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3223098b17a83d8db5ef6032dbdbfb87cdcdf403 | subsection | 48 | 110 | Quadratic action for the perturbations of Minkowski | Let us start by writing down the quadratic action (REF ) for Minkowski perturbations respecting the symmetries of the sphere.
For such perturbations, we take all fields to be independent of the coordinates on the sphere and seth^0_{ab}=h^0_{(ab)}+\frac{1}{p+2}\eta _{ab}H^0,\qquad h_{ai}=0,\qquad h_{ij}=\frac{1}{n+1}\si... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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d358fbdb4846e27d103e371b82937cdb232b55a7 | subsection | 49 | 110 | Quadratic action for the perturbations of Minkowski | \\
&
+\frac{p(p+1)}{4(p+2)^2}{\partial }_aH{\partial }^aH
-\frac{n+1}{2r}h_{(ar)}{\partial }^aH
+\frac{1}{4r^2}\frac{n(n+1)}{p+2}H^2
-\frac{n+1}{2}{\partial }^bh_{(ab)}{\partial }^a\pi \\
&\qquad +\frac{n+1}{2}\frac{p+1}{p+2}{\partial }_aH{\partial }^a\pi -\frac{n^2-1}{2r}h_{(ar)}{\partial }^a\pi +\frac{n+1}{2r}\frac{n... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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59532a4db3a1c815c5a3246619a92b89a80aadab | subsection | 50 | 110 | Quadratic action for the perturbations of Minkowski | The full action for the gauge invariant fields \hat{h}^{I_\mathsf {s}}_{ab}, \hat{B}^{I_\mathsf {v}}_{\mathsf {(v)}a}, \hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}, and \hat{\pi }^{I_\mathsf {s}} defined in (REF )-() can then be recovered by hatting the original fields h^{I_\mathsf {s}}_{ab}, B^{I_\mathsf {v}}_{\mathsf {(... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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70251b9ede5e6f05a62be0a5bf38655b90d4af0a | subsection | 51 | 110 | Quadratic action for the perturbations of Minkowski | After integration over the internal sphere {\mathcal {S}}^{n+1} we obtainS_0 & =\int d^{p+2}y\,r^{n+1}\left\lbrace
\frac{1}{2}{\partial }_a\hat{h}^{I_\mathsf {s}}_{bc}{\partial }^b\hat{h}^{I_\mathsf {s}}{}^{ac}-\frac{1}{4}{\partial }_a\hat{h}^{I_\mathsf {s}}_{bc}{\partial }^a\hat{h}^{I_\mathsf {s}}{}^{bc}
+\frac{\Lamb... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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c93602e496cdb503055038f7ff0d953ba2033be0 | subsection | 52 | 110 | Quadratic action for the perturbations of Minkowski | First, the variation \delta _{\hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}}S_0 of the action with respect to the field \hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}} yields the equation \left.E^{(0)}_{ij}\right|_{\mathbb {T}^{I_\mathsf {t}}_{ij}}=0. The equation \left.E^{(0)}_{ai}\right|_{\mathbb {V}^{I_\mathsf {v}}_i}=0 is re... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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3f41701b674aea26938d2625f2bcf057c374da78 | subsection | 53 | 110 | Quadratic action for the perturbations of Minkowski | One can indeed check that the variation \delta _{\phi ^{I_\mathsf {v}}_{\mathsf {v}}}S_0 gives equation \left.E^{(0)}_{ij}\right|_{{(i}\mathbb {V}^{I_\mathsf {v}}_{j)}}=0, and that the variation with respect to \phi ^{I_\mathsf {s}}_{\mathsf {s}} and B^{I_\mathsf {s}}_{\mathsf {(s)}a}, give the other equations
\left.E^... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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acfd86833331b41f7f851e0b2d91dbe24648a824 | subsection | 54 | 110 | Quadratic action for AdS perturbations | We work now with the AdS background written in Poincaré coordinates \zeta ^m=\lbrace r,z^\alpha \rbrace , with metric (REF ) and dimension D=d+1. Using\nabla _m h_{np}={\partial }_m h_{np}+\frac{1}{r}\left(2\delta _{m}{}^rh_{np}+\delta _{n}{}^rh_{mp}+\delta _{p}{}^rh_{mn}-\eta _{mn}h_{rp}-\eta _{pm}h_{n r}\right)we can... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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fba5b44b3c97a4c67b195d5ae3231f064439b84a | subsection | 55 | 110 | Quadratic action for AdS perturbations | Also, we work in De Donder-Lorentz gauge{\partial }^ih_{(ij)}=0,\qquad {\partial }^ih_{ia}=0;the action for the gauge invariant combinations \hat{h}^{\mathbf {m}_\mathsf {s}}_{ab}, \hat{C}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}a}, \hat{\varpi }^{\mathbf {m}_\mathsf {s}}, and \hat{\psi }^{(k,l,{\mathbf {m}_\math... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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7158a56f23d195e8f2c6c14106c6898a23efb612 | subsection | 56 | 110 | Quadratic action for AdS perturbations | \\
&\qquad \qquad \qquad \qquad \qquad +\left[\vphantom{\left(\frac{2}{p}\right)}
-\frac{1}{4}\hat{F}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}ab}\hat{F}^{(k,{\mathbf {m}_\mathsf {v}})ab}_{\mathsf {(v)}}
-\frac{1}{2}{\mathbf {m}^2_\mathsf {v}}\hat{C}^{(k,{\mathbf {m}_\mathsf {v}})a}_{\mathsf {(v)}}\hat{C}^{(k,{\ma... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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397ad6e28a10a65d1abee732ce935627bfd6cf38 | subsection | 57 | 110 | Quadratic action for AdS perturbations | More precisely, \delta _{\hat{\psi }^{(k,l,{\mathbf {m}_\mathsf {t}})}_{\mathsf {t}}} S_\Lambda =0 is the equation \left.E_{ij}^{(\Lambda )}\right|_{\mathbb {T}^{(k,l,{\mathbf {m}_\mathsf {t}})}_{(ij)}}=0, and
\delta _{\hat{C}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}a}} S_\Lambda =0 is the equation \left.E_{ai}^{... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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8a4feb2b71123a75ff872b4a2b7803869be59844 | subsection | 58 | 110 | Quadratic action for AdS perturbations | The equations \delta _{\hat{h}^{\mathbf {m}_\mathsf {s}}_{ab}} S_\Lambda =0 and \delta _{\hat{\varpi }^{\mathbf {m}_\mathsf {s}}} S_\Lambda =0 are equivalent to the equations \left.E_{ab}^{(\Lambda )}\right|_{\mathbb {S}^{\mathbf {m}_\mathsf {s}}}=0 and \left.E_{ij}^{(\Lambda )}\right|_{\delta _{ij}\mathbb {S}^{\mathbf... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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bb6b97815fdfbe0edf1315ace1c578399a996697 | subsection | 59 | 110 | Quadratic action for AdS perturbations | The precise relation between them can be found by thinking of the action S_\Lambda as a functional of the original fields h^{\mathbf {m}_\mathsf {s}}_{ab}, C^{\mathbf {m}_\mathsf {s}}_{\mathsf {(s)}a}, C^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}a}, \psi ^{\mathbf {m}_\mathsf {s}}_{\mathsf {s}}, \psi ^{(k,{\mathbf ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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ae12de27eb15d69819e2d029479b05440bdacb6c | subsection | 60 | 110 | Quadratic action for AdS perturbations | The field equation for C^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}a} indeed reproduces (REF ). | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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b36d42989c28defe2dc10e733363cf3eb3c93ba9 | subsection | 61 | 110 | Decoupling the Kaluza-Klein equations | In this section we look closely at the structure of the Kaluza-Klein equations, both for perturbations of Minkowski and for perturbations of AdS, and show that we can decouple them (almost) completely. | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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fd33fd898939080216b9a9ae83cbdd769f107501 | subsection | 62 | 110 | Perturbations of Minkowski spacetime reduced on | The field equations for the metric perturbation modes of Minkoswki are obtained in § REF . First, we notice that the scalar mode \hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}, associated with a tensor harmonic with eigenvalue \Lambda ^{I_\mathsf {t}} given by (), decouples from all the other fields. Its equation of motion,... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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fa203f6548bc7a485c2c53404eec24c97313bf2b | subsection | 63 | 110 | Perturbations of Minkowski spacetime reduced on | ()). We use this relation to eliminateThe n=1 case is special because we are reducing over a two-sphere, on which there are no tensor harmonics. Therefore, there is no field \hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}. Furthermore, the trace \hat{h}^{I_\mathsf {s}} is automatically vanishing by equation (REF ).
