The Kac Moody Approach to Supergravity
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1 Miami, December p. 1/3 The Kac Moody Approach to Supergravity Eric Bergshoeff Centre for Theoretical Physics, University of Groningen based on arxiv:hep-th/ ,arxiv:hep-th/ with I. De Baetselier, J. Gomis, T. Nutma and D. Roest
2 Miami, December p. 2/3 Supergravity Supergravity (SUGRA) is the gauge theory of the supersymmetry algebra { Q α,q β } = ( γ µ C 1) αβ P µ
3 Miami, December p. 3/3 Representations physical states D=11 supergravity : states
4 Miami, December p. 3/3 Representations physical states D=11 supergravity : states deformation potentials d F (10) (A (9) ) = 0 F (10) (A (9) ) m phenomenology, cosmology
5 Miami, December p. 3/3 Representations physical states D=11 supergravity : states deformation potentials d F (10) (A (9) ) = 0 F (10) (A (9) ) m phenomenology, cosmology top forms string theories with broken supersymmetries
6 Miami, December p. 4/3 D=11 Supergravity L = g { R 1 2( F(4) ) 2 + } + C(3) C (3) C (3) g µν = η µν + h µν, δh µν = µ ξ ν + ν ξ µ : : 44 of SO(9) F (4) = C (3), δc (3) = λ (2) : : 84 of SO(9) p form potentials : brane interpretation
7 Miami, December p. 5/3 Six form Potential L 11 C (3) C (3) C (3) : C (3) or C (6) plus C (3) C (6) C (3) C (3) = }{{} F (6) C (3) }{{} F (4) δc (3) = λ (2) [3,3] = 0 δc (6) = λ (5) + C (3) λ (2) Λ λ constant
8 Miami, December p. 6/3 Six form Potential L 11 C (3) C (3) C (3) : C (3) or C (6) plus C (3) C (6) C (3) C (3) = }{{} F (6) C (3) }{{} F (4) δc (3) = λ (2) [3,3] = 0 δc (6) = Λ (6) + C (3) Λ (3) [3,3] = 6! Cremmer, Julia, Lu, Pope
9 Miami, December p. 7/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West
10 Miami, December p. 7/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West STEP 1 : Reduce to D=3 dimensions scalar coset G/K(G)
11 Miami, December p. 7/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West STEP 1 : Reduce to D=3 dimensions scalar coset G/K(G) STEP 2 : Take the very extension G +++
12 Miami, December p. 7/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West STEP 1 : Reduce to D=3 dimensions scalar coset G/K(G) STEP 2 : Take the very extension G Cartan matrix A rs
13 Miami, December p. 8/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West STEP 1 : Reduce to D=3 dimensions scalar coset G/K(G) STEP 2 : Take the very extension G E 8 det A > 0
14 Miami, December p. 9/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West STEP 1 : Reduce to D=3 dimensions scalar coset G/K(G) STEP 2 : Take the very extension G E 9 det A = 0
15 Miami, December p. 10/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West STEP 1 : Reduce to D=3 dimensions scalar coset G/K(G) STEP 2 : Take the very extension G E 10 det < 0
16 Miami, December p. 11/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West STEP 1 : Reduce to D=3 dimensions scalar coset G/K(G) STEP 2 : Take the very extension G E 11 det A < 0
17 Miami, December p. 12/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West STEP 1 : Reduce to D=3 dimensions scalar coset G/K(G) STEP 2 : Take the very extension G E 11 det A < 0 STEP 3 : Oxidize back to 3 D D max
18 Miami, December p. 13/3 Kac Moody Approach I Houart, Englert; Damour, Henneaux, Kleinschmidt, Nicolai; Schnakenburg, West STEP 1 : Reduce to D=3 dimensions scalar coset G/K(G) STEP 2 : Take the very extension G +++ /K(G +++ ) duality disabled node gravity line STEP 3 : Oxidize back to 3 D D max
19 Miami, December p. 14/3 Different Group Disintegrations D= E 8
20 Miami, December p. 15/3 Different Group Disintegrations D= E 7
21 Miami, December p. 16/3 Different Group Disintegrations D= E 6
22 Miami, December p. 17/3 Different Group Disintegrations D= SO(5, 5)
23 Miami, December p. 18/3 Different Group Disintegrations D= SL(5, R)
24 Miami, December p. 19/3 Different Group Disintegrations D= SL(2, R) SL(3, R)
25 Miami, December p. 20/3 Different Group Disintegrations D= SL(2, R) R +
