Simon Salamon. Turin, 24 April 2004

Transcription

1 G 2 METRICS AND M THEORY Simon Salamon Turin, 24 April 2004 I Weak holonomy and supergravity II S 1 actions and triality in six dimensions III G 2 and SU(3) structures from each other 1

2 PART I The exceptional geometry of conical singularities, based on Alfred Gray s notion of weak holonomy: If (Y, g 7 ) has weak holonomy G 2 then dr 2 + r 2 g 7 on X = R + Y has holonomy in Spin 7. Examples: Y = S 7, S 7 sq ( S3 S 4 ) Ber 7 = SO(5) SO(3), AW7 = SU(3) U(1) p,q If (Z, g 6 ) is nearly-kähler (weak hol SU(3)) then has holonomy in G 2. ds 2 + s 2 g 6 on R + Z Examples: The 3-symmetric spaces Z = S 6, CP 3, F = SU(3) T 2, S 3 S 3. 2

3 11-dimensional equations Maximal D = 11 supergravity involves a metric g of signature (10, 1), and a 4-form F = da satisfying R im = 1 12 ( Fijkl F ijk m 1 12 g imf 2) df = 0 d F = F F M-theory adds conjectural correction terms that we shall ignore. Supersymmetry requires Killing spinor(s): m η [ Γ ijkl m 8δ i mγ jkl] F ijkl η = 0 and holonomy reduction of a suitable connection. Let ν = 1 32 dim(k spinors). There are theories of generalized holonomy groups H SO(10, 1) [Duff], [Hull]. 3

4 First solutions Identifying F with the volume form of a 4-manifold gives an Einstein product M 4 M 7. With ν = 1: AdS 7 S 4 or AdS 4 S 7. M2 brane solution with ν 1 2 : ) 2/3 ) 1/3 (1 + a6 g2,1 + (1 + a6 (dr 2 + r 2 g r 6 r 6 7 ) with F = vol 2,1 f(r)dr and F = 6a 6 vol 7. Interpolates between the asymptotic (r = ) metric g 2,1 + (dr 2 + r 2 g 7 ) on M 2,1 X 8 and the near-horizon limit (r a) ( r 4 ) a g 4 2,1 + a2 r 2 dr2 + a 2 g 7 on AdS 4 Y 7. If g 7 has weak holonomy G 2 then these two limits have ν = 1 16 and ν = 1 8 respectively. 4

5 A weak G 2 orbifold Suppose that X 8 = S 1 Y 7 has a product metric g 8 = dx 2 + (dy 2 + y 2 g 6 ) with holonomy {1} G 2 Spin 7. Setting x=r cos t and y =r sin t gives g 8 = dr 2 + r 2 (dt 2 + sin 2 t g 6 ). Corollary [Acharya et al] If (Z, g 6 ) is nearly-kähler, the spherical metric dt 2 + sin 2 t g 6 has weak holonomy G 2 (so Einstein); its singularities at t = 0, π approximate G 2 holonomy cones. If the nearly-kähler structure is defined by a 2-form ω and (3, 0)-form ψ + +iψ then the G 2 3-form is ϕ = s 2 ω dt + s 3 (cψ + +sψ ) where c=cos t and s = sin t. 5

6 PART II If (Y, g 7 ) has holonomy G 2 then Q = Y/S 1 has a symplectic SU(3) structure with g 7 = f(q) 2 (dx + θ) f(q) k 6 with dilation factor f. The metric k 6 forms part of the dual 10-dimensional type IIA string theory. In certain cases, (Q, k 6 ) is Kähler [Apostolov-S]. Theorem [Atiyah-Witten] Each 3-symmetric space Z admits a S 1 action for which Z/S 1 = S 5. Moreover, (R + Z)/S 1 = R 6 contains the fixed points (f = 0) as Lagrangian submanifolds. Example: F = U(3) U(1) U(1) U(1) CP 2. A standard S 1 action has fixed point set S 2 S 2 S 2, one fibre for each of the three projections to CP 2. Then (R + F)/S 1 has three R 3 s intersecting transversally at a point. 6

7 Nearly-Kähler 6-manifolds as quotients? Example: S 3 S 3 = SU(2) SU(2) SU(2) SU(2) admits an action of S 3 = σ, τ. The diagonal S 3 is invariant by σ, and its images by τ give a 120 o configuration of S 3 s intersecting at z. These give rise to intersecting R 4 s in R + (S 3 S 3 ). Suggests existence of a Spin 7 total space X = R + Y π S 1 R + (S 3 S 3 ) z where Y is a weak G 2 -orbifold, π maps fixed points to the three S 3 s, and π 1 (z) is the vertex of a cone modelled on R + F. Another topic: metrics with holonomy G 2 on X 8 /S 1 via hyper-kähler reduction [Acharya-Witten]. 7

8 Lagrangian triality and quaternionic geometry Let N 5 ( T 2 T 3 ) be a nilmanifold of type (0, 0, 0, 12, 13), so de 4 =e 12 and de 5 = e 13. Then N 6 = N 5 S 1 has a symplectic form with trivial e 6 factor. ω = e 16 + e 25 + e 34 Theorem [Giovannini] The compact manifold N 6 has a configuration of invariant Lagrangian submanifolds at 120 o through each point. Proof. Follows from the existence of closed 3-forms e 1 (e 2 3e 5 ) (e 3 + 3e 4 ) (e 1 + 3e 6 ) e 2 (e 3 3e 4 ) (e 1 3e 6 ) (e 2 + 3e 5 ) e 3. Note that Sp(3, R) acts transitively on positive-definite Lagrangian triples (L 1, L 2, L 3 ) in R 6. Corollary N 6 T 2 has a closed 4-form Ω with stabilizer Sp(2)Sp(1). 8

9 PART III In 6 dimensions, an SU(3) structure is characterized by a 2-form ω and a complex (3, 0)-form ψ + + iψ. A G 2 structure is defined in dimensions by The holonomy reduces iff ϕ = ω dt + ψ + ϕ = ψ dt ω2. dϕ=0 and d ϕ = 0 implying the half-flat conditions dψ + = 0 and d( 1 2 ω2 ) = 0. Example: N 6 = N 5 S 1 has a half-flat structure: ψ + (0) = e e e 356 e ω(0)2 = e e e Problem: Find ω(t), ψ + (t) satisfying. Hamiltonian theory guarantees solution [Hitchin]. 9

10 Solution: Can split off e 6 to get a Calabi-Yau orbifold: Theorem N 5 ( π 2, π 2 ) has a metric k 6 with holonomy equal to SU(3). Proof. Let c = cos t, s = sin t. Then k 6 has an orthonormal basis ce 1, (1+s) 1/2 e 2, (1 s) 1/2 e 3, (1+s) 1/2 e 4, (1 s) 1/2 e 5, c 2 dt. The closed foms are Ω = c 3 e 1 dt e 24 + e 35 Ψ + = c(1+s)e 25 dt c(1 s)e 34 dt + c 2 e e 145 Ψ = ce 45 dt c 3 e 23 dt (1 s)e 134 (1 + s)e 125. Since e 145 and c 3 e 23 dt are also closed, exp(θj) e 145 generates a pencil {L θ,z } of special Lagrangian submanifolds through each point z, each with phase θ. 10

11 Back to M theory One can seek solutions g 10,1 = e A (y)g 3,1 + e B (y)g 7 F = f(y)vol 3,1 + F 7 η = θ ε 1 + iγθ ε 2 on AdS 4 Y in which the spinors ε 1, ε 2 determine an SU(3) structure on T y Y = R C 3. Equations link F to the intrinsic torsion [Chiossi-S]. Theorem [Lukas-Saffin] (i) f = 0 so F F = 0. (ii) Y is foliated by 6-manifolds on which the induced SU(3) structure is half-flat. (iii) if F is SU(3)-invariant ( singlet flux ) then g 7 = dt 2 + e 2α(t) g 6 is a warped product with a nearly-kähler metric g 6. Exact nature of G 2 structures and singularities unclear. 11

12 Selected references Acharya-Denef-Hofman-Lambert, hep-th/ Acharya-Witten, hep-th/ Apostolov-Salamon, math.dg/ Atiyah-Witten, hep-th/ Chiossi-Salamon, math.dg/ Duff, hep-th/ Freund, hep-th/ Giovannini, PhD thesis, Turin, 2004 Hitchin, math.dg/ Hull, hep-th/ Lukas-Saffin, hep-th/