Modular flow for bulk
reconstruction and the
QNEC
Strings 2018
Work with: Zuhair Khandker, Huajia Wang
Srivatsan Balakrishnan, Min Li
arXiv:1706.09432

arXiv:1806.XXXXX

Modular flow
• Consider a bi-partite quantum system:| i 2 H = HA ⌦ HĀ
<latexit sha1_base64="f/HrXkxo+WJQS6i1SAD6J3m0eWc=">AAACAHicbVA9SwNBEN2LXzF+RS0sbBaDYBXuRNBKgjaWEUwM5I6wt5lLluztHbtzQohp/Cs2ForY+jPs/Ddukis08cHA470ZZuaFqRQGXffbKSwtr6yuFddLG5tb2zvl3b2mSTLNocETmehWyAxIoaCBAiW0Ug0sDiXch4PriX//ANqIRN3hMIUgZj0lIsEZWqlTPvAlRPhI/dQI6mvR6+Ml9YV1Km7VnYIuEi8nFZKj3il/+d2EZzEo5JIZ0/bcFIMR0yi4hHHJzwykjA9YD9qWKhaDCUbTB8b02CpdGiXalkI6VX9PjFhszDAObWfMsG/mvYn4n9fOMLoIRkKlGYLis0VRJikmdJIG7QoNHOXQEsa1sLdS3meacbSZlWwI3vzLi6R5WvXcqnd7Vqld5XEUySE5IifEI+ekRm5InTQIJ2PyTF7Jm/PkvDjvzsesteDkM/vkD5zPHwlolgc=</latexit><latexit sha1_base64="9lAv0noQn9/tIDbVkq6frUSdF1Y=">AAACIXicbVDLSsNAFJ3UV62vqEs3g0VwVRIR7EZoddNlBfuAJoTJdNIOnWTCzEQoIb/ixl9x40KR7sSfcdJm0YcHBs6ccy/33uPHjEplWT9GaWt7Z3evvF85ODw6PjFPz7qSJwKTDuaMi76PJGE0Ih1FFSP9WBAU+oz0/Mlj7vdeiJCUR89qGhM3RKOIBhQjpSXPrDshUmOMWNrK7uHSx2tChysaErmipo6PRNrMMs+sWjVrDrhJ7IJUQYG2Z86cIcdJSCKFGZJyYFuxclMkFMWMZBUnkSRGeIJGZKBphPRoN51fmMErrQxhwIV+kYJzdbkjRaGU09DXlfmyct3Lxf+8QaKCupvSKE4UifBiUJAwqDjM44JDKghWbKoJwoLqXSEeI4Gw0qFWdAj2+smbpHtTs62a/XRbbTwUcZTBBbgE18AGd6ABWqANOgCDV/AOPsGX8WZ8GN/GbFFaMoqec7AC4/cPzNKkiA==</latexit>• Observations restricted to OA described by ⇢A<latexit sha1_base64="3Ap41f3zTowzZmo6uGHvt1eR0NE=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bj14jGCeUCyhNnJbDJmdmaZ6RVCyD948aCIV//Hm3/jJNmDJhY0FFXddHdFqRQWff/bW1ldW9/YLGwVt3d29/ZLB4cNqzPDeJ1pqU0ropZLoXgdBUreSg2nSSR5MxreTv3mEzdWaPWAo5SHCe0rEQtG0UmNjhno7nW3VPYr/gxkmQQ5KUOOWrf01elpliVcIZPU2nbgpxiOqUHBJJ8UO5nlKWVD2udtRxVNuA3Hs2sn5NQpPRJr40ohmam/J8Y0sXaURK4zoTiwi95U/M9rZxhfhWOh0gy5YvNFcSYJajJ9nfSE4QzlyBHKjHC3EjaghjJ0ARVdCMHiy8ukcV4J/Epwf1Gu3uRxFOAYTuAMAriEKtxBDerA4BGe4RXePO29eO/ex7x1xctnjuAPvM8fX7qO+w==</latexit><latexit