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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
03df1f7c0801e563a1b0ded27a445cb8326fabad | subsection | 91 | 120 | Body | As described in Section REF ,
this is “order” optimal among all ULDP mechanisms.Consider the low privacy regime where \epsilon =\ln |\mathcal {X}| and |\mathcal {X}_S| \ll |\mathcal {X}|.
By Proposition REF ,
the expected l_2 loss of the (\mathcal {X}_S,\epsilon )-blackuRAP
mechanism is given by:\mathbb {E}\left[ l_2^2... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.01697070710361004,
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0.019168352708220482,
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... | |
d82d652cc413f9a544f5548205fe6de33360fd35 | subsection | 92 | 120 | Body | \hspace{9.34998pt}(\text{by $|\mathcal {X}_S| \ll |\mathcal {X}|$})When |\mathcal {X}_S| \ll \sqrt{|\mathcal {X}|},
the right side of (REF ) is simplified as:\frac{1}{n} \Bigl ( {\textstyle 1 + \frac{|\mathcal {X}_S|+1}{\sqrt{|\mathcal {X}|}}} \Bigr )
\approx \!\frac{1}{n}. ~~~~(\text{by $|\mathcal {X}_S| / \sqrt{|\mat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 892,
"openalex_id": "https://openalex.org/W2964074929",
"raw": "P. Kairouz, K. Bonawitz, and D. Ramage. Discrete distribution estimation under local privacy. In Proc. 33rd International Conference on Machine Learning (ICML'16), pa... | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... | |
c3f2891fc299f2826aeb513530a7719f6d0dde6f | subsection | 93 | 120 | Maximum of the | Next we show that
when 0 < \epsilon < \ln (|\mathcal {X}_N|+1),
the l_1 loss
is maximized by the uniform distribution
\mathbf {p}_{U_{\!N}} over \mathcal {X}_N.*Here we do not present the general result for |\mathcal {X}_S| > |\mathcal {X}_N|, because in this section later, we are interested in using this proposition t... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.0035174740478396416,... | |
cfff4eb15ec8399a47691887d5c25d022d59fa4e | subsection | 94 | 120 | Maximum of the | Then:\frac{d \sqrt{A(w)}}{dw}
&= \frac{1}{2\sqrt{A(w)}} \frac{dA(w)}{dw} \\
&= \frac{1}{2\sqrt{A(w)}} \Bigl (
- 2 w + |\mathcal {X}_S| \bigl ( v - {\textstyle \frac{2}{u^{\prime }}} \bigr )
\Bigr ) \\
\frac{d \sqrt{B(w)}}{dw}
&= \frac{1}{2\sqrt{B(w)}} \frac{dB(w)}{dw} \\
&= \frac{1}{2\sqrt{B(w)}} \Bigl (
- 2 w + \bigl ... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.015971621498465538,
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-0.011509634554386139,
-0.0011126361787319183,
... | |
538cd260e7dd33a2319de1d5f9d7a0c2464124e1 | subsection | 95 | 120 | Maximum of the | By e^\epsilon < |\mathcal {X}_N|+1, we have:\frac{1}{|\mathcal {X}_N|} - \frac{1}{u^{\prime }}
= \frac{1}{|\mathcal {X}_N|} - \frac{1}{e^{\epsilon }-1}
< \frac{1}{|\mathcal {X}_N|} - \frac{1}{|\mathcal {X}_N|}
= 0.Hence \frac{dF(w)}{dw} < 0.
Therefore for w\in [0,1), F(w) is decreasing in w.Now we prove Proposition REF... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03981376811861992,
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-0.016169575974345207,
0.... | |
4d7450f0ae28c9e19503d972b81dcb74fd990bfa | subsection | 96 | 120 | Maximum of the | Hence by (REF ) we obtain:&\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right] \\
&\lesssim \sqrt{\frac{2}{n\pi }} \biggl ( \sqrt{|\mathcal {X}_S| \sum _{x \in \mathcal {X}_S} \bigl ( \mathbf {p}(x) + 1/u^{\prime } \bigr ) \bigl ( v - \mathbf {p}(x) - 1/u^{\prime } \bigr )} \\
&\hspace{42.5pt} + \sqrt{|\mathca... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04416041076183319,
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0.04400781914591789,
-0.01297040469944477,
0.... | |
29bdcd34597b618f814c25eda1f717b787bb7108 | subsection | 97 | 120 | Maximum of the | Note that in (REF ), \approx holds iff \mathbf {p}\in \mathcal {C}_{SN}.By Lemma REF and
0 < \epsilon < \ln (|\mathcal {X}_N|+1),
F(\mathbf {p}(\mathcal {X}_S)) is maximized when \mathbf {p}(\mathcal {X}_S)=0.
