Exceptional geometry for affine and other groups Axel Kleinschmidt (Albert Einstein Institute, Potsdam) Exceptional Field Theory, Strings and Holography Texas A&M, 23 April 2018 Joint work with Guillaume Bossard, Martin Cederwall, Jakob Palmkvist and Henning Samtleben [arXiv:1708.08936, Phys. Rev. D96 (2017) 106022] Exceptional geometry for affine and other groups – p.1
Motivation In theories of gravity and matter Geometry Matter diffeomorphisms gauge transformations δ ξ g µν = 2 ∂ ( µ ξ ν ) δ λ A µ = ∂ µ λ ( x µ space-time coordinates) Exceptional geometry for affine and other groups – p.2
Motivation In theories of gravity and matter Geometry Matter diffeomorphisms gauge transformations δ ξ g µν = 2 ∂ ( µ ξ ν ) δ λ A µ = ∂ µ λ ( x µ space-time coordinates) Two sides of the same coin (waffle)? Exceptional geometry for affine and other groups – p.2
Motivation In theories of gravity and matter Geometry Matter diffeomorphisms gauge transformations δ ξ g µν = 2 ∂ ( µ ξ ν ) δ λ A µ = ∂ µ λ ( x µ space-time coordinates) Two sides of the same coin (waffle)? Idea: Find common structure for both ⇒ Generalised/exceptional geometry Exceptional geometry for affine and other groups – p.2
Example: Generalised geometry [Gualtieri, Hitchin 2004] Metric g µν , diffeomorphisms with vector ξ µ Two-form B µν , gauge parameter co-vector λ µ Structure: TM ⊕ T ∗ M over space-time M with coords. x µ Exceptional geometry for affine and other groups – p.3
Example: Generalised geometry [Gualtieri, Hitchin 2004] Metric g µν , diffeomorphisms with vector ξ µ Two-form B µν , gauge parameter co-vector λ µ Structure: TM ⊕ T ∗ M over space-time M with coords. x µ Generalised Lie derivative (a.k.a. Dorfman derivative) w.r.t. joint parameter Λ M = ( ξ µ , λ µ ) � ∂ M Λ N − ∂ N Λ M � L Λ V M = Λ N ∂ N V M + V N Raise/lower with η MN = ( 0 1 O ( d, d ) structure 1 0 ) ⇒ ∂ µ = 0 Derivatives ∂ M = ( ∂ µ , ˜ ∂ µ ) and set ˜ Algebra closes: [ L Λ 1 , L Λ 2 ] = L [Λ 1 , Λ 2 ] C Exceptional geometry for affine and other groups – p.3
Example: Generalised geometry (II) Considering generalised metric � � g µν − g µρ B ρν H MN = ∈ O ( d, d ) B µρ g ρν g µν − B µρ g ρσ B σν gives standard transformations from L Λ H MN . Exceptional geometry for affine and other groups – p.4
Example: Generalised geometry (II) Considering generalised metric � � g µν − g µρ B ρν H MN = ∈ O ( d, d ) B µρ g ρν g µν − B µρ g ρσ B σν gives standard transformations from L Λ H MN . In closed string (field) theory (and Kaluza–Klein theory) g µν couples to momentum modes B µν couples to winding modes T-duality mixes these. Natural to introduce also new coordinates ˜ x µ for string fields [Siegel; Hull, Zwiebach] Exceptional geometry for affine and other groups – p.4
Example: Double geometry If fields and parameters depend on X M = ( x µ , ˜ x µ ) dimension of base manifold M is doubled ∂ µ . X M fundamental of O ( d, d ) non-trivial derivative ˜ can still use same formula for generalised Lie derivative But: Closure now only with (strong) section constraint η MN ∂ M A ( X ) ∂ N B ( X ) = 0 η MN ∂ M ⊗ ∂ N = 0 ⇔ for any A ( X ) , B ( X ) . Constraint is singlet of O ( d, d ) Exceptional geometry for affine and other groups – p.5
Example: Double geometry If fields and parameters depend on X M = ( x µ , ˜ x µ ) dimension of base manifold M is doubled ∂ µ . X M fundamental of O ( d, d ) non-trivial derivative ˜ can still use same formula for generalised Lie derivative But: Closure now only with (strong) section constraint η MN ∂ M A ( X ) ∂ N B ( X ) = 0 η MN ∂ M ⊗ ∂ N = 0 ⇔ for any A ( X ) , B ( X ) . Constraint is singlet of O ( d, d ) Field theories constructed on doubled space with constraint double field theory (DFT) [Aldazabal, Andriot, Berman, Betz, Blumenhagen, Deser, Geissb¨ uhler, Hassler, Hohm, Hull, Larfors, Kwak, L¨ ust, Marques, Musaev, Nunez, Park, Patalong, Penas, Perry, Plauschinn, Rudolph, Zwiebach,...] Exceptional geometry for affine and other groups – p.5
Exceptional geometry String theory also has U-duality ⊃ T-duality. Type II on T d t E d +1 E d +1 ⊃ O ( d, d ) t t t t Exceptional geometry and exceptional field theory (ExFT): ⇒ Redo DFT analysis for U-duality groups Exceptional geometry for affine and other groups – p.6
Exceptional geometry String theory also has U-duality ⊃ T-duality. Type II on T d t E d +1 E d +1 ⊃ O ( d, d ) t t t t Exceptional geometry and exceptional field theory (ExFT): ⇒ Redo DFT analysis for U-duality groups coord rep. section rep. n E n 5 SO (5 , 5) 16 10 momentum and 6 E 6 27 27 brane winding 7 1 ⊕ 133 E 7 56 8 1 ⊕ 248 ⊕ 3875 E 8 248 Exceptional geometry for affine and other groups – p.6
