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61f72bb34f41af13ffb571255e05c244d9e0fb0a | subsection | 71 | 289 | Almost Complex Structure | We here very briefly recap some standard results of complex geometry that we will need in the following.Definition II.1 An almost complex structure on a differentiable manifold M is a differentiable endomorphism in the tangent bundle,
J: TM\mapsto TM, such that J^2=-Id .A differentiable manifold with some fixed almost... | {
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029ec9f4e7323b93ff3aebdb0e2cabcc663f80b1 | subsection | 72 | 289 | Almost Complex Structure | Of course if M is a complex manifold with holomorphic coordinates \lbrace z^I\rbrace then
T^{1,0}M=\left\lbrace \frac{\partial }{\partial z^I}\right\rbrace ,\quad T^{0,1}M=\left\lbrace \frac{\partial }{\partial \overline{z}^I}\right\rbrace ,\quad \Omega ^{1,0}\left(M\right)=\left\lbrace dz^I\right\rbrace ,\quad \Omega... | {
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e57b643f7e601d7f91a08d82a310649cb1fd7e2b | subsection | 73 | 289 | Almost complex structure on a Riemannian manifold | We are now interested in the almost complex structures on a Riemannian manifold that are compatible with the metric structure.Definition II.2 An almost complex structure J is said to be compatible with a metric g if for any vector fields X,Yg\left( J(X) , J(Y) \right)= g\left( X , Y \right).As J^2=-Id, this is equivale... | {
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12286f3698fcb048ec88b9397c3b95e5e1151560 | subsection | 74 | 289 | Almost complex structure on a Riemannian manifold | This convention also naturally allows to interpret self-dual two-forms as representation of \mathfrak {su}(2) see in the appendix eq (REF ) and below.\omega &= -\left(\alpha _{(A^{\prime }} \hat{\alpha }_{A^{\prime })}\;\epsilon _{AB} + \beta _{(A} \hat{\beta }_{A)}\;\epsilon _{A^{\prime }B^{\prime }}\right)\;\frac{e^{... | {
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bfac4c842a08066ce766da0720b440af660c9e32 | subsection | 75 | 289 | The Twistor Space of a General Riemannian 4-manifold; essential results | We are now in a position to describe the essential structure of the twistor space of a Riemannian manifold. See , and reference therein for the original results. We however presents these results in a way that is suitable for our `pure connection' generalisation. | {
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c18975a597d03c8a9ec0b90c15c2e25d3227383f | subsection | 76 | 289 | The Twistor Space of a Riemannian manifold | Given a Riemannian manifold \left(M, g\right) the Twistor space of M, M), is the total space of the primed bundle, i.e locally M) \simeq S^{\prime } \times M. The associated projective Twistor space \mathbb {PT}(M) is just M) with projectivised fibres, locally \mathbb {PT}(M) \simeq \mathbb {C}{P}^1 \times M.The discus... | {
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741f7f7393b79c879a73bb9c805b181a4335e757 | subsection | 77 | 289 | Body | We here emphasise the geometry induced on M) by a {\rm {SU}}(2)-connection only. It will serve as a starting point for our `connection approach' to Twistor theory.Accordingly, we now take `space-time' to be a SU(2)-principal bundle{\rm {SU}}(2) \hookrightarrow P \rightarrow Mover a four dimensional manifold M equipped ... | {
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Three-Forms | [
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7c46efd18a1ec59609c54ae2060b45b74baf7ff6 | subsection | 78 | 289 | Body | Having SU(2) structure group, the 2 fibres of this bundle come equipped with a hermitian metric we represent by an anti-linear, anti-involutive map,\text{⌃}\colon \left\lbrace \begin{array}{ccc}
2 &\rightarrow &2 \\ \pi _{A^{\prime }} &\mapsto &\hat{\pi }_{A^{\prime }}
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27026017065870187025a99ce7ffa77d4f1a9f05 | subsection | 79 | 289 | Body | The main advantage with this notation is that section of \mathcal {O}(n,m)-bundle over \mathbb {C}{P}^1 (and by extension over \mathbb {PT}(M)) are equivalent to functions f(x,\pi _{A^{\prime }}) with homogeneity n in \pi _{A^{\prime }} and m in \hat{\pi }_{A^{\prime }} .The \mathcal {O}\left(n,m\right) bundles are `na... | {
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e9f480b1dbbbeaae5eb606ac427d5b3f5d023821 | subsection | 80 | 289 | Body | Then we can define its covariant derivative asd_{(n,m)} f df + n\;\frac{\hat{\pi }_{A^{\prime }} D \pi ^{A^{\prime }}}{\pi .\hat{\pi }}\;f - m\;\frac{\pi _{A^{\prime }} D \hat{\pi }^{A^{\prime }}}{\pi .\hat{\pi }}\;fIt is a simple exercise to verify that E d_{(n,m)} f=0, \overline{E}d_{(n,m)} f =0, \mathcal {L}_{E} d_{... | {
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af3646668c50059ca6093aaaf3e8f30d396dbbe2 | subsection | 81 | 289 | Body | In other terms, they respectively reduce the structure group to {\rm {SL}}\left(3,)\right. and {\rm {Sp}}\left(6,\mathbb {R}\right).In order for \left( J_{C} , B\right) to be an almost hermitian structure i.e reduce the structure group to {\rm {SU}}(3) = {\rm {SL}}(3, \cap {\rm {Sp}}\left(6,\mathbb {R}\right), one need... | {
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96c773ee00ad05f09540a20c31bc1391bfb4a449 | subsection | 82 | 289 | Body | This is obviously an open condition.There is however more to the doublet \left(C,B\right) than an hermitian structure \left(J_{C}, B, g\right): suppose that (REF ) is verified and take \left(\alpha ^i\right)_{i \in 1,2,3} a basis of (1,0)-form adapted with the hermitian structure i.eWe recall that our notation it that ... | {