Hence the right-hand side of (REF ) is maximized when \mathbf {p}(\mathcal {X}_S) = 0 and \mathbf {p}\in \math... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05913354456424713,
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0.009016492404043674,
-0.02540179342031479,
0.039788514375686646,
-0.02436436153948307,
0.026... | |
fa16785dfd18ef27bcb484730bcdf725dd143798 | subsection | 98 | 120 | Maximum of the | If x\in \mathcal {X}_S then we have:\mathbf {m}(x)
&= \frac{e^\epsilon - 1}{|\mathcal {X}_S| + e^\epsilon - 1} \mathbf {p}^*(x) + \frac{1}{|\mathcal {X}_S| + e^\epsilon - 1} \\
&= \frac{e^\epsilon - 1}{|\mathcal {X}_S| + e^\epsilon - 1} \frac{1 - \frac{|\mathcal {X}_N|}{e^{\epsilon }-1}}{|\mathcal {X}_S| + |\mathcal {X... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.023055411875247955,
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-0.051024023443460464,
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0.045531004667282104,
0.02072088047862053,
-0.03083718568086624,
0.0036314944736659527,
-0.00482546258717775... | |
0be1a22fdba242facb7bd5b9e36dea50b0874085 | subsection | 99 | 120 | Maximum of the | Hence:\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right]
&\lesssim \mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}^*) \right] \\
&= \sqrt{\frac{2}{n\pi }} \cdot v \sqrt{|\mathcal {X}| \sum _{x \in \mathcal {X}} \frac{1}{|\mathcal {X}|} \Bigl (1 - \frac{1}{|\mathcal {X}|}\Bigr )} \\
&= \sqrt{\frac{2(|\math... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0589120090007782,
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0.016818920150399208,
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0.014682215638458729,
-0.014422758482396603,
... | |
2cf7f7892e5791341a3b602b893dbc3a68b60d4f | subsection | 100 | 120 | Maximum of the | Then:\frac{d \sqrt{A(w)}}{dw}
&= \frac{1}{2\sqrt{A(w)}} \frac{dA(w)}{dw} \\
&= \frac{1}{2\sqrt{A(w)}} \Bigl (
- 2 w + |\mathcal {X}_S|\bigl ( v_N - {\textstyle \frac{1}{u^{\prime }}} \bigr )
\Bigr ) \\
\frac{d \sqrt{B(w)}}{dw}
&= \frac{1}{2\sqrt{B(w)}} \frac{dB(w)}{dw} \\
&= \frac{1}{2\sqrt{B(w)}} \Bigl (
- 2 w + \bigl... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03873853757977486,
0.03104577772319317,
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0.02390250191092491,
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-0.005681804846972227,
-0.004632445052266121,
... | |
c31dbe835163758354f4b833815d0c79beddfa14 | subsection | 101 | 120 | Maximum of the | By \epsilon < 2\ln \bigl (\frac{|\mathcal {X}_N|}{2}+1\bigr ), we have e^{\epsilon /2} < \frac{|\mathcal {X}_N|}{2}+1, hence:\frac{2}{|\mathcal {X}_N|} - \frac{1}{u^{\prime }}
= \frac{2}{|\mathcal {X}_N|} - \frac{1}{e^{\epsilon /2}-1}
< \frac{2}{|\mathcal {X}_N|} - \frac{2}{|\mathcal {X}_N|}
= 0.Hence \frac{dF(w)}{dw} ... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05581647902727127,
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0.026855384930968285,
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-0.00420759804546833,
-0.02378837764263153,
0.... | |
3ba7af66473a0837ee785443a41743386756d3e7 | subsection | 102 | 120 | Maximum of the | Hence by (REF ) we obtain:\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right]
\lesssim & \sqrt{\frac{2}{n\pi }} \Biggl (\!\sqrt{|\mathcal {X}_S| \sum _{j=1}^{|\mathcal {X}_S|} \bigl ( \mathbf {p}(x_j) + 1/u^{\prime } \bigr ) \bigl ( v_N - \mathbf {p}(x_j) \bigr )} \\
&\hspace{25.5pt} + \!\sqrt{|\mathcal {X}_N|... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.051130883395671844,
0.0058800517581403255,
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0.018422380089759827,
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0.043316252529621124,
-0.027030684053897858,
... | |
99187dfa8d888ef69ad3ba6d23871da8dbdc2e9b | subsection | 103 | 120 | Maximum of the | Note that in (REF ), \approx holds iff \mathbf {p}\in \mathcal {C}_{SN}.By Lemma REF and
0 < \epsilon < 2\ln (\frac{|\mathcal {X}_N|}{2}+1),
F(\mathbf {p}(\mathcal {X}_S)) is maximized when \mathbf {p}(\mathcal {X}_S)=0. | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03299921751022339,
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0.005208595190197229,
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0.006788345053792,
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0.03069446049630642,
-0.014675646089017391,
0.00... | |
9fbca4b28e3b75eb43d411c661145e5f6058a9f5 | subsection | 104 | 120 | Maximum of the | Hence the right-hand side of (REF ) is maximized when \mathbf {p}(\mathcal {X}_S) = 0 and \mathbf {p}\in \mathcal {C}_{SN}, i.e., when \mathbf {p} is the uniform distribution \mathbf {p}_{U_{\!N}} over \mathcal {X}_N.Therefore we obtain:\mathbb {E}\left[ l_1(\hat{\mathbf {p}},\mathbf {p}) \right]
&\lesssim \!\mathbb {E... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.06660372763872147,
0.022130019962787628,
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0.0190928652882576,
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0.03029523231089115,
-0.01570468209683895,
0.010... | |
d9d02fe79b09b167bf80bfb7f5d6762f357a2a0a | subsection | 105 | 120 | Maximum of the | Then e^{\epsilon }-1 < |\mathcal {X}_N|.\frac{d F(w)}{dw}
&=
- \frac{2}{n (e^{\epsilon }-1)}
+ \frac{1}{n} \Bigl ( - \frac{2w}{|\mathcal {X}_S|} - \frac{2w-2}{|\mathcal {X}_N|} \Bigr )&<
- \frac{2}{n |\mathcal {X}_N|}
+ \frac{1}{n} \Bigl ( - \frac{2w}{|\mathcal {X}_S|} - \frac{2w}{|\mathcal {X}_N|} + \frac{2}{|\mathcal... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.035217612981796265,
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0.020233342424035072,
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0.01583877019584179,
-0.03671298921108246,
-... | |
c2ccd678c1f66ebb9e40f66887b9c2e7dfb7348a | subsection | 106 | 120 | Maximum of the | An analogous inequality holds for \mathcal {X}_N.
Therefore we obtain:\sum _{x\in \mathcal {X}} \mathbf {p}(x)^2
&\ge \! \@root |\mathcal {X}_S| \of {\prod _{x\in \mathcal {X}_S} \mathbf {p}(x)^2} \cdot |\mathcal {X}_S|
+\!\@root |\mathcal {X}_N| \of {\prod _{x\in \mathcal {X}_N} \mathbf {p}(x)^2} \cdot |\mathcal {X}_N... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.011972719803452492,
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0.008783047087490559,
-0.025105321779847145,
0.007844459265470505,
-0.03635311499238014,
... | |
4903319d5416bdf40d4cd20165345670c19712f3 | subsection | 107 | 120 | Maximum of the | Therefore \mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}) \right] is maximized when \mathbf {p}(\mathcal {X}_S) = 0 and \mathbf {p}\in \mathcal {C}_{SN}, i.e., when \mathbf {p} is the uniform distribution \mathbf {p}_{U_{\!N}} over \mathcal {X}_N.Therefore we obtain:\mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mat... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0554305836558342,
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0.033942073583602905,
-0.02127484418451786,
... | |
18c73f2f2daea284665f55b96e1ab75057ac30ae | subsection | 108 | 120 | Maximum of the | (Note that by \epsilon \ge \ln (|\mathcal {X}_N|+1), \mathbf {p}^*(x)\ge 0 holds for all x\in \mathcal {X}.)