Correspon... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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1fbd1281d67a1f0e4779e9e781597007002fceca | subsection | 64 | 110 | Perturbations of Minkowski spacetime reduced on | Finally, we introduce a new scalar field \hat{\varphi }^{I_\mathsf {s}} defined by\hat{\varphi }^{I_\mathsf {s}}=\frac{1}{r^2}\left(\hat{h}^{I_\mathsf {s}}_{(rr)}-\frac{n+p+1}{p+2}\hat{\pi }^{I_\mathsf {s}}\right).The third equation () and the rr component of the first one (REF ) become respectively,\Box \hat{\varphi }... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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3bdc13561d822e9c953e34332681c03b19e902c7 | subsection | 65 | 110 | Perturbations of Minkowski spacetime reduced on | (), give the field equations for the symmetric traceless mode \hat{h}^{I_\mathsf {s}}_{(ab)},&\Box \hat{h}^{I_\mathsf {s}}_{(ab)}+\frac{n+1}{r}{\partial }_r\hat{h}^{I_\mathsf {s}}_{(ab)}
-\frac{2}{r}\left({\partial }_a\hat{h}^{I_\mathsf {s}}_{(br)}+{\partial }_b\hat{h}^{I_\mathsf {s}}_{(ar)}\right)-\frac{n-1}{r^2}\left... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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aa5a6a5178b6b4299ff5a0eef84bd036b1b4a687 | subsection | 66 | 110 | Perturbations of Minkowski spacetime reduced on | Together with the solutions to the decoupled equations for \hat{B}^{I_\mathsf {v}}_{\mathsf {(v)}a} and \hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}, this solves completely the problem. We explicitly solve these equations in § REF . | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
69f54149e37b3743867589b1c2ae1cdae4a7c76b | subsection | 67 | 110 | Perturbations of AdS spacetime reduced on | The field equations for the metric perturbation modes of AdS are obtained in § REF . Similarly to what we just saw for in the Ricci-flat case, the scalar field \hat{\psi }^{(k,l,{\mathbf {m}_\mathsf {t}})}_{\mathsf {t}} associated with a tensor harmonic decouples from the other fields, and must solve the equation\Box \... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.000796695239841938,
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95ebe13a63b6b2be9cf860c3c46de19443897d56 | subsection | 68 | 110 | Perturbations of AdS spacetime reduced on | When {\mathbf {m}^2_\mathsf {v}}=0, equation (REF ) still holds but for unhatted variable C^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}a}, and equation (REF ) does not hold anymore (but we have an additional gauge invariance).The remaining modes are \hat{h}^{\mathbf {m}_\mathsf {s}}_{ab} and \hat{\varpi }^{\mathbf {... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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0... |
4c2f7de10ec2aee4555251504307d5594d4823a8 | subsection | 69 | 110 | Perturbations of AdS spacetime reduced on | \hat{h}^{\mathbf {m}_\mathsf {s}} in favor of \hat{\varpi }^{\mathbf {m}_\mathsf {s}},\hat{h}^{\mathbf {m}_\mathsf {s}}=-(d-p-3)\hat{\varpi }^{\mathbf {m}_\mathsf {s}}.Following the Minkowski steps, we decompose the metric perturbation \hat{h}^{\mathbf {m}_\mathsf {s}}_{ab} into its trace \hat{h}^{\mathbf {m}_\mathsf {... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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7443ed3059da34d9c307774579fa9f78d5b3085b | subsection | 70 | 110 | Perturbations of AdS spacetime reduced on | The resulting equations can be completely decoupled by defining the new fields \hat{\varphi }^{\mathbf {m}_\mathsf {s}} and \hat{\chi }^{\mathbf {m}_\mathsf {s}} as (the transformation is invertible as long as d\ne 2),\hat{\varpi }^{\mathbf {m}_\mathsf {s}}=\hat{\varphi }^{\mathbf {m}_\mathsf {s}}-\frac{r^2}{\ell ^2}\h... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
a758bfa0ab8e6fbe00869065129f0529ad5a59cd | subsection | 71 | 110 | Solving the Kaluza-Klein equations | In this section we solve the Kaluza-Klein equations for linearized metric perturbations for both Minkowski and AdS. | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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3f63e606f39aa692234f9ba499d6ac84390cd014 | subsection | 72 | 110 | Solving for Minkowski perturbations | We first solve equation (REF ) for the linearized scalar perturbation \hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}} associated with tensor harmonics of {\mathcal {S}}^{n+1},\Box \hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}+\frac{n+1}{r}{\partial }_r\hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}-\frac{l(l+n)}{r^2}\hat{\phi }^{I_\ma... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
f2d23d801ddf323a1c396dff3a7ad0801b3a84fa | subsection | 73 | 110 | Solving for Minkowski perturbations | Then instead of presenting the solutions as in (REF ), we will use\hat{\phi }^{I_\mathsf {t}}_{\mathsf {t}}=r^{-\frac{n}{2}}\left(
\phi _1 J_{l+\frac{n}{2}}(k_r r)+ (1 \rightarrow 2, J \rightarrow Y)
\right)e^{i\mathbf {k}\cdot \mathbf {x}}.All solutions in this and the next subsection will be presented in this way.Nex... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
ff6434ee5bbd92653ad8d4954663fc72d4c947f4 | subsection | 74 | 110 | Solving for Minkowski perturbations | The equation for the remaining components is obtained by setting a=\mu in (REF ), and using () to eliminate the r-derivatives of \hat{B}^{I_\mathsf {v}}_{\mathsf {(v)}r},\Box \hat{B}^{I_\mathsf {v}}_{\mathsf {(v)}\mu }+\frac{n+3}{r}{\partial }_r\hat{B}^{I_\mathsf {v}}_{\mathsf {(v)}\mu }-\frac{1}{r^2}(l-1)(l+n+1)\hat{B... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
017b3d838515448998279dd8a3b2572c3e862f01 | subsection | 75 | 110 | Solving for Minkowski perturbations | We use this freedom to simplify the solution of the divergence equation ().