26 Miami, December p. 21/3 Different Group Disintegrations D=10, IIA R +
27 Miami, December p. 22/3 Different Group Disintegrations D=10, IIB SL(2, R)
28 Miami, December p. 23/3 Different Group Disintegrations D=
29 Miami, December p. 24/3 Kac Moody Approach II STEP 4 : Calculate the spectrum
30 Miami, December p. 24/3 Kac Moody Approach II STEP 4 : Calculate the spectrum 8 =
31 Miami, December p. 24/3 Kac Moody Approach II STEP 4 : Calculate the spectrum 8 = Use computer program SimpLie
32 Miami, December p. 24/3 Kac Moody Approach II STEP 4 : Calculate the spectrum 8 = Use computer program SimpLie One finds at low levels physical states plus dual p form potentials (but no duality) deformation and top form potentials plus dual gravity (D > 3) and more... Riccioni, Steele, West
33 The p form Algebra Miami, December p. 25/3
34 Miami, December p. 25/3 The p form Algebra Kac Moody algebra Bosonic gauge algebra (1 p D) form truncation Constant gauge parameters Λ = λ λ (2n 1) = x Λ (2n) p form algebra
35 Miami, December p. 25/3 The p form Algebra Kac Moody algebra Bosonic gauge algebra (1 p D) form truncation Constant gauge parameters Λ = λ λ (2n 1) = x Λ (2n) p form algebra [A µ1 µ p,b ν1 ν q ] = C µ1 µ p ν 1 ν q [p,q] = r r = p + q
36 Miami, December p. 25/3 The p form Algebra Kac Moody algebra Bosonic gauge algebra (1 p D) form truncation Constant gauge parameters Λ = λ λ (2n 1) = x Λ (2n) p form algebra [A µ1 µ p,b ν1 ν q ] = C µ1 µ p ν 1 ν q [p,q] = r r = p + q [q,...,[q[q,q]]...] }{{} l times = p fundamental generator
37 Miami, December p. 25/3 The p form Algebra Kac Moody algebra Bosonic gauge algebra (1 p D) form truncation Constant gauge parameters Λ = λ λ (2n 1) = x Λ (2n) p form algebra [A µ1 µ p,b ν1 ν q ] = C µ1 µ p ν 1 ν q [p,q] = r r = p + q [q,...,[q[q,q]]...] }{{} l times = p fundamental generator [3,3] = 6 6 occurs at level l = 2
38 Example : IIB SUGRA Miami, December p. 26/3
39 Miami, December p. 26/3 Example : IIB SUGRA
40 Miami, December p. 26/3 Example : IIB SUGRA level l [2 α,2 β ] = 4ǫ αβ 2 [2 α,4] = 6 α 3 [2 α,6 β ] = 8 αβ 4 [2 α,8 βγ ] = 10 αβγ + ǫ α(β 10 γ) 5
41 Miami, December p. 27/3 Example : IIB SUGRA δa α (2) = Λ α (2) δa (4) = Λ (4) + ǫ γδ Λ γ (2) Aδ (2) δa α (6) = Λ α (6) + Λ (4) A α (2) 2Λ α (2) A (4) Λ (2n) λ (2n 1) δa αβ (8) = Λαβ (8) + Λ(α (6) Aβ) (2) 3Λ(α (2) Aβ) (6) δa αβγ (10) = Λαβγ (10) + Λ(αβ (8) Aγ) (2) 4Λ(α (2) Aβγ) (8) δa α (10) = Λ α (10) ǫ βγλ αβ (8) Aγ (2) ǫ βγλ β (2) Aγα (8) + Λ (4) A α (6) 2 3 Λα (6) A (4)
42 Deformation Potentials Miami, December p. 28/3
43 Miami, December p. 28/3 Deformation Potentials type p deformations : [p,(d p 1)] = (D 1)
44 Miami, December p. 28/3 Deformation Potentials type p deformations : [p,(d p 1)] = (D 1) massive fundamental p form gauge field
45 Miami, December p. 28/3 Deformation Potentials type p deformations : [p,(d p 1)] = (D 1) massive fundamental p form gauge field D Fundamental p forms 11 3 IIA 2,1 IIB
46 Miami, December p. 28/3 Deformation Potentials type p deformations : [p,(d p 1)] = (D 1) massive fundamental p form gauge field D Fundamental p forms 11 3 IIA 2,1 IIB IIA : [7,2] = 9 [8,1] = 0
47 Miami, December p. 28/3 Deformation Potentials type p deformations : [p,(d p 1)] = (D 1) massive fundamental p form gauge field D Fundamental p forms 11 3 IIA 2,1 IIB IIA : [7,2] = 9 [8,1] = 0 IIB and D=11 : no massive deformation
48 Top form Potentials Miami, December p. 29/3
49 Miami, December p. 29/3 Top form Potentials 3 m αβ : SO(2) SO(1, 1) SL(2, R) R + - IIB (and miia) origin 2 m α : R + - IIA origin
50 Miami, December p. 29/3 Top form Potentials 3 m αβ : SO(2) SO(1, 1) SL(2, R) R + - IIB (and miia) origin 2 m α : R + - IIA origin quadratic constraint: m αβ m γ = = 4 + 2
51 Miami, December p. 29/3 Top form Potentials 3 m αβ : SO(2) SO(1, 1) SL(2, R) R + - IIB (and miia) origin 2 m α : R + - IIA origin quadratic constraint: m αβ m γ = = Lagrange multipliers or independent top forms
