M = 0
43 The Lie algebra of SL(2, R): [h, e] = 2e [h, f] = 2f [e, f] = h
44 The Lie algebra of SL(2, R): [h, e] = 2e [h, f] = 2f [e, f] = h Matrix realization: e = ( ) f = ( ) h = ( )
45 A Kac-Moody algebra g of rank r is generated by r copies of SL(2, R), modulo the Chevalley-Serre relations:
46 A Kac-Moody algebra g of rank r is generated by r copies of SL(2, R), modulo the Chevalley-Serre relations:
47 A Kac-Moody algebra g of rank r is generated by r copies of SL(2, R), modulo the Chevalley-Serre relations: i j i j
48 A Kac-Moody algebra g of rank r is generated by r copies of SL(2, R), modulo the Chevalley-Serre relations: i j i j [h i, e j ] = e j [h i, f j ] = f j [e i, e j ] 0 [f i, f j ] 0 [h i, e j ] = 0 [h i, f j ] = 0 [e i, e j ] = 0 [f i, f j ] = 0 [h i, h j ] = [e i, f j ] = 0
49 A Kac-Moody algebra g of rank r is generated by r copies of SL(2, R), modulo the Chevalley-Serre relations: i j i j [h i, e j ] = e j [h i, f j ] = f j [[e i, e j ], e j ] = 0 [[f i, f j ], f j ] = 0 [h i, e j ] = 0 [h i, f j ] = 0 [e i, e j ] = 0 [f i, f j ] = 0 [h i, h j ] = [e i, f j ] = 0
50 The Kac-Moody algebra g is spanned by all multiple (k 1) commutators [ [[e i1, e i2 ], e i3 ],..., e ik ] [ [[f i1, f i2 ], f i3 ],..., f ik ] together with the r Cartan elements h i.
51 For any i = 1, 2,..., r, we can write g = + g 1 + g 0 + g 1 + where each subspace g l is spanned by all multiple commutators at level l = number of e i number of f i.
52 For any i = 1, 2,..., r, we can write g = + g 1 + g 0 + g 1 + where each subspace g l is spanned by all multiple commutators at level l = number of e i number of f i. This gives a grading: [g m, g n ] g m+n
53 For any i = 1, 2,..., r, we can write g = + g 1 + g 0 + g 1 + where each subspace g l is spanned by all multiple commutators at level l = number of e i number of f i. This gives a grading: [g m, g n ] g m+n The adjoint representation of g decomposes into representations of g 0.
54 The maximal compact subalgebra K(g) of a Kac-Moody algebra g is invariant under the Chevalley involution
55 The maximal compact subalgebra K(g) of a Kac-Moody algebra g is invariant under the Chevalley involution ω(e i ) = f i ω(f i ) = e i ω(h i ) = h i
56 The maximal compact subalgebra K(g) of a Kac-Moody algebra g is invariant under the Chevalley involution ω(e i ) = f i ω(f i ) = e i ω(h i ) = h i It is generated by all elements e i f i modulo the Chevalley-Serre relations.
57 Consider the E 10 /K(E 10 ) coset model L = n(t) 1 P(t) P(t) where Q K(E 10 ) and P Q = 0.
58 Consider the E 10 /K(E 10 ) coset model L = n(t) 1 P(t) P(t) where Q K(E 10 ) and P Q = 0. The local K(E 10 ) invariance makes it possible to choose the Borel gauge:
59 Consider the E 10 /K(E 10 ) coset model L = n(t) 1 P(t) P(t) where Q K(E 10 ) and P Q = 0. The local K(E 10 ) invariance makes it possible to choose the Borel gauge: P + Q g 0 + g 1 + g 2 +
60 Level decomposition of E 10 under SL(2, R) E 8 up to level l = 2:
61 Level decomposition of E 10 under SL(2, R) E 8 up to level l = 2: Level SL(2, R) E 8 Components l representation of P + Q 0 (1 3, 1) P ab, Q ab (1, 248) P A, Q IJ 1 (2, 248) P a A 2 (1, 1) P (1, 3875) P AB (3, 248) P ab A
62 Level decomposition of E 10 under SL(2, R) E 8 up to level l = 2: Level SL(2, R) E 8 Components l representation of P + Q 0 (1 3, 1) P ab, Q ab (1, 248) P A, Q IJ 1 (2, 248) P a A 2 (1, 1) P (1, 3875) P AB
63 P ab P A P a A Q ab Q IJ P P AB
64 P ab P A Q ab Q IJ P a A P a IJ P P AB
65 P ab P A Q ab Q IJ P a A P a IJ P P AB
66 The dictionary of the E 10 /supergravity correspondence: P ab (t) P ab (t, x 0 ) Q ab (t) Q ab (t, x 0 ) P A (t) P t A (t, x 0 ) Q IJ (t) Q t IJ (t, x 0 ) P a A (t) Nε ab P b A (t, x 0 ) P a IJ (t) Nε ab Q b IJ (t, x 0 ) P (t) Ngθ(t, x 0 ) P AB (t) 1 28 Ng T AB (t, x 0 )
67 With this identification, the equations of motion coincide, up to...
68 With this identification, the equations of motion coincide, up to... The gauge choices on both sides
69 With this identification, the equations of motion coincide, up to... The gauge choices on both sides Higher order spatial derivatives
70 With this identification, the equations of motion coincide, up to... The gauge choices on both sides Higher order spatial derivatives Higher levels in the decomposition
71 With this identification, the equations of motion coincide, up to... The gauge choices on both sides Higher order spatial derivatives Higher levels in the decomposition Some other mismatches...
72 Open questions:
73 Open questions: No interpretation of P ab A
74 Open questions: No interpretation of P ab A Second order spatial gradient? Trombone gauging?
75 Open questions: No interpretation of P ab A Second order spatial gradient? Trombone gauging? What about the higher levels?
76 Open questions: No interpretation of P ab A Second order spatial gradient? Trombone gauging? What about the higher levels? How the reconcile the Killing form with the indefinite potential?
77 Open questions: No interpretation of P ab A Second order spatial gradient? Trombone gauging? What about the higher levels? How the reconcile the Killing form with the indefinite potential? To be continued...
78 Thank you!
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