Then, given a solution \hat{B}^{I_\mathsf {v}}_{\mathsf {(v)}r} determined by b_1 and b_2, as given by (REF ), the general solution to the full equation for \hat{B}^{I_\mathsf {v}}_{\mathsf {(v)}\mu } is given by\hat{B}^{I_\mathsf {v}}_{\mathsf... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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5d3fec4e21ea4180351e0da50cef021861adae5e | subsection | 76 | 110 | Solving for Minkowski perturbations | We have partially decoupled the equations they satisfy by introducing the scalar field \hat{\varphi }^{I_\mathsf {s}} in equation (REF ), resulting in equations (REF ) and (), that we report here once more for convenience,\Box \hat{\varphi }^{I_\mathsf {s}}+\frac{n+1}{r}{\partial }_r\hat{\varphi }^{I_\mathsf {s}}-\frac... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
0d5cf640f16bab264433f6ef657af5e8e91b1a74 | subsection | 77 | 110 | Solving for Minkowski perturbations | In (REF ) we fixed \alpha so that no multiple of the homogeneous solution appears in the particular solution.
sourced by \hat{\varphi }^{I_\mathsf {s}},\hat{\pi }^{I_\mathsf {s}}=r^{-\frac{n}{2}}\left(
\pi _1 J_{l+\frac{n}{2}}(k_r r) +\frac{r}{k_r} \varphi _1 J_{l+\frac{n}{2}+1}(k_r r)+ (1 \rightarrow 2, J \rightarrow ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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0... |
e61885693fdaefd73ed0cbbb7ac51356701d9eb9 | subsection | 78 | 110 | Solving for Minkowski perturbations | Then the r\mu component reads,\Box \hat{h}^{I_\mathsf {s}}_{(r\mu )}+\frac{n-1}{r}{\partial }_r\hat{h}^{I_\mathsf {s}}_{(r\mu )}-\frac{(l+1)(l+n-1)
}{r^2} \hat{h}^{I_\mathsf {s}}_{(r\mu )} = 2 r \partial _\mu \hat{\varphi }^{I_\mathsf {s}},while taking the trace of (REF ) leads to\partial ^\mu \hat{h}^{I_\mathsf {s}}_{... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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fcc14868e0a6b6eac194da90809a0260ba64510c | subsection | 79 | 110 | Solving for Minkowski perturbations | Finally, the solution to (REF ) is constrained by equation (REF ),{\partial }^\nu \hat{h}^{I_\mathsf {s}}_{(\mu \nu )}+{\partial }_r\hat{h}^{I_\mathsf {s}}_{(\mu r)}+\frac{n-1}{r}\hat{h}^{I_\mathsf {s}}_{(\mu r)}-\frac{n+p+1}{p+2}{\partial }_\mu \hat{\pi }^{I_\mathsf {s}}=0.We now solve these equations. The homogenous ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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6d4e3c17d0431450cb9f2445ffdaaded2cd18140 | subsection | 80 | 110 | Solving for Minkowski perturbations | The choice in (REF ) simplifies the solution of (REF ).\hat{h}^{I_\mathsf {s}}_{(r\mu )\mathsf {part}}[\hat{\varphi }^{I_\mathsf {s}}] = \frac{i k_\mu }{\bf {k}^2}\left(r^2 {\partial }_r \hat{\varphi }^{I_\mathsf {s}}+ (n+1) r \hat{\varphi }^{I_\mathsf {s}}\right).Using (REF ), we find that the solution of (REF ) is gi... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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824f9a918bd0ddac8cc3f849a8dd37a2918f1dc4 | subsection | 81 | 110 | Solving for Minkowski perturbations | The particular solutionConsider the differential operator
\mathcal {L} ={\partial }_r^2+\frac{n+1}{r}{\partial }_r+\left(k_r^2-\frac{l(l+n)}{r^2}\right);
it satisfies the identities
&\mathcal {L}\left[r^{1-\frac{n}{2}}J_{l+\frac{n}{2}+1}(k_rr)\right]=2k_rr^{-n/2}J_{l+\frac{n}{2}}(k_rr),\\
&\mathcal {L}\left[-k_rr^{2-... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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a60fc612d2efef42f042b5b0fa56adfca0b172f9 | subsection | 82 | 110 | Solving for Minkowski perturbations | \left.\left.\quad +\left(n\frac{k_\mu k_\nu }{k_r^2}+\frac{n-1}{p+2}\eta _{\mu \nu } \right)\varphi _1
\right]
J_{1+l+\frac{n}{2}}(k_r r)\right) + (1 \rightarrow 2, J \rightarrow Y)
\right\rbrace
e^{i\mathbf {k}\cdot \mathbf {x}},where\hat{\mathfrak {h}}^s_{\mu \nu }&=\mathfrak {h}^s_{\mu \nu }+ \frac{1}{{\bf k}^2} \l... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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24b7202f7c05299196e692020c84e819b04b53ed | subsection | 83 | 110 | Solving for Minkowski perturbations | It will be useful to also record the combination that is traceless in p+1 dimensions:\tilde{h}^{I_\mathsf {s}}_{(\mu \nu )} &=& \hat{h}^{I_\mathsf {s}}_{(\mu \nu )} - \frac{1}{p+1} \eta _{\mu \nu } \left(\eta ^{\kappa \lambda } \hat{h}^{I_\mathsf {s}}_{(\kappa \lambda )}\right)=\hat{h}^{I_\mathsf {s}}_{(\mu \nu )} + \f... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
ddcb49209ef7706d82a9af3e3bfb7a67f5544322 | subsection | 84 | 110 | Solving for AdS perturbations | Anti-de Sitter perturbations are much simpler. It is possible to decouple completely the system of equations controlling them, and we can easily find all modes. Like in the case of perturbations of Minkowski, all equations are of the Bessel type.Equation (REF ) for the scalar perturbations associated with tensor harmon... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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4212a0f683f2932464429d19695ef306ed8ece8f | subsection | 85 | 110 | Solving for AdS perturbations | They satisfy equations (REF ) and (REF ),& \partial ^a\hat{F}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}ab}-\frac{d-1}{r}\hat{F}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}rb}- {\mathbf {m}^2_\mathsf {v}}\hat{C}^{(k,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}b}=0,
\\
&{\partial }^a\left(r^{-(d-1)}\hat{C}^{(k,{\m... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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0.... |
