ELEMENTARY DEFORMATIONS AND THE HYPERKAEHLER / QUATERNIONIC KAEHLER CORRESPONDENCE

Transcription

1 ELEMENTARY DEFORMATIONS AND THE HYPERKAEHLER / QUATERNIONIC KAEHLER CORRESPONDENCE Oscar Macia (U. Valencia) (in collaboration with Prof. A.F. Swann) Aarhus, DK March 2, 2017

2 1 Planning Introduction and motivation. The c-map Interlude: Differential-geometric construction of the c-map (pre-twist version) Twist and hk/qk correspondence

3 2 Planning 1/3 Introduction and motivation. The c-map

4 3 Berger s List Theorem Let M be a Riemannian, oriented, simply-connected n- dimensional manifold, which is not locally a product, nor symmetric. Then its holonomy group belongs to the following list: M. Berger (1955) image credit: S.M. Salamon (1989)

5 4 Kähler manifolds (K) (M 2m, g) + {Hol U(m)} K U(m) = SO(2m) Sp(m, R) GL(m, C) U(m) GL(m, C) I End(T M) : I 2 = Id dλ 1,0 Λ 2,0 Λ 1,1 U(m) SO(2m) g(ix, IY ) = g(x, Y ) U(m) Sp(m, R) ω(x, Y ) = g(ix, Y ) Λ 2 T M dω = 0

6 5 Quaternion-Kähler manifolds (QK) (M 4k, g) + {Hol Sp(k)Sp(1)} QK I, J, K ΓEndT M : I 2 = J 2 = K 2 = IJK = Id g(ax, AY ) = g(x, Y ) A = I, J, K ω A (X, Y ) = g(ax, Y ) Λ 2 T M Ω = A ω 2 A Λ4 T M Ω = 0. Sp(k)Sp(1) U(m) QK K { Hol Sp(k) Sp(k)Sp(1) HK Hol Sp(k)Sp(1) M symmetric space

7 6 Hyper-Kähler manifolds (HK) (M 4k, g) + {Hol Sp(k)} HK Sp(k) U(m) HK K I, J, K ΓEndT M : I 2 = J 2 = K 2 = IJK = Id Curvature ω A Λ 2 T M dω I = dω J = dω K = 0. QK Einstein QK + {s = 0} HK

8 7 (First) Motivation Wolf spaces G 2 HP n, Gr 2 (C n+2 ), Gr 4 (R n+4 ) SO(4), F 4 Sp(3)Sp(1), E 6 SU(6)Sp(1), E 7 Spin(12)Sp(1), E 8 E 7 Sp(1) Alekseevsky spaces Homogeneous, non-symmetric QK, with s < 0.

9 8 (First) Motivation Wolf spaces G 2 HP n, Gr 2 (C n+2 ), Gr 4 (R n+4 ) SO(4), F 4 Sp(3)Sp(1), E 6 SU(6)Sp(1), E 7 Spin(12)Sp(1), E 8 E 7 Sp(1) Alekseevsky spaces Homogeneous, non-symmetric QK, with s < Find explicit examples of QK manifolds The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics. Sir Michael F. Atiyah Collected Works Vol. 1 (1988), 19, p.13

10 9 M-Theory

11 10 T-Duality & c-map T-Duality c-map IIA IIB K 1 QK 2 CH(1) K 2 QK 1 CH(1) K 2m c-map QK 4(m+1) Rigid c-map Cecotti, Ferrara & Girardello (1989), Ferrara & Sabharwal (1990) K 2m rigid c-map HK 4m

12 11 c-map & homogeneous QK classification Completely solvable Lie groups admitting QK metrics. 2. All known homogeneous non-symmetric QK manifolds. Alekseevsky (1975) 1989, 1990 K 2m c QK 4(m+1) Cecotti, Ferrara & Girardello (1989), Ferrara & Sabharwal (1990) 1992 Use c-map to complete Alekseevsky s classification. de Witt & Van Proeyen (1992) 1996 Complete classification (without c-map). Cortes (1996)