sha1_base64="kt2PKk834GQ2BKfwZNs6G+dycjE=">AAAB9HicbVDLSgMxFL3js9ZX1aWbYBFclRkRdFl1484K9gHtUDJp2oZmMmNyp1CGfocbF4q49WPc+Tdm2llo64HA4Zx7uScniKUw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmCjRjNdZJCPdCqjhUiheR4GSt2LNaRhI3gxGt5nfHHNtRKQecRJzP6QDJfqCUbSS3wkpDhmV6f20e90tld2KOwNZJl5OypCj1i19dXoRS0KukElqTNtzY/RTqlEwyafFTmJ4TNmIDnjbUkVDbvx0FnpKTq3SI/1I26eQzNTfGykNjZmEgZ3MQppFLxP/89oJ9q/8VKg4Qa7Y/FA/kQQjkjVAekJzhnJiCWVa2KyEDammDG1PRVuCt/jlZdI4r3huxXu4KFdv8joKcAwncAYeXEIV7qAGdWDwBM/wCm/O2Hlx3p2P+eiKk+8cwR84nz/LbpIZ</latexit>• Assuming ⇢A is invertible, define Modular flow:<latexit sha1_base64="3Ap41f3zTowzZmo6uGHvt1eR0NE=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bj14jGCeUCyhNnJbDJmdmaZ6RVCyD948aCIV//Hm3/jJNmDJhY0FFXddHdFqRQWff/bW1ldW9/YLGwVt3d29/ZLB4cNqzPDeJ1pqU0ropZLoXgdBUreSg2nSSR5MxreTv3mEzdWaPWAo5SHCe0rEQtG0UmNjhno7nW3VPYr/gxkmQQ5KUOOWrf01elpliVcIZPU2nbgpxiOqUHBJJ8UO5nlKWVD2udtRxVNuA3Hs2sn5NQpPRJr40ohmam/J8Y0sXaURK4zoTiwi95U/M9rZxhfhWOh0gy5YvNFcSYJajJ9nfSE4QzlyBHKjHC3EjaghjJ0ARVdCMHiy8ukcV4J/Epwf1Gu3uRxFOAYTuAMAriEKtxBDerA4BGe4RXePO29eO/ex7x1xctnjuAPvM8fX7qO+w==</latexit>isis
OA (s) = ⇢A OA ⇢A<latexit sha1_base64="fXbqX/B98iGRQ7Bu1Pc4FuK0xt8=">AAACInicbZDLSsNAFIYn9VbrLerSzWAR6sKSiKAuhFY37qxgL9DEMJlO2qGTCzMToYQ8ixtfxY0LRV0JPoyTNohtPTDw83/nMOf8bsSokIbxpRUWFpeWV4qrpbX1jc0tfXunJcKYY9LEIQt5x0WCMBqQpqSSkU7ECfJdRtru8Crj7QfCBQ2DOzmKiO2jfkA9ipFUlqOfWz6SA4xYcpM69Yo4hBfQ4oPQqd8nVKRwCv+SI4UcvWxUjXHBeWHmogzyajj6h9ULceyTQGKGhOiaRiTtBHFJMSNpyYoFiRAeoj7pKhkgnwg7GZ+YwgPl9KAXcvUCCcfu34kE+UKMfFd1ZhuLWZaZ/7FuLL0zO6FBFEsS4MlHXsygDGGWF+xRTrBkIyUQ5lTtCvEAcYSlSrWkQjBnT54XreOqaVTN25Ny7TKPowj2wD6oABOcghq4Bg3QBBg8gmfwCt60J+1Fe9c+J60FLZ/ZBVOlff8Ao1+juw==</latexit>• Why? in some situations
thermal/Gibbs state<latexit sha1_base64="HQXoe/5SaXdsPR8wI4oJBqEw3kQ=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBi2VXBD1WvXisYG2hu5Rsmm1Ds0lIskJZ+je8eFDEq3/Gm//GtN2Dtj4YeLw3w8y8WHFmrO9/e6WV1bX1jfJmZWt7Z3evun/waGSmCW0RyaXuxNhQzgRtWWY57ShNcRpz2o5Ht1O//US1YVI82LGiUYoHgiWMYOuk8AyFXKBQD2Xvulet+XV/BrRMgoLUoECzV/0K+5JkKRWWcGxMN/CVjXKsLSOcTiphZqjCZIQHtOuowCk1UT67eYJOnNJHidSuhEUz9fdEjlNjxmnsOlNsh2bRm4r/ed3MJldRzoTKLBVkvijJOLISTQNAfaYpsXzsCCaauVsRGWKNiXUxVVwIweLLy+TxvB749eD+ota4KeIowxEcwykEcAkNuIMmtICAgmd4hTcv8168d+9j3lryiplD+APv8wfGMpDa</latexit>ln ⇢A ~ Hamiltonian, e.g. a

Obscure …
• Still seems like an obscure operation … maximally mixed
state?