f1869fbfe7196cdd1dffc5695aa36629e0a67da4 | subsection | 86 | 110 | Solving for AdS perturbations | Equation () is identical with (REF ) and thus is solved in the same way, while the extra term in (REF ) simply shifts the order of the Bessel function:&\hat{C}^{(j,{\mathbf {m}_\mathsf {v}})}_{\mathsf {(v)}r}=r^{\frac{d}{2}}\left(\gamma _1 J_{\frac{d}{2}-1} (k_rr) + (1 \rightarrow 2, J \rightarrow Y)
\right)e^{i\mathbf... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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629fa7fcebfa015778e1e8daa22b6a901a1a9073 | subsection | 87 | 110 | Solving for AdS perturbations | Thus, the general solution is simply given by () with a null momentum k^a,&C^0_{(\mathsf {v})r}=0,\qquad C^0_{(\mathsf {v})\mu }=r^{\frac{d}{2}}\left(\gamma ^1_\mu J_{\frac{d}{2}} (k_rr)+ (1 \rightarrow 2, J \rightarrow Y)
\right)e^{i\mathbf {k}\cdot \mathbf {x}},
\\
&k^ak_a=k_r^2+k^\mu k_\mu =0.and one can use the gau... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
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5cf226ac959359ac2df1a0256a766175c8a88978 | subsection | 88 | 110 | Solving for AdS perturbations | The equations for the scalars \hat{\chi }^{\mathbf {m}_\mathsf {s}} and \hat{\varphi }^{\mathbf {m}_\mathsf {s}},&\Box \hat{\chi }^{\mathbf {m}_\mathsf {s}}-\frac{d-5}{r}{\partial }_r\hat{\chi }^{\mathbf {m}_\mathsf {s}}-{\mathbf {m}^2_\mathsf {s}}\hat{\chi }^{\mathbf {m}_\mathsf {s}}=0,
\\
&\Box \hat{\varphi }^{\math... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
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5f8d3d73b890d3b24d6c0b3266e5b3a707b0c72b | subsection | 89 | 110 | Solving for AdS perturbations | Setting a=\mu , b=r in (REF ) we obtain\Box \tilde{h}^{\mathbf {m}_\mathsf {s}}_{(\mu r)}-\frac{d-1}{r}{\partial }_r\tilde{h}^{\mathbf {m}_\mathsf {s}}_{(\mu r)}-\left({\mathbf {m}^2_\mathsf {s}}-\frac{d-1}{r^2}\right)\tilde{h}^{\mathbf {m}_\mathsf {s}}_{(\mu r)} =0which is solved by\tilde{h}^{\mathbf {m}_\mathsf {s}}_... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
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4285e6d14dd0f95ac33798f8c338895e970d913e | subsection | 90 | 110 | Solving for AdS perturbations | Additionally, we need to impose the divergence equation (), and this implies
that the polarization vectors are given by&\hat{h}^s_\mu = h^{s}_\mu - i \frac{k_\mu k_r}{{\bf k}^2}\left(f_s+\frac{d-1}{\ell ^2}\chi _s\right),
\qquad k^\mu h^{s}_\mu =0, \qquad s=1,2with h^1_\mu and h^2_\mu transverse polarization vectors.Se... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
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] | 2,018 | en | Physics | [
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d797e19e264cd6134bf5682a2a43ab13842dd27f | subsection | 91 | 110 | Solving for AdS perturbations | \\
&\left.\hphantom{=}
+\frac{1}{p} \left( (p+1) k_\mu k_\nu - \eta _{\mu \nu } {\bf k}^2 \right) \left(\frac{d-2}{p+1} f_s - \frac{k_r^2}{\mathbf {k}^2} \left(f_s+\frac{d-1}{\ell ^2}\chi _s\right) \right)
\right)
\\
k^\mu h^s_{\mu \nu }&=k^\nu h^s_{\mu \nu } = 0, \qquad s=1, 2with h^1_{\mu \nu }, h^2_{\mu \nu } are tr... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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e1474284bba988a50c7541f629509a5147a463ff | subsection | 92 | 110 | AdS/RF map at large | The explicit solutions we found, both in the vacuum Einstein gravity and in the AdS gravity theories, are linear combinations of terms that take the general formr^a\left(c_1J_b(k_rr)+c_2Y_b(k_rr)\right)e^{i\mathbf {k}\cdot \mathbf {x}}times some polarization vector or tensor where appropriate. We can tabulate the disti... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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b8bf62a92dd31bf0ab89fbd787b76b92fcf08ae5 | subsection | 93 | 110 | AdS/RF map at large | Effectively, this behavior decompactifies the torus in the large d limit, reducing the mass gap to zero and making the AdS momentum lightlike, (k^{(\mathbf {m})}_r)^2+\mathbf {k}^2\sim 1/d^2\rightarrow 0 (equivalently, we could have considered the limit \mathbf {k}^2\gg \mathbf {m}^2, for which k^{(\mathbf {m})}_r\sim ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
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dd29c8697bf94e8acebfac64c3291eac49741379 | subsection | 94 | 110 | Body | {\rm Mink:} &\quad \mathfrak {h}^s_{\mu \nu }+ \frac{1}{{\bf k}^2}
\left(\vphantom{\frac{n^2}{kp}}
i n\left(\mathfrak {h}^s_\mu k_\nu + \mathfrak {h}^s_\nu k_\mu \right)\right. \\
&\quad \hphantom{\mathfrak {h}^s_{\mu \nu }}\left.
+\frac{1}{p} \left((p+1) k_\mu k_\nu - \eta _{\mu \nu } {\bf k}^2 \right)\left(-\frac{n^2... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
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198b8a7b054d9aee11e174f2de2f8bcf1805b922 | subsection | 95 | 110 | Body | \\
&\quad \hphantom{h^s_{\mu \nu }}\left.
+\frac{1}{p} \left( (p+1) k_\mu k_\nu - \eta _{\mu \nu } {\bf k}^2 \right) \left(\frac{d}{p+1} f_s - \frac{(k^{(\mathbf {m})}_r)^2}{{\bf k}^2} \left(f_s+\frac{d}{\ell ^2}\chi _s\right) \right)
\right)Thus we conclude that we need& n \mathfrak {h}^{s}_\mu =- k_r^{(\mathbf {m})} ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
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] | [
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a1fb117775d9a37d024a9f1fce5e5d9e0463bacc | subsection | 96 | 110 | Body | This signals that the harmonics start to mix under the AdS/RF, a phenomenon that we explore in more depth in . | {
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... | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
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8cde66ae1720d2a8cb5f74f5fc51e80e9d8f7b56 | subsection | 97 | 110 | Discussion and Outlook | In this paper we performed a Kaluza-Klein reduction of Minkowski over a sphere and of AdS over a torus.
