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8fe081fd0a25903c38826e6d8f76793b396ff601 | subsection | 171 | 289 | The first variation and Euler-Lagrange equations | The expression (REF ) makes it clear that the first variation of the pure connection action (REF ) is given by\delta S[{{w}}]= - \Lambda \;\int \,\textrm {Tr}(\delta {{e}}_{{f}}\wedge {{e}}_{{f}}\wedge {{e}}_{{f}}) = - \Lambda \;
\int \textrm {Tr}(\delta ({{e}}_{{f}}\wedge {{e}}_{{f}}) \wedge {{e}}_{{f}}) =
\int \textr... | {
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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",
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c37ce9d8b9331fdb9b73040b57f25d116241deb6 | subsection | 172 | 289 | Hitchin Functional, the Chern-Simons three-form and the Pure Connection Formulation of 3D Gravity | Let us now come back to the 6D notations. We consider again the three-form (REF ) which we rewrite here for convenience.C = -2\frac{\textrm {Tr}}{3}\left({{A}}\wedge {{A}}\wedge {{A}}+ {{\widetilde{A}}}\wedge {{\widetilde{A}}}\wedge {{\widetilde{A}}}\right).Our initial question was What is `Hitchin Theory' for this par... | {
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Three-Forms | [
"Yannick Herfray"
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e2d13b526cc901b57b40722bea0f5279310ae981 | subsection | 173 | 289 | Closing | In order to get a geometric interpretation of the constraints dC =0 it is best to open up again the Chern-Simons connections in terms of connection and triad:{{A}}&{{W}}+ \sqrt{\Lambda } {{E}}= g^{-1}dg + g^{-1} \; {{a}}\; g \\
{{\widetilde{A}}}&{{W}}- \sqrt{\Lambda } {{E}}= g^{-1}dg + g^{-1} \; {{\widetilde{a}}}\; g.R... | {
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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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6f05e78d618ba5a5e5e54264381bd4b91739d199 | subsection | 174 | 289 | Closing | Once the constraints (REF ) are satisfied we have a theory of a {\rm {SU}}(2) connection only!Coming back to the three-forms (REF ), it can now be rewritten (just making use of (REF ))C = 4\;\textrm {Tr}\left(-\frac{1}{3} {{W}}\wedge {{W}}\wedge {{W}}+ {{W}}\wedge {{F}}_{{{W}}} \right)or equivalentlyC = -2\;CS\left({{W... | {
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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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5790d7a019e3d2b0f2b35ed5af4c381bbfdf9e22 | subsection | 175 | 289 | Closing | Rather, it turns out to be useful to repackage \alpha and \widetilde{\alpha } and parametrise them in terms of two scalar fields k and \rho :\alpha = \frac{k}{2} \left(1 + \sqrt{\Lambda }\rho \right), \qquad \widetilde{\alpha }= \frac{k}{2} \left(1 - \sqrt{\Lambda }\rho \right)and shift {{W}}{{W}}^{\prime } = {{W}}+ \r... | {
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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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e620919504b4c984a02fe0027379b8dd49e9fedd | subsection | 176 | 289 | Closing | We however already did this computation in the pure 3D gravity case, see (REF ):
l
dC =dkk C + k Tr( -2 W' W' (F'W' + EE) - W' (dW' [EE]) )
From the above, one easily reads off the constraints asdC=0 \qquad \Leftrightarrow \qquad k=cst,\quad {{F}}^{\prime }_{{{W}}^{\prime }} +\frac{\Lambda }{2}\; \left[{{E}}\wedge {{E}... | {
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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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c36c559411475533e33103eacd88afaabff4e3c2 | subsection | 177 | 289 | Closing | We obtaind\Omega = \left[ - \frac{\Lambda }{3} \alpha ^3 + 4 \left( \alpha \beta ^2 \right)^{\prime } \right] \left( y^i
d_{\widetilde{A}} y^i \right) \wedge \left( d_{\widetilde{A}} y^i \wedge \Sigma ^j \right) \, .Thus, we must have4 \left( \alpha \beta ^2 \right)^{\prime } = \frac{\Lambda }{3} \alpha ^3in order for ... | {
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Three-Forms | [
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136da6afd2d2f5a0c96ee7f50635f09308e7821e | subsection | 178 | 289 | Hitchin Functionnal and The Pure Connection Action | We now compute the Hitchin's action (REF ),(REF ) on our three-form \Omega . We use the expressions (REF ),(REF ) and rewrite \hat{\Omega }\hat{C}= -2\left(\sqrt{\Lambda }\right)^{3}\frac{\textrm {Tr}}{3}\left({{A}}\wedge {{A}}\wedge {{A}}- {{\widetilde{A}}}\wedge {{\widetilde{A}}}\wedge {{\widetilde{A}}}\right).Note t... | {
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Three-Forms | [
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225cc5499a4d733139b400733376e62615940133 | subsection | 179 | 289 | Hitchin Functionnal and The Pure Connection Action | We can also rewrite the last term here as a multiple of the first, using some simple properties of the Lie algebra generators \sigma ^i\textrm {Tr}\left( {{W}}{{E}}^2\right) \textrm {Tr}\left( {{W}}^2 {{E}}\right)= \frac{1}{3} \textrm {Tr}({{W}}^3) \textrm {Tr}({{E}}^3).Thus, overall\Phi \left[C\right] = \frac{1}{2} \h... | {
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814e70be9654616014f3de5490fb4ca857573525 | subsection | 180 | 289 | A Very Brief Introduction to Cartan Geometry | Cartan geometry is a generalisation both of Riemannian geometry and Klein geometry.In Riemannian geometry a d-dimensional manifold can be infinitesimally identified with \mathbb {R}^d. This is indeed the role of the metric. The Riemann curvature tensor is then the obstruction to make this identification local, we won't... | {
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Three-Forms | [
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8160b3b6d6dbefc9b0c3c5fea427cf84f626dbf2 | subsection | 181 | 289 | Non Abelian Generalisation of the Fundamental Theorem of Calculus | Let G be a lie group, \mathfrak {g} its Lie algebra. We note {{m}}_G the Maurer-Cartan form on G. We take M to be a smooth manifold.If f \colon M \rightarrow G is a smooth map we can define its `Darboux derivative' {{\omega }}_f, a \mathfrak {g}-valued one-form on M, as{{\omega }}_f = f^* ({{m}}_G).Reciprocally we will... | {
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22f8ce670288ea3e20cd7c0d98c08769f152b0cc | subsection | 182 | 289 | Non Abelian Generalisation of the Fundamental Theorem of Calculus | Then, for each point p\in M, there is a neighborhood U of p and a smooth map f\colon U \rightarrow G such that {{\omega }}\big |_U = {{\omega }}_f.A related theorem isTheorem V.6 "Monodromy representation of {{\omega }}"Let p and q be two point in M. Let \sigma be a path in M starting at p and ending at q. If {{\omega ... | {
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Three-Forms | [
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3cbe4272d48f9e8aa37840de3622ee8f91f84f5c | subsection | 183 | 289 | Cartan Geometry and the Tractor Connection | Let M be a d dimensional manifold and let H \hookrightarrow P \rightarrow M be a principal H bundle.In general the topology of M (respectively P) will be very different from the topology of G/H (respectively G). However one can follow the philosophy of Riemannian geometry and try to make an infinitesimal identification... | {
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269685d29115ff01fbc047515802a4c3e373a84a | subsection | 184 | 289 | The Flat Case | Before we get to the full fledge definition of Cartan geometry it is good to take some times to look at the flat model:H\hookrightarrow P=G \rightarrow M=G/H.Accordingly G is though of as the total space of a H bundle over G/H.On G we have the Maurer-Cartan one-form {{m}}_G. It's fundamental geometrical meaning is to i... | {