• We will study it for several reasons:• Universality in QFT ~ like a boost generator close to
the entangling surface for any state• Satisfies powerful constraints - analyticity andunitarity• For AdS/CFT ~ tool for revealing bulk locality andcausality from the boundary

In QFT• Consider a geometric partition:A<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexitĀ<latexit sha1_base64="PjTI1X3daC/ZwUfCQ052fYaiOHo=">AAAB7nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPVi8cK9gPaUDbbTbt0swm7E6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px3cxvP3FtRKIecZLyIKZDJSLBKFqp3Qupzm+mfbfq1bw5yCrxC1KFAo2++9UbJCyLuUImqTFd30sxyKlGwSSfVnqZ4SllYzrkXUsVjbkJ8vm5U3JmlQGJEm1LIZmrvydyGhsziUPbGVMcmWVvJv7ndTOMroNcqDRDrthiUZRJggmZ/U4GQnOGcmIJZVrYWwkbUU0Z2oQqNgR/+eVV0rqo+V7Nf7is1m+LOMpwAqdwDj5cQR3uoQFNYDCGZ3iFNyd1Xpx352PRWnKKmWP4A+fzB1Blj4o=</latexit>

In QFT• Consider a geometric partition:ĀA<latexit sha1_base64="PjTI1X3daC/ZwUfCQ052fYaiOHo=">AAAB7nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPVi8cK9gPaUDbbTbt0swm7E6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px3cxvP3FtRKIecZLyIKZDJSLBKFqp3Qupzm+mfbfq1bw5yCrxC1KFAo2++9UbJCyLuUImqTFd30sxyKlGwSSfVnqZ4SllYzrkXUsVjbkJ8vm5U3JmlQGJEm1LIZmrvydyGhsziUPbGVMcmWVvJv7ndTOMroNcqDRDrthiUZRJggmZ/U4GQnOGcmIJZVrYWwkbUU0Z2oQqNgR/+eVV0rqo+V7Nf7is1m+LOMpwAqdwDj5cQR3uoQFNYDCGZ3iFNyd1Xpx352PRWnKKmWP4A+fzB1Blj4o=</latexit><latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit• Not really a tensor factorization SEE = 1<latexit sha1_base64="Wjns9zipHkY3eFZj8kN4KAluFbc=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoZbEInkoigl6EohQ8VrQf0Iaw2W7apZtN2N0IMfSXePGgiFd/ijf/jds2B60+GHi8N8PMvCDhTGnH+bJKK6tr6xvlzcrW9s5u1d7b76g4lYS2Scxj2QuwopwJ2tZMc9pLJMVRwGk3mFzP/O4DlYrF4l5nCfUiPBIsZARrI/l29c7Pm80pukQDJkKd+XbNqTtzoL/ELUgNCrR8+3MwjEkaUaEJx0r1XSfRXo6lZoTTaWWQKppgMsEj2jdU4IgqL58fPkXHRhmiMJamhEZz9edEjiOlsigwnRHWY7XszcT/vH6qwwsvZyJJNRVksShMOdIxmqWAhkxSonlmCCaSmVsRGWOJiTZZVUwI7vLLf0nntO46dff2rNa4KuIowyEcwQm4cA4NuIEWtIFACk/wAq/Wo/VsvVnvi9aSVcwcwC9YH9+3wpJ4</latexit>

In QFT• Consider a geometric partition:ĀA<latexit sha1_base64="PjTI1X3daC/ZwUfCQ052fYaiOHo=">AAAB7nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPVi8cK9gPaUDbbTbt0swm7E6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px3cxvP3FtRKIecZLyIKZDJSLBKFqp3Qupzm+mfbfq1bw5yCrxC1KFAo2++9UbJCyLuUImqTFd30sxyKlGwSSfVnqZ4SllYzrkXUsVjbkJ8vm5U3JmlQGJEm1LIZmrvydyGhsziUPbGVMcmWVvJv7ndTOMroNcqDRDrthiUZRJggmZ/U4GQnOGcmIJZVrYWwkbUU0Z2oQqNgR/+eVV0rqo+V7Nf7is1m+LOMpwAqdwDj5cQR3uoQFNYDCGZ3iFNyd1Xpx352PRWnKKmWP4A+fzB1Blj4o=</latexit><latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit• Not really a tensor factorization SEE = 1• Rather think about algebra of operators in spacetime<latexit sha1_base64="Wjns9zipHkY3eFZj8kN4KAluFbc=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoZbEInkoigl6EohQ8VrQf0Iaw2W7apZtN2N0IMfSXePGgiFd/ijf/jds2B60+GHi8N8PMvCDhTGnH+bJKK6tr6xvlzcrW9s5u1d7b76g4lYS2Scxj2QuwopwJ2tZMc9pLJMVRwGk3mFzP/O4DlYrF4l5nCfUiPBIsZARrI/l29c7Pm80pukQDJkKd+XbNqTtzoL/ELUgNCrR8+3MwjEkaUaEJx0r1XSfRXo6lZoTTaWWQKppgMsEj2jdU4IgqL58fPkXHRhmiMJamhEZz9edEjiOlsigwnRHWY7XszcT/vH6qwwsvZyJJNRVksShMOdIxmqWAhkxSonlmCCaSmVsRGWOJiTZZVUwI7vLLf0nntO46dff2rNa4KuIowyEcwQm4cA4NuIEWtIFACk/wAq/Wo/VsvVnvi9aSVcwcwC9YH9+3wpJ4</latexit>