In the case of Minkowski the sphere was the celestial sphere transverse to a flat p-brane located at the origin of Minkowski
and in the case of AdS the torus compactified part of the boundary directions. The reason f... | {
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2973af72364a6df602672a770df2799db0370082 | subsection | 98 | 110 | Discussion and Outlook | We were able to integrate all equations and obtain the general solution as linear combination of Bessel functions.Having succeeded in finding the general solution of linear perturbations, we can then turn to the original question about the AdS/RF correspondence. On general grounds, such a map may involve general linear... | {
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"Marco M. Caldarelli",
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490d3ecff4bc38931f59b0a86a5bfd968864e26c | subsection | 99 | 110 | AdS/Ricci-flat map for the zero modes | Consider perturbations that respect the AdS/RF Ansatz (REF ) and (REF ), i.e. we only allow for perturbations that respect the sphere/torus symmetries of the original Ansatz. In the case of the perturbations of Minkowski spacetime, this means that we keep the perturbation h^{0}_{ab} of the reduced (p+2)-dimensional met... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
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32fc6e0748d54bc40a93412c755888c6c841aff6 | subsection | 100 | 110 | AdS/Ricci-flat map for the zero modes | Again, y^a=\lbrace r,x^\mu \rbrace indicate collectively the p+2 coordinates in the reduced theory.ds^2_\Lambda =\frac{\ell ^2}{r^2}\left(dr^2+\eta _{\alpha \beta }dz^\alpha dz^\beta \right)+\frac{\ell ^2}{r^2}h_{ab}^{\Lambda }(y;d)\,dy^ady^b+\frac{\ell ^2}{r^2}\varpi (y;d)\delta _{ij}\,d\chi ^id\chi ^j.Again, this is ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
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a1ed894714e83f856735a169da61c9ff188e9a91 | subsection | 101 | 110 | AdS/Ricci-flat map for the zero modes | Concretely, the AdS/Ricci-flat prescription (REF ) applied to the perturbations (REF ) and (REF ) gives – at linear level – the following action of the map on the perturbation,h^{0}_{(ab)}(y;n) & = h^\Lambda _{(ab)}(y;-n),
\\
H^{0}(y;n) & = H^\Lambda (y;-n)-(p+2)\varpi (y;-n),
\\
\pi (y;n) & =-\varpi (y;-n).It can be ... | {
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61d5c1a3b42e412a6467cfb87780935b0bb20c6f | subsection | 102 | 110 | The sphere | Consider the (n+1)-dimensional sphere {\mathcal {S}}^{n+1} with unit metric d\Omega _{n+1}^2=\sigma _{ij}\,d\theta ^id\theta ^j. It is a constant curvature manifold, and its Riemann and Ricci tensors readR_{ijkl}=\sigma _{ik}\sigma _{jl}-\sigma _{jk}\sigma _{il},\qquad R_{ij}=n\sigma _{ij}.It follows that expressions i... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
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0b320ebbaf36a240df446f74bacc140fbb907b49 | subsection | 103 | 110 | The sphere | This leads to the decomposition () for the metric perturbation: scalars can be decomposed into scalar harmonics, vectors are decomposed into a vector harmonics that is divergent-free and the gradient of scalars as a consequence of the Hodge decomposition theorem, and symmetric tensors are decomposed in traceless symmet... | {
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"raw": "P. van Nieuwenhuizen, “An introduction to simple supergravity and the Kaluza-Klein program” in Relativity, groups and topology: Proceedings, 40th Summer School of Theoretical Physics - Session 40 : ... | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
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4b53c3fb86bdd6664f18858413c9bc265e7bf3c3 | subsection | 104 | 110 | The sphere | Moreover, \mathbb {V}^{I_\mathsf {v}}_i are Killing vectors of {\mathcal {S}}^{n+1} when l=1, so {(i} \mathbb {V}^{I_\mathsf {v}}_{j)}=0, and the harmonic expansion contains terms proportional to {(i} \mathbb {V}^{I_\mathsf {v}}_{j)}=0 only for l\ge 2.These form a complete set for functions on the sphere. Any scalar fu... | {
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} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
"hep-th"
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4288ed1e4a9553ff3cfd2e657dc649ae199fb8cd | subsection | 105 | 110 | The sphere | This leads to the decomposition
(REF )-() for perturbations of Minkowski spacetime.To conclude, we display a few properties of the spherical harmonics that we used to simplify the calculations, and can be easily obtained using the identities (REF ),\Box i\mathbb {S}^{I_\mathsf {s}}=\left(\Lambda ^{I_\mathsf {s}}+n\righ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
] | [
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f8317d41f0440f999e870b45e31e13a002bcb1a5 | subsection | 106 | 110 | The torus | Consider a flat N-dimensional torusIn the main text, the torus is taken to be of dimension N=d-p-1.
with metric d\sigma ^2=\delta _{ij}\,d\chi ^id\chi ^j, with the coordinates \chi ^i periodic, with period \tau . We want to decompose our fields in scalar, vector, and traceless symmetric tensor eigenmodes of the Laplace... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
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a7b57c6a5a07bcf3fef2ecab1325510de64fd956 | subsection | 107 | 110 | The torus | We decide to label them with the pair (k,{\mathbf {m}_\mathsf {v}}) comprising a coordinate index k, singling out a polarization, and a wave vector {\mathbf {m}_\mathsf {v}}=(m_1/\tau ,\ldots ,m_N/\tau ) with m_1,\ldots , m_N\in \mathbb {Z} and m_k=0:&\mathbb {V}^{(k,{\mathbf {m}_\mathsf {v}})}_j(\chi )=\delta ^k{}_j e... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
"Kostas Skenderis"
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a905334d3fc882e888b6dac08517f4da472bcfc6 | subsection | 108 | 110 | The torus | We decide to label them with the triplet (k,l,{\mathbf {m}_\mathsf {t}}) formed by two coordinate indices k<l singling out a polarization, and a wave vector {\mathbf {m}_\mathsf {t}}=(m_1/\tau ,\ldots ,m_N/\tau ), with m_1,\ldots , m_N\in \mathbb {Z} and m_k=m_l=0,&\mathbb {T}^{(k,l,{\mathbf {m}_\mathsf {t}})}_{(ij)}(\... | {
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} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
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"Kostas Skenderis"
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d9b63fe7ccdfb9a7c96c50acf6053569f81035bd | subsection | 109 | 110 | The torus | In addition, when {\mathbf {m}^2_\mathsf {t}}=0 and N \ge 2, there are special tensor harmonics given by the constant symmetric traceless tensors,
\mathbb {T}^{(k,l, 0)}_{(ij)} = \frac{1}{2}\left(\delta ^k{}_i\delta ^l{}_j+\delta ^l{}_i\delta ^k{}_j\right)-\frac{1}{N} \delta _{ij} \delta ^{kl}, k \le l.It is worth reme... | {
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} | 10.1140/epjc/s10052-018-6058-8 | 1802.06085 | Kaluza-Klein reductions and AdS/Ricci-flat correspondence | [
"Marco M. Caldarelli",
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c1885f96212419784a0f0a0cc0e8dd94cc438194 | abstract | 0 | 69 | Abstract | Andromeda is an LCF-style proof assistant where the user builds derivable
judgments by writing code in a meta-level programming language AML. The only
trusted component of Andromeda is a minimalist nucleus (an implementation of
the inference rules of an object-level type theory), which controls
construction and decompo... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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92c1939ac53c5e715925cc30e96811fd47017c85 | subsection | 1 | 69 | Introduction | A type theory can be interesting and very useful, yet lack metatheoretic
properties (e.g., decidability) that permit a straightforward implementation.