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26be1f4c09cd524b19fd4c64c9679ee5e3343d2b | subsection | 185 | 289 | General Cartan Geometry | The generalisation to curved (non-flat) homogeneous space is now straightforward.Let M be a d-dimensional manifold. A Cartan geometry on M modelled on (G,H), where G is a Lie group of dimension n and H a subgroup of dimension n-d, consists of the following data:A principal H bundle over MH\hookrightarrow P \rightarrow ... | {
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206e2dd51ac0fb3f1ba7f450f8d113f178cda374 | subsection | 186 | 289 | Curvature | One can now define the curvature of a Cartan connection as{{\Omega }}= d{{\omega }}+ \frac{1}{2} [{{\omega }},{{\omega }}]or equivalently as the curvature of the tractor connection. The two being related by restriction from P^{\prime } to P. The generalised version of the fundamental theorem of calculus asserts that wh... | {
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dfb23b845789a5bf12ea5da82b749be5459e4576 | subsection | 187 | 289 | A canonical example: 3d gravity | Gravity in three dimensions has no propagating degrees of freedom and Einstein's equations are just the statement that locally M^3 is an homogeneous space. This is exactly what Cartan connections are good for! Depending on the sign of the cosmological constant all we need to do is find a Cartan connection associated wi... | {
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af3e3f1604e66aa0102afd19237c7ae0c05186cf | subsection | 188 | 289 | 3D Gravity as SU(2) Reduction of 6D Hitchin Theory | We here would like to show that having the Cartan point of view on geometry in mind drastically simplifies the proof given in that the general SU(2) reduction of Hitchin theory is 3D gravity. | {
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d870d6773c8dc9516d5cd6ab32d07cd9734b5c7c | subsection | 189 | 289 | 6d Hitchin theory | As we have already seen in section , in six dimensions a general complex stable three-form C \in \Gamma _{^3(P^6) is equivalent to two independent triples of complex-valued one-forms \left(A^i , \widetilde{A}^i \right)_{i\in 1,2,3} defined up to SL(3, \times SL(3, gauge transformations, where each SL(3, transform acts ... | {
"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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] | 2,018 | en | Physics | [
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b8664ae82a35dba928e9d6942b41315a733386aa | subsection | 190 | 289 | SU(2) reduction of 6d Hitchin field equations | We now consider the case where we have a free SU(2) action on P^6. We note M^3 the 3d quotient manifold M = P^6/SU(2). In particular the infinitesimal version of this action gives us an identification of the Lie algebra \mathfrak {su}(2) with vertical right invariant vector fields:
llll
R* : su(2) [TP6].
Taking a cano... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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ec27aad9cb9d2473ec58d78bd8d74fe8f704a08a | subsection | 191 | 289 | SU(2) reduction of 6d Hitchin field equations | For concreteness{{A}}= A^i \sigma ^i,\qquad {{\widetilde{A}}}= \widetilde{A}^i \sigma ^i.ThenC = -2\frac{\textrm {Tr}}{3}\left( \alpha \; {{A}}\wedge {{A}}\wedge {{A}}+ \widetilde{\alpha }\; {{\widetilde{A}}}\wedge {{\widetilde{A}}}\wedge {{\widetilde{A}}}\right)With this new parametrisation the Hitchin field equations... | {
"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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020017470c26e3d278fd83d54a563d99f1bd9e43 | subsection | 192 | 289 | SU(2) reduction of 6d Hitchin field equations | In the negative case {{A}} is an {\rm {SU}}(2)-equivariant \mathfrak {sl}(2,-valued one-form and (REF ) is a Cartan geometry modelled on \left({\rm {SL}}(2, , {\rm {SU}}(2)\right).The field equations (REF ) then implies that these Cartan geometries are flat,{{\Omega }}= \left(d{{A}}+ \frac{1}{2}\left[{{A}},{{A}}\right]... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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29dec9555de576cb10c46e471c27db724777553e | subsection | 193 | 289 | SU(2) reduction of Hitchin Theory | We just saw how the {\rm {SU}}(2) reduction of a three-form C together with the field equations d C=0, d\hat{C}=0 is just 3d-gravity. We now want to consider the associated variational principle. We again consider a {\rm {SU}}(2) invariant three-formC = -2\frac{\textrm {Tr}}{3}\left( \; \alpha \;{{A}}\wedge {{A}}\wedge... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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46c92ce3031ee8661defa2a6e1ca3eeb9905eb65 | subsection | 194 | 289 | Hitchin Functional and The Pure Connection Action, again. | We now turn to the Hitchin functional.\Phi \left[C\right]= 4\;\alpha \; \widetilde{\alpha }\; \left(\sqrt{\Lambda }\right)^3\frac{1}{3}\textrm {Tr}\left( {{A}}\wedge {{A}}\wedge {{A}}\right)\wedge \frac{1}{3}\textrm {Tr}\left( {{\widetilde{A}}}\wedge {{\widetilde{A}}}\wedge {{\widetilde{A}}}\right)We essentially alread... | {
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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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58831255028a4a2c18661c86cebe9235c2ab1d21 | subsection | 195 | 289 | Variations on Hitchin Theory in Seven Dimensions | Variations on Hitchin Theory in Seven Dimensions | {
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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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3e62c885ba35b4cb01c2445d2962970a3edcfc86 | subsection | 196 | 289 | Introduction to Part 3: Hitchin Theory and Seven Dimensions | equationsection
thmcnterchapter
Introduction to Part 3In the first part of this thesis, quaternion geometry and its consequences played a major role. First, because of the identification \mathbb {R}^4 \simeq \mathbb {H} and the isomorphism{\rm {SO}}(4) \simeq {\rm {U}}(1,\mathbb {H}) \times {\rm {U}}(1,\mathbb {H})th... | {
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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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b737b424ec404b5d9b7ded3f01ce6fb2e913805d | subsection | 197 | 289 | Introduction to Part 3: Hitchin Theory and Seven Dimensions | Then critical points in Hitchin theory have the following metric interpretation: Three-forms that are solutions of the seven-dimensional Hitchin theory then give metric with holonomy G_2 (see below for a general discussion on holonomy in Riemannian geometry).This has the following important consequences. When consideri... | {
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d9b78b7f4c78577c46db7cb93e89b3067fb4fa89 | subsection | 198 | 289 | Introduction to Part 3: Hitchin Theory and Seven Dimensions | While the author of this thesis was not directly involved in those works, some of the aspects will be reviewed here, in order to present as complete a picture as possible. Most details, however, will be left aside and the interested reader should consult , .This last part of the thesis is organised as follows. In the f... | {
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113d409ad9d6d56ec98910a851189e334a7995b0 | subsection | 199 | 289 | Hitchin Theory in Seven Dimensions | In this chapter we first review the geometry of stable three-form in seven dimensions: A globally defined stable positive three-form in seven dimensions gives a G_2 structure. Just like almost complex structure identified the tangent space of a real, 2n-dimensional, manifold with n, a G_2 structure identifies the tange... | {
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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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12d5121403ab986b12f24b6ab60fcdde151157a5 | subsection | 200 | 289 | Octonions | We take octonions \mathbb {O} to be the algebra defined by the basis \left\lbrace 1,e_1,e_2,e_3,e_4,e_5,e_6,e_7 \right\rbrace and the multiplication table REF .