regions:<latexit sha1_base64="siZHz4HJWYBDdnrQlLOaJhYB1lw=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5CRahbkoigi7rY+Gygn1AE8rNdNIOnUzCzESoIfgrblwo4tb/cOffOGmz0NYDA4dz7uWeOX7MqFS2/W2UlpZXVtfK65WNza3tHXN3ry2jRGDSwhGLRNcHSRjlpKWoYqQbCwKhz0jHH1/nfueBCEkjfq8mMfFCGHIaUAxKS33zwA1BjTCw9CaruT6I9DI76ZtVu25PYS0SpyBVVKDZN7/cQYSTkHCFGUjZc+xYeSkIRTEjWcVNJIkBj2FIeppyCIn00mn6zDrWysAKIqEfV9ZU/b2RQijlJPT1ZJ5Vznu5+J/XS1Rw4aWUx4kiHM8OBQmzVGTlVVgDKghWbKIJYEF1VguPQABWurCKLsGZ//IiaZ/WHbvu3J1VG1dFHWV0iI5QDTnoHDXQLWqiFsLoET2jV/RmPBkvxrvxMRstGcXOPvoD4/MHI7GVAA==</latexit>

D(Ā)<latexit sha1_base64="qjr96GpjDRjk7uZalxjHEdneOjc=">AAAB9XicbVDLSgMxFL3xWeur6tJNsAh1U2ZE0GV9LFxWsA9ox5JJM21oJjMkGaUM/Q83LhRx67+482/MtLPQ1gOBwzn3ck+OHwuujeN8o6XlldW19cJGcXNre2e3tLff1FGiKGvQSESq7RPNBJesYbgRrB0rRkJfsJY/us781iNTmkfy3oxj5oVkIHnAKTFWeuiGxAwpEenNpHJ50iuVnaozBV4kbk7KkKPeK311+xFNQiYNFUTrjuvExkuJMpwKNil2E81iQkdkwDqWShIy7aXT1BN8bJU+DiJlnzR4qv7eSEmo9Tj07WSWUs97mfif10lMcOGlXMaJYZLODgWJwCbCWQW4zxWjRowtIVRxmxXTIVGEGltU0Zbgzn95kTRPq65Tde/OyrWrvI4CHMIRVMCFc6jBLdShARQUPMMrvKEn9ILe0cdsdAnlOwfwB+jzB88Jkgo=</latexit>

D(A)<latexit sha1_base64="siZHz4HJWYBDdnrQlLOaJhYB1lw=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5CRahbkoigi7rY+Gygn1AE8rNdNIOnUzCzESoIfgrblwo4tb/cOffOGmz0NYDA4dz7uWeOX7MqFS2/W2UlpZXVtfK65WNza3tHXN3ry2jRGDSwhGLRNcHSRjlpKWoYqQbCwKhz0jHH1/nfueBCEkjfq8mMfFCGHIaUAxKS33zwA1BjTCw9CaruT6I9DI76ZtVu25PYS0SpyBVVKDZN7/cQYSTkHCFGUjZc+xYeSkIRTEjWcVNJIkBj2FIeppyCIn00mn6zDrWysAKIqEfV9ZU/b2RQijlJPT1ZJ5Vznu5+J/XS1Rw4aWUx4kiHM8OBQmzVGTlVVgDKghWbKIJYEF1VguPQABWurCKLsGZ//IiaZ/WHbvu3J1VG1dFHWV0iI5QDTnoHDXQLWqiFsLoET2jV/RmPBkvxrvxMRstGcXOPvoD4/MHI7GVAA==</latexit>