In fact, the more flexible and expressive the theory, the less likely these
properties will hold. Nevertheless, even very expressive type theories deserve
automated supp... | {
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"... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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848ab5a723bfc7d51bca5702d2a540a3c0a30076 | subsection | 2 | 69 | Introduction | Second, applications of equality reflection are not recorded in the conclusion,
and omitting the explicit equality eliminators keeps
terms smaller and simpler.The proof assistant NuPRL validated equality reflection by
implementing so-called computational type theory, a specific
interpretation of type theory akin to re... | {
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"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
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f70beb826a14216d8ddc8ad8a62a3f7eaf0671d3 | subsection | 3 | 69 | Contributions | The present paper should be read as a progress report on the development of
Andromeda; the system and the underlying type theory may evolve as we gain more
experience and consider a wider variety of applications. We focus
on the following points of interest:the goals of Andromeda and the structure of the system (§);
t... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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78698a5e83524904ea0fd556a523f6e976e9fb19 | subsection | 4 | 69 | An overview of Andromeda | Andromeda follows design principles that are similar to those of other proof assistants:The system should work well in the common case. Equality reflection
affords many possibilities for complicating one's life, but we expect most applications to be very
reasonable. If the user introduces new computation and extensiona... | {
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... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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bed354615b8095ab49d21b45d95830f0ef6cc99b | subsection | 5 | 69 | An overview of Andromeda | Bugs in AML code either
prevent code from constructing the desired judgments or
construct an unintended judgment, but the abstract type of
judgments and run-time checks in the nucleus ensure that only derivable
judgments are ever constructed. Any other memory-safe metalanguage (e.g., one modeled
on Python or Scheme) wo... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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2d9761ff9c40dcbed9709f47a09b2ff1b3627cf5 | subsection | 6 | 69 | Andromeda in action | Before looking at the three constituent parts of Andromeda in more detail, we provide a small worked example. At this point we cannot explain all the
technical details, so we focus on emphasizing the important points and showcasing what
Andromeda can do.We begin by declaring some constants that Andromeda adds to the am... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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71cd0fac2c0adb1b16e809e5a21c5a568e7eec65 | subsection | 7 | 69 | Andromeda in action | This tracking process becomes
apparent if we temporarily hypothesize an equality a ≡ b and use it as a hint while constructing a judgment that v above has type P b,
assume ζ : a ≡ b in
now hints = addhint ζ in (v : P b)
This AML expression causes the nucleus to build the hypothetical judgment:
ζ₀ : a ≡ b ⊢ v : P b
... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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bc72f03ea30890914901d3ebd6655cbd9d162db8 | subsection | 8 | 69 | The nucleus | The nucleus is the part of the system that implements the object-level type theory. Its
functionality includes the following:formation and decomposition of term and type judgments,
construction of equality judgments,
substitution and syntactic equality checking,
pretty-printing of judgments and export to JSON.Before... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
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db0efa3e0be067ab1a4c61e41e0529578b69da84 | subsection | 9 | 69 | Type theory with equality reflection | The Andromeda nucleus implements an extensional Martin-Löf type
theory , with dependent products \mathop {\textstyle \prod _{(x {:} A)}}{B}
and equality types \mathsf {Eq}_{A}(s,t). Complete rules
are provided in Appendix .
Fundamentally, the system is not too far removed from the more common
intensional Martin-Löf typ... | {
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"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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c3c3ddc3f42b59473c07b379bb87ca41b0bfaf5c | subsection | 10 | 69 | Type theory with equality reflection | These annotations ensure
that terms have unique types up to equality: working again under the assumption (REF ), we can apply the identity function
at type \mathsf {Nat}\rightarrow \mathsf {Nat} to get
(\lambda x {:} \mathsf {Nat}.{\mathsf {Nat}}\,.\,{x})\mathbin {@^{x{:}\mathsf {Nat}.\mathsf {Nat}}} 0 of type \mathsf ... | {
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"Gaëtan Gilbert",
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"Matija Pretnar",
"Christopher A. Stone"
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6e06110a130f8d16999ae5b54a8b1c7d9b150d3c | subsection | 11 | 69 | Type theory with equality reflection | (For the same reason, both Coq
and Agda allow the assumption \mathsf {Type}: \mathsf {Type} as an option.) Nevertheless, although users
are unlikely to stumble into inconsistencies by accident, we ultimately
want a sound foundation, and plan to remove \text{\textsc {ty-type}}, as discussed
in §. | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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6acf1eecb469edce76d26f07451b5ab103b3cd78 | subsection | 12 | 69 | Implementation of type theory | The type theory implemented in the nucleus differs from the one presented in several ways.