To show this, we recall that if \mathbf {p}= \mathbf {p}^* then \mathbf {m} is the uniform distribution over \mathcal {Y}, as shown
in the proof for PropositionREF .Let v = \frac{|\mathcal {X}_S|+e^{\epsilon }-1... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.03854657709598541,
-0.0035307658836245537,
-0.013741564936935902,
-0.04379599168896675,
0.02893282286822796,
-0.03442639857530594,
... | |
515e4e41e464eb96114a15734fa996eb3bc6c8d3 | subsection | 109 | 120 | Maximum of the | Hence:\mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}) \right]
&\le \mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}^*) \right] \\
&= \frac{v^2}{n} \Bigl ( 1 - \@root |\mathcal {X}| \of {\textstyle \prod _{x\in \mathcal {X}} \bigl (\frac{1}{|\mathcal {X}|}\bigr )^2} \cdot |\mathcal {X}| \Bigr ) \\
&= \frac{v... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05683859810233116,
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0.01611749641597271,
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0... | |
a122d24f3dcc297f815cb3d3fc48c38172fe2a17 | subsection | 110 | 120 | Maximum of the | \hspace{42.5pt}\bigl (\text{by ${\textstyle \epsilon < 2\ln (\frac{|\mathcal {X}_N|}{2}+1)}$ and $w\ge 0$}\bigr )Therefore, F(w) is decreasing in w.Now we prove Proposition REF as follows.Let
M = 1 + {\textstyle \frac{(|\mathcal {X}_S|+1)e^{\epsilon /2} - 1}{(e^{\epsilon /2}-1)^2}}.
By Proposition REF , we have:\mathbb... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.042585551738739014,
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-0.006437439471483231,
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... | |
461e7b455a9ea32a2bba926b0bf5017aa0c3792c | subsection | 111 | 120 | Maximum of the | \@root |\mathcal {X}_S| \of {\prod _{j=1}^{|\mathcal {X}_S|} \mathbf {p}(x_j)^2} \cdot |\mathcal {X}_S|
+\!\@root |\mathcal {X}_N| \of {\prod _{j=|\mathcal {X}_S|+1}^{|\mathcal {X}|} \mathbf {p}(x_j)^2} \cdot |\mathcal {X}_N| \\
&={\textstyle \frac{\mathbf {p}(\mathcal {X}_S)^2}{|\mathcal {X}_S|} + \frac{\mathbf {p}(\m... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.... | |
32a7e66b7331d489d9172708413d69dc1c4b3e1c | subsection | 112 | 120 | Maximum of the | Therefore, \mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\mathbf {p}) \right] is maximized when \mathbf {p}(\mathcal {X}_S) = 0 and \mathbf {p}\in \mathcal {C}_{SN}, i.e., when \mathbf {p} is the uniform distribution \mathbf {p}_{U_{\!N}} over \mathcal {X}_N.Therefore we obtain:\mathbb {E}\left[ l_2^2(\hat{\mathbf {p}},\ma... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
cbba16a85641e50299034abf598c625ad3764b19 | subsection | 113 | 120 | L2 loss of the utility-optimized Mechanisms | In this section we
theoretically analyze
the l_2 loss of the utility-optimized RR and the utility-optimized RAPPOR.
Table REF summarizes the l_2 loss of each obfuscation mechanism.
We also show the results of the MSE in our experiments.
[Table: l_2 loss of each obfuscation mechanism in the worst case (RR: randomized re... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.... | |
b50a6e348b5241b6e2e172daf11f887d10e3a148 | subsection | 114 | 120 | Experimental Results of the MSE | Figures REF , REF , REF , and REF
show the results of the MSE corresponding to Figures REF , REF , REF , and REF , respectively.
It can be seen that a tendency similar to the results of the TV is obtained for the results of the MSE, meaning that our proposed methods are effective in terms of both the l_1 and l_2 losse... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0... | |
e5c554471e98415b6d8a44e2b5ef7d68e379d757 | subsection | 115 | 120 | Privacy Analysis of PUMs | Below we show the proof of
blackPropositions REF and REF .*Since \mathbf {Q}_{cmn} provides (\mathcal {Z}_S,\mathcal {Y}_P,\epsilon )-ULDP,
for any output data y \in \mathcal {Y}_I, there exists intermediate data
x \in \mathcal {X}_N such that
\mathbf {Q}_{cmn}(y|x) > 0 and \mathbf {Q}_{cmn}(y|x^{\prime }) = 0 for any ... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.0... | |
8f791273221711c7a4f22156b7a9f9163c8f0713 | subsection | 116 | 120 | Privacy Analysis of PUMs | Thus,
(REF ) holds for any x,x^{\prime } \in \mathcal {X} and any y \in \mathcal {Y}.*
blackSince \mathbf {Q}^{(i)} = \mathbf {Q}_{cmn} \circ f_{pre}^{(i)} and f_{pre}^{(i)} is given by (REF ), we have:\mathbf {Q}^{(i)}(y|x) =
{\left\lbrace \begin{array}{ll}
\mathbf {Q}_{cmn}(y|\bot _k) & \text{(if $x \in \mathcal {X}... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0... | |
8fcc1de149e1778191d45f635d7d4a863d933199 | subsection | 117 | 120 | Utility Analysis of PUMs | Below we show the proof of Theorem REF .*Let \hat{\mathbf {p}}^* be the estimate of \mathbf {p} in the case where the exact distribution \pi _k is known to the analyst; i.e., \hat{\pi }_k = \pi _k for any
k = 1, \cdots , \kappa . | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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... | |
b8094065bf4589feb3978acbfb64693367138efc | subsection | 118 | 120 | Utility Analysis of PUMs | Then the l_1 loss of \hat{\mathbf {p}} can be written, using the triangle inequality, as follows:l_1(\hat{\mathbf {p}}, \mathbf {p}) \le l_1(\hat{\mathbf {p}}, \hat{\mathbf {p}}^*) + l_1(\hat{\mathbf {p}}^*, \mathbf {p}).Since \pi _k(x) is the conditional probability that
personal data is x \in \mathcal {X} given that ... | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.01... | |
2c32f4118e3158635960a925851c5c3e72952c3e | subsection | 119 | 120 | Utility Analysis of PUMs | MSE when \epsilon = 0.1 or \ln |\mathcal {X}|.][Figure: \epsilon vs. MSE (personalized-mechanism) ((I): w/o background knowledge, (II) POI distribution, (III) true distribution).] | {
"cite_spans": []
} | 1807.11317 | Utility-Optimized Local Differential Privacy Mechanisms for Distribution
Estimation | [
"Takao Murakami",
"Yusuke Kawamoto"
] | [
"cs.DB",
"cs.CR",
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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46a7a5256c4959ccf6e5d1298241273c5bdf7678 | abstract | 0 | 289 | Abstract | In this thesis we take Einstein theory in dimension four seriously, and
explore the special aspects of gravity in this number of dimension. Among the
many surprising features in dimension four, one of them is the possibility of
`Chiral formulations of gravity' - they are surprising as they typically do not
rely on a me... | {
"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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5dc254a8ba5dbb34a81f49c7b66e9dd7a9ed267d | subsection | 1 | 289 | Résumé de la Thèse (English version below) | Dans cette thèse nous explorons les aspects de la gravité d'Einstein qui sont propres à la dimension quatre.