13 12 Field content of N = 2, D = 4 SIMPLE SUGRA a = 0,..., 3, ˆµ = 0,... 3, Λ = 1, 2 Gravity supermultiplet m Vector supermultiplets (V aˆµ, ψλ, A 0ˆµ ) φ i a (A iˆµ, λi Λ, φ i ), i = 1,..., m m C-scalars = 2m R-scalars Coordinates on a 2m-dimensional, PSK Σ-model

14 13 4D 3D Kaluza Klein compactification, I Metric ˆµ = 0,... 3 (0, µ), µ = 1, 2, 3 η ab V aˆµ V bˆν = gˆµˆν KK = 4-vectors ( e 2σ e 2σ A ν e 2σ A µ e iσ g µν + e 2σ A µ A ν ) A 0ˆµ KK = (A 0 0, A 0 µ) (ζ 0, A 0 µ) A iˆµ KK = (A i 0, A i µ) (ζ i, A i µ) σ, ζ 0, ζ i m + 2 extra scalar fields A µ, A 0 µ, A i µ m + 2 extra 3D-vector vields.

15 14 4D 3D Kaluza Klein compactification, II Dualization of 3D-vector fields (A µ, A 0 µ, A i µ) Duality (a, ζ 0, ζ i ) a, ζ 0, ζ i m + 2 extra scalar fields.

16 15 SUGRA N = 2, D = 4 Kaluza-Klein 4D 3D Gravity supermultiplet 4D #s 3D #s * #s V iˆµ Vµ, i A µ, e 2σ 1 Vµ, i a, φ 2 ψ Λ ψ Λ ψ Λ A 0ˆµ A 0 0, A0 µ 1 ζ 0, ζ 0 2 m Vector supermultiplet 4D #s 3D #s * #s A iˆµ A i 0, Ai µ m ζ i, ζ i 2m λ i Λ λ i Λ λ i Λ φ i 2m, R φ i 2m φ i 2m

17 16 Planning 2/3 Introduction and motivation: The c-map. Interlude: Differential-Geometric construction of the c-map (pre-twist version)

18 17 (Second) Motivation 2. To provide a geometric construction to the c-map. Actually... K 2m c-map QK 4(m+1) PSK 2m c-map QK 4(m+1)

19 18 Special Kähler manifolds, I Special Kähler manifolds, SK 1. s ω = 0, 2. R( s ) = T( s ) = 0, 3. s X IY = s Y SK = (K, s ) IX ( special condition ) Conic special Kähler manifolds, CSK 1. g(x, X) 0; CSK = (SK, X) 2. ω X = I = g X. X XM

20 19 Special Kähler manifolds, II * Moment mapping µ : M R : p X p 2 Projective special Kähler manifolds, PSK PSK = CSK// c X = µ 1 (c)/x c R CSK 0 = µ 1 (c) CSK CSK// c X = PSK

21 20 General picture arising from physics CSK 2(m+1) rigid c-map HK 4(m+1) // c X PSK 2m c-map QK 4(m+1) HK 4(m+1) = T CSK 2(2(m+1)) rigid c-map CSK 2(m+1) R 2(m+1)

22 CSK 2(m+1) R 2(m+1) HK 4(m+1) // c X? PSK 2m c-map QK 4(m+1)

23 21 Planning 3/3 Introduction and Motivation. The c-map. Interlude: Differential-Geometric construction of the c-map. (pre-twist version) Twist and HK/QK correspondence

24 22 The Twist construction (sketch) The twsit construction associates to a manifold M with a S 1 -action, a new space W of the same dimension, with a distinguished vector field. This construction fits into a double fibration S 1 π P π W S 1 M n twist so W is M twisted by the S 1 -bundle P. W n A. Swann, (2007, 2010)

25 23 Twists & HK/QK correspondence 1992, 2001 Instanton twists (Hypercomplex, Quaternionic, HKT) D.Joyce (1992), G.Grantcharov & Y.S. Poon (2001) 2007, 2010 General twists (T-duality, HKT, KT, SKT,...) A.F. Swann (2007, 2010) 2008 HK/QK correspondence {HK + symmetry fixing one ω} {QK + circle action} 2013 A. Haydys (2008) Twistor interpretation N.J. Hitchin (2013)

26 24 Idea CSK 2(m+1) R 2(m+1) HK 4(m+1) πm // c X P PSK 2m c-map QK 4(m+1) π W

27 25 The Twist construction (in detail) Y XP : S 1 π P π W X XM : S 1 M n W n = P/ X twist 1. P (M, S 1 ) : connection θ, curvature π M F = dθ. 2. L X F = 0 A. Swann, (2007, 2010) 3. X = X θ + ay XP : such that L X θ = L X Y = W = P/ X with induced action by Y.

