Connection description of 3D Gravity joint work with Y. Herfray - - PowerPoint PPT Presentation

Connection description

• f 3D Gravity

Kirill Krasnov (Nottingham)

joint work with

• Y. Herfray and C. Scarinci

arXiv:1605.07510

Workshop on Teichmueller theory and geometric structures on 3-dimensional manifolds

We usually describe geometry using metrics Natural constructions: Einstein metrics Ricci flow Works in any dimension

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)

Many have previously suggested that metric may not be the best “variable” to describe gravity 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

interpretation via a certain 3-form in the total space of the SU(2) bundle over space(time) connection on space(time) rather than metric In the approach to be described 3D geometry is encoded by

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

Discovered in:

Einstein-Cartan, Chern-Simons descriptions of 3D gravity Plan 3D gravity in terms of connections 6D interpretation Volume gradient flow on connections

• torsion flow

3D gravity in first order formalism

For concreteness - hyperbolic case Riemannian Let e be (co)frame field Compatibility equation (zero torsion)

• unique solution of the algebraic equation for w

Λ < 0 e ∈ Λ1 ⊗ su(2) Metric ds2 = −2 Tr(e ⊗ e) The associated SU(2) spin connection w ∈ Λ1 ⊗ su(2) (locally) re = 0 re ⌘ de + w ^ e + e ^ w w = w(e) (1)

3D Einstein equations Metric has constant curvature Both (1), (2) follow as EL equations from If substitute f = e ∧ e Λ = −1 f ≡ f(w) = dw + w ∧ w curvature 2-form f ∈ Λ2 ⊗ su(2) (2) S[e, w] = Z Tr ✓ f ∧ e − 1 3e ∧ e ∧ e ◆ w = w(e) get EH Lagrangian for the metric

(1),(2) arise as real, imaginary parts of connection Chern-Simons functional Chern-Simons description f(a) = 0 a := w + i e SL(2, C) S[e, w] = Im(SCS[a]) where SCS[a] = Z Tr ✓ a ∧ da + 2 3a ∧ a ∧ a ◆

“Pure connection” description Instead of solving for the connection, can solve e = e(w) f = e ∧ e algebraic equation for e

w

• nly possible for non-

zero scalar curvature!

To describe the solution, need some notions Given f ∈ Λ2 ⊗ su(2) and choosing a volume form v get a map φf : Λ1 → su(2) φf(α) := α ∧ f/v ∀α ∈ Λ1 Can apply this map to the curvature itself λ(f) := 4 3Tr (φf ⊗ φf(f))

Note that the sign here is invariantly defined! must be special in order there to be a solution also possible in higher D

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 For a positive definite connection, can similarly solve e = e(f) In both cases, the associated metric is a Riemannian 3D metric f = −e ∧ e Proof: Define vf := p −λ(f) v does not depend on v, only on its orientation The solution explicitly iξef = (f ∧ iξf − iξf ∧ f)/2vf Can be checked to satisfy f = ef ∧ ef

For a connection that comes from frame a simple calculation shows that ef = √ detR R−1e where R is Ricci gives a Riemannian metric, degenerate where R is degenerate

does not change under constant rescaling of the

• riginal frame

forgets about the scale of the original metric!

f = e ∧ e

the scale is introduced when solving this equation

Λ = −1

Pure connection action Substituting e = ef into the first-order action gives S[w] = Z vf

Volume of the space computed using the metric defined by the connection

Functional on the space of connections of definite sign Its critical points - “constant curvature” connections vf := p −λ(f) v λ(f) := 4 3Tr (φf ⊗ φf(f)) φf(α) := α ∧ f/v

Euler-Lagrange equations S[w] = −2 3 Z Tr (ef ∧ ef ∧ ef) δS[w] = − Z Tr (δ(ef ∧ ef) ∧ ef) = Z Tr (δ(f) ^ ef) = Z Tr (rδw ^ ef) ⇒ ref = 0 second-order PDE on w

says that w is the spin connection compatible with the frame defined by w

• nce this equation is satisfied, the

metric is automatically constant curvature since by construction f = ef ∧ ef

Associated gradient flow Recall the flow that plays role in Floer homology

• the gradient flow of Chern-Simons

da dt = ✓δSCS[a] δa ◆∗ where need a metric to define the * 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 where now the star is defined using the metric defined by the connection Alternatively dw dt = (ref)∗ If decompose w = ˜ w + t ˜ ref = 0 with by definition t-torsion Then ref = t ^ ef + ef ^ t and (ref)∗ is basically torsion Parabolic equation

Flow by torsion - Possibly useful as an alternative to Ricci flow

dw dt = − ✓δS[w] δw ◆∗

For positive connection needs to change the sign

Homogeneous case

w1 = g1e1 f 1 = (2g1 + g2g3)e2 ∧ e3 a1 = s −(2g2 + g3g1)(2g3 + g1g2) 2g1 + g2g3 ds2 =

3

X

i=1

(ai)2(ei)2

If start with metric with

de1 = 2e2 ∧ e3

Get connection

g1 = (a1)2 − (a2)2 − (a3)2 a2a3

Note that in the round case

a1 = a2 = a3 g1 = g2 = g3 = g = −1

Now start with connection Get metric

w1 = g1e1

connection forgets about the scale!