[Figure: NO_CAPTION]For X \in \mathbb {O} we will write,X= \sum _{0}^{7} \;X^i e_i, \qquad \text{with}\; e_0 1.The identity element of octonions plays somewhat... | {
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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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697ad84682233c15e24e5f6365a13a002a3cca91 | subsection | 201 | 289 | Octonions | This is convenient to introduce the tensor notation:e_i e_ j \times _{kij} \;e_ k .Multiplying on both side by e_k we obtain,\left(e_i e_j\right) e_k = -\times _{kij}By alternativity we can get rid of the parenthesis and making use of anti-commutativity of unit quaternions, one sees that the tensor \times is completely... | {
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Three-Forms | [
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24931548c9c28218f1564d6470f6c8e8447a7ba2 | subsection | 202 | 289 | Split-Octonion | Similarly to Octonions, Split Octonions \mathbb {O}^{\prime } are defined by a basis \left\lbrace 1,e_1,e_2,e_3,e_4,e_5,e_6,e_7 \right\rbrace but with a different multiplication rule see table REF , see also figure REF for the associated Fano plane.
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Three-Forms | [
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f8f83c73a6002bb33026a582d37d99f8fb25a234 | subsection | 203 | 289 | The exceptional group | The exceptional group G_2 is a Lie group with dimension 14. It is best thought as the automorphism group of octonions, i.e \phi \in G_ 2 if and only if \phi \in End(\mathbb {O}) and is such that for all X,Y \in \mathbb {O}\phi \left(XY\right) = \phi \left(X\right) \phi \left(Y\right).In particular \phi must stabilise t... | {
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81ca8f5cbbdd485f7594dd22dbb44868299fa055 | subsection | 204 | 289 | Stable 3-Forms in Seven Dimensions | Following we here apply the methods from to seven dimensions. Let E be a seven dimensional vector space. A three-form \Omega \in \Lambda ^3 E^* is called stable if it lies in a open orbit under the action of {\rm {GL}}(7).For real three-forms, there are exactly two distinct open orbits of stable forms, each of which is... | {
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84eeba6c0962395f8172f6b90c1fb53b95f5a5b7 | subsection | 205 | 289 | Stable 3-Forms in Seven Dimensions | It also follows from this discussion that the space of positive stable three-forms is the homogeneous group manifold {\rm {GL}}(7)/G_2.One then generalise the notion of stable forms to three-forms on a 7-dimensional differentiable manifold M. Stable differential forms then are differential forms that are stable at ever... | {
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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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784dfe9b45bda22dd08313eb42dfb09e7765861b | subsection | 206 | 289 | The metric and Hitchin volume form | A basic fact about stable three-forms on a 7-dimensional manifold M is that they naturally define a metric on M. For positive three-forms this metric is definite and of signature \left(3,4\right) for negative ones. Of course none of this comes as a surprise if one has in mind the preceding discussion on octonions.This ... | {
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4fc8a5ee890df583ca438b5bfe8bdd99e4270b98 | subsection | 207 | 289 | The Hat Operator | Having a metric in hand one can consider *\Omega , the hodge dual of \Omega . This is a stable four-form. Taking, \Omega of the form (REF ), it has the following form*\Omega = e^4 \wedge e^5 \wedge e^6 \wedge e^7 - \frac{1}{2}\epsilon ^{ijk} e^i \wedge e^j \wedge \widetilde{\Sigma }^k \qquad i,j,k \in 1,2,3then\Psi _{\... | {
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"Yannick Herfray"
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e752f60ff2db427c1d6a7cf186a30c42073ae44d | subsection | 208 | 289 | A more direct construction of the functional | Given a stable three-form, we construct the metric and the corresponding Hitchin volume form as described above. Integrating this volume form over the manifold we get the functionalS[\Omega ] = \int _M \Psi _{\Omega } \, .This functional can also be computed explicitly, without computing the metric, via the following c... | {
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{
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"raw": "Agricola, I. (2008). Old and new on the exceptional group G_2.",
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"start": 1076
}
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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",
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4015297c0ba6f10cdc0b3399a9619a68a236af90 | subsection | 209 | 289 | Holonomy reduction | The fundamental result due to M.Fernandez and A.Gray, , states: Let \Omega \in \Omega ^3\left(M\right)
be a three-form on a 7-manifold. Then \Omega is parallel with respect to the Levi-Civita
connection of g_\Omega iff d\Omega =0 and d{}^*\Omega =0. In other words, the
condition of \Omega being parallel with respect to... | {
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{
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"raw": "Fernandez, M. and Gray, A. (1982). Riemannian manifolds with structure group g_2. Annali di Math. Pura Appl, 32.",
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"start": 0... | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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d5e2d1d15ce78b9a791bf926809c010befffa6f4 | subsection | 210 | 289 | Bryant–Salamon construction | We now review the construction of using a notation compatible with ours. | {
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"raw": "Bryant, R. and Salamon, S. (1989). On the construction of some complete metrics with exceptional holonomy. Duke Math. J., 58:829–850.",
... | 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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0853209175446c4dd7f6abc82e8dc2fb198483d2 | subsection | 211 | 289 | Ansatz | Let (M,g) be a self-dual Einstein 4-manifold, and let \widetilde{\Sigma }^i, i=1,2,3, be a basis of anti-self-dual two-forms of the form (REF ). They satisfy\widetilde{\Sigma }^i \wedge \widetilde{\Sigma }^j = -2 \delta ^{ij} e^0\wedge e^1 \wedge e^2 \wedge e^3Let \widetilde{A}^i be the anti-self-dual part of the Levi-... | {