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2e7c5550e5e2487ef48095f73fc01e430c166b6c | subsection | 83 | 289 | Body | This can be seen in different ways but we believe that the most convincing proof is by using concept from Cartan Geometry. Cartan geometry is a beautiful framework generalising both Klein geometry, i.e the geometry of homogeneous spaces and the essential idea of Riemannian geometry which is the make the geometry local.... | {
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9cc17645c5c1fe5ca60883bb35f67851957238b0 | subsection | 84 | 289 | Body | Here the B field just implements the constraint that C is closed, on the other hand the field equations for C say that \hat{C} is exact:dB=\hat{C}.In the previous chapter we saw (see (REF )) that for three-forms in seven dimension, \hat{C}= \frac{1}{3} *C. Here the hodge dual is given by the metric g_{C} constructed fr... | {
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f055287b292a48bf2ec4d03796159ac0f93eec8b | subsection | 85 | 289 | Body | In fact for any (x,y,z) such that x^2+y^2+ z^2=1, xI + yJ + zK is a parallel almost complex structure so that we have a whole S^2 of them. As the notation suggest, Hyperkähler manifolds are related to the geometry of quaternions (see our discussion on quaternionic structure in section REF ) , i.e there is as sense in w... | {
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
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8901e8882b1b2b968dadfb70ea18fbc88438de7c | subsection | 86 | 289 | The Almost Complex Structure on | We now come back to a metric context. We take \left(M, g\right) to be a Riemannian manifold and M) the associated twistor space. As we already explained the self-dual part of the Levi-Civita connection gives a \mathcal {O}(2)-valued one-form on \mathbb {PT}(M) and a connection on the \mathcal {O}(n,m)-bundle over \math... | {
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434d3582beb48783f3ff679bd63f216c6a96f168 | subsection | 87 | 289 | The Almost Complex Structure on | For example,\Sigma ^{A^{\prime }B^{\prime }}\big |_{(2,0)} = \Sigma \pi \pi \, \frac{\hat{\pi }^{A^{\prime }}\hat{\pi }^{B^{\prime }}}{(\pi .\hat{\pi })^2}, \qquad \Sigma ^{A^{\prime }B^{\prime }}\big |_{(0,2)} = \Sigma \hat{\pi }\hat{\pi }\, \frac{\pi ^{A^{\prime }}\pi ^{B^{\prime }}}{(\pi .\hat{\pi })^2},\Sigma ^{A^{... | {
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06278465ea51f3cf89080eb343a5f05f79289b88 | subsection | 88 | 289 | The Contact Structure on | In the context of an almost complex manifold, it is natural to introduce Dolbeault operators on the space \Omega ^{p,q}\left[n,m\right] of \mathcal {O}\left(n,m\right)-valued (p,q)-forms as\begin{array}{lll}
\partial : \left|
\begin{array}{ccc}
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
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d271cb35b9bbd07c08e0107956f7cf583aa4d380 | subsection | 89 | 289 | The Kähler Structure on | Let us now consider the following hermitian structure on \mathbb {PT}(M) (compare with the flat case ())g &= 4R^2\frac{\pi _{A^{\prime }}D\pi ^{A^{\prime }} \odot \hat{\pi }_{B^{\prime }}D\hat{\pi }^{B^{\prime }}}{2\left(\pi .\hat{\pi }\right)^2} + \frac{1}{2}e^{AA^{\prime }}\odot e_{AA^{\prime }} \\
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dd82bc33756cb1cef489fa5db663a55f9d05ce16 | subsection | 90 | 289 | Example: | As we already saw, in the flat case the projective twistor space is the projective space \mathbb {C}{P}^3. The above proposition just says that the Fubini-Study metric () is Kähler. We here recall the form of this metric for convenienceg_{\mathbb {PT}} = \frac{\pi .D \hat{\pi }\odot \hat{\pi }.D\pi }{2\left(\pi .\hat{\... | {
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Three-Forms | [
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afb7a9ba09ec21ab5d147ced6fa397dbf0424a5f | subsection | 91 | 289 | The Non-Linear Graviton Theorem | In its original form, see , the aim of twistor theory was to realise solutions of complicated differential equations on space-time in terms of simpler, essentially free, geometrical data on the associated Twistor space. The key insight was holomorphicity. The original success of twistor theory takes the form of three t... | {
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8f234796abb35a9e22923b9a961698bd4113a434 | subsection | 92 | 289 | Euclidean Twistor Theory Revisited: a Connection Point of View | We now come back on some of the preceding results but from an unusual `connection point of view', the presentation and results from this section are taken from .Accordingly, we now take `space-time' to be a SU(2)-principal bundle{\rm {SU}}(2) \hookrightarrow P \rightarrow Mover a four dimensional manifold M equipped wi... | {
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Three-Forms | [
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3310fc799ba8c8fac048c37e2e18775270da74f7 | subsection | 93 | 289 | Symplectic and Almost Hermitian Structure on | We now restrict ourselves to the case of definite connections (REF ), ie the case where \tilde{X}^{ij} = F^i\wedge F^j /_{d^4x} is a definite 3x3 conformal metric. This is in fact equivalent to the requirement that no real 3-vector \left(v^i\right)_{i\in 1,2,3} is such that v^i\;F^i is a simple two-form:A\; \text{is a ... | {
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7d8e6b84f9e1f2672ffb6015a5243226c54a592f | subsection | 94 | 289 | Symplectic and Almost Hermitian Structure on | In general this triplet is neither Hermitian (J_A is not integrable) nor almost Kähler (\omega _A is non degenerate but generically not closed).We first describe how to construct the almost complex structure J_A on \mathbb {PT}(M) from a definite connection:
Because the connection is definite, one can make sense of the... | {
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Three-Forms | [
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9052e2df4e43c0f12c5635d2c05d2d04abf9fa0d | subsection | 95 | 289 | Symplectic and Almost Hermitian Structure on | This gives a metric on the horizontal tangent space (as defined by A), on the other hand the vertical tangent space comes equipped with a metric and altogether this gives the following metric on \mathbb {PT}(M) Here A \odot B = A \otimes B + B \otimes A:g_A &= 4R^2\frac{\pi _{A^{\prime }}D\pi ^{A^{\prime }} \odot \hat{... | {
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ff5e7940a9810577811434d381dcee470b18cd04 | subsection | 96 | 289 | Symplectic and Almost Hermitian Structure on | As compared to the classical construction from there are however small differences:First the conformal structure is obtained from the connection.Second one does not use the notion of horizontality associated with the (Levi-Civita connection of the) conformal structure but the one given by our original SU(2)-connection.... | {
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388cf4138eda76fc0cc93a52d46facecb5279bc7 | subsection | 97 | 289 | Symplectic and Almost Hermitian Structure on | Thus we can writeF^{A^{\prime }B^{\prime }} = \Psi ^{A^{\prime }B^{\prime }}{}_{C^{\prime }D^{\prime }} \Sigma ^{C^{\prime }D^{\prime }} + \lambda (x)\Sigma ^{A^{\prime }B^{\prime }}\quad \text{with} \quad \Psi ^{A^{\prime }B^{\prime }C^{\prime }D^{\prime }} = \Psi ^{(A^{\prime }B^{\prime }C^{\prime }D^{\prime })}.It w... | {
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45b1819f3ade9ce56e267d72310410d5f2a14d35 | subsection | 98 | 289 | Symplectic and Almost Hermitian Structure on | On the other hand, if the connection is perfect thend\left(e^{BB^{\prime }}\pi _{B^{\prime }}\right)\big |_{(0,2)} = \left(d_A e^{BB^{\prime }} \right)\big |_{(0,2)} \pi _{B^{\prime }} + e^{BB^{\prime }} \wedge D\pi _{B^{\prime }}\big |_{(0,2)} = \left(d_A e^{BB^{\prime }} \right)\big |_{(0,2)} \pi _{B^{\prime }}holds ... | {
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25704495f3f12ace53745677d5fdc928e2789f39 | subsection | 99 | 289 | The Mason-Wolf Action for Self-Dual Gravity | In , L.Mason and M.Wolf described a twistor action for self-dual gravity. It is an action for an \mathcal {O}(2)-valued one-form \tau and a \mathcal {O}(-6)-valued one-form b on some 6d real manifold, the `projective twistor space'. It essentially used a new version of the non linear graviton theorem relying on the equ... | {
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d4e462c1e5f2e7eeb6360b62914a0c4c3b6eebe1 | subsection | 100 | 289 | The Mason-Wolf Action for Self-Dual Gravity | We recall from the previous discussion that\tau \wedge d\tau \wedge d\tau = \tau \wedge F^{A^{\prime }B^{\prime }}\pi ^{A^{\prime }}\pi ^{B^{\prime }}\wedge F^{C^{\prime }D^{\prime }}\pi ^{C^{\prime }}\pi ^{D^{\prime }}.From this one readily sees that the Mason-Wolf action (REF ) is the immediate generalisation of (REF... | {
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
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2c32ba72baa97a619fc68a96d0e8ebd328cc18ba | subsection | 101 | 289 | The Non-Linear-Graviton Theorem Revisited | Up to now we constructed different geometrical structure on \mathbb {PT}(M) from a definite connection. In particular we saw that \mathbb {PT}(M) can be given a Kähler structure when \tau \wedge d\tau \wedge d\tau =0, with \tau = \pi _{A^{\prime }} \left( d\pi ^{A^{\prime }} + A^{A^{\prime }}{}_{B^{\prime }}\pi ^{B^{\p... | {
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8def320390341567cdb301d3108ccb3fb10e678f | subsection | 102 | 289 | Spacetime from | Having constructed an almost complex structure, J_{\tau } on \mathcal {PT} we are now in a similar situation as in where the almost complex structure is taken as a starting point.Following the same steps as in this reference we can construct a Euclidean `space-time' M from \left(\mathcal {PT}, J_{\tau }\right). Then \m... | {
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ce72600db16f5bad67e9a532f8d0b22e16ae00bb | subsection | 103 | 289 | Spacetime from | This will be the case if we construct \tau by a small deformation of the standard holomorphic one-form with values in \mathcal {O}(2) on \mathbb {C}{P}^3.We then define the associated twistor space \mathcal {T} to be the fourth root of the canonical bundle. It is thus a complex line bundle over \mathcal {PT}, \mathcal ... | {
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
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e87fb0b23bfc472d3258e01a27891ee9e8120b41 | subsection | 104 | 289 | A non linear graviton theorem | We now give a new proof of the (euclidean) non-linear-graviton theorem. As explained in introduction, the essential result of this theorem already appeared in but the presentation that we make here is original.Introduce coordinates that form a trivialisation of \mathcal {T}, \lbrace x^{\mu }, \pi _{A^{\prime }} \rbrace... | {
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3d490f331aa18a8eed470177eb1d7bf1a66024dd | subsection | 105 | 289 | A non linear graviton theorem | However this is in contradiction with \tau \wedge \left( d_{a}\tau + \lambda d_{\bar{a}} \bar{\tau }\right) \in \Omega ^{3,0}.(ii) \Rightarrow (iii)If \tau \wedge d\tau \wedge d\tau =0 then by construction \tau \wedge d\tau \in \Omega ^{3,0}. We now take \zeta to be coordinates on \mathbb {C}{P}^1, \partial _{\overline... | {