D(Ā)

In QFT• Modular flow still well defined, associated to some | i<latexit sha1_base64="MXH9tO9D+WBNwXA27Dr/ENCQ0co=">AAAB/HicbVDLSgNBEJz1GeNrNUcvg0HwFHZF0JMEvXiMYB6QDWF20psMmX0w0yssa/wVLx4U8eqHePNvnCR70MSChqKqm+4uP5FCo+N8Wyura+sbm6Wt8vbO7t6+fXDY0nGqODR5LGPV8ZkGKSJookAJnUQBC30JbX98M/XbD6C0iKN7zBLohWwYiUBwhkbq2xVPQoCP1Eu0oJ4SwxFe9e2qU3NmoMvELUiVFGj07S9vEPM0hAi5ZFp3XSfBXs4UCi5hUvZSDQnjYzaErqERC0H38tnxE3pilAENYmUqQjpTf0/kLNQ6C33TGTIc6UVvKv7ndVMMLnu5iJIUIeLzRUEqKcZ0mgQdCAUcZWYI40qYWykfMcU4mrzKJgR38eVl0jqruU7NvTuv1q+LOErkiByTU+KSC1Int6RBmoSTjDyTV/JmPVkv1rv1MW9dsYqZCvkD6/MHVLqUjA==</latexit>

1

A (= ⇢A ⌦ ⇢Ā )Modular operator:<latexit sha1_base64="dosQVxZXS7l+3q1SQ5rINUBtfTo=">AAACGXicbVDLSgMxFM34rPU16tJNsAh1YZkRQTdCqy5cVrAP6Iwlk6ZtaGYyJHeEMsxvuPFX3LhQxKWu/BvTx0JbDwTOPedebu4JYsE1OM63tbC4tLyymlvLr29sbm3bO7t1LRNFWY1KIVUzIJoJHrEacBCsGStGwkCwRjC4GvmNB6Y0l9EdDGPmh6QX8S6nBIzUth3vmgkg7QouXmBP9aVhngQeMj0pUy8gKq1k2X167Gb4qG0XnJIzBp4n7pQU0BTVtv3pdSRNQhYBFUTrluvE4KdEAaeCZXkv0SwmdEB6rGVoRMxqPx1fluFDo3RwVyrzIsBj9fdESkKth2FgOkMCfT3rjcT/vFYC3XM/5VGcAIvoZFE3ERgkHsWEO1wxCmJoCKGKm79i2ieKUDBh5k0I7uzJ86R+UnKdknt7WihfTuPIoX10gIrIRWeojG5QFdUQRY/oGb2iN+vJerHerY9J64I1ndlDf2B9/QBjWp9L</latexit>

OA (s) =<latexit sha1_base64="VZXrqEaYzwCZ3Ngh5J9XiNe4kzc=">AAACJnicbVDLSsNAFJ34rPUVdelmsAh1YUlE0E2hVRfurGAf0MQwmU7aoZMHMxOhhHyNG3/FjYuKiDs/xUmbRR8eGDiccy5z73EjRoU0jB9tZXVtfWOzsFXc3tnd29cPDlsijDkmTRyykHdcJAijAWlKKhnpRJwg32Wk7Q5vM7/9QrigYfAkRxGxfdQPqEcxkkpy9KrlIznAiCUPqVMvizNYhdYdYRI59eeEihTOBWa8c2U6esmoGBPAZWLmpARyNBx9bPVCHPskkJghIbqmEUk7QVxSzEhatGJBIoSHqE+6igbIJ8JOJmem8FQpPeiFXL1Awok6O5EgX4iR76pktrNY9DLxP68bS+/aTmgQxZIEePqRFzMoQ5h1BnuUEyzZSBGEOVW7QjxAHGGpmi2qEszFk5dJ66JiGhXz8bJUu8nrKIBjcALKwARXoAbuQQM0AQav4B2Mwaf2pn1oX9r3NLqi5TNHYA7a7x9wJqUl</latexit><latexit sha1_base64="3KYxvphc7CZnnmxeVJrjm4nuEY8=">AAACIXicbVDJSgNBEO2JW4xb1KOXxiB4CjMimIsSl4PHCGaBTAw9nZqkSc9Cd40QxvyKF3/FiwdFchN/xs5y0JgHBY/3qqiq58VSaLTtLyuztLyyupZdz21sbm3v5Hf3ajpKFIcqj2SkGh7TIEUIVRQooRErYIEnoe71r8d+/RGUFlF4j4MYWgHrhsIXnKGR2vmSewMSWfvyIRVUD6krwccn6sZaUFeJbg8v6PkitZ0v2EV7AvqfODNSIDNU2vmR24l4EkCIXDKtm44dYytlCgWXMMy5iYaY8T7rQtPQkAWgW+nkwyE9MkqH+pEyFSKdqL8nUhZoPQg80xkw7Ol5bywu8poJ+qVWKsI4QQj5dJGfSIoRHcdFO0IBRzkwhHElzK2U95hiHE2oOROCM//yf1I7KTp20bk7LZSvZnFkyQE5JMfEIWekTG5JhVQJJ8/klbyTD+vFerM+rdG0NWPNZvbJH1jfP0uKovY=</latexit>