L'une des propriétés surprenantes liées à cette dimension est la possibilité de formuler la gravité de manière 'Chirale'. Dans ce type de reformulations, typiquement, la métrique perd son rôle centrale. La corre... | {
"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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... | |
daa7cccda50bb89b6febbb873d11886399c303bb | subsection | 2 | 289 | Summary of the Thesis | In this thesis we take Einstein theory in dimension four seriously, and explore the special aspects of gravity in this number of dimension.
Among the many surprising features in dimension four, one of them is the possibility of `Chiral formulations of gravity' - they are surprising as they typically do not rely on a me... | {
"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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80c35034c4da516e7b467ff78c4d64c478123215 | subsection | 3 | 289 | Introduction | Introduction
equationsection
thmcnterchapter | {
"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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68cf2155d419f56ae6d17eb3358aac6e330228c7 | subsection | 4 | 289 | Name the problem: Quantum Gravity | The essential field equations of the general theory of relativity are now more than one hundred years old. Less than a year after the celebration of this centenary, the LIGO cooperation offered the theory its most triumphal confirmation with the first detection of gravitational waves . The other pillar of contemporary ... | {
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"raw": "Einstein, A. (1915). The Field Equations of Gravitation. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1915:844–847.",
"source_ref_id": "dc0dcea87b34fe213812524aacf9a7ccb5898bfb",
... | 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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350d8bab86ef57abc372d4c500c6ce80437c5f57 | subsection | 5 | 289 | Name the problem: Quantum Gravity | From a more pragmatic perspective, the different infinities (or singularities) that plague both theories are another motivation: singularities seems to be ubiquitous to GR , and even though in QFT most infinities were tamed during the tortuous development of the standard model, when it is applied to gravity they prolif... | {
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"doi": "10.1103/physrevlett.14.57",
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"raw": "Penrose, R. (1965). Gravitational collapse and space-time singularities. Phys. Rev. Lett., 14:57–59.",
"source_ref_id": "871929045357bc... | 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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b07d969386b8b45f7807db09dcbe4cc86dbe8c4f | subsection | 6 | 289 | Gravity, Quantum and a Matter of attitude | Even though not directly tied up to quantum gravity the work presented in this thesis took its motivation from this problem and we thus wish to take some more time to consider the different possible attitudes towards it.One of the few essential aspects they all agree on is that something new should happen at a fundamen... | {
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"raw": "Weinberg, S. (1980). ULTRAVIOLET DIVERGENCES IN QUANTUM THEORIES OF GRAVITATION. In General Relativity: An Einstein Centenary Survey, pages 790–831.",
"source_r... | 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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f9567b3298170876203b8e8b91b691273f8db3bf | subsection | 7 | 289 | Gravity, Quantum and a Matter of attitude | What is more, if this space of `asymptotically safe' theories is small enough -and if one assumes that our world is described by such a theory- only a finite number of measurement might be enough to know the theory at any scale. In fact this is probably the only approach to quantum gravity were essentially nothing part... | {
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"source_ref_id": "0f1c23e58cabb3f5d4bd827ab01295c... | 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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f904e787bdaa19e0ad5e86acc575b06760229776 | subsection | 8 | 289 | Gravity, Quantum and a Matter of attitude | Finally the asymptotic safety of gravity is still an open -interesting- problem. As for now -and once again- `Quantum gravity' is therefore less the name of a theory to come than the name of a problem. | {
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Three-Forms | [
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9d2f2216c0351d92a085f1568520e3b1dfec0464 | subsection | 9 | 289 | Following Penrose and Hitchin: Geometry as a guiding line | This thesis is not about quantum gravity. Rather, its most obvious unifying theme is the description of classicali.e non-quantum gravity in terms of unusual geometrical structures. The motivation for looking for and studying such alternative descriptions of gravity however takes its root in the `Penrose's approach' to ... | {
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88f79ab7dceed33bc2ae9ca0b351aaf03054a9be | subsection | 10 | 289 | Following Penrose and Hitchin: Geometry as a guiding line | Associated with these chiral formulations is a natural family of chiral deformations of GR, that were first studied by Bensgtsson (see e.g ) and dubbed `neighbours of GR'. We prefer the term `chiral deformations of gravity' to emphasis the following facts: all those theories describe spin-2 particles with only 2 propag... | {
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97275649ee49d2a9f31c98a7927f420d8ecb9353 | subsection | 11 | 289 | Following Penrose and Hitchin: Geometry as a guiding line | This is our motivations for considering dimensional reduction of Hitchin theory from six to three and seven to four dimension. Accordingly we will consider fibre bundle over 3D and 4D manifold such that the total space is a 6D or 7D manifold.In the second part of this thesis we show that the dimensional reduction of 6D... | {
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a6f4892de0bb0d63f5312903b47356e34877a087 | subsection | 12 | 289 | Chiral Formulations of 4D Gravity and Twistors | Chiral Formulations of 4D Gravity and Twistors | {
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138638866ab248fe7b20052783a86d84eef537e3 | subsection | 13 | 289 | Introduction to Part 1: Chiral and Twistor Formulations of Four Dimensional Gravity | equationsection
thmcnterchapter
Introduction to Part 1It is well known that gravity can be given `chiral formulations' For the most striking ones see , , . See also section REF for a review of chiral Lagrangians for gravity, i.e formulations where the full local isometry group{\rm {SO}}(4,= {\rm {SL}}(2, \times {\rm ... | {
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5f3ffb168ffbbf4e8eff6eb655df8da3b5ffde50 | subsection | 14 | 289 | Introduction to Part 1: Chiral and Twistor Formulations of Four Dimensional Gravity | This reduces the group of local symmetries to {\rm {SL}}(4, which is also the 4d (complex)conformal group {\rm {SO}}(6, \simeq {\rm {SL}}(4,/\mathbb {Z}^2 and indeed the first half of the non-linear graviton theorem asserts that, under some generic conditions, integrability of this almost complex structure is equivalen... | {
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340b52a2c4a8edf598451f5879c0117aa68000a9 | subsection | 15 | 289 | Introduction to Part 1: Chiral and Twistor Formulations of Four Dimensional Gravity | In usual twistor theory this is just taken to be some additional data that complements the almost complex structure, the latter being fundamental. However the pure connection formulation of GR suggests that it is the one-form \tau ( loosely related again to the chiral connection) that should be taken as the starting po... | {