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9ddc96cd7fe454ce525dd4a15b4ba7820807242a | subsection | 106 | 289 | Discussion on the would be `Twistor action for Einstein gravity' | In two new variational principles for Yang-Mills theory and conformal gravity based on fields living on twistor space were presented. The fact that the fields which appear in this action live on a 6d manifold (`projective twistor space') is compensated by new symmetries of the action and the propagating degrees of free... | {
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c5462cad27377226cad03ab86124fbbc2552c5d6 | subsection | 107 | 289 | The Twistor action for Yang-Mills from the Chalmer-Siegel action | In , the Chalmer-Siegel action for Yang-Mills was taken as a starting point on the way to a twistor action:S\left[{{A}}, {{B}}\right] = \int _M Tr\left( {{B}}\wedge {{F}}- \frac{\epsilon }{2} {{B}}\wedge {{B}}\right)where B is taken to be a lie algebra valued self-dual two-form, ie {{B}}= {{B}}_{A^{\prime }B^{\prime }}... | {
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Three-Forms | [
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db013c940c69bd87c75e8bbb84568e39b930406c | subsection | 108 | 289 | The Twistor action for Yang-Mills from the Chalmer-Siegel action | By Liouville theorem, holomorphicity on a compact manifold ensure that this choice of frame is unique up to a global `rotation'.{{B}}_{A^{\prime }B^{\prime }} = \int _{\mathbb {C}{P}^1} \pi _{A^{\prime }}\pi _{B^{\prime }} \;{{b}}\wedge \tau(where {{b}} is a Lie algebra valued (0,1)-form on \mathbb {PT} with values in ... | {
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6c14249b76669105a6177e7a1f7aa762cd0efedb | subsection | 109 | 289 | A first twistor ansatz... and why it fails | We now would like to take as starting point the following space-time actionS[{{A}},\Psi ] = \frac{1}{2}\int \left(\left(\Psi +\frac{\Lambda }{3} \delta \right)^{-1}\right)^{ij} F^i\wedge F^j.This is an action for gravity and can be obtained from Plebanski's action by integrating out the {{B}} field (see section REF ). ... | {
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Three-Forms | [
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51676ca163d5865a879cb270cf61e3706bd56934 | subsection | 110 | 289 | A first twistor ansatz... and why it fails | As we already described in REF such a one-form is enough to construct an almost complex structure J_{\tau } on \mathcal {PT} and to give it a fibre bundle structure over some space-time M, \mathbb {C}{P}^1 \hookrightarrow \mathcal {PT}\rightarrow M.This action also contains \psi \in \Omega ^1_{ \otimes \mathcal {O}\lef... | {
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Three-Forms | [
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0baa08a9434f6efc1e3d7ddd7c68d959cd51d1d7 | subsection | 111 | 289 | A new action for Gravity as a background invariant generalisation of the Chalmers-Siegel action. | Let us now come back to the Chalmer-Siegel action and consider the special case of a SU(2)-connection:S\left[A, B\right] = \int _M B^i_{A^{\prime }B^{\prime }} \Sigma ^{A^{\prime }B^{\prime }}\wedge F^i - \frac{\epsilon }{2} B^i_{A^{\prime }B^{\prime }}B^i{}_{C^{\prime }D^{\prime }} \Sigma ^{A^{\prime }B^{\prime }}\wed... | {
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Three-Forms | [
"Yannick Herfray"
] | [
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8d1f3652ae357299d57c9c502afc17ddf518fb51 | subsection | 112 | 289 | A new action for Gravity as a background invariant generalisation of the Chalmers-Siegel action. | What is more for \epsilon =0 this action describes anti-self-dual gravity.By construction the \Sigma _A's are such that,F^i = M^{ij} \Sigma _A^j.Our choice of volume form,\frac{1}{3}\Sigma ^i \wedge \Sigma ^i = \frac{1}{\Lambda ^2}\left(tr \sqrt{F\wedge F}\right)^2,is such that Tr M = \Lambda is a constant.Now, varying... | {
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e01a196afe6dc2522721b16974ad2620a4f7c849 | subsection | 113 | 289 | Discussion on a second ansatz | The action in Proposition (REF ) looks like a promising starting point to construct ansatz for twistor action for gravity. It indeed has many appealing features. First it explicitly separates the self-dual sector (\epsilon =0) of the theory from the full theory (\epsilon \ne 0). Second it superficially looks like the s... | {
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ecfdb14e369f65ec88732b9f438b3c422e9f1353 | subsection | 114 | 289 | Discussion on a second ansatz | \end{equation}Where in this last line one should integrate over \pi ^{-1}(x) \simeq \mathbb {C}{P}^1.An appealing feature of actions of this type is that, linearising around a given background (let say describing flat space-time) we obtain \delta \psi \in H^{0,1}\left(\mathcal {PT}, \mathcal {O}(-6)\right) and \delta \... | {
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Three-Forms | [
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818e944570e1513838c48c204c6dc170f544de7e | subsection | 115 | 289 | Discussion on a second ansatz | From this we can follow the same procedure as in the first section and construct \Sigma :\Sigma ^i\left( x,\zeta \right)= X^{-\frac{1}{2}}{}^{ij}B^j \qquad \Sigma ^{A^{\prime }B^{\prime }}=\sigma ^{A^{\prime }B^{\prime }}_i \Sigma ^isuch that \Sigma ^i \wedge \Sigma ^j \propto \delta ^{ij}.