"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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aae24bec93a93fad1fa8c7653d4273777b7acc44 | subsection | 212 | 289 | Canonical form | We now compute the metric defined by \Omega , as well as its Hodge dual. The easiest way to do this is to write the three-form in the canonical form, so that the metric and the dual form are immediately written. Thus, let \theta ^1, \ldots , \theta ^7 be a set of one-forms such that the three-form \Omega is\Omega = \th... | {
"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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9f948b2b17fffca21810d321048396e35326ed51 | subsection | 213 | 289 | Calculation of the metric and the dual form | We now put ansatz (REF ) into the canonical form (REF ), and compute the associated metric and the dual form. The canonical frame is easily seen to be\theta ^{i} = \alpha d_A y^i, \qquad \theta ^{4+I} = \beta \sqrt{2} e^I, \qquad I=1,2,3,4\, ,where \lbrace e^I\rbrace _{ I\in 0,1,2,3} is an orthonormal frame on the base... | {
"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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393d790e1da51fe2d409623fedb38b109d2c1442 | subsection | 214 | 289 | Co-closing | We now demand the 4-form (REF ) to be closed as well. The first point to note is that when we apply the covariant derivative to the factor \beta ^2 \alpha ^2 in the second term, we generate a 5-form proportional to the volume form of the fibre. There is no such
term arising anywhere else, and we must demand\alpha \beta... | {
"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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e21ac3bc80b71f71063f6815acaca13e5aa72464 | subsection | 215 | 289 | Determining | The overdetermined system of equations (REF ), (REF ) and
(REF ) is nevertheless compatible. Without loss of generality, we can simplify
things and rescale y^i (and therefore \alpha ) so that\alpha \beta =1 \, .With this choice, we have only one remaining equation to solve, which gives\beta ^4 = k + \frac{\Lambda }{3} ... | {
"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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1e2f94d62b1c68ece8cffbca1082aa1cc6ee6269 | subsection | 216 | 289 | A `natural' alternative to Einstein gravity | At the end of the first chapter of this thesis, see section , we described a family of modified theories of gravity that we referred to as `chiral deformations of GR'. This infinite family of theories was parametrised by a function f, see REF . Even though there exists freedom in choosing this function, there exists 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"
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45e5673adaf8a8b977a2a9242865adfb4b2b6def | subsection | 217 | 289 | A `natural' alternative to Einstein gravity | Thenv_{{{F}}} = \frac{1}{2}\left(det\left(\widetilde{X}\right)\right)^{1/3} d^4xIn order to prove the proposition above we will need the followingLemma VI.3 If \left\lbrace \Sigma ^i\right\rbrace _{i\in 1,2,3} is an orthonormal basis of self-dual two-forms i.e * \Sigma ^i = \Sigma ^i\; \forall i and \Sigma ^i \wedge \S... | {
"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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5083ebaf6e8667ff3b07ce30b2d4aae391704e91 | subsection | 218 | 289 | A `natural' alternative to Einstein gravity | This implies in particularF^i \wedge F^j = \widetilde{X}^{ij} \;\frac{\Sigma ^k \wedge \Sigma ^k}{3}.With this in hand (REF ) can be rewritten\tilde{g}_{{{F}}}\left(X,Y\right) = \sigma \; det\left(\sqrt{\widetilde{X}}\right) \textrm {Tr}\left( {\Sigma }\wedge \left[ {\Sigma }_{X}\wedge {\Sigma }_{Y}\right]\right)Then, ... | {
"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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825088cd72f0817c9340e89ae897b167f79caf17 | subsection | 219 | 289 | New local example of | We now give details of our generalisation of the Bryant–Salamon construction. The construction presented here is a local one. Global aspects will not be discussed here. | {
"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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7182f0db36a633a72122c6fef7ba1eeee45ebe43 | subsection | 220 | 289 | Ansatz and closure | We parametrise the three-form by an {\rm {SO}}(3) connection in an \mathbb {R}^3 bundle over M^4:\Omega = \frac{1}{6} \alpha ^3\epsilon ^{ijk} d_A y^i \wedge d_A y^j \wedge d_A y^k +
2\sigma \alpha \beta ^2 d_A y^i \wedge F^i\, ,where the factor \sigma =\pm 1 is the sign of the definite connection (see REF ). It is 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",
"math.MP"
] | 2,018 | en | Physics | [
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22075f9276c5bcb89da5c7731b9e25e4d7207071 | subsection | 221 | 289 | The canonical form and the metric | We now put (REF ) into the canonical form (REF ). To this end, we use
the parametrisation (REF ) of the curvature,F^i = \sigma X^{ij} \Sigma ^jUp to this point we do not have to choose any scale i.eF^i \wedge F^j = X^{ij} \frac{\Sigma ^k \wedge \Sigma ^k}{3} = X^{ij} d^4xbut d^4x is some volume form which is left unspe... | {
"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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a49b41080689bb3249c58b737dbd04bc86d7b926 | subsection | 222 | 289 | The dual form and the co-closure | The dual form reads*\Omega = -\frac{2}{3} \beta ^4 \left( \det X \right)^{1/3} \left( X^{-1} F \right)^i
\wedge F^i - \sigma \beta ^2 \alpha ^2 \left( \det X \right)^{1/3} \left( X^{-1} F
\right)^i \epsilon ^{ijk} \wedge d_A y^j \wedge d_A y^k\, ,where again we expressed all anti-self-dual two-forms on the base in term... | {
"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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5c6b68ac6378de4491bae2b9e0916a8193c718b4 | subsection | 223 | 289 | G2 holonomy and `gravity' | It follows from the above considerations that, starting from\Omega = \frac{1}{6}(1+\sigma y^2)^{-3/4} \epsilon ^{ijk} d_A y^i \wedge d_A y^j \wedge d_A y^k + 2\sigma (1+\sigma y^2)^{1/4} d_A y^i \wedge F^i.we have the followingTheorem VI.4