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eb11e89d0536d3afa3ee2253b858005d86aed7bf | subsection | 16 | 289 | Introduction to Part 1: Chiral and Twistor Formulations of Four Dimensional Gravity | However, only in the integrable case does the symplectic structure described in this reference coincides with our \omega _A. {\rm {SL}}(2,-connections which are the self-dual connection of a self-dual Einstein metric with non zero cosmological constant were called `perfect' in and are the one such that their curvature ... | {
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8e204e7924ca6b03b9bbbcd922356803d18b83d3 | subsection | 17 | 289 | Introduction to Part 1: Chiral and Twistor Formulations of Four Dimensional Gravity | This will serve as a model for our `connection version' of the non-linear-graviton theorem. At the end of this chapter, see section we take some time to discuss `chiral deformations of GR'. This is an infinite family of spin-two theories with only two propagating degrees of freedom which is naturally related with chira... | {
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44d6011aa7d4b467db06bdf8d592653459db74e2 | subsection | 18 | 289 | Chiral Gravity | In this chapter we review `chiral formulations of gravity'. In section , we tried to adopt a broad perspective and describe the conceptual elements that are common to all these formulations rather than describe a particular Lagrangian. We also tried to avoid hiding the simplicity of the geometrical concepts involved un... | {
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9cba98fc7a582cec0d4d7fb1013ab59bde91e7cd | subsection | 19 | 289 | Chiral Formulations of Gravity : Geometrical Foundations | Chiral formulations of gravity exploit the fact that Einstein equations can be stated using only `one half' of the decomposition {\rm {SO}}(4, = {\rm {SL}}(2, \times {\rm {SL}}(2,/\mathbb {Z}^2. We here briefly review why this is possible.The whole discussion in this section could be treated in complexified terms but f... | {
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d5e16de351eff9099b4badee68f774c6e53e87e9 | subsection | 20 | 289 | Chiral decomposition of the curvature tensor | Let us consider a Riemannian manifold \left(M, g\right). We note \left\lbrace e^I\right\rbrace _{I\in 0..3} an orthonormal frame and \left\lbrace e_I\right\rbrace _{I\in 0..3} a dual co-frame, they are defined up to {\rm {SO}}(4) transformations. In order to see that Einstein equations can be stated using only one half... | {
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ae0e3a82bb39102964f07ebe5f79363fbfed9469 | subsection | 21 | 289 | Chiral decomposition of the curvature tensor | It splits into two {\rm {SU}}(2)-connections D and \widetilde{D},\nabla = D + \widetilde{D}.They naturally act as connections on the bundle of self-dual two-forms and anti-self-dual two-forms respectively.As a consequence of (REF ) the curvature \nabla ^2 two-form can be rewritten\nabla ^2 = D^2 + \widetilde{D}^2.At th... | {
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934669a9f8d313469f10110384b309744b7c6419 | subsection | 22 | 289 | Chiral decomposition of the curvature tensor | The torsion-free condition indeed implies that \nabla ^2 has to be symmetric, i.eG{}^t = \widetilde{G},\qquad \Psi ^t= \Psi ,\qquad \widetilde{\Psi }^t = \widetilde{\Psi }.What is more, for the torsion-free connection tr F = tr \widetilde{F}. Using coordinates, one can indeed immediately see that this last identity is ... | {
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6fd08bccbac3797edcd7ad338f2d9c9fe2f6a94d | subsection | 23 | 289 | Urbantke metric | In the above, we explained how Einstein equations can be stated in an essentially chiral way, i.e in terms of \mathfrak {su}(2)-valued fields. This general principle underlies any chiral formulation of gravity. However this was still very classical in spirit as we considered the metric as the fundamental field. We now ... | {
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28d9268eb6bb2ee6addf39195ea405becc029e59 | subsection | 24 | 289 | Urbantke metric | The signature of the internal metric \tilde{X} however is enough information to fix this ambiguity: for an Euclidean conformal metric \tilde{g} the metric \tilde{X} on self-dual two-forms given by wedge product is Euclidean while for a Kleinian signature it would be Lorentzian.Thus if we start with a triplet \left(B^i\... | {
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345e4fd443cb10048cf770b91fd570ba5de2b2be | subsection | 25 | 289 | Two useful tensors: the sigma two-forms | We already made the remark that a metric allows to identify the Lie algebra \mathfrak {so}(4) with the space of two-forms \Omega ^2. We denote by\Phi \colon \mathfrak {so}(4)= \mathfrak {su}(2)\oplus \mathfrak {su}(2) \rightarrow \Omega ^2=\Omega ^2_+\oplus \Omega ^2_-this isomorphism.We choose a basis \left( \,\sigma ... | {
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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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01fb1f60ce956cd0abf764b343c2fefb8e63f06d | subsection | 26 | 289 | Two useful tensors: the sigma two-forms | This basis is also defined (up to SU(2) transformations) by the orthogonality relations\Sigma ^i \wedge \Sigma ^j = -\widetilde{\Sigma }^i \wedge \widetilde{\Sigma }^j = 2 \delta ^{ij} Vol_g, \qquad \Sigma ^i \wedge \widetilde{\Sigma }^j = 0.The awkward factor of one half in the definition is there for it to fit with t... | {
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Three-Forms | [
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7a172bbb53e584051dc283d22723725d5aeba78a | subsection | 27 | 289 | Two useful tensors: the sigma two-forms | If D=d + {{A}} is the `left' or `self-dual' connection and D^2 = {{F}} its curvature, then we can rewrite the first half of the bloc decomposition (REF ) asD^2 = F^i\,\sigma ^i = \left( F^{ij} \Sigma ^j + G^{ij}\widetilde{\Sigma }^j \right)\sigma ^i.Then, as we already discussed, the self-dual part of Weyl curvature is... | {
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Three-Forms | [
"Yannick Herfray"
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e5aa3f22ac5e10dad19d78b258e1af4ed96a1252 | subsection | 28 | 289 | Definite Connections and Gravity | We review here how to write equations for Einstein-anti-self-dual metric in terms of connections. This is a well known construction (cf ) and we here use the terminology of ,. We also briefly recall how to write equations for full Einstein gravity in terms of connections from ,.We now take {{A}}= A^i \,\sigma ^i to be ... | {
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Three-Forms | [
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3a82fd35d2e980d03d0b427dc28f2311eb962ce2 | subsection | 29 | 289 | Definite Connections | We mainly consider definite connections, i.e connections such that the curvature two-form is a definite triplet:Definition I.4 Definite ConnectionsA SU(2)-connection D = d + A^i \,\sigma ^i, is called definite if the conformal metric, \tilde{X}^{ij} = F^i\wedge F^j \big /d^4x, constructed from its curvature, D^2=F^i \,... | {
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c50d1160cf7d1a298deab7c453879f8d6d42e5ef | subsection | 30 | 289 | Definite Connections | Finally we also have two scaling transformations, one acting on \widetilde{X} and the other one on \Sigma , we identify them by requiring that F^i \wedge F^j = \widetilde{X}^{ij} \; \frac{1}{3} \Sigma ^k \wedge \Sigma ^k.In what follows these identifications will always be assumed unless we explicitly specify otherwise... | {
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Three-Forms | [
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26952008670ca2c2b96ee1c56763adc02429bb9c | subsection | 31 | 289 | Anti-self-dual gravity and Perfect Connections | A metric is said to be `anti-self-dual' if the self-dual part of its Weyl curvature vanishes ie, if W_+=0 in (REF ).