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
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f7f170a79ce6ca2f6bd6eec4491f51cdc6a0418f | subsection | 116 | 289 | Discussion on a second ansatz | It can then be checked that, under such conditions, the twistor action (REF ) coincides with the original space-time action from proposition (REF ).Therefore we could hope that with this definition for \Sigma _{\tau }, the action (REF ) would describe gravity: all we need are the field equations for \psi to imply the e... | {
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Three-Forms | [
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7ea861877bfd6d5d1c755a4eea688ed74b298ff4 | subsection | 117 | 289 | Discussion on a second ansatz | On the other hand\bar{\partial }_{\overline{\zeta }} \left(d\tau \big |_{0,2}\right) =0 \qquad \Leftrightarrow \qquad \bar{\partial }_{\overline{\zeta }}\left(A_{\mu }\right) dx^{\mu }\big |_{0,1} = 0are just not enough field equations to conclude that \bar{\partial }_{\overline{\zeta }}\left(A_{\mu }\right)=0. In this... | {
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Three-Forms | [
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54f68ee2f6c89b45823a238c53c8a9ee9ac53037 | subsection | 118 | 289 | Variations on Hitchin Theory in Six Dimensions | Variations on Hitchin Theory in Six Dimensions | {
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f71e0d502f2a7af65310d4e55851b2903a90b8d9 | subsection | 119 | 289 | Introduction to Part 2: Hitchin Theory and Six Dimensions | equationsection
thmcnterchapter
Introduction to Part 2In the preceding part we saw that, solutions of self-dual gravity are naturally described in the following terms. Start with a {\rm {SU}}(2)-principal bundle{\rm {SU}}(2) \hookrightarrow \rm {P}^7 \rightarrow \rm {M}^4together with a connection {{A}}. Then, constr... | {
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d4830ff66532f69464e08f901b1e8b26864185b2 | subsection | 120 | 289 | Introduction to Part 2: Hitchin Theory and Six Dimensions | Theories of the form (REF ) are very well-known see , and we will refer to these as `Schwarz type' theories. These are obviously diffeomorphism invariant and they are known to be topological. The partition function of Schwarz type theories is a variant of Ray-Singer analytic torsion of a manifold.Accordingly, in our si... | {
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f22f56739d0792cda2f9d9866e07c1c127b28a05 | subsection | 121 | 289 | Introduction to Part 2: Hitchin Theory and Six Dimensions | The resulting theory on M^3 is (Euclidean) 3D gravity with non-zero cosmological constant coupled to a (constant) scalar field.In particular, the Hitchin functional is then simply related to the pure connection action of 3D gravity as\int _{p^6} \phi \left[C\right] \propto \int _{M^3} \left(1+ \Lambda \rho ^2 \right) v... | {
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835c8353253ee056647c14393094bec943233e03 | subsection | 122 | 289 | Introduction to Part 2: Hitchin Theory and Six Dimensions | When \Omega is taken to be {\rm {SU}}(2)-invariant this identification `fits^{\prime } with the {\rm {SU}}(2) action in such a way that
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afc951e822f8fc08a9a1969c2d865511c05b3865 | subsection | 123 | 289 | Hitchin Theory: An Overall Picture | The aim of this chapter is two-fold. One the one hand we review after , , the notion of stability for differential forms and the related geometrical construction, in particular Hitchin's volume and functional. On the other hand we make use of these notions to introduce action functionals for differential forms in six d... | {
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3c29865f24a2f010419e03210eebad93803c87fa | subsection | 124 | 289 | Stable Forms | The notion of `stability' of a skew-linear form is, in essence, a purely algebraic concept. It is thus simpler to start at a linear algebra level before considering generalisation to differential geometry. Accordingly, everywhere in this section, let E be a n-dimensional vector space. Stability of skew linear form is a... | {
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d07948abb50796ae66302ecab037f2a2c2dd1268 | subsection | 125 | 289 | Non-degenerate two-forms | For any skew-bilinear forms \alpha \in \wedge ^2 E^* there is a well-known notion of non-degeneracy. Consider the endomorphism \iota \alpha \colon E \rightarrow E^* defined by the interior product,\iota _{X} \alpha \alpha (X,.).One says then that \alpha is non-degenerate (or pre-symplectic) if the kernel of \iota \alph... | {
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
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ef0b89aef947e4d604736ecdd35a45314c30bf37 | subsection | 126 | 289 | Action of the General Linear Group on Forms | Let us now consider the group of linear isomorphisms End\left(E\right) \simeq GL(n) on E. Its natural action on E extends to k-skew-linear form: Let \alpha \in \wedge ^k E^* , then for any g \in End\left(E\right) \simeq GL(n) we defineg.\alpha \left(X_1, ..., X_k\right) \alpha \left(g(X_1), ..., g(X_k)\right), \qquad \... | {
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Three-Forms | [
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09d2d7d7c84c7a0d7fb4bf2732f12187b124be6b | subsection | 127 | 289 | Stable forms | One might wonder if there is a way of generalising these notions to any k-form.
In , Hitchin considered the followingDefinition III.4 A k-skew-linear form of E is said to be stable if and only if \mathcal {O}_{\alpha } is an open subset of \wedge ^{k} E^*Stable 2-skew-linear forms coincide with the maximally non-degene... | {
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49421950c8f8bbdb9b1d75e18814f70f488fd6c1 | subsection | 128 | 289 | Definition | We now take \left(k,n\right) \in \mathbb {N}^2 such that k-forms exist in n dimensions. Let \rho be any stable k-form and take \mathcal {O}_{\rho } its orbit. By definition, this is an open subset \mathcal {O}_{\rho } \subset \wedge ^k E^*. Looking at the list (REF ) one sees that all stabilisers are subgroups of {\rm ... | {
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Three-Forms | [
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a91f46bcc79701c2f540db3d6662c718fe72ad71 | subsection | 129 | 289 | Hitchin functional for stable 2 form | Let \omega \in \wedge ^2E^* be a stable form in dimension n= 2m. Then\Phi (\omega ) = \frac{1}{m!}\omega ^m.As a slightly less trivial example, one can look at the dual case i.e ( 2m-2)-forms: | {
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Three-Forms | [
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7824118bb59e5653a77de4f94ab97131e8da9b75 | subsection | 130 | 289 | Hitchin functional for stable | Let \rho \in \wedge ^{2m-2}E^* be stable. As already discussed this implies that \rho = \frac{1}{(m-1)!}\omega ^{m-1} with \omega a stable 2 form. Then Hitchin functional is again the Liouville form for \omega (REF ).This can be constructed directly from \rho as follows. First we make use of the isomorphism \wedge ^k E... | {
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Three-Forms | [
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592b8ca773bf44d35f1d287a4280d0dfd10e15d2 | subsection | 131 | 289 | The `hat' operator on stable forms | The differential of \Phi at \rho is\delta \Phi \in \left(\wedge ^k E^* \right)^*\otimes \wedge ^n E^* \simeq \wedge ^k E \otimes \wedge ^n E^*Using again the canonical isomorphisms \wedge ^k E \otimes \wedge ^n E^* \simeq \wedge ^{n-k} E^* , one can thus define an operator\text{⌃}\colon \wedge ^k E^* \rightarrow \wedge... | {
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Three-Forms | [
"Yannick Herfray"
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3fa4c5c5d6435ddff6da226efd68a776fa6637f4 | subsection | 132 | 289 | Critical Points | Let us now consider a n dimensional manifold M^n. We say that \rho \in \Omega ^k(M^n) is a stable k-form on M^n if it is globally defined and stable at each point. In particular, it reduces the structure group of the tangent space to the Stab_{\rho } \subset GL(n).We can now make use of Hitchin volumes to set up a vari... | {
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5e36c29d67a05285fd5c49840f30fc07dbcf8441 | subsection | 133 | 289 | Critical Points | Taking the exterior derivative of\alpha ^i \wedge \alpha ^1\wedge \alpha ^2 \wedge \alpha ^3 =0 \quad \forall i \in \lbrace 1,2,3\rbraceone indeed obtainsd\alpha ^i \wedge \alpha ^1\wedge \alpha ^2 \wedge \alpha ^3=0 \quad \forall i \in \lbrace 1,2,3\rbracewhich is just the integrability condition for the distribution ... | {
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6f90b44183f7a11244cf406c6be51579b1ffbee5 | subsection | 134 | 289 | On the `Background independence' of Hitchin Theories | A subtle point in `Hitchin theory' is the prescription of varying inside a cohomology class. It means that one must first choose a cohomology class \left[\rho _0\right] and only then vary the form. Practically this often implies to write the form as\rho = \rho _0+ dB.Then \rho _0 will effectively play the role of a bac... | {
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a99812e6df99a2d7fe6cceb2146c7477ee5abe02 | subsection | 135 | 289 | `Background Independent' Hitchin theories in Six Dimensions | In the setting of a 6-dimensional manifold let's consider the following topological theory:S\left[B,C\right] =\int _{M^6} \;B \wedge dC\qquad B\in \Omega ^2(M^6),\quad C\in \Omega ^3(M^6).As was already pointed out in the introduction to this part, this is a particular example of Schwarz theory , that are genealogicall... | {
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84d940beda7a918521d92460887c8c22a93bab02 | subsection | 136 | 289 | `Background Independent' Hitchin theories in Six Dimensions | As will be explicitly described in the next chapter (see in particular section), a stable three-forms in six dimensions defines a linear operator on the tangent space J_C \colon TM^6 \rightarrow TM^6 such that J_C^2 = \pm \mathbb {Id}. The exact sign in this expression depends on the three-forms considered. Whatever th... | {
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d51f823f465760b9e993953496decf02b7aa741f | subsection | 137 | 289 | A New Action Functional for Nearly-Kähler Manifolds | In the above we successively added to the kinetic term (REF ) the Hitchin volume for two-forms and the Hitchin volume for three-forms. We obtained diffeomorphism invariant topological theories that we coined `background invariant' Hitchin Theories.At this point it is natural to put the two Hitchin volumes together and ... | {
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81c25f4d591be5e909e95a62d6b147b2999b0953 | subsection | 138 | 289 | Remarks | In Hitchin described a generalisation of the volume functional C\wedge \hat{C} to all odd or even polyforms in 6D. There is thus a generalisation of all 3 theories (REF ), (REF ) and (REF ) to polyforms, necessarily involving forms of all degree. It would be interesting to study these theories, and characterise them in... | {
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df54611383cdf9de54be2ca9e7625262fb0afc5f | subsection | 139 | 289 | Hitchin Theories in Six Dimensions | In this chapter we look into the geometrical details of different constructions related to stable forms in six dimensions. The aim is to allow to understand the content of the theories described at the end of the preceding chapter and give a precise sense to the different statements that were made. Accordingly we first... | {
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3c0081cd066da336b4fdf7f7dcacb10f0bf28e60 | subsection | 140 | 289 | Geometry of Stable 3-Forms in Six Dimensions | This section is a review of the geometry of stable forms in six dimensions. In particular we recall how to explicitly construct Hitchin volumes. See , for the modern references. | {
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654dd8062d9d455a73992571629d335dc7a7aebd | subsection | 141 | 289 | Stable three-forms over | A stable three-form in six dimensions is a form that lies in an open {\rm {GL}}(6) orbit. For complex three-forms, there is a single open orbit. On the other hand for real three-forms there are two distinct orbits that can be distinguished by a sign. It is thus easier to start with the description of the situation over... | {
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4a45fe14ccbc416837baac970fbb3d68904153b5 | subsection | 142 | 289 | Stable three-forms over | As a consequence, the distributions they define span the whole tangent space:
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3a66689596e18fe6e1c4386f1c32ee12dd7c4db0 | subsection | 143 | 289 | Stable three-forms over | Rather, making use of the decomposition (\ref {Stable forms in 6d: om-canonical}) and taking a basis of tangent vector \left(\xi _i,\zeta _i \right)_{i\in \lbrace 1,2,3\rbrace } dual to \left(\alpha ^i,\beta ^i \right)_{i\in \lbrace 1,2,3\rbrace } this endomorphism can be rewritten
\begin{equation}
\widetilde{K}_{C} =... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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3f8805aa87965f586fb37605409406381d692123 | subsection | 144 | 289 | Stable three-forms over | A related problem is that in general there is no way to make a distinction between D and \widetilde{D}, e.g eq (REF ) is symmetric under permutation of \alpha 's and \beta 's. Therefore there is no way to choose between one of the two possible `square roots' \Phi _C and -\Phi _C in (). In what follows we will however c... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
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4f2efa481989d7f460c447df59344efbd65e7105 | subsection | 145 | 289 | Stable three-forms over | We now consider real three-forms where this ambiguity is less drastic. | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
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c2d840dfc5b2854d0a5d57832f180095521b0158 | subsection | 146 | 289 | Stable forms over | There are exactly two {\rm {GL}}(6,\mathbb {R}) stable orbits of three-forms, characterised by the sign of\frac{1}{6} \textrm {Tr}(\widetilde{K}_C^2) \in \left(\Omega ^6(M)\right)^2.Note that, despite the fact that (REF ) is not really a number but rather a section of \left(\Omega ^6(M)\right)^2, its sign is invariantl... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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7a4452b6f6247ba7a5014f0491ad4edeedcb1895 | subsection | 147 | 289 | Stable forms over | We suppose that such a choice has been made and choose \Phi _{C} such that it is oriented accordingly.Practically, if v is an oriented volume form we can write\Phi _{C} := \sqrt{\lambda } \;v.Where \lambda is defined by (REF ).Integrating this volume form we get the Hitchin functional for the positive orbitS[C] := \int... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
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192d6809237db0ed10e1389e44d2296214c16dbd | subsection | 148 | 289 | Stable forms over | As in the positive sign case solving this ambiguity amounts to making a choice of orientation.We now define the Hitchin volume as\Phi _{C} \pm \;\sqrt{-\frac{1}{6} \textrm {Tr}(\widetilde{K}_C^2)}Integrating this form we obtain the Hitchin functional for the negative orbitS[C] := \int _M \Phi _C \quad \in \mathbb {R}.N... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
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6da4d948ecd5152c63161a9eae2d2f80fd32304f | subsection | 149 | 289 | Hitchin Functional | Let us consider a three-form C on M. We formally write the Hitchin functional as\Phi \colon \left|\begin{array}{ccc}
\Omega ^3_{(M) & \rightarrow & \\
C & \mapsto & \int _M \Phi _C.