If A is a definite connection of sign \sigma which is a critical point ofv_{{{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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3347e125f8e7718b58577215199fbb696070f2ac | subsection | 224 | 289 | Complete indefinite | We can modify our construction by not putting the sign \sigma in front of the second term in (REF ). Then all of the construction goes unchanged except that \sigma does not appear either in \Omega or in *\Omega . The differential equations for \alpha and \beta then give \beta =(1+y^2)^{1/4}, and the metric is then comp... | {
"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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c43f699449ce71de09fa74ca490f0314adb7a9a3 | subsection | 225 | 289 | Metric induced on the base | The three-form (REF ) defines the metric (REF ) on the total space of the bundle. The metric induced on the base is of Urbantke type i.e it that makes the curvature two-forms F^i anti-self-dual. Its exact form can be read off from (REF ). The corresponding volume form isv_\Omega = 4 \left( 1 + \sigma y^2 \right) \left(... | {
"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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83f86abd5dedd0ce2335e73161307221563f529a | subsection | 226 | 289 | Relation between the 7D and 4D action functionals | As we already discussed in REF , the co-closure condition d{}^*\Omega is naturally obtained as the equations for Hitchin theory: critical point of the Hitchin functional are closed and co-closed.For practical purpose, this functional is just the volume of the 7D manifold computed using the metric defined by \Omega . Fo... | {
"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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e9a3f9afb921d47cc56f4b4c05790a52c62969ec | subsection | 227 | 289 | Some more Variations on Hitchin Theory in 7d | In this final chapter we describe a new functional for three-forms in seven dimensions describing nearly parallel G_2 structures and consider its reduction to four dimensions. The resulting theory is some scalar-tensor gravity in BF type formulation. These results were originally presented in , . Note that the author o... | {
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fe7d78101644e1dfd63344466c473d69b3d6a12f | subsection | 228 | 289 | Example: The weak holonomy | We now give a nice example of solutions to the above equations related to construction discussed in the second chapter. This can be thought as a variant of the twistor construction. Let us consider the {\rm {SU}}(2)(spin) bundle over a four dimensional manifold M^4{\rm {SU}}(2) \hookrightarrow P^7 \rightarrow M^4.We ta... | {
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Three-Forms | [
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18d8e753c1f4d59ea7f55a494e9cb3322a8c5c8e | subsection | 229 | 289 | Example: The weak holonomy | \right)On the other hand,dC =k^3\left( R\; F^i \wedge \Sigma ^i + \left(\frac{\epsilon ^{ijk}}{2} e^j \wedge e^k \right) \left( R^3 F^i + R \Sigma ^i \right)\right)It follows that dC = k *C if and only if the base manifold is an instanton {{F}}= \frac{\Lambda }{3} {\Sigma }. What is more one needs,2R \Lambda = k/ \wide... | {
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Three-Forms | [
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8f39803406420cf6f6383de2373249aed5ca0aae | subsection | 230 | 289 | Looking back: holonomy as a unifying theme | In the preceding chapters, we respectively encountered nearly Kahler structures, G_2 holonomy manifolds and nearly parallel G_2 structures. All these structures are natural in the context of holonomy on a Riemannian manifold that we now review.The material described in this subsection is standard see for example , or . | {
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ccfa0f80433bc8c998934bd7a5088a74a3cff2f4 | subsection | 231 | 289 | Holonomy on a Riemannian manifold | Associated with any Riemannian manifold \left(M,g\right) we have a unique torsionless connection, the Livi-Civita connection \nabla . By means of this connection one can define parallel transport as follows:
Let \gamma \colon [0,1] \rightarrow M be a path connecting two points x_0 = \gamma (0) and x_1 = \gamma (1). The... | {
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f4623635f48e9f0d9f03ad66d2eb9540b8bd740f | subsection | 232 | 289 | Fundamental Principle of holonomy | Consider E \rightarrow M a tensor or spinor bundle. We have a natural action of the Livi-Cevita connection \nabla on this bundle. We define invariant section (i.e under parallel transport) as sections \alpha such that for every loop \gamma :[0,1] \rightarrow M, \gamma (0)=\gamma (1)=x, P_{\gamma }\alpha = \alpha (x). B... | {
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5442e82463d8453d8ea847f528d02825457c9b47 | subsection | 233 | 289 | Parallel and Killing Spinors | It is also natural to look for Manifolds admitting covariantly constant spinors \nabla \Psi =0. The question then arises: When does a Riemannian manifold admits such spinors ? By the holonomy principle such manifolds must have a non-generic holonomy group and should thus appear as a sub-case of Berger's List. We indeed... | {
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bdc691a221ce6be5f0262fe7507232651b8fb1fd | subsection | 234 | 289 | Reduction of the three-form | We now briefly consider the {\rm {SU}}(2)-reduction of (REF ), see , for details.We take P^7 to be a seven dimensional manifold and, just as in section , we suppose that {\rm {SU}}(2) acts freely on P^7 such that P^7 has the structure of a principal bundle with base M^4 = P^7 / {\rm {SU}}(2):{\rm {SU}}(2) \hookrightarr... | {
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ce424a60616ecbb48f7af3366377f1e4253c08b2 | subsection | 235 | 289 | Reduction of the three-form | Had we start from another connection, this would just have shifted the connection-form {{W}}\rightarrow {{W}}+ {{a}} by a {\rm {SU}}(2)-equivariant, \mathfrak {su}(2)-valued one-form {{a}}\in \Omega ^1\left(P^7\right) \times \mathfrak {su}(2) (then all other forms should be shifted accordingly).Let us now suppose that ... | {
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48fe551aea5945b5e71b35980735c3766cac6659 | subsection | 236 | 289 | Reduction of the Action | Evaluating the kinematic term of (REF ) on (REF ), we obtainHere, {{m}}= g^{-1}dg is the Maurer-Cartan frame on {\rm {SU}}(2).\int _{P^7} \frac{1}{2}\;C dC = \left(\int _{{\rm {SU}}(2)} \frac{-2}{3} \textrm {Tr}({{m}}^3)\right) \times \int _{M^4} -2\textrm {Tr}\left(\phi ^4\; {{B}}{{F}}+ \frac{\phi ^2}{2}\; {{B}}\wedge... | {