Exceptional geometry String theory also has U-duality ⊃ T-duality. Type II on T d t E d +1 E d +1 ⊃ O ( d, d ) t t t t Exceptional geometry and exceptional field theory (ExFT): ⇒ Redo DFT analysis for U-duality groups coord rep. section rep. n E n 5 SO (5 , 5) 16 10 momentum and 6 E 6 27 27 brane winding 7 1 ⊕ 133 E 7 56 8 1 ⊕ 248 ⊕ 3875 E 8 248 � Closure of generalised Lie derivative and ExFT for n ≤ 8 [Berman, Blair, Cederwall, Ciceri, Coimbra, Godazgar 2 , Guarino, Hohm, Hull, Inverso, AK, Malek, Musaev, Nicolai, Palmkvist, Park, Perry, Rosabal, Samtleben, Strickland-Constable, Suh, Thompson, Waldram,...] Exceptional geometry for affine and other groups – p.6
Why is this interesting? E n covariant formulations for different duality frames ‘geometric’ origin of gauged supergravities via generalised Scherk–Schwarz reduction [Baguet, du Bosque, Hassler, Hohm, Inverso, L¨ ust, Malek, Samtleben] Uplift formula for lower-dimensional solutions [Godazgar 2 , Guarino, Kr¨ uger, Nicolai, Pilch, Varela] Non-geometric solutions and orbits of exact solutions [Bakhmatov, Berman, Hassler, Jensen, AK, L¨ ust, Rudolph, Musaev] Construction of M-theory effective action [Bossard, AK] Exceptional geometry for affine and other groups – p.7
Why is this interesting? E n covariant formulations for different duality frames ‘geometric’ origin of gauged supergravities via generalised Scherk–Schwarz reduction [Baguet, du Bosque, Hassler, Hohm, Inverso, L¨ ust, Malek, Samtleben] Uplift formula for lower-dimensional solutions [Godazgar 2 , Guarino, Kr¨ uger, Nicolai, Pilch, Varela] Non-geometric solutions and orbits of exact solutions [Bakhmatov, Berman, Hassler, Jensen, AK, L¨ ust, Rudolph, Musaev] Construction of M-theory effective action [Bossard, AK] More speculatively on the horizon Fundamental symmetries of M-theory, e.g. E 10 [Julia; Damour, Henneaux, Nicolai] E 11 [West] New variables for quantisation? Exceptional geometry for affine and other groups – p.7
Aim of this talk Exceptional geometry for affine and other groups – p.8
Aim of this talk Extend construction of exceptional geometry to E 9 discuss infinite-dimensional affine group E 9 identify correct coordinate representation identify appropriate section condition check closure of algebra elementary check of generalised Scherk–Schwarz ansatz vs. gauged supergravity in D = 2 Exceptional geometry for affine and other groups – p.8
Aim of this talk Extend construction of exceptional geometry to E 9 discuss infinite-dimensional affine group E 9 identify correct coordinate representation identify appropriate section condition check closure of algebra elementary check of generalised Scherk–Schwarz ansatz vs. gauged supergravity in D = 2 Along the way: Find different view on existing constructions! Exceptional geometry for affine and other groups – p.8
What is E 9 ? 8 ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ 0 1 2 3 4 5 6 7 E 9 = extension of E 8 loop group. Current algebra formulation: e 8 -valued Laurent series in z (horizontal) e 8 generators: T A ( A = 1 , . . . , 248 ) � T A , T B � = f ABC T C , � T A | T B � = η AB m = T A ⊗ z m ( m ∈ Z ), T A Current modes T A = T A 0 � � � T A m , T B = f ABC T C m + n + η AB m δ m + n, 0 K n � � K , T A K is the central extension: = 0 m Exceptional geometry for affine and other groups – p.9
What is E 9 ? (II) So far: T A m , K . Also derivation d � � d, T A = − mT A [ d , K ] = 0 m , m � � T A ⇒ Kac–Moody algebra e 9 = span m , K , d Exceptional geometry for affine and other groups – p.10
What is E 9 ? (II) So far: T A m , K . Also derivation d � � d, T A = − mT A [ d , K ] = 0 m , m � � T A ⇒ Kac–Moody algebra e 9 = span m , K , d Irreducible highest representations labelled by e 8 irrep r ( λ ) , level k and weight h . Use Fock space notation: ‘Ground states’ | v � ∈ r ( λ ) T A for n > 0 , n | v � = 0 K | v � = k | v � , d | v � = h | v � The T A 0 act as e 8 rotations in r ( λ ) . Excited states are combinations of � T A i (irrep=remove sing. vectors) | V � = − n i | v � Exceptional geometry for affine and other groups – p.10
E 9 representations From e 8 label λ and level k ⇒ e 9 label Λ Denote e 9 irrep by R (Λ) h . Often h = 0 or irrelevant here. Exceptional geometry for affine and other groups – p.11
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