is
A OA
is
A|

is
A

i=| i

in D(A)<latexit sha1_base64="WAFiq2FJJ5VQutp9c5//RmgLF0U=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBahQimJCLqsj4XLCvYBTSiT6bQdOjMJMxOhhGzc+CtuXCji1n9w5984abPQ1gMXDufcy733BBGjSjvOt1VYWl5ZXSuulzY2t7Z37N29lgpjiUkThyyUnQApwqggTU01I51IEsQDRtrB+Drz2w9EKhqKez2JiM/RUNABxUgbqWcfJp7kkIoUelWvCj2O9AgjltyklcuTnl12as4UcJG4OSmDHI2e/eX1QxxzIjRmSKmu60TaT5DUFDOSlrxYkQjhMRqSrqECcaL8ZPpFCo+N0oeDUJoSGk7V3xMJ4kpNeGA6syvVvJeJ/3ndWA8u/ISKKNZE4NmiQcygDmEWCexTSbBmE0MQltTcCvEISYS1Ca5kQnDnX14krdOa69Tcu7Ny/SqPowgOwBGoABecgzq4BQ3QBBg8gmfwCt6sJ+vFerc+Zq0FK5/ZB39gff4AeJyXQQ==</latexit>(Tomita-Takesaki)

In QFT• Modular flow still well defined, associated to some | i<latexit sha1_base64="MXH9tO9D+WBNwXA27Dr/ENCQ0co=">AAAB/HicbVDLSgNBEJz1GeNrNUcvg0HwFHZF0JMEvXiMYB6QDWF20psMmX0w0yssa/wVLx4U8eqHePNvnCR70MSChqKqm+4uP5FCo+N8Wyura+sbm6Wt8vbO7t6+fXDY0nGqODR5LGPV8ZkGKSJookAJnUQBC30JbX98M/XbD6C0iKN7zBLohWwYiUBwhkbq2xVPQoCP1Eu0oJ4SwxFe9e2qU3NmoMvELUiVFGj07S9vEPM0hAi5ZFp3XSfBXs4UCi5hUvZSDQnjYzaErqERC0H38tnxE3pilAENYmUqQjpTf0/kLNQ6C33TGTIc6UVvKv7ndVMMLnu5iJIUIeLzRUEqKcZ0mgQdCAUcZWYI40qYWykfMcU4mrzKJgR38eVl0jqruU7NvTuv1q+LOErkiByTU+KSC1Int6RBmoSTjDyTV/JmPVkv1rv1MW9dsYqZCvkD6/MHVLqUjA==</latexit>

1

A (= ⇢A ⌦ ⇢Ā )Modular operator:<latexit sha1_base64="dosQVxZXS7l+3q1SQ5rINUBtfTo=">AAACGXicbVDLSgMxFM34rPU16tJNsAh1YZkRQTdCqy5cVrAP6Iwlk6ZtaGYyJHeEMsxvuPFX3LhQxKWu/BvTx0JbDwTOPedebu4JYsE1OM63tbC4tLyymlvLr29sbm3bO7t1LRNFWY1KIVUzIJoJHrEacBCsGStGwkCwRjC4GvmNB6Y0l9EdDGPmh6QX8S6nBIzUth3vmgkg7QouXmBP9aVhngQeMj0pUy8gKq1k2X167Gb4qG0XnJIzBp4n7pQU0BTVtv3pdSRNQhYBFUTrluvE4KdEAaeCZXkv0SwmdEB6rGVoRMxqPx1fluFDo3RwVyrzIsBj9fdESkKth2FgOkMCfT3rjcT/vFYC3XM/5VGcAIvoZFE3ERgkHsWEO1wxCmJoCKGKm79i2ieKUDBh5k0I7uzