Generalising Calabi–Yau Geometries Daniel Waldram Stringy Geometry MITP, 23 September 2015 Imperial College, London with Anthony Ashmore, to appear 1
Introduction
Supersymmetric background with no flux ∇ m ǫ = 0 = ⇒ special holonomy Classic case: Type II on Calabi–Yau Geometry encoded in pair of integrable objects d ω = 0 symplectic structure d Ω = 0 complex structure 3
Supersymmetric background with flux (eg type II)? 16 e φ � � 8 H mnp γ np � ε ± + 1 F ( i ) γ m ε ∓ = 0 ∇ m ∓ 1 / i γ m � � 24 H mnp γ np − ∂ m φ ε ± = 0 ∇ m ∓ 1 What is the geometry? • special holonomy? analogues of ω and Ω? • integrability? • deformations? moduli spaces? . . . (n.b. no-go means non-compact/bdry for Minkowski) 4
G structures Killing spinors ε ± invariant under i G = Stab( { ε ± i } ) ⊂ SO(6) ⊂ GL(6) define G -structure and flux gives lack of integrability, eg � d ω ≃ flux Sp(6 , R ) structure G = SU(3) d Ω ≃ flux SL(3 , C ) structure • classification, new solutions, . . . • global questions: G can change, . . . , moduli hard, . . . [ Gauntlett, Martelli, Pakis & DW; Gauntlett, Gutowski, Hull, Pakis & Reall; Gauntlett & Pakis; . . . , ] 5
Is there some integrable geometry? supersymmetry ⇔ integrable G-structure in generalised geometry 6
Generalised Calabi–Yau structures Generalised tangent space E ≃ TM ⊕ T ∗ M with Stab( { ε ± i } ) = SU(3) × SU(3) ⊂ SO(6) × SO(6) ⊂ O(6 , 6) × R + gives class of pure NS-NS backgrounds � d Φ + = 0 SU(3 , 3) + structure G = SU(3) × SU(3) d Φ − = 0 SU(3 , 3) − structure for generalised spinor Φ ± ∈ S ± ( E ) ≃ Λ ± T ∗ M Φ + = e − φ e − B − i ω Φ − = e − φ e − B (Ω 1 + Ω 3 + Ω 5 ) [ Hitchin, Gualtieri; Gra˜ na, Minasian, Petrini and Tomasiello ] 7
Generic N = 2 backgrounds Warped compactification � M 6 type II d s 2 = e 2∆ d s 2 ( R 3 , 1 ) + d s 2 ( M ) M-theory M 7 • “exceptional generalised geometry” with E 7(7) × R + • spinors in SU(8) vector rep ǫ = ( ε + , ε − ) • so for N = 2 we have Stab( ǫ 1 , ǫ 2 ) = SU(6) ⊂ SU(8) ⊂ E 7(7) × R + 8
The problem Generalised ω and Ω � ??? “H structure” G = SU(6) ??? “V structure” • how do we define structures? • what are integrability conditions? [ cf. Gra˜ na, Louis, Sim & DW; Gra˜ na & Orsi; Gra˜ na & Triendl ] 9
Generalised geometry
E d(d) × R + generalised geometry ( d ≤ 7) Unify all symmetries of fields restricted to M d − 1 in type II δ C ± = L v C ± + d λ ∓ + . . . δ g = L v g δ ˜ B = L v ˜ B + d ˜ δ B = L v B + d λ λ + . . . gives generalised tangent space E ≃ TM ⊕ T ∗ M ⊕ Λ 5 T ∗ M ⊕ Λ ± T ∗ M ⊕ ( T ∗ M ⊗ Λ 6 T ∗ M ) V M = ( v m , λ m , ˜ λ m 1 ··· 5 , λ ± , . . . ) Transforms under E d(d) × R + rep with R + weight (det T ∗ M ) 1 / (9 − d ) [ Hull; Pacheco & DW ] 11
In M-theory E ≃ TM ⊕ Λ 2 T ∗ M ⊕ Λ 5 T ∗ M ⊕ ( T ∗ M ⊗ Λ 7 T ∗ M ) V M = ( v m , λ m , ˜ λ m 1 ··· 5 , . . . ) Generalised Lie derivative L V = diffeo + gauge transf = V · ∂ − ( ∂ × ad V ) where type IIA, IIB and M-theory distinguished by ∂ M f = ( ∂ m f , 0 , 0 , . . . ) ∈ E ∗ [ Coimbra, S-Constable & DW ] 12
Generalised tensors: E d(d) × R + representations For example, adjoint includes potentials ad ˜ F ≃ R ⊕ ( TM ⊗ T ∗ M ) ⊕ Λ 2 T ∗ M ⊕ Λ 2 TM ⊕ Λ 6 TM ⊕ Λ 6 T ∗ M ⊕ Λ ± TM ⊕ Λ ± T ∗ M , N = ( . . . , B mn , . . . , ˜ A M B m 1 ... m 6 , . . . , C ± ) Gives “twisting” of generalised vector and adjoint B + C ± ˜ B + C ± ˜ V = e B + ˜ R = e B + ˜ R e − B − ˜ B − C ± V 13
Generalised geometry and supergravity Unified description of supergravity on M • Generalised metric invariant under max compact H d ⊂ E d(d) × R + G MN equivalent to { g , φ, B , ˜ B , C ± , ∆ } • Generalised Levi–Civita connection D M V N = ∂ M V N + Ω M N P V P exists gen. torsion-free connection D with DG = 0 but not unique 14
• Analogue of Ricci tensor is unique gives bosonic action � S B = | vol G | R eom = gen. Ricci flat M where | vol G | = √ g e 2∆ • Leading-order fermions and supersymmetry δρ = / δψ = D � ǫ D ǫ etc unique operators, full theory has local H d invariance [ CSW ] (c.f [ Berman & Perry;. . . ;Aldazabal, Gra˜ na, Marqu´ es & Rosabal;. . . ] and [ Siegel; Hohm, Kwak & Zweibach; Jeon, Lee & Park ] for O( d , d )) 15
H and V structures
H and V structures Generalised structures in E 7(7) × R + G = Spin ∗ (12) H structure “hypermultiplets” V structure G = E 6(2) “vector-multiplets” [ Gra˜ na, Louis, Sim & DW ] Invariant tensor for V structure Generalised vector in 56 1 K ∈ Γ( E ) such that q ( K ) > 0 where q is E 7(7) quartic invariant, determines second vector ˆ K 17
Invariant tensor for H structure Weighted tensors in 133 1 J α ( x ) ∈ Γ(ad ˜ F ⊗ (det T ∗ M ) 1 / 2 ) forming highest weight su 2 algebra [ J α , J β ] = 2 κǫ αβγ J γ tr J α J β = − κ 2 δ αβ where κ 2 ∈ Γ(det T ∗ M ) 18
Compatible structures and SU(6) The H and V structures are compatible if � 2 κ 2 q ( K ) = 1 J α · K = 0 6 ω 3 = 1 8 i Ω ∧ ¯ analogues of ω ∧ Ω = 0 and 1 Ω the compatible pair { J α , K } define an SU(6) structure J α and K come from spinor bilinears. 19
Example: CY in IIA ad ˜ F ≃ ( TM ⊗ T ∗ M ) J + = 1 2 κ Ω − 1 2 κ Ω ♯ ⊕ Λ 2 T ∗ M ⊕ Λ 2 TM ⊕ R ⊕ Λ 6 TM ⊕ Λ 6 T ∗ M 2 κ vol ♯ J 3 = 1 2 κ I + 1 2 κ vol 6 − 1 6 ⊕ Λ − TM ⊕ Λ − T ∗ M , where κ 2 = vol 6 = 1 8 i Ω ∧ ¯ Ω and I is complex structure E ≃ TM ⊕ T ∗ M ⊕ Λ + T ∗ M K + i ˆ K = e − i ω ⊕ Λ 5 T ∗ M ⊕ ( T ∗ M ⊗ Λ 6 T ∗ M ) 20
Example: CY in IIB ad ˜ F ≃ ( TM ⊗ T ∗ M ) 2 κ e − i ω − 1 2 κ e − i ω ♯ J + = 1 ⊕ Λ 2 T ∗ M ⊕ Λ 2 TM ⊕ R ⊕ Λ 6 TM ⊕ Λ 6 T ∗ M 2 κ ω ♯ − 1 2 κ vol ♯ J 3 = 1 2 κ ω + 1 2 κ vol 6 − 1 6 ⊕ Λ + TM ⊕ Λ + T ∗ M , where κ 2 = vol 6 = 1 6 ω 3 E ≃ TM ⊕ T ∗ M ⊕ Λ − T ∗ M K + i ˆ K = Ω ⊕ Λ 5 T ∗ M ⊕ ( T ∗ M ⊗ Λ 6 T ∗ M ) 21
Example: D3-branes in IIB Smeared branes on M SU(2) × R 2 d s 2 = e 2∆ d s 2 ( R 3 , 1 ) + d ˜ s 2 ( M SU(2) ) + ζ 2 1 + ζ 2 2 , with integrability d ( e ∆ ζ i ) = 0 , d ( e 2∆ ω α ) = 0 , d ∆ = − 1 4 ⋆ F , where ω α triplet of two-forms defining SU(2) structure. (Can also add anti-self-dual three-form flux.) 22
The H and V structures are ad ˜ F ≃ ( TM ⊗ T ∗ M ) ˜ J α = − 1 2 κ I α − 1 2 κ ω α ∧ ζ 1 ∧ ζ 2 ⊕ Λ 2 T ∗ M ⊕ Λ 2 TM ⊕ R ⊕ Λ 6 TM ⊕ Λ 6 T ∗ M α ∧ ζ ♯ 1 ∧ ζ ♯ + 1 2 κ ω ♯ 2 , ⊕ Λ + TM ⊕ Λ + T ∗ M , where κ 2 = e 2∆ vol 6 and I α are complex structures K + i ˜ ˜ ˆ K = n i e ∆ ( ζ 1 − i ζ 2 ) E ≃ TM ⊕ T ∗ M ⊕ Λ − T ∗ M ⊕ Λ 5 T ∗ M ⊕ ( T ∗ M ⊗ Λ 6 T ∗ M ) + i n i e ∆ ( ζ 1 − i ζ 2 ) ∧ vol 4 where n i = ( i , 1) is S-duality doublet, then twist by C 4 K = e C 4 ˜ J α = e C 4 ˜ J α e C 4 K 23
Generic form? Complicated but • interpolates between symplectic, complex, product and hyper-K¨ ahler structures • can construct from bilinears and twisting 24
Integrability
GDiff and moment maps Symmetries of supergravity give generalised diffeomorphisms GDiff = Diff ⋉ gauge transf. acts on the spaces of H and V structures ⇔ integrability vanishing moment map Ubiquitous in supersymmetry equations • flat connections on Riemann surface (Atiyah–Bott) • Hermitian Yang–Mills (Donaldson-Uhlenbeck-Yau) • Hitchin equations, K¨ ahler–Einstein, . . . 26
Space of H structures, A H Consider infinite-dimensional space of structures, J α ( x ) give coordinates A H has hyper-K¨ ahler metric inherited fibrewise since at each x ∈ M J α ( x ) ∈ W = E 7(7) × R + Spin ∗ (12) and W is HK cone over homogenous quaternionic-K¨ ahler (Wolf) space (n.b. A H itself is HK cone by global H + = SU(2) × R + ) 27
Triplet of moment maps Infinitesimally parametrised by V ∈ Γ( E ) ≃ gdiff and acts by δ J α = L V J α ∈ Γ( T A H ) preserves HK structure giving maps µ α : A H → gdiff ∗ ⊗ R 3 � µ α ( V ) = − 1 2 ǫ αβγ tr J β ( L V J γ ) M functions of coordinates J α ( x ) ∈ A H 28
Integrability integrable H structure ⇔ µ α ( V ) = 0 , ∀ V for CY gives d ω = 0 or d Ω = 0 Moduli space Since structures related by GDiff are equivalent / GDiff = µ − 1 1 (0) ∩ µ − 1 2 (0) ∩ µ − 1 M H = A H / / 3 (0) / GDiff . moduli space is HK quotient, actually HK cone over QK space of hyper- multiplets (as for CY 4 h 1 , 1 + 4 or 4 h 2 , 1 + 4) 29
Space of V structures, A V Consider infinite-dimensional space of structures, K ( x ) give coordinates A H has (affine) special-K¨ ahler metric (explains ˆ K ) inherited fibrewise since at each x ∈ M K ( x ) ∈ P = E 7(7) × R + E 6(2) and P is homogenous special-K¨ ahler space 30
Moment map Infinitesimally gdiff and acts as δ K = L V K ∈ Γ( T A V ) preserves SK structure giving maps µ : A V → gdiff ∗ � µ ( V ) = − 1 tr s ( K , L V K ) 2 M where s ( · , · ) is E 7(7) symplectic invariant 31
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