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Connection Description of 3D Gravity Connection description of 3D Gravity joint work with Y. Herfray and C. Scarinci arXiv:1605.07510 Workshop on Teichmueller theory and geometric structures on 3-dimensional manifolds Kirill Krasnov (Nottingham) School of Mathematical Sciences First Year Research Report PhD. in Mathematical Sciences Student: Supervisor: Johnny Espin Prof. Kirill Krasnov July 11, 2013 We usually describe geometry using metrics Works in any dimension Natural constructions: Einstein metrics Ricci flow There are of course other types of geometric structures (reductions of the structure group of the frame bundle) and other types of geometry (e.g. complex, symplectic) In specific dimensions other descriptions possible 2D - complex rather than conformal structure 3D - Cartan formalism (Chern-Simons) 4D - Penrose encodes metric into an almost complex structure on twistor space plus contact form Many have previously suggested that metric may not be the best “variable” to describe gravity In the approach to be described 3D geometry is encoded by connection on space(time) rather than metric Discovered in: Connection formulation of (2+1)-dimensional Einstein gravity and topologically massive gravity Peter Peldan (Goteborg, ITP). Oct 1991. 33 pp. Published in Class.Quant.Grav. 9 (1992) 2079-2092 Cited by 9 records interpretation via a certain 3-form in the total space of the SU(2) bundle over space(time) Plan Einstein-Cartan, Chern-Simons descriptions of 3D gravity 3D gravity in terms of connections Volume gradient flow on connections - torsion flow 6D interpretation 3D gravity in first order formalism ⇤ < 0 For concreteness - hyperbolic case Riemannian Let e be (co)frame field Metric 2 e ⇤1 su(2) ds = 2Tr(e e) 2 ⌦ − ⌦ The associated SU(2) spin connection w ⇤1 su(2) (locally) 2 ⌦ Compatibility equation (zero torsion) (1) e =0 r e de + w e + e w r ⌘ ^ ^ w = w(e) -unique solution of the algebraic equation for w 3D Einstein equations ⇤ = 1 Metric has constant curvature − (2) f = e e f f(w)=dw + w w ^ ⌘ ^ curvature 2-form f ⇤2 su(2) 2 ⌦ Both (1), (2) follow as EL equations from 1 S[e, w]= Tr f e e e e ^ − 3 ^ ^ Z ✓ ◆ If substitute w = w(e) get EH Lagrangian for the metric Chern-Simons description (1),(2) arise as real, imaginary parts of f(a)=0 a := w + i e SL(2, C) connection S[e, w]=Im(SCS[a]) where 2 S [a]= Tr a da + a a a CS ^ 3 ^ ^ Z ✓ ◆ Chern-Simons functional “Pure connection” description only possible for non- zero scalar curvature! Instead of solving for the connection, can solve e = e(w) also possible f = e e algebraic equation for e in higher D ^ w must be special in order there to be a solution To describe the solution, need some notions Given f ⇤2 su(2) and choosing a volume form v get a map 2 ⌦ φ : ⇤1 su(2) f ! 1 ↵ ⇤ φf (↵):=↵ f/v 8 2 ^ Can apply this map to the curvature itself 4 Note that the sign here is λ(f):= Tr (φ φ (f)) invariantly defined! 3 f ⌦ f Definition: Connection w is called definite if map φf is an isomorphism Connection w is called positive (negative) definite if λ(f) > 0(λ(f) < 0) Can be defined at a point, then everywhere For a connection that comes from a metric map φf is just the Ricci curvature definiteness asks the Ricci to have no zero eigenvalues similarly λ(f) is a multiple of the determinant of Ricci note that our definiteness is a weak condition that does not require the eigenvalues to have the same sign Proposition: Given a negative definite connection, can solve f = e e for e = e(f) ^ For a positive definite connection, can similarly solve f = e e − ^ In both cases, the associated metric is a Riemannian 3D metric Proof: Define v := λ(f) v f − does not dependp on v, only on its orientation The solution explicitly i e =(f i f i f f)/2v ⇠ f ^ ⇠ − ⇠ ^ f Can be checked to satisfy f = e e f ^ f For a connection that comes from frame a simple calculation shows that does not change under 1 constant rescaling of the ef = pdetRR− e where R is Ricci original frame gives a Riemannian metric, degenerate where R is degenerate forgets about the scale of the original metric! f = e e the scale is introduced ^ when solving this equation ⇤ = 1 − Pure connection action Substituting e = ef into the first-order action gives Volume of the space computed using the metric S[w]= vf defined by the connection Z vf := λ(f) v 4 − λ(f):= Trp (φf φf (f)) 3 ⌦ φ (↵):=↵ f/v f ^ Functional on the space of connections of definite sign Its critical points - “constant curvature” connections Euler-Lagrange equations 2 S[w]= Tr (e e e ) −3 f ^ f ^ f Z δS[w]= Tr (δ(e e ) e ) − f ^ f ^ f Z = Tr (δ(f) e )= Tr ( δw e ) − ^ f − r ^ f Z Z e =0 second-order PDE on w ) r f says that w is the spin connection compatible with the frame defined by w once this equation is satisfied, the metric is automatically constant curvature since by construction f = e e f ^ f