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Three-Forms | [
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b57c1998fc625ca3bc87dd5a4ff3106d51e89ec0 | subsection | 32 | 289 | Anti-self-dual gravity and Perfect Connections | It follows that D= d + {{A}} is the self-dual connection of the Urbantke metric with volume form \mu =\frac{3}{2 \Lambda ^2} F^k \wedge F^k. With this observation (REF ) are just the field equations for Einstein anti-self-dual gravity (REF ) with cosmological constant s\,|\Lambda |. | {
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Three-Forms | [
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395a24ab3dce5c091bf42bf26fb0e9409fe51280 | subsection | 33 | 289 | Pure connection formulation of Einstein equations | At this point it is hard to resist writing down the pure connection formulation of Einstein equations.Consider a definite SU(2)-connection D=d + {{A}} with curvature {{F}}=F^i\,\sigma ^i. As already explained, it is associated with an orientation, a sign s and conformal class of metric \tilde{g}_{\text{\tiny (F)}}. We ... | {
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Three-Forms | [
"Yannick Herfray"
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4d34a110cd8e3963cb1ad35b269b31bb2077f15b | subsection | 34 | 289 | Pure connection formulation of Einstein equations | Note that for perfect connections, X^{ij} = \delta ^{ij}\frac{\Lambda ^2}{9}, as a result of which perfect connections are special case of Einstein connections with the `definite square root' convention (note that perfect connections are not Einstein connections for the `indefinite square root' as d_A \left( \left(\sqr... | {
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b60acab3bd1d394008699bd7ec7f316aba7d9440 | subsection | 35 | 289 | Pure connection formulation of Einstein equations | We call them the `definite identification' and the `indefinite identification' depending whether or not the resulting matrix M^{ij} is definite or not.As a rule, we now take the identification corresponding to the square root that we chose, ie if one chooses the `definite square root', we take the `definite identificat... | {
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650395c8fd6d6ed7a98f54b0a4fcbde48ce491bf | subsection | 36 | 289 | Chiral Deformations of Gravity | While chiral formulations of GR described above are certainly known to differential geometers specialising in Einstein manifolds, the related `chiral deformations' of the Einstein theory are almost completely unknown to the community. It is however an interesting fact that the four-dimensional Einstein condition can be... | {
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fcf8c96d4871f8153ab1f29a1536caeb8b51045f | subsection | 37 | 289 | Pure connection formulation of the Chiral Deformations of GR | Let again {{A}}= A^i \sigma ^i be a definite {\rm {SU}}(2)-connection for a {\rm {SU}}(2) principal bundle over a 4-dimensional manifold M,{\rm {SU}}(2) \hookrightarrow P \rightarrow Mand {{F}}= F^i \sigma ^i be its curvature two-form. The definiteness of the connection, as defined in REF , amounts to the definiteness ... | {
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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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589fe7b03a75c3672df9f5feab75776cd98f56fa | subsection | 38 | 289 | The anti-self-dual sector: Instanton solutions | In general, solutions to (REF ) strongly depend on the theory. There is however a sector which is shared by all Chiral deformations, mainly the anti-self-dual sector of gravity i.e anti-self dual Einstein metrics:As we already discussed in REF perfect connection i.e satisfying X^{ij}\sim \delta ^{ij} give rise to metri... | {
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Three-Forms | [
"Yannick Herfray"
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2861a829fa4f368b47422bca055eb84e4162efb8 | subsection | 39 | 289 | Metric interpretation. | As already discussed in Proposition REF , a {\rm {SU}}(2)-connection that satisfies the rather weak requirement of being definite defines a conformal Riemannian metric on M. As already discussed, the triple of curvature two-forms is anti-self-dual with respect to this (conformal) metric and this property defines it uni... | {
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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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f04aee9a68e71856db0dae3221c8984d60643ab0 | subsection | 40 | 289 | More general solutions and propagating degrees of freedom. | Even though we are far from understanding all Einstein metrics on 4-manifolds, some intuition as to how many solutions exist comes from the Lorentzian version of the theory. Indeed, GR with Lorentzian signature is a theory with local degrees of freedom, and so the space of solutions is infinite-dimensional. For example... | {
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8da1395386fa99889a13568ec7f88a807d57e163 | subsection | 41 | 289 | Lorentzian signature and the physical significance of these deformations. | Once GR gets embedded into an infinitely large class of gravity theories all with similar properties, one is forced to ask a number of questions: What makes GR unique as compared to all these other theories? In fact, as all the chiral deformations of GR approximately look the same around DeSitter space could it be that... | {
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cebcb50f185c7cc3fdb01ed30db66e81ad3290a8 | subsection | 42 | 289 | Variational Principles | In the above we gave a pure connection formulation of the Chiral deformation of GR (REF ). This deformations where parametrised by a choice of deformation-function f, GR itself being given by a particular representative (REF ). We chose this presentation as it is the most compact one but many other descriptions of thes... | {
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Three-Forms | [
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dcf82e8169d6caf6e9cf7b3492b15887b7335693 | subsection | 43 | 289 | Plebanski-like actions, | The most basic way to describe these chiral deformations is in terms of the following LagragianS[{{A}},{{B}},\Psi ] = \int B^i \wedge F^i - \left(\Psi ^{ij} + \frac{\Lambda \left(\Psi \right)}{3}\delta ^{ij}\right) \frac{1}{2}B^i \wedge B^j.It is not a very economical action as is contains a lot of fields: a {\rm {SU}}... | {
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5b7c2d4bce7e1795bc822ef706429f91635709a1 | subsection | 44 | 289 | Intermediate actions of the type | Starting from the Plebanski-like action (REF ), the most direct way to see the equivalence with the pure connection action (REF ) is to integrat out {{B}} and \Psi in this order. The resulting intermediate action isS[{{A}},\Psi ] = \frac{1}{2}\int \left(\left(\Psi +\frac{\Lambda \left(\Psi \right)}{3} \delta \right)^{-... | {
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"raw": "Herfray, Y., Krasnov, K., and Shtanov, Y. (2016b). Anisotropic singularities in chiral modified gravity. Class. Quant. Grav., 33:235001.... | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