28 26 Twist data (M, X, F, a) = π M P π W M W Twist data 1) M, a C manifold. 2) X XM, generating the S 1 -action. 3) F Ω 2 M, X-invariant, with integral periods. 4) a C M such that da = X F

29 27 H-related tensors M H P π M π W T p P = H p + V p H p = Tπ(p) M = T πw (p) W W α TM, α W TW α H α W (π M α) H = (π W α W ) H Lemma α Ω p M X!α W Ω p W : α W H α π W α W = π α θ π (a 1 X α)

30 28 Computing the Twist dα W α Ω p M X, dα W H dα 1 F (X α). a Twisted exterior differential, d W α W H α, dα W H d W α d W := d 1 a F X

31 29 Twist vs Integrability Complex case Let I an invariant complex structure on M, H-related to an almost-complex structure I W on W, I W is integrable iff F Ω 1,1 I M. Kähler case K C π M S 2n+1 T 2 π W CP n T 2 S 2n+1 S 1

32 30 Need for a deformation Problem dα = 0 dα W = 0 dα W d W α = dα 1 a F (X α) = 1 F (X α) 0 a 31 Simetríes of the structure HK = (M, g, I, J, K) Rotating symmetry X XM 1. L X (g) = L X (I) = I, J, K 3. (a) L X I = 0, (b) L X J = K, (c) L X K = J

33 32 Elementary deformations of the HK metric g α HX = X, IX, JX, KX α 0 = g(x, ), α A = g(ax, ) (A = I, J, K) g α := α A α 2 A g HX g N, Elementary deformation of the metric g N = fg + hg α, f, g C M

34 33 Uniqueness Theorem (M 4k, g, I, J, K) HK, k 2 X XM Rotating symmetry µ Moment mapping! Elementary deformation! Twist data g N = 1 µ c g + 1 (µ c) 2g α W F = kg = k(dα 0 + ω I ), a = k(g(x, X) µ + c). QK O.Macia, A.F.Swann (2015)

35 34 Idea of proof 1. From g N and (I, J, K), construct ω N A y ΩN. 2. Impose arbitrary twist of Ω N to be QK. 3. Decompose equation with respect to the splitting T M = HX HX. 4. All this leads to f = f(µ), h = h(µ) y h = f. 5. Impose da = X F to determine a. 6. Imposing df = 0 leads to ODE s which determine f.

36 MANGE TAK

37 35 Planning 4/3 (!) Introduction and Motivation. The c-map. Interlude: Differential-Geometric construction of the c-map. (pre-twist version) Twist and HK/QK correspondence Bonus Track: A worked-out example

38 36 The hyperbolic plane RH(2) - CH(1) : Real, 2-dimensional solvable Lie group with Kähler metric of constant curvature. - Local basis of one-forms on S CH(1) : {a, b} Ω 1 S : da = 0, db = λa b - Metric: g S = a 2 + b 2 - Almost complex structure: Ia = b - Kähler two-form: ω S = a b.