The changes are inessential from a theoretical point of view, but have significant
practical impact. We describe them in this section. | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
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1d225fb31c621986d6bd8b83bbdf80255aa4c1cb | subsection | 13 | 69 | Signatures | In Andromeda the user extends the type theory by postulating constants, i.e., they work in
type theory over a signature. In this respect Andromeda is much like other proof
assistants that allow the user to state axioms and postulates. The signature is controlled
by the nucleus through an abstract datatype whose interfa... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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-0... | |
05769e8ce6ef6014cf217dd92e07d0ff4f192d2b | subsection | 14 | 69 | Inversion principles and natural types | The nucleus implements inversion principles for deconstruction of judgments into sub-judgments;
these are exposed in AML through pattern
matching, cf. §REF . For example, an application
\Gamma \vdash s\mathbin {@^{x{:}A.B}} t : C can be decomposed into
\Gamma \vdash s : \mathop {\textstyle \prod _{(x {:} A)}}{B}, \Gamm... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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... | |
4fdde06e73e78300074da84e22a3253653f13734 | subsection | 15 | 69 | Assumption sets | The nucleus is responsible for decomposing judgments into their component parts,
a facility used by pattern matching in AML. For example, we can combinef : \mathsf {Nat}\rightarrow \mathsf {Nat} \vdash f : \mathsf {Nat}\rightarrow \mathsf {Nat}
\qquad \text{and}\qquad x: \mathsf {Nat} \vdash x : \mathsf {Nat}(using wea... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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... | |
d70e369b2649fbe1dcab5280d13e54a6e169fa14 | subsection | 16 | 69 | Context joins | The standard rules of inference require the contexts of the premises to match, for
instance the application rule \text{\textsc {term-app}} does not allow a change of the
context:{\Gamma \vdash s : \mathop {\textstyle \prod _{(x {:} A)}} B \\
\Gamma \vdash t : A
}
{\Gamma \vdash s\mathbin {@^{x{:}A.B}} t : B[t/x]}If we ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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... | |
19123b3e74cf8f9a961249ac1371004e1d9501cf | subsection | 17 | 69 | The Andromeda meta-language | The Andromeda meta-language (AML) is a programming language in the style of ML .
We review its structure and capabilities, focusing on the parts that are peculiar to
Andromeda. For constructs that are standard in the ML-family of languages, such as type
definitions, |let|-bindings, recursive functions, etc., we refer t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.7551/mitpress/2319.001.0001",
"end": 80,
"openalex_id": "https://openalex.org/W1829244603",
"raw": "Robin Milner, Mads Tofte, and Robert Harper. The Definition of Standard ML. MIT Press, 1990.",
"source_ref_id": "2122025e5ee163137c... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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... | |
f55e9d9725f93ef481e4c548280b40b0c1151956 | subsection | 18 | 69 | ML-types | AML is equipped with static type inference in the style of Hindley-Milner
parametric polymorphism . It supports definitions of
parametric ML-types, including inductive types. The only non-standard aspect of
the ML-type inference arises from the fact that application is overloaded, as it
is used both for invoking ML-lev... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/582153.582176",
"end": 99,
"openalex_id": "https://openalex.org/W2163976959",
"raw": "Luis Damas and Robin Milner. Principal type-schemes for functional programs. In Proceedings of the 9th ACM SIGPLAN-SIGACT Symposium on Principles ... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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0.0... | |
5a1436267527a9cf7f9132cee993abc53b7c00a8 | subsection | 19 | 69 | Pattern matching | AML pattern matching in |match| statements and |let|-bindings is more flexible than that
of Standard ML and related languages. AML patterns need not be linear (i.e.,
a pattern variable may appear several times in a pattern) and variables may be
interpolated into patterns. Pattern variables are prefixed with |?| so
that... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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0.01770162396132946,
... | |
daa0c669f0a881189e4f693846154564fa182000 | subsection | 20 | 69 | Operations and handlers | During evaluation of a computation of ML-type \mathtt {judgment} the interpreter may need evidence of
equality between two types (in order to present it to the nucleus), which it gets by passing control back to user code, together
with information on what needs to be done, and how to resume the evaluation once the
evid... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2168/lmcs-9(4:23)2013",
"end": 435,
"openalex_id": "https://openalex.org/W2129902163",
"raw": "Gordon D. Plotkin and Matija Pretnar. Handling algebraic effects. Logical Methods in Computer Science, 9(4), 2013.",
"source_ref_id": "9... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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0.01783786527812481,
0.0409... | |
dce7254aafb3b67d2a351522eb7b19ee56795d0c | subsection | 21 | 69 | The datatype | In Andromeda the user always works with an entire judgment
\Gamma \vdash t : A, and never a bare term t. Similarly a type A never
stands by itself, but always in a judgment \Gamma \vdash A : \mathsf {Type}. The judgments are
represented by values of a special primitive type \mathtt {judgment}. | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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0.0195193849503994,
0.012773827649652958,
-0.004170189145952463,
... | |
8317afcbd620268542dc554b4e01a61e500e8ce6 | subsection | 22 | 69 | Judgment forms | The OCaml interface for the nucleus uses distinct abstract datatypes to represent the
different judgment forms. These distinctions are not visible to the user, because AML exposes all
forms through the single datatype \mathtt {judgment} whose values are judgments of the form
\Gamma \vdash t : A. This is possible becaus... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03325639292597771,
0.013310184702277184,
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0.002885144669562578,
0.0064758434891700745,
0.0005868495209142566,
0.00944298505783081,
... | |
b31bbb5de270e3ef318960bb12d475291a05eda4 | subsection | 23 | 69 | Inferring and checking modes of evaluation | There are two
modes of AML evaluation, inferring and checking. In inferring mode the type of
the result is unconstrained. In checking mode the type is prescribed in advance: there is
given a type A (or more precisely, a judgment \Gamma \vdash A : \mathsf {Type}) and the
computation must evaluate to a judgment of the fo... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.00649117399007082,
-0.007242502644658089,
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0.03908434137701988,
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0.04399658367037773,
0.011319508776068687,
0.02431711181998253,
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-0.022669529542326927,
0.029702268540859222,
-0... | |
880a66cd6b0a0eb28b5a1e5f6ce0b52b56845ebb | subsection | 24 | 69 | Judgment computations | The following primitives for computing judgments are provided:Primitives for term and type formation:
\mathtt {Type}\qquad \Pi (x {:} c_1),\, c_2 \qquad c_1 \, c_2 \qquad \lambda (x {:} c_1),\, c_2 \qquad c_1 \equiv c_2 \qquad \mathtt {refl}\,c.