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Three-Forms | [
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2b557c2f106f87ef946238f14c2cd04a4315be63 | subsection | 237 | 289 | Reduction of the Action | Here{{B}}\wedge {{B}}= -2\;Vol_{{\widetilde{\Sigma }}} \; \left(X^{ij}\right) \sigma _i \otimes \sigma _jandVol_{{\widetilde{\Sigma }}} -6\;\widetilde{\Sigma }^i \wedge \widetilde{\Sigma }^iis a four-form with density weight -1.It is also convenient to parametrise the three-form c \in \Omega ^3\left(M^4\right) in terms... | {
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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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b8b43a82cff05f1015722691ebe429a7d083c18f | subsection | 238 | 289 | Reduction of the Action | It follows that |V|^2 =\tilde{g}_{{{B}}}\left(v,v\right) has density weight 0 and is therefore a proper scalar.A direct calculation of the seven dimensional conformal metric using (REF ) then gives the following matrix form for \tilde{g}_C,\ \begin{pmatrix} \phi ^(5/2) \sqrt{X} & 0 \\
0 & \phi ^{3/2} \left(det(X)\right... | {
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dead3ff6cefbaa602e4c76c5a522229df30a7581 | subsection | 239 | 289 | Reduction of the Action | Second the last term \left(det(X)\right)^{1/3}\; Vol_{{\Sigma }} looks just like (REF ).Combining the kinematic term (REF ) and the potential term (REF ) and factoring out by the volume of {\rm {SU}}(2) the reduction of (REF ) is\begin{array}{ll}
S\left[{{W}}, {{B}}, \phi , v\right] = \\ \\
\begin{array}{ll}
\int _{M^4... | {
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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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238e6918856cf9b0f42057c859823cd4b5e65c22 | subsection | 240 | 289 | Interpretation | A first point is that if one takes \phi =cst, the above action is a particular chiral deformation of GR of the general form BF plus potential V(BB) (see (REF ) and ). This therefore describes two propagating degrees of freedom of gravity type (spin two).Another interesting feature of the above equation is that for \phi... | {
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762193ddfb2e4dba2adf518f543913f3e099f1f8 | subsection | 241 | 289 | Interpretation | We now take \Psi to be small in Planck units, \Psi \ll 1.Then\left(det(X) \right)^{1/3} = \left( det\left( \delta ^{ij}+ \Psi ^{ij}\right) \right)^{2/3} \simeq 1 + \mathcal {O}(\Psi ^2)so that to first order in \Psi and taking \phi =cst, v=0 the action (REF ) can be rewrittenS\left[{{W}}, {{B}}\right] \propto \int _{M^... | {
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4f3254a65b47de69fce5daa53c6eee9667c78f46 | subsection | 242 | 289 | Conclusion | ConclusionAll the approaches developed in this thesis were aspects of a search for a new perspective on gravity. On the one hand, we reconsidered the twistor formulation of (Euclidean) four dimensional GR and clarified that it fits in the broader scheme of chiral formulations of gravity. On the other hand, we proposed ... | {
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56099b2e3a6c23d069e04daad41453302cff8527 | subsection | 243 | 289 | Conclusion | Unlike the original case of Hitchin where the Lagrangian only depends on a three-form field, these are theories of two- and three-form fields. We named these theories `background independent Hitchin theories'. Here `background independent' refers to the fact that one does not pick by hand a particular cohomology class ... | {
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7dc1708c66860d5ede11ba55070b03a296e6a1a1 | subsection | 244 | 289 | Conclusion | What is more the construction naturally extends to {\rm {SU}}(2)-principal bundle over self-dual Einstein solutions.Considering the {\rm {SU}}(2)-reduction of this theory one obtains what can be interpreted as another chiral deformation of gravity together with a scalar field, see , .The physical interpretation -if any... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.physletb.2017.06.025",
"end": 285,
"openalex_id": "https://openalex.org/W2553957103",
"raw": "Krasnov, K. (2016). General Relativity from Three-Forms in Seven Dimensions.",
"source_ref_id": "d07879179f64dbb3d3bbd3082450189f4... | 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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] | 2,018 | en | Physics | [
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bad445ae22b5dd202355c23891443ba8001a48b6 | subsection | 245 | 289 | Conclusion | This is as opposed to the Euclidean case where, as we emphasised all along this thesis, the notion of `definite connections' is the good reality condition.All the results described, while not giving a fully satisfactory higher dimensional perspective on 4D General Relativity, do show convincingly that the four-dimensio... | {
"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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] | 2,018 | en | Physics | [
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e3535c4e7cc78354eaa4f2bba8b4f74b9f0a8198 | subsection | 246 | 289 | Density and All That | Let Id^n \in \wedge ^n TM \otimes \Omega ^n(M) be the invariantly defined tensor given by the identity of n-vectorId_{\wedge ^n TM} \colon \wedge ^n TM \rightarrow \wedge ^n TM.Alternatively this is the only tensor Id^n \in \wedge ^n TM \otimes \Omega ^n(M) that gives one when contracted with itself.In tensorial notati... | {
"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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bd1bc4b554d798473daf8299c21396848bf06f1e | subsection | 247 | 289 | Density and All That | \nu _n } \partial _{\nu _1}\otimes ...\otimes \partial _{\nu _n}This tensor is useful to give a concrete form to the isomorphism \Omega ^k(M) \simeq TM^{n-k}(M) \otimes \Omega ^n(M):\begin{array}{ccc}
\Omega ^k(M) & \rightarrow & TM^{n-k}(M) \otimes \Omega ^n(M) \\ \\
\rho & \mapsto & \widetilde{\rho }= \frac{n!}{k!}\;... | {
"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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f26ac5d83b962d38626923b502e842417b9f33b1 | subsection | 248 | 289 | Intrinsic definition | \mathfrak {su}2 is the three dimensional Lie algebra defined by