39 37 Local cone structure - The PSK S is the Kaehler quotient of a CSK manifold C (C, g, ω,, X) C 0 S π i C S = C// c X - Locally C = R >0 C 0. - C 0 is the level set µ 1 (c) for the moment map of X. - C 0 S is a bundle with connection 1-form ϕ : dϕ = 2π ω S

40 38 Metric and Kaehler form on C - Write t for the standard coordinate on R >0 and ˆψ = dt. - Write â = tπ a, ˆb = tπ b, ˆϕ = tϕ Lemma The (2, 2) pseudo Riemannian metric and the Kaehler form of C = R >0 C 0 are g C = â 2 + ˆb 2 ˆϕ 2 ˆψ 2 = dt 2 + t 2 g C0 ω C = â ˆb ˆϕ ˆψ. The conic isometry satisfies IX = t t

41 39 LC connection - Coframe s θ = (â, ˆb, ˆϕ, ˆψ) Exterior differential d(s θ) dâ = 1 t dt â d ˆϕ = 1 t (dt ˆϕ + 2â ˆb) d ˆψ = 0 dˆb = 1 t (dt ˆb λâ ˆb) LC connection s ω LC 0 ˆϕ + λˆb ˆb â s ω LC = 1 ˆϕ λˆb 0 â ˆb t ˆb â 0 ˆϕ â ˆb ˆϕ 0

42 40 Curvature 2-form Curvature 2-form s Ω LC s Ω LC = 4 λ2 t 2 0 â ˆb 0 0 â ˆb Lemma The pseudo Riemannian metric g C is flat iff λ 2 = 4.

43 41 Conic special Kaehler condition Symplectic connection in terms of LC connection Conditions on η. 1. η s θ. 2. i η = η i. 3. t η s = s η. 4. X η = IX η = 0. Proposition s ω = s ω LC + η. The cone (C, g C, ω C ) over S RH(2) is CSK iff λ 2 = 4 3 or λ 2 = 4.

44 42 HK structure on T CSK HK metric ( g HK = â 2 + ˆb 2 ˆϕ 2 ˆψ 2) + HK structure (Â2 + ˆB 2 ˆΦ 2 ˆΨ 2) ω I = â ˆb ˆϕ ˆψ Â ˆB + Φ ˆΨ ω J = Â â + ˆB ˆb + ˆΦ ˆϕ + ˆΨ ˆψ ω K = Â ˆb ˆB â + ˆΦ ˆψ ˆΨ ˆϕ.

45 43 Elementary deformation of g α HK 0,... α K α I = I X g H = t ˆψ, α 0 = Iα I = t ˆϕ µ α J = I X g H = tˆφ, α K = Iα J = t ˆΨ µ = 1 2 X 2 = t2 2 Elementary deformation g N = 1 µ g H + 1 µ 2(α2 0 + α2 I + α2 J + α2 K ) = 2 t 2(â2 + ˆb 2 + ˆϕ 2 + ˆψ 2 + Â2 + ˆB 2 + ˆΦ 2 + ˆΨ 2 )

46 44 Twist data Twisting two-form F = â ˆb + ˆϕ ˆψ Â ˆB + ˆΦ ˆΨ Twisting function a = µ = t2 2

47 45 Flat case λ 2 = 4. - The LC and the special connection coincide. - X-invariant coframe on CSK γ = s θ/t = (ã, b, ϕ, ψ) (we can compute the twisted differentials immediatelly) d W ã = 0 d W b = 2 b ã d W ϕ = 2ã b + 2 t 2F d W ψ = 0 - The (vertical) coframe δ = (s α)/t = is NOT X-invariant. (Ã, B, Φ, Ψ)

48 δ = δe iτ ɛ = 1 2 (δ 1 + δ 4, δ 2 δ 3, δ 2 δ 3, δ 1 + δ 4 ) ψ ã 0 2 b 0 0 ψ ã 0 2 b d W ɛ = ɛ 0 0 ψ ã ψ + ã d W ϕ = 2(ϕ ψ + ɛ 13 + ɛ 4 ɛ 2 ) The resulting Lie algebra is isomorphic to the non-compact symmetric space Gr 2 + U(2, 2) (C2,2 ) = U(2) U(2)

49 46 Non-flat case λ 2 = Same g HK, g N and same twist data F, a. - Twisted differentials for γ = (ã, b, ϕ, ψ) : 2 d W ã = 0 d W b = 3ã b d W ψ = 0 d W ϕ = 2(ϕ ψ Ã B + Φ Ψ)

50 - Adjusting the vertical coframe we arrive to d W ɛ = ɛ ψid ɛ ã ɛ b We see the structure of solvable algebra associated to G 2 SO(4)

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