Note that the notation c_1 \equiv c_2 is used for the equality type, rat... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.022966932505369186,
0.03216896578669548,
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0.029300007969141006,
0.0015441637951880693,
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0.05182439088821411,
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0.01416930090636015,
-0.012902685441076756,
-0.03482428193092346,
0.001987669849768281,
0.0... | |
d1055329c83ed8017ac6c509338850947ce5e4a3 | subsection | 25 | 69 | Judgment patterns | Apart from computations that form judgments, we also need flexible ways of analyzing and
deconstructing them. In AML this is done with the \mathtt {match} statement and judgment
patterns of the form \vdash p_1 : p_2, where p_2 may be omitted, and p_1 and p_2
are among the following:Anonymous pattern ||, pattern variabl... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.024416610598564148,
0.002813632832840085,
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0.030337639153003693,
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0.004810835234820843,
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-0.006542888469994068,
0.01713740825653076,
0.03... | |
7271f4bbbf8456b521934ff38d2635f87658b252 | subsection | 26 | 69 | Judgment patterns | Instead, the primitive computation \mathtt {context}\,c evaluates c to a judgment
\Gamma \vdash t : A and gives the list of all hypotheses in \Gamma , sorted so that each
hypothesis is preceded by its dependencies. | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.006968090310692787,
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... | |
c0307d7a5c714d2cb774f86c4e183f6b61d12162 | subsection | 27 | 69 | Equality checks and coercions | AML only verifies syntactic equality automatically. It delegates any other equality
\Gamma \vdash s ~ \equiv ~ t : A by triggering the operation
\mathtt {equal}\,(\Gamma \vdash s : A)\,(\Gamma \vdash t : B), which passes control back
to the user-level AML code. The operation may go unhandled, in which case an error is
... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.01034328993409872,
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0.008077834732830524,
... | |
4c035b10468e65785cab6535d00c82f33e2e0b1d | subsection | 28 | 69 | Equality checks and coercions | If the head c_1 of an application c_1 \, c_2 evaluates to a term
\Gamma \vdash t : A where A is not a product type, the interpreter asks the user code to
convert t to a function by triggering the operation
\mathtt {coerce{_}fun}\,(\Gamma \vdash t : A). The handler should yield
\mathtt {NotCoercible},
\mathtt {Convertib... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
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... | |
22ad30df15bfb89b314938627805a528835fc550 | subsection | 29 | 69 | References and dynamic variables | As a convenience, AML provides ML-style mutable references. They are used to store the
current state of implicit arguments in the standard library
(see §REF ).AML also supports dynamic variables. These are globally defined mutable values with
dynamic binding discipline. A dynamic variable x is declared and initialized ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.07036571949720383,
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... | |
4f1a017cefe41c6c06f41fea4291f070cb26fe92 | subsection | 30 | 69 | Soundness of Andromeda | Soundness in Andromeda has both theoretical and engineering aspects.Theoretical soundness pertains to the differences between the original type theory,
(Appendix ) and the type theory implemented in the nucleus
(§REF ), which uses assumption sets, context joins,
and natural types. In the following we write s^\sigma for... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.0196703989058733,
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-0.0... | |
995d27e99ab93ce8f921ba5ed934a8dc0bccd5f5 | subsection | 31 | 69 | Soundness of Andromeda | As soon as we
remove \mathsf {Type}: \mathsf {Type} the theory becomes consistent, since what remains are just bare
products and equality types with reflection, and these are consistent in virtue of having
a model (such as the hereditarily finite sets). We discuss removal of
\mathsf {Type}: \mathsf {Type} in §.
The sta... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/1183278.1183281",
"end": 899,
"openalex_id": "https://openalex.org/W2161017670",
"raw": "Christopher A. Stone and Robert Harper. Extensional equivalence and singleton types. ACM Transactions on Computational Logic, 7(4):676–722, Oct... | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.026865117251873016,
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0.02216639183461666,
-0.0371321365237236,
0.01... | |
4bb0d5519f50da0d14070d48fa550762dacddd83 | subsection | 32 | 69 | Soundness of Andromeda | There are three kinds:
an \eta -hint, or an extensionality hint, is a term whose type has the form
\mathop {\textstyle \prod _{(x_1 {:} A_1)}} \cdots \mathop {\textstyle \prod _{(x_n {:} A_m)}}
\mathop {\textstyle \prod _{(y_1 {:} B)}} \mathop {\textstyle \prod _{(y_2 {:} B)}}
\mathsf {Eq}_{C_1}(t_1,s_1) \rightarrow ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03469240665435791,
-0.0026507501024752855,
-0.028406886383891106,
0.007826386950910091,
-0.01077844575047493,
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0.011457342654466629,
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0.020092304795980453,
0.0006550594698637724,
-0.06651666760444641,
0.014500938355922699,
-0.016064690425992012,
... | |
4ad8197dacfd27aefacb0c63a13cb4a1de2429df | subsection | 33 | 69 | Soundness of Andromeda | Another example is the recursor for natural numbers, which should eagerly
reduce the number at which it is applied.
Examples of equality hints and uses of reduction strategies will be shown in
§. Let us only remark that the hints and reduction strategies may be
installed locally, even under a binder using a temporary e... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.05165906250476837,
0.0016696510137990117,
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0.03585315868258476,
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-0.00295788561925292,
-0.01878095231950283,
... | |
2dd7116e5fbca2022487f9e6d8b1d1b3ba7cdf01 | subsection | 34 | 69 | Soundness of Andromeda | To see how this works, let us walk through a computation that
constructs a term witnessing symmetry of equality (without the standard library
installed):
λ (A : Type) (x y : A) (p : x ≡ y),
(handle
refl x : y ≡ x
with
| equal x y ⇒ yield (Some p)
end)
The \lambda -abstraction introduces a type A, elements x,
y of typ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03862493112683296,
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0.013103798031806946,
-0.006437488365918398,
... | |
238c7d2e4d1f1e06fe217db6364ed14b17389156 | subsection | 35 | 69 | Soundness of Andromeda | The \beta -rules are
straightforward, except that we must install the \beta -rule for the first projection
before we postulate the second projection, or else Andromeda does not know why the second
projection is well typed:
constant π₁β :
Π (A : Type) (B : A → Type) (a : A) (b : B a),
(π₁ A B (existT A B a b) ≡ a)
now ... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.05182420462369919,
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-0.008538175374269485,
-0.024676011875271797,
... | |
b97d03b163078a168fc3685ecc94a5a5e15b9598 | subsection | 36 | 69 | Soundness of Andromeda | For example, we may define addition as follows:
constant ( + ) : nat → nat → nat
constant plusdef :
∏ (n m : nat), n + m ≡ natrect (λ , nat) n (λ x, S x) m
Note that plusdef could be written as
constant plusdef' :
( + ) ≡ (λ (n m : nat), natrect (λ , nat) n (λ x, S x) m)
The difference between the two is visible wh... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.00696394219994545,
-0.007566517684608698,
-0.03234076127409935,
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-0.047809407114982605,
0.02715403400361538,
-0.06577988713979721,
-0.00... | |
a48f913f2247de1d2506c9d5c3ad6cf02cb491a2 | subsection | 37 | 69 | Soundness of Andromeda | Notice that the proof of equality between 3 \times 4 and 12 is a reflexivity
term, even though the normalization procedure generated the proof by stringing together a
large number of reduction steps. In order to keep equality proofs small, the standard
library aggressively replaces equality proofs with reflection terms... | {
"cite_spans": []
} | 1802.06217 | Design and Implementation of the Andromeda Proof Assistant | [
"Andrej Bauer",
"Gaëtan Gilbert",
"Philipp G. Haselwarter",
"Matija Pretnar",
"Christopher A. Stone"
] | [
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.020999547094106674,
0.022724073380231857,
-0.023761842399835587,
-0.0038439396303147078,
0.0007730801589787006,
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0.030141064897179604,
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0.03256761282682419,
-0.017703106626868248,
-0.0338495597243309,
-0.014177746139466763,
0.01591753400862217,
... |
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