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16f253b8b2bbb81afe2325c3cc73dfb279e463da | subsection | 45 | 289 | Intermediate actions of the type | This action was first discussed in .The particular potential necessary to describe GR is however not so easy to derive from the Plebanski-like action and was rather guessed and first discussed in (see however for a derivation from a more complicated Lagrangian):S_{GR}\left[{{A}}, {{B}}\right] = \int B^i \wedge F^i + \f... | {
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Three-Forms | [
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4b368b1af3e07fdd4a33c0c669751e6a70e18008 | subsection | 46 | 289 | The self-dual Palatini-like action | In order to integrate out \widetilde{M} from (REF ), it is convenient to introduce the following parametrisation for the curvature {{F}}({{A}}),{{F}}= \left(\sqrt{\widetilde{X}}^{ij} \Sigma ^i + G^{ij} \widetilde{\Sigma }^j \right) \sigma _iWhere \widetilde{X} is again a square-matrix \widetilde{X}= \widetilde{X}^{ij} ... | {
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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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d1f08fcd1a3c18b374451c8213bda47baee237f3 | subsection | 47 | 289 | Intermediate action of type | Starting back at (REF ) one can instead integrate out the metric. The resulting field equations say that e is a tetrad for the Urbantke metric \tilde{g}_{{{F}}} constructed from the curvature and with volume form \left(\textrm {Tr}\sqrt{ {{F}}\wedge {{F}}} \right)^2:S\left[{{A}}, \widetilde{M}\right] = \int \widetilde{... | {
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Three-Forms | [
"Yannick Herfray"
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da5a6d520ac8aaa4e9d7ab600f45ac9249f037d7 | subsection | 48 | 289 | Twistors | In this chapter, we wish to clarify that twistor theory is closely related to the above chiral formulations of GR. In particular, the description of self-dual Einstein metrics in terms of {\rm {SU}}(2)-connections has a very nice twistor counterpart in the form of the Non-Linear-Graviton theorem. This perspective leads... | {
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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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d66ff3ccd1c94b0c11640dd0c8f6f7d6b814a66f | subsection | 49 | 289 | Euclidean Twistor Space: Traditional Approach | We now review the geometry of the Twistor space M) associated with a Riemannian manifold \left(M, g\right). In this Euclidean signature setting, this is just the primed spinor bundle over M:\pi _{A^{\prime }} \hookrightarrow M) \xrightarrow{} M.The correspondence between space-time points and twistor-space points is th... | {
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Three-Forms | [
"Yannick Herfray"
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190d08fa9354a8bf59555d0a38c1d962a9b0c16f | subsection | 50 | 289 | The Flat Case from Quaternions | We first describe the flat twistor space i.e the twistor space associated with the conformal compactification S4 of the four dimensional Euclidean space. In the flat case, the twistor space has a beautiful interpretation in terms of quaternion geometry that we now review.
Our presentation will be non-standard in the fo... | {
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Three-Forms | [
"Yannick Herfray"
] | [
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2a945468fc8173294e8c5c38a126bc9c00c16a74 | subsection | 51 | 289 | Quaternion Geometry | They are many descriptions of a quaternions \mathbb {H} (also called Hamilton's numbers) : in terms of matrix, spinors, or in more abstract terms. The most common starting point is to describe quaternions as hyper-complex numbers:q = q_0 + {\mathrm {i}}\;q_1 + {\mathrm {j}}\;q_2 + {\mathrm {k}}\;q_3 \in \mathbb {H}Here... | {
"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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83936f81e658b1db1566c06867d636bf6e8866b2 | subsection | 52 | 289 | Quaternion Geometry | They, however preserve associativity (contrary to octonions) this allows for a matrix representation as the matrix group U(2,: In terms of the notation (), this is easily done asq \simeq \begin{pmatrix}
\alpha & -\beta ^* \\
\beta & \alpha ^*
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"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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389c1d35a4e1acf4a5702231f415437a3f054a4d | subsection | 53 | 289 | Quaternionic Structure | In most situations we will not be working directly with quaternions but rather with vector spaces equipped with a quaternion structure.A quaternionic structure on a complex vector space V \simeq {2n} is given by an operator J on V which is both anti-linear, J(\lambda v) = \lambda ^* J(v), \forall \lambda \in and an ant... | {
"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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542730389e4f90d3bc1b60a67e20455791fbc17e | subsection | 54 | 289 | Quaternionic Structure | This bilinear product is said to be compatible with the quaternionic structure J if\omega (J(.),J(.)) = \left(\omega (.,.)\right)^*.This is a useful condition because theng(.,.) = \omega (J(.), .)is a Hermitian product on V \simeq {2n}. In general it will not be definite. Through the identification V \simeq {2n}\simeq ... | {
"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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e21c8507c78c38a3f5f386fd055859099c216a14 | subsection | 55 | 289 | Example: Euclidean Spinors | The simplest example of quaternion structure is that of a two dimensional complex vector space S \simeq 2 together with an anti-linear, anti-involutive operator J. It is best to think of S as the space of spinors \omega ^{A}\in S. In this context we will write J = \text{⌃}. Choosing an adapted basis identifies S with \... | {
"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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7e0e96dc9381e1762d6fe8b98bee776d7a5cedf6 | subsection | 56 | 289 | Hopf Bundles | The Hopf bundles are fibre bundles made up of spheres only i.e of the form {S}^p \hookrightarrow {S}^q \rightarrow {S}^r. It turns out that the only possible cases are{S}^1 &\hookrightarrow {S}^3 \rightarrow {S}^2 \\
{S}^3 &\hookrightarrow {S}^7 \rightarrow {S}^4 \\
{S}^7 &\hookrightarrow {S}^{15} \rightarrow {S}^8. \\... | {
"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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b94f1f34a371d3a322db2508e73e6fec14a1b746 | subsection | 57 | 289 | Hopf Bundles | Of particular importance for us are:
\begin{}[h]
\centering {white}
\begin{}{ >{{blue!40!white}}l >{{blue!20!white}}c >{{blue!20!white}}c}
{blue!40!white}
\end{}Action on {S}^n& n= 2 & n=4 \\
Conformal group & SO(3,1) \simeq PSL(2, & SO(5,1) \simeq PSL(2,\mathbb {H}) \\
Isometry group & SO(3) = SU(2,/_{\pm \mathbb {Id}... | {