}
\end{array}
Because of the sign ambiguity for \right.\Phi _{C} at each points of the manifold this does not really make sense for comple... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
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2f80afb43f7028bc64c17e2cc493bbe829cb9480 | subsection | 150 | 289 | Stable forms, Hermitian structure and Nearly Kähler Manifold | In the preceding section we saw how a negative stable three-form in six dimensions C defines an almost complex structure J_C. Recall, that stable two-forms B are the non-degenerate ones. We now come to the case where the two structures interact to give an (almost) hermitian structure, i.e a compatible triplet \left(J_C... | {
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"Yannick Herfray"
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aa0e05a3f18dc2a81fa4c3277e8a109cf86c983a | subsection | 151 | 289 | Calabi-Yau manifold | Let B and C respectively be two-forms and three-forms satisfying the compatibility equations (REF ) and (REF ) and satisfying the necessary open conditions for\left(J_{C}, B, g \right)to be a good almost hermitian structure. I.e such that J_{C} is an almost complex structure and g is definite.Let us further suppose tha... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
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8fc5609030fb464fa928027ae45ebb53c6373a62 | subsection | 152 | 289 | Nearly Kähler Manifold | At the end of chapter we considered the following variational principles\left[C, B\right] = \int _{M^6} BdC + \Phi \left(B\right) + \Phi \left(C\right),\qquad C\in \Omega ^3\left(M^6\right),\qquad B \in \Omega ^2\left(M^6\right).The resulting field equations are (after rescaling the fields)d\hat{C}= B^2,\qquad dB = C.A... | {
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Three-Forms | [
"Yannick Herfray"
] | [
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f991cc07ecc3ef76e4251de51dba37ab605c1ba1 | subsection | 153 | 289 | Constrained Hitchin functional | As a point of comparison, we briefly described how nearly Kähler manifold were obtained in .Let, C be a stable exact three-form, and let \rho be a stable exact 4-formC = d \alpha , \qquad \rho =B^2 = d \beta .In , Hitchin considered a constrained variational principle. Mainly he looked for the critical points (in a coh... | {
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... | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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dd661fd3c61591e813fa2952194774fb1946d2aa | subsection | 154 | 289 | Example: The nearly Kähler structure of | As a neat example of nearly Kähler structure, and in order to relate this part with the first one, we briefly describe here the nearly Kähler structure on the twistor space of an anti-self-dual Einstein metric.In the first part of this thesis we considered a first almost hermitian structure on \mathbb {PT}(M^4)\Omega ^... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0894a3eebb46c18ded52015374eb21074c5b45cc | subsection | 155 | 289 | Example: The nearly Kähler structure of | In particular on \mathbb {C}{P}^3 this gives the Fubini-Study metric (which is indeed known to be Kähler-Einstein)However we here wish to now consider the alternative `non-integrable' almost hermitian structure:C = \Omega + \bar{\Omega }with\Omega = \frac{1}{\left(\pi .\hat{\pi }\right)^2}\;\pi _{A^{\prime }}D\pi ^{A^{... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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d25f7affda1753f9961eb33f567228d4f39840a8 | subsection | 156 | 289 | Three Dimensional Gravity as a Dimensional Reduction of Hitchin Theory in Six Dimensions | In this chapter, we want to establish that {\rm {SU}}(2) reduction of the `background independent' version of Hitchin theory is 3D gravity coupled with a constant scalar field. We first show how 3D gravity can be naturally understood from a 6D point of view and, what is more, is a subset of solutions to Hitchin theory.... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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ecbef1c0ab2ae69b83708b6f8521db013f62071e | subsection | 157 | 289 | 3D Gravity in Terms of 3-Forms on the Principal Bundle of Frames | There are several alternative ways of writing the canonical form of C. We already consideredC = \alpha ^1\wedge \alpha ^2 \wedge \alpha ^3 + \beta ^1\wedge \beta ^2 \wedge \beta ^3.Which is given by the proposition .Yet another way of writing C arises if we set\alpha ^i = W^i+ \sqrt{\Lambda } \;E^i, \qquad \beta ^i = W... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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676b360127c7764950602cffd817563d4065a75c | subsection | 158 | 289 | Lift from 3D gravity to 6D | We now recall the relationship between the Riemannian geometry of a base manifold M^3 with that of the total space P^6 of the related {\rm {SU}}(2) principal bundle. Our notations for 3D gravity are standard ones so we don't feel that it is necessary to detail them in the main body of this thesis. See however appendix ... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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b73bfd650819ec0fee1051ef7d9670279367c4fb | subsection | 159 | 289 | Connection forms and forms on a principal | Thus, we want to establish the classical relation between the potential one-form {{w}} of a {\rm {SU}}(2) connection on M^3 and a more geometrical description in terms of Lie algebra valued one-forms on the total space P^6 of the principal bundle.Consider the total space P^6 of the principal {\rm {SU}}(2) bundle over M... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
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a1048f759fc1dfe11257e1cfae404ff1db881a2e | subsection | 160 | 289 | Connection forms and forms on a principal | Therefore, a change of trivialisation amounts to a change of coordinate described by the following diagram.x1) at (0,0)
\begin{array}{c}
U_p\\
p \in \end{array};y0) at (6,1.)
\begin{array}{c}
{\rm {SU}}(2) \times V_x\\
\left(g, x\right) \in \end{array};y2) at (6,-1.)
\begin{array}{c}
{\rm {SU}}(2) \times V_x\\
\left(h^... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
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901eb692ae082a396188e40ffa0b84d0c90fb008 | subsection | 161 | 289 | Connection forms and forms on a principal | Accordingly, this is a geometrically simple object on P^6 - a Lie-algebra valued one-form.In general terms a connection one-form is a Lie-algebra valued one-form in the total space of the bundle, whose kernel defines the notion of horizontal vector fields. Importantly it reduces to the Maurer Cartan frame when restrict... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
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f0b08c422fd4c70b8f9dc1a2a8627a16f812e82d | subsection | 162 | 289 | Chern-Simons Connection forms and Hitchin Equations | A standard approach to 3d gravity is in terms of the so called Chern-Simons formulation. See appendix for conventions.Let {{a}}{{w}}+ \sqrt{\Lambda } {{E}} and {{\widetilde{a}}}{{w}}- \sqrt{\Lambda } {{E}} be the potential of the Chern-Simons connections. Here \sqrt{\Lambda } is a mnemonic standing for \sqrt{|\Lambda |... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
255db71e2b5ccfe6006961c88c5dfe48596da622 | subsection | 163 | 289 | Chern-Simons Connection forms and Hitchin Equations | What is more, the sign of its orbit corresponds to the sign of \Lambda !Indeed when \Lambda is positive both families \left(A^i\right)_{ \in 1,2,3} and \left(\widetilde{A}^i\right)_{ \in 1,2,3} are real forms which implies that (REF ) is in the positive orbit. On the other hand, when \Lambda is negative they are comple... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
fb874e0b03b14c711753db57d989379b1502de25 | subsection | 164 | 289 | Chern-Simons Connection forms and Hitchin Equations | In fact a closer look at the equations shows that this is an equivalence:Proposition V.1For three-forms on P^6 constructed from a frame field on M^3 and a {\rm {SU}}(2)-connection as in (REF ) and (REF ), the following system of equations are equivalentdC=0, d\hat{C}=0 `Hitchin's Equations'
{{F}}=0, {{\widetilde... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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-0... | |
fcc5bbd747b1df86f261ce47dc04f542682f4337 | subsection | 165 | 289 | The pure connection formulation of 3D gravity | In this section we review the pure connection description of 3D gravity. It seems that the pure connection formulation of 3D gravity was first worked out in , starting from the Hamiltonian point of view. A simpler description, directly at the level of the Lagrangian, appears in Section 3.4 of . We here only give the La... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/0264-9381/9/9/014",
"end": 203,
"openalex_id": "https://openalex.org/W2080756086",
"raw": "Peldan, P. (1992). Connection formulation of (2+1)-dimensional Einstein gravity and topologically massive gravity. Class. Quant. Grav., 9:207... | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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687eef0b0600906fc5d5d6dbcc14aa735df6ec0c | subsection | 166 | 289 | Definite | For now let us consider again a {\rm {SU}}(2)-principal bundle P^6 on a 3-manifold M^3. Let {{w}} be the potential of a {\rm {SU}}(2)-connection. Let {{f}}=d{{w}}+{{w}}\wedge {{w}} be its curvature two-form. It is convenient to see the curvature as a map from bi-vectors to \mathfrak {su}(2):{{f}}\colon \Lambda ^2\;TM \... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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-... | |
89f091b2e9712888ce3d26c6e0c9fb5d4c0ea6f2 | subsection | 167 | 289 | Definite | For any volume form \nu ,\widetilde{\alpha }\wedge \widetilde{\beta }\wedge \tilde{\gamma }= - \lambda \left(\nu \right)^2, \quad \lambda \in \mathbb {R}Then \left(\alpha , \beta , \gamma \right) is oriented according to the above rules if and only if \lambda >0. Note the minus sign necessary here. That the two orienta... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0... | |
52f217916d5e4ab125ce9588cbe62a347a7b9e1f | subsection | 168 | 289 | Definite | Explicitly:&\left(dx^3\right)^3 \;\frac{1}{3}\textrm {Tr}\left( {{\widetilde{f}}}^{\mu } \left[{{\widetilde{f}}}^{\nu }, {{\widetilde{f}}}^{\rho } \right] \right) \;\partial _{\mu }\otimes \partial _{\nu } \otimes \partial _{\rho }\\ &= -\left(dx^3\right)^3\;\frac{\epsilon ^{ijk}}{6} \widetilde{f}_i^{\mu } \widetilde{f... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
9389adee1f770b0f40edf107fbd1c66f2d5f90cf | subsection | 169 | 289 | The pure connection formulation | Consider a definite connection {{w}} with sign \Lambda =\pm 1. The pure connection formulation gravity action is just the total volumeS_{GR}[{{w}}] = -\Lambda \; \int _{M^3} \; v_{{f}}.An interesting property of definite connections with sign \Lambda = \pm 1 is that they define a frame field {{e}}_{{f}}\in \Omega ^1\le... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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cd1909e29d24e21dc87ec1d96b0b592d89eec3d7 | subsection | 170 | 289 | The pure connection formulation | More explicitly,{{\widetilde{e}}}= \frac{1}{2} \epsilon ^{ijk} \widetilde{f}^{\mu j} f^k{}_{\mu \nu } \;dx^{\nu } \otimes \left(d^3x\right) \otimes \; \sigma _i .{{\widetilde{e}}}&= \frac{1}{2} \epsilon ^{ijk} \widetilde{f}^{\mu j} f^k{}_{\mu \nu } \;dx^{\nu } \left(dx^3\right) \; \sigma _i \\
&= \frac{1}{2} \epsilon ^... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... |
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