Associated gradient flow Recall the flow that plays role in Floer homology - the gradient flow of Chern-Simons da δSCS[a] ⇤ = where need a metric to define the * dt δa ✓ ◆ dimensional reduction of the 4D self-duality equations plays role in Donaldson-Witten theory The volume gradient flow similarly define the gradient flow for our connection functional dw δS[w] ⇤ where now the star is defined using the = dt − δw metric defined by the connection ✓ ◆ Parabolic equation Alternatively dw =( ef )⇤ For positive dt r connection needs to change the sign If decompose w = w˜ + t t-torsion with ˜ e =0 by definition r f Then e = t e + e t and ( e )⇤ is basically torsion r f ^ f f ^ r f Flow by torsion - Possibly useful as an alternative to Ricci flow Homogeneous case 3 If start with metric ds2 = (ai)2(ei)2 i=1 Get connection w1 = g1e1 X de1 =2e2 e3 ^ (a1)2 (a2)2 (a3)2 with g1 = − − a2a3 connection forgets about the scale! Note that in the round case a1 = a2 = a3 g1 = g2 = g3 = g = 1 Now start with connection − 1 1 1 w = g e f 1 =(2g1 + g2g3)e2 e3 ^ f 1 = ✓2 ✓3 ✓1 = a1e1 Get metric − ^ ⇤ =+1 (2g2 + g3g1)(2g3 + g1g2) a1 = s− 2g1 + g2g3 The associated volume is vol2 = (2g1 + g2g3)(2g2 + g3g1)(2g3 + g1g2) − If parametrise g1 = 1+x, g2 = 1+y, g3 = 1+z − − − So that x = y = z =0 is the round metric 3-sphere The contour plots of the volume function are vol2 =0 vol2 =1/2 concave function! so the flow will maximum at the origin return to the round metric 3-sphere The gradient flow explicitly a1(2a1 + g2a3 + g3a2) g˙1 = − 2a2a3 For the round sphere g = 1 1 2 3 g = g = g = g 1+x for connection that comes− from the metric ⌘ a1 = a2 = a3 = (2g + g2) 1 x2 − ⌘ − p p vol(g)=(1 x2)3/2 metric sphere has the largest volume − gradient flow x˙ = x − returns to the metric sphere Compare Ricci flow 3 ds2 = a2 (ei)2 i=1 ! X dg Ricci flow gives =Riccig Not a gradient flow! dt d 2 (a2)= collapses in finite time dt −a2 Similarly, for negative curvature the Ricci flow will expand the manifold, while the volume flow just returns to the metric connection Towards 6D interpretation: 3-forms in 6D Let P be a 6-manifold dim(⇤3P )=6 5 4/3! = 20 · · Consider a 3-form ⌦ ⇤3P 2 This form is called generic or stable of positive (negative) type if at each point it is in the orbit GL(6, R)/SL(3, R) SL(3, R) dim = 36 8 8 = 20 ⇥ − − ⌦ = ↵ ↵ ↵ + β β β , ↵ ↵ ↵ β β β =0 1 ^ 2 ^ 3 1 ^ 2 ^ 3 1 ^ 2 ^ 3 ^ 1 ^ 2 ^ 3 6 GL(6, R)/SL(3, C) ⌦ =2Re(↵ ↵ ↵ ) , ↵ , ↵ , ↵ T ⇤P 1 ^ 2 ^ 3 1 2 3 2 C Negative type case: almost complex structure Given a volume form v, can define an endomorphism K⌦ : T ⇤P T ⇤P i (K (↵)) := ↵ i ⌦ ⌦/v ! ⇠ ⌦ ^ ⇠ ^ This endomorphism squares to a multiple of identity 2 K⌦(↵) = λ(⌦)I For negative type (stable) 3-forms λ(⌦) < 0 Can define 1 2 J⌦ := K⌦ J = I λ(⌦) ⌦ − − p Almost-complex structure For ⌦ in the canonical form ↵1,2,3 are (0,1) forms Hitchin functional invariantly defined form in v := λ(⌦)v ⌦ − given orientation class p S[⌦]= v⌦ ZP Theorem (Hitchin): Critical points of S[⌦] under variations in a given cohomology class are integrable J⌦ Proof: 1 ˆ is the result of acting with v⌦ = ⌦ˆ ⌦ where ⌦ 2 ^ J⌦ in all 3 slots of ⌦ δS[⌦]= ⌦ˆ δ⌦ = ⌦ˆ dB d⌦ˆ =0 ^ ^ ) Z Z ⌦c = ⌦ +i⌦ˆ closed d⌦c =0 ) ⌦c is (0,3) form integrable ACS ) 6D interpretation of the connection formulation of 3D gravity Let P be the principal SU(2) bundle over a 3-dimensional M P M ! This is necessarily a trivial bundle P = SU(2) M ⇥ Let ⌦ be an SU(2) invariant stable 3-form in P Proposition: ⌦ defines an SU(2) connection and metric on M (with some genericity assumption) Proof: Take J⌦ Define the J ⌦ image of vertical vector fields to be horizontal connection in P ) Define the pairing of horizontal vector fields to be the Killing-Cartan pairing of their vertical J ⌦ images In turn an SU(2) connection on M defines an SU(2) invariant closed 3-form in the total space of the bundle ⌦ = CS(W ) d⌦ =0 where 2 CS(W )=Tr W dW + W W W ^ 3 ^ ^ ✓ ◆ W is an SU(2) connection in the total space of the bundle Proposition: Then the connection defined by ⌦ = CS ( W ) is W When ⌦ is stable of positive (negative) type W is positive (negative) definite The metric defined by ⌦ coincides with the one defined by W All 3D pure connection notions introduced earlier are induced by 6D notions for the Chern-Simons 3-form in the total space of the SU(2) bundle Proposition: Hitchin functional for ⌦ = CS(W ) is a (constant) multiple of the pure connection 3D functional S[CS(W )] S[w] ⇠ in particular critical points are constant curvature connections Pure connection 3D gravity is the 6D Hitchin theory for the Chern-Simons 3-form in the total space of the SU(2) bundle Corollary: the total space of the frame bundle over a negative scalar curvature 3D Einstein manifold is naturally a complex manifold Summary of the 6D construction All 3D geometric information can be encoded into
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