"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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b436450a7450329c09c8feaa9aaa53f4de2a05be | subsection | 58 | 289 | Example: Euclidean Spinors and the Riemann Sphere | As a baby example, let us briefly consider the simplest case 2 \rightarrow \mathbb {C}{P}^1. The idea is to construct a metric (resp a conformal metric) on the quotient space \mathbb {C}{P}^1 such that the action of the isometry group {\rm {SU}}(2, (resp the conformal group {\rm {PSL}}(2,) acts linearly on the total sp... | {
"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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ca721a317d59d53dc16d906d088e08d00877be6e | subsection | 59 | 289 | Example: Euclidean Spinors and the Riemann Sphere | \omega \right)^2}.As compare to (REF ), the Lie derivative of this metric in the Euler directions (REF ) vanishes and it therefore descends to the round metric on \mathbb {C}{P}^1. Note that it does not depend on the precise scaling of \epsilon . The precise numerical factor in front of this expression has been chosen ... | {
"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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36b99991e7edf1e8f81992d22c4df4608132ed34 | subsection | 60 | 289 | Example: Euclidean Spinors and the Riemann Sphere | Then the quaternionic metric readsg_{S} = \frac{1}{2}d\bar{\omega } \odot d \omega = dr^2 + r^2\left(dg\right)^2.The two formulas are related byr^2= \hat{\omega }.\omega ,\qquad \left(g^{-1} dg\right)^{AB} = \frac{1}{\hat{\omega }.\omega }\left(\omega ^{(A}d\hat{\omega }^{B)}- \hat{\omega }^{(A}d\omega ^{B)}\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",
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585207280b7d69f7a09952296d4c99c36c53c2c4 | subsection | 61 | 289 | The Twistor space of | We now come to the twistor space of the four-sphere {S}^4. This is just the total space of the Hopf bundle\mathbb {H}\hookrightarrow \mathbb {H}^2 \rightarrow S^4 \simeq \mathbb {H}{P}^1.Just as in the above example we want to construct a metric (resp a conformal metric) on \mathbb {H}{P}^1 such that the isometry group... | {
"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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47a66e84e9620ae311fb477b0bfe0beedd091fdc | subsection | 62 | 289 | Conformal Structure on | The four-sphere can now be given a conformally flat metric as follows. Making use of the four dimensional skew-symmetric tensor \epsilon _{\alpha \beta ,\gamma \delta } (defined up to a scale), we introduce a metric (also defined up to a global rescaling) on as
\begin{equation}
\tilde{g}_{S^4} = \epsilon _{\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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0.04801328107714653,
0.014124696142971516,
-0.00365... | |
0862a88e5a4897abdf90b3f7ea99c930d4bafb2c | subsection | 63 | 289 | Conformal Structure on | Making use of this coordinates and restricting () to \pi = cst, one indeed obtains the conformally flat metric on S^4,ds^2 \propto d\bar{X} \odot dX. | {
"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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0.009757909923791885,
0.04116785153746605,
0.019... | |
8975f1c7a0df578ba7b3324367a173ffccf13bda | subsection | 64 | 289 | Infinity Twistor | If, on top of the quaternionic structure, one is given an `infinity twistor' i.e a compatible complex-bilinear form on , I 2( such that I(Y,Z) = (I(Y,Z))* one obtains an hermitian structure g(X,Y) = I(X,Y) on . In a suitable basis,g(Z,Z)= I_{\alpha \beta } \hat{Z}^{\alpha } Z^{\beta } = \bar{\pi }\pi + \Lambda \;\bar{\... | {
"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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... | |
8d59f4f2db525bf575a99dd80a88fc6681dd2e51 | subsection | 65 | 289 | Infinity Twistor | The metric (REF ) is the hyperbolic metric on the four-dimensionnal Poincarré `four-ball'. Then the action of {\rm {U}}\left(1,1;\mathbb {H}\right) on \mathbb {H}^2 preserves this metric and it follows that {\rm {U}}\left(1,1,\mathbb {H}\right)/_{\pm \mathbb {Id}} is the group of isometry of the hyperbolic space {{H}}^... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.040675606578588486,
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-... | |
a105e6af3aeedced1327e24985263a475b07db9c | subsection | 66 | 289 | Infinity Twistor | \lbrace I\rbrace \\ \\
\end{array}Z^{\alpha } = \frac{1}{\sqrt{1+\Lambda |X|^2}} \begin{pmatrix} \pi \\ X \pi \end{pmatrix} & \mapsto & \left(\; \pi \;,\; X\; \right)
\right.
\end{equation}
Then we can put the above metric in a form adapted to the fibre bundle structure,\footnote {Here D \pi = d \pi + {\mathrm {i}}A \p... | {
"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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0.00108... | |
623f4e729a2e809bdbaa5e5795a60d90e22f7a68 | subsection | 67 | 289 | Infinity Twistor | \end{equation}Because we have a quaternionic structure, PTCP3 itself is fibre bundle
\begin{equation}
\mathbb {C}{P}^1 \hookrightarrow \mathbb {C}{P}^3 \rightarrow {S}^4
\end{equation}
and the Fubiny-Study metric on PT can be refined into
\begin{equation}
g_{\mathbb {PT}} =\frac{Z dZ \odot \hat{Z}d\hat{Z}}{2\left(\hat{... | {
"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 | [
-0.03539694473147392,
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-... | |
47712b7454cf9cc4c7129cc8ba96d32794dcdcfc | subsection | 68 | 289 | 4D Euclidean Space-Time, Complex structure and Spinor Conventions | We now consider a general 4D Riemannian manifold \left(M,g\right). Let \left(e^{I}\right)_{I\in 0,1,2,3} be a orthonormal co-frame and \left(e_{I}\right)_{ I\in 0,1,2,3} be a dual orthonormal frame.
Everywhere Latin indices are raised and lowered with the Euclidean metric \eta _{IJ} = diag\left(1,1,1,1\right).One here ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1083,
"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": 9... | 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 | [
-0.020053284242749214,
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0.021960940212011337,
-0.0025810592342168093,
0... | |
9b6e5e7f42283158a818750a57fdf3544d783558 | subsection | 69 | 289 | Spinor notation | We already described in the previous sections how the isomorphism {\rm {SO}}\left(4\right) \simeq {\rm {SU}}(2) \times {\rm {SU}}(2) /_{\pm \mathbb {Id}} can be made explicit by using identification \mathbb {H}\simeq \mathbb {R}^4. Then {\rm {SU}}(2) actions are just multiplications by unitary quaternions on the 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 | [
-0.023128150030970573,
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0.003129394259303808,
-0.0... | |
d873dccfcebecb89cac7f9be068ac786bcb9ddd6 | subsection | 70 | 289 | Spinor notation | It will be of the form (\ref {Euclidean metric Twistor Space: Twistor- V^AA^{\prime } def}) and thus describe a \emph {real} tangent vector if and only if for any \pi ^{A^{\prime }} \in S^{\prime } there exists \omega ^{A} \in S such that
\begin{equation}
V^{AA^{\prime }} = \frac{1}{\pi .\hat{\pi }}\left(\omega ^{A}\ha... | {
"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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0.019560351967811584,
0.008605333976447582,
-0... |
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