f 1 = −θ2 ∧ θ3 θ1 = a1e1 Λ = +1

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 = 1/2 vol2 = 0

concave function! maximum at the origin

so the flow will return to the round metric 3-sphere

The gradient flow explicitly

g1 = g2 = g3 = g ≡ −1 + x a1 = a2 = a3 = p −(2g + g2) ≡ p 1 − x2 vol(g) = (1 − x2)3/2

gradient flow

metric sphere has the largest volume returns to the metric sphere

g = −1

for connection that comes from the metric

˙ g1 = −a1(2a1 + g2a3 + g3a2) 2a2a3

For the round sphere

˙ x = −x

Compare Ricci flow

ds2 = a2 3 X

i=1

(ei)2 ! d dt(a2) = − 2 a2

Ricci flow gives collapses in finite time

Similarly, for negative curvature the Ricci flow will expand the manifold, while the volume flow just returns to the metric connection

dg dt = Riccig

Not a gradient flow!

Towards 6D interpretation: 3-forms in 6D

Let P be a 6-manifold Consider a 3-form Ω ∈ Λ3P This form is called generic or stable of positive (negative) type it is in the orbit if at each point GL(6, R)/SL(3, R) × SL(3, R) GL(6, R)/SL(3, C)

Ω = α1 ^ α2 ^ α3 + β1 ^ β2 ^ β3, α1 ^ α2 ^ α3 ^ β1 ^ β2 ^ β3 6= 0

Ω = 2 Re (α1 ∧ α2 ∧ α3) , α1, α2, α3 ∈ T ∗

CP

dim(Λ3P) = 6 · 5 · 4/3! = 20

dim = 36 − 8 − 8 = 20

Negative type case: almost complex structure Given a volume form v, can define an endomorphism KΩ : T ∗P → T ∗P iξ(KΩ(α)) := α ∧ iξΩ ∧ Ω/v This endomorphism squares to a multiple of identity KΩ(α)2 = λ(Ω)I For negative type (stable) 3-forms λ(Ω) < 0 Can define JΩ := 1 p −λ(Ω) KΩ J2

Ω = −I

Almost-complex structure For in the canonical form Ω α1,2,3 are (0,1) forms

Hitchin functional vΩ := p −λ(Ω)v S[Ω] = Z

P

vΩ invariantly defined form in given orientation class in a given cohomology class Theorem (Hitchin): Critical points of are integrable S[Ω] under variations JΩ Proof: vΩ = 1 2 ˆ Ω ∧ Ω where ˆ Ω is the result of acting with in all 3 slots of JΩ Ω δS[Ω] = Z ˆ Ω ∧ δΩ = Z ˆ Ω ∧ dB ⇒ dˆ Ω = 0 ⇒ Ωc = Ω + iˆ Ω closed dΩc = 0 Ωc 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 Proposition: Ω defines an SU(2) connection and metric on M Ω Proof: Take JΩ Define the image of vertical vector JΩ fields to be horizontal ⇒ connection in P Define the pairing of horizontal vector fields to be the Killing-Cartan pairing of their vertical images JΩ Let be an SU(2) invariant stable 3-form in P

(with some genericity assumption)

In turn an SU(2) connection on M defines an SU(2) invariant closed 3-form in the total space of the bundle where CS(W) = Tr ✓ W ∧ dW + 2 3W ∧ W ∧ W ◆ W is an SU(2) connection in the total space of the bundle Ω = CS(W) dΩ = 0

Proposition: Then the connection defined by is W When is stable of positive (negative) type Ω W is positive (negative) definite The metric defined by coincides with the one defined by W Ω 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 Ω = CS(W)

Proposition: Hitchin functional for Ω = 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

All 3D geometric information can be encoded into an SU(2) invariant 3-form in the total space of the principal SU(2) bundle Summary of the 6D construction Hitchin theory reduces to 3D gravity in its pure connection formulation Geometrically, the metric arises from the connection because W gives rise to CS(W) in the total space of the bundle, in turn to an almost complex structure, and finally to the metric A particularly useful closed 3-form is Chern-Simons

3D gravity as the dimensional reduction of 6D Hitchin theory A variant of the above story is obtained by considering S[B, Ω] = Z B ∧ dΩ + vΩ ⇒ dΩ = 0 dB = ˆ Ω ⇒ dˆ Ω = 0 defines an integrable ACS Ω The dimensional reduction of this theory to 3D, assuming SU(2) invariance of both 2- and 3-forms, is 3D gravity in Einstein-Cartan formalism

arXiv:1705.04477

Summary

It is possible to describe 3D gravity using SU(2) connections rather than metrics The “explanation” for why the pure connection formulation exists is the 6D Hitchin theory for the Chern-Simons 3- form in the total space of the principal SU(2) bundle 3D gravity can be thought of as a particular dimensional reduction of the 6D Hitchin theory Similar game can be played in 4D!

Interesting open questions Thank you!

Usefulness of the volume gradient flow for connections Since 3D gravity sits inside 6D Hitchin theory, and we know how to quantise 3D gravity, can we quantise 6D Hitchin?