52 Miami, December p. 30/3 Maximal Supergravity D IIA IIB de-forms top-forms Riccioni, West; De Baetselier, Nutma + E.B.
53 Miami, December p. 31/3 Relation to Embedding Tensor Nicolai, Samtleben de Wit, Samtleben, Trigiante
54 Miami, December p. 31/3 Relation to Embedding Tensor Nicolai, Samtleben de Wit, Samtleben, Trigiante D=3 ) L sugra = 1 4 e R ep µa P µa V g ǫµνρ Θ MN (A M µ ν A N ρ gθ KSf NS LA M µ A K ν A L ρ
55 Miami, December p. 31/3 Relation to Embedding Tensor Nicolai, Samtleben de Wit, Samtleben, Trigiante D=3 ) L sugra = 1 4 e R ep µa P µa V g ǫµνρ Θ MN (A M µ ν A N ρ gθ KSf NS LA M µ A K ν A L ρ constraints : Θ is constant and Θ KP Θ L(M f KL N) = 0
56 Miami, December p. 32/3 Relation to Embedding Tensor Equivalent Lagrangian Hohm, Nutma + E.B. L sugra = 1 4 e R ep µa P µa V g ǫµνρ Θ MN ( µ C MN νρ + A M µ ν A N ρ gθ KSf NS LA M µ A K ν A L ρ + Θ O(P f MO Q) C N(P Q) )
57 Miami, December p. 32/3 Relation to Embedding Tensor Equivalent Lagrangian Hohm, Nutma + E.B. L sugra = 1 4 e R ep µa P µa V g ǫµνρ Θ MN ( µ C MN νρ + A M µ ν A N ρ gθ KSf NS LA M µ A K ν A L ρ + Θ O(P f MO Q) C N(P Q) ) E.O.M. of Θ duality between deformation potential C MN µνρ and Θ MN
58 Massive IIA SUGRA Miami, December p. 33/3
59 Miami, December p. 33/3 Massive IIA SUGRA δa (1) = λ (0) δa (2) = λ (1) [2,1] = 3 δa (3) = λ (2) + 3 λ (0) A (2)
60 Miami, December p. 33/3 Massive IIA SUGRA δa (1) = λ (0) mλ (1) δa (2) = λ (1) δa (3) = λ (2) + 3 ( λ (0) mλ (1) ) A(2)
61 Miami, December p. 33/3 Massive IIA SUGRA δa (1) = λ (0) mλ (1) δa (2) = λ (1) δa (3) = λ (2) + 3 ( λ (0) mλ (1) ) A(2) [2, 1] = m 1 massive 2 form West; Lavrinenko, Lu, Pope, Stelle
62 Miami, December p. 33/3 Massive IIA SUGRA δa (1) = λ (0) mλ (1) δa (2) = λ (1) δa (3) = λ (2) + 3 ( λ (0) mλ (1) ) A(2) [2, 1] = m 1 massive 2 form West; Lavrinenko, Lu, Pope, Stelle [2,1] = 3 [2, 1] = m1 [2,2] = m(3,1) [(3,1), 1] = 3
63 Miami, December p. 33/3 Massive IIA SUGRA δa (1) = λ (0) mλ (1) δa (2) = λ (1) δa (3) = λ (2) + 3 ( λ (0) mλ (1) ) A(2) [2, 1] = m 1 massive 2 form West; Lavrinenko, Lu, Pope, Stelle [2,1] = 3 [2, 1] = m1 [2,2] = m(3,1) [(3,1), 1] = 3 compare to E 10 approach of Kleinschmidt,Nicolai
64 Miami, December p. 33/3 Massive IIA SUGRA δa (1) = λ (0) mλ (1) δa (2) = λ (1) δa (3) = λ (2) + 3 ( λ (0) mλ (1) ) A(2) [2, 1] = m 1 massive 2 form West; Lavrinenko, Lu, Pope, Stelle [2,1] = 3 [2, 1] = m1 [2,2] = m(3,1) [(3,1), 1] = 3 compare to E 10 approach of Kleinschmidt,Nicolai see also Riccioni,West, arxiv : hep th/ [E 11, L] = L
65 Miami, December p. 34/3 Conclusions E 11 gives non-trivial information about supergravity theories in D < 11 dimensions!
66 Miami, December p. 34/3 Conclusions E 11 gives non-trivial information about supergravity theories in D < 11 dimensions! Relation with Embedding Tensor Approach?
67 Miami, December p. 34/3 Conclusions E 11 gives non-trivial information about supergravity theories in D < 11 dimensions! Relation with Embedding Tensor Approach? Dual Gravity with Matter?
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