[Table: NO_CAPTION]For convenience, everywhere in the text we take the overall factor of the Killing metric on \mathfrak {su}(2) to be defined byK\left(\sigma _i, \sigma _j\right) = \delta _{ij}.We have the alternative basis:\sigma _+=-i\sigma _1-\sigma _2... | {
"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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... | |
1af28c63b0060cbbb5ae06a8a55cf7718cdbfb32 | subsection | 249 | 289 | Spin | In the fundamental representation, elements of \mathfrak {su}2 corresponds to hermitian tracefree matrices. The explicit isomorphism is:\tau \text{:}\left\lbrace
\begin{array}{llc}
su2 & \longmapsto & \text{Hermitian tracefree}\\
v^i \sigma _i &\qquad & v^i \sigma _i{}^{A^{\prime }}{}_{B^{\prime }}
\end{array} \right.... | {
"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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0.0... | |
76e28708c2cdbe08b54e046f26931191ee4bfd9b | subsection | 250 | 289 | Spin | It corresponds to the 3\times 3 antisymmetric matrices such that:exp(T^k{}_i) \; \delta _{kl} \; exp( T^l{}_j) = \delta _{ij} \Rightarrow T_{ij}+T_{ji}=0 .The explicit isomorphism now is:\sigma \text{:}\left\lbrace
\begin{array}{llc}
su2 & \longmapsto & \text{3$\times $3 antisymmetric}\\
v^i \sigma _i &\qquad & v^i \s... | {
"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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0.0... | |
6a502b9e8bc0934a7201eac4ec3c863518c8da14 | subsection | 251 | 289 | NB: | With this notation the spin \frac{1}{2} representation, M^{A^{\prime }}{}_{B^{\prime }}=M^i \sigma _i{}^{A^{\prime }}{}_{B^{\prime }}, of a Lie algebra element M= M^i \sigma _i coincide with its coordinates in the creation-annihilation basis, M= M^{A^{\prime }B^{\prime }} \left( 2\tau _{A^{\prime }B^{\prime }}\right).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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... | |
d47db2e78aae7d55659797ad02cf53f1a38dac63 | subsection | 252 | 289 | Self-dual two-forms in 4D as | One can also represent \mathfrak {su}(2) elements by self-dual two-forms in four dimension. The explicit isomorphism now is:\Sigma \text{:}\left\lbrace
\begin{array}{llc}
su2 & \longmapsto & \text{self-dual two-forms}\\
v^i \tau _i &\qquad & v^i \frac{\Sigma ^i }{2}
\end{array} \right.where\Sigma ^i = -e^0 \wedge e^i ... | {
"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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... | |
1dc1f1f94dc21bf990ec54479708ff311ed2dbd8 | subsection | 253 | 289 | Self-dual two-forms in 4D and spinor notation. | Let \Lambda = \Lambda ^i \sigma ^i \in \mathfrak {su}(2), let \Lambda ^i \frac{\Sigma ^i}{2} be the associated self-dual two-form.Depending on the notation (tetras or spinor notation) it can be written\Lambda = \Lambda ^i \frac{\Sigma ^i}{2} = \Lambda _{A^{\prime }B^{\prime }} \;\Sigma ^{A^{\prime }B^{\prime }} = -\Lam... | {
"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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... | |
0a8eccbe5480087424a76ed254371302e82bc189 | subsection | 254 | 289 | Self-dual two-forms in 4D and spinor notation. | \end{array}It follows from this discussion that the self-dual basis \left\lbrace \frac{1}{2}\Sigma ^i = \Sigma ^{A^{\prime }B^{\prime }} \sigma ^i_{A^{\prime }B^{\prime }} \right\rbrace can be thought of as the isomorphism between \mathfrak {su}2 and self-dual two-forms (which is the 1/2 \otimes 0 representation of \ma... | {
"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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ca7d14978f5aa2daf318dd1f08323c9038ea28a7 | subsection | 255 | 289 | Decomposition of the Riemann Curvature Tensor in Coordinates | In this appendix we prove, using coordinates, the different claims made in the first part of chapter .In this appendix we use freely the isomorphism \mathfrak {so}(4)\simeq \Omega ^2 to represent elements of \mathfrak {so}(4, \mathbb {R}) as two-forms. I.e, we pick up a basis of one-forms, \left\lbrace e^I\right\rbrace... | {
"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... | |
4d2ee37f336f5c60ab92d54586eb60eeb624ce49 | subsection | 256 | 289 | Decomposition of the Riemann Curvature Tensor in Coordinates | This basis is orthogonal for the wedge product:\Sigma ^i \wedge \Sigma ^j = -\widetilde{\Sigma }^i \wedge \widetilde{\Sigma }^j = 2 \delta ^{ij} e^0\wedge e^1 \wedge e^2 \wedge e^3, \qquad \Sigma ^i \wedge \widetilde{\Sigma }^j = 0.As was already stated in the main part of this thesis, the decomposition of Lie algebra ... | {
"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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f49bc4398b4a183af0d754774d519a69c51a9363 | subsection | 257 | 289 | Decomposition of the Curvature tensor in coordinates | Consider a 4d Riemannian manifold \lbrace g, M\rbrace , \lbrace e^I\rbrace _{I\in 0..4} an orthonormal co-frame. The Levi-Civita connection, \nabla , then naturally is a SO(4)-connection. We will write its potential one-form {a} and curvature two-form {f} asa^I{}_J = a^{I}{}_{J\;K} e^K, \qquad f^I{}_J = d a^I{}_J + a^I... | {
"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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382ba816616e3c5ff0143006d70593f5d299dbc0 | subsection | 258 | 289 | Decomposition of the Curvature tensor in coordinates | In chapter we stated that these connections are compatible with \Sigma ^i, \widetilde{\Sigma }^i in the following sense:d_A \Sigma ^i = 0, \qquad d_{\widetilde{A}} \widetilde{\Sigma }^i=0.We can prove this by a direct computation:d_A \Sigma ^i &= \frac{1}{2} d_A \left( \Sigma ^i_{IJ} e^I \wedge e^J \right)\\
&=\frac{1}... | {
"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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0.018282286822795868,
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0.... | |
512158306afecb6aaffabd20850d5917ea7f8db9 | subsection | 259 | 289 | Decomposition of the Curvature tensor in coordinates | It follows thatd_{A^{\prime }} \Sigma ^i - d_A \Sigma ^i = \epsilon ^{ijk} M^j \Sigma ^k =0,or equivalently, by making use of the self-duality of \Sigma ,M_{\nu }^{[i} \Sigma ^{j]}{}^{\mu \nu }=0.By multiplying this expression by another sigma symbol we have0&= \epsilon ^{ijk} M^j_{\nu } \Sigma ^k{}^{\mu \nu } \Sigma ^... | {
"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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0.0... | |
887ba36f389d139b2012e0b5d10ffeb727847377 | subsection | 260 | 289 | Decomposition of the Curvature tensor in coordinates | I.e, the (anti-)self-dual part of the curvature is the curvature of the (anti-)self-dual connection.Now F^i, \widetilde{F}^i being (\mathfrak {su}(2)-valued) two-forms, we can decompose them into self-dual and anti-self-dual pieces:F^i = F^{ij} \Sigma ^j + G^{ij} \widetilde{\Sigma }^j, \qquad \widetilde{F}^i = \widetil... | {
"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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0.0... | |
b2c49dfbcd771bfa781ddf94641c659ed6e13929 | subsection | 261 | 289 | Decomposition of the Curvature tensor in coordinates | Finally, we can write:f^I{}_J &= \underbrace{\frac{\widetilde{\Lambda }}{3} \;\widetilde{\Sigma }^i \; \frac{\widetilde{\Sigma }^i{}^I{}_J}{2} + \frac{\Lambda }{3} \;\Sigma ^i \; \frac{\Sigma ^i{}^I{}_J}{2}}_{\text{Scalar Part}} \\
&+ \underbrace{ \frac{1}{2}\left(G^{ij} \; \Sigma ^j \;\widetilde{\Sigma }^i{}^I{}_J + \... | {
"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.... | |
9d0318e3077c66c1ce0cf379bc23613eb61a6e52 | subsection | 262 | 289 | Decomposition of the Curvature tensor in coordinates | One can identify the following elementary brick of the Riemann tensor:\begin{array}{ll}
W_{IJKL}= \frac{1}{2}\Psi ^{(ij)} \;\Sigma ^i_{IJ} \; \Sigma ^j{}_{KL} \quad &\text{is the self-dual part}\\ & \text{of the Weyl~tensor}, \\ \\
\widetilde{W}_{IJKL} = \frac{1}{2}\widetilde{\Psi }^{(ij)} \;\widetilde{\Sigma }^i_{IJ} ... | {
"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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0.03870701... | |
ee5a2cde884a678ec7d734c4c16aecda4138e6ec | subsection | 263 | 289 | Decomposition of the Curvature tensor in coordinates | It leads to further simplifications:f_{IJKL} = f_{KLIJ} \quad \Rightarrow \quad \Psi ^{ij}=\Psi ^{(ij)},\quad \widetilde{\Psi }^{ij}=\widetilde{\Psi }^{(ij)}, \quad \widetilde{G}^{ij}=G^{ji}.f_{I[JKL]}=0 \quad \Leftrightarrow \quad f_{NIKL} \epsilon ^{NJKL}=0 \quad \Rightarrow \quad \Lambda = \widetilde{\Lambda }.The s... | {
"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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0.0... | |
9c3b360e778153f10eb662a18754ca22bb8ccd1e | subsection | 264 | 289 | Spinors and | We convert \mathfrak {su}(2) lie algebra indices into spinor notations according to the rule:B As a convention, we use \sigma _i^{AB} to convert "spatial indices" i\in \lbrace 1,2,3\rbrace into "unprimed spinor indices" and \bar{\sigma }_i^{A^{\prime }B^{\prime }}, its complex conjugate, to convert "spatial indices" in... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 925,
"openalex_id": "",
"raw": "Penrose, R. and Rindler, W. (1985). Spinors and space-time. Vol 1. Two spinor calculus and relativistic fields.",
"source_ref_id": "6c2c9d2216bfcbacf547f619d3a8442015514fd5",
"start": 87... | 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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0.038035910576581955,
-0.028908511623740196,
... | |
d7a8fa244e4ea2979231dee025e1af0bd4143f9e | subsection | 265 | 289 | Spinors and | This is equivalent to equipped spinors with a Hermitian metric:\left\langle \alpha , \beta \right\rangle \hat{\alpha }_{A^{\prime }}\beta ^{A^{\prime }} = \hat{\alpha }.\beta \ge 0, \qquad \left\langle \alpha , \beta \right\rangle \hat{\alpha }_{A}\beta ^{A} = \hat{\alpha }.\beta \ge 0 .We go from one type of indices t... | {
"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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... | |
79e1dcaef9b8ed987ead81bc7186914a4e87dba1 | subsection | 266 | 289 | Null tetrad and spinors | In order to convert space-time indices into spinor ones we use the convention:V^I e_I^{AA^{\prime }} = \frac{1}{i\sqrt{2}}\begin{pmatrix} -it+z & x-iy \\ x+iy & -it-z \end{pmatrix}, \qquad V^I = \lbrace t,x,y,z\rbrace .With this convention,V^{AA^{\prime }}= V^I e_I^{AA^{\prime }} \qquad \Leftrightarrow \qquad V^I = e^I... | {
"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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-... | |
7d67d1a3065aad1a9653d0dd94a0bb71a77b1d40 | subsection | 267 | 289 | Two-forms and spinors | Let \Lambda be a general two-form, in spinor notation:\begin{array}{llll}
\Lambda &= \Lambda _{AA^{\prime }BB^{\prime }} \frac{e^{AA^{\prime }}\wedge e^{BB^{\prime }}}{2} \\ \\
&=\left( \Lambda {}_{E}{}^{E}{}_{A^{\prime }B^{\prime }} \;\epsilon _{AB}+ \Lambda {}_{E^{\prime }}{}^{E^{\prime }}{}_{AB} \;\epsilon _{A^{\pri... | {
"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... | |
32790b6202825359a6bdd728eb38ae9c2933b586 | subsection | 268 | 289 | Two-forms and spinors | \end{array}Converting the AB indices into spatial indices:,\widetilde{\Sigma }^i&=2\widetilde{\Sigma }^{AB}\sigma ^i{}_{AB}& & \widetilde{\Sigma }^{AB}=\widetilde{\Sigma }^i \sigma _i{}^{AB}=-\frac{e^{A}{}_{C^{\prime }}\wedge e^{BC^{\prime }}}{2},one can rewrite this decomposition in a usual tetrad:\Lambda =\Lambda _{I... | {
"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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... | |
09a27980ee4e7767d5f06bf70390a7aa954d5263 | subsection | 269 | 289 | The Riemann sphere and its Fubini-Study metric | The Riemann Sphere is the one dimensional complex projective space \mathbb {C}{P}^1, i.e the projective version of S^{\prime } \simeq 2 (here S^{\prime } stands for the space of primed spinors),\mathbb {C}{P}^1 \left\lbrace \text{One dimensional subspaces of } S^{\prime } \right\rbrace .as such it is at the same time t... | {
"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 | [
-0.043765291571617126,
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0.046206869184970856,
0.006042906083166599,
0.0147257... | |
b1709e249eb21fc4937bda7c7e15a52d40c47cd4 | subsection | 270 | 289 | The Riemann sphere and its Fubini-Study metric | \end{array}With transition map:\begin{array}{ll}
\phi ^{\prime -1} \circ \phi & \left\lbrace
\begin{array}{llc}
\mapsto &
\zeta &\mapsto & \zeta ^{\prime }=\frac{1}{\zeta }.
\end{array} \right.
\end{array}In explicit computations we will mostly use homogeneous coordinates \left[\pi ^{A^{\prime }}\right] to emphasize ... | {
"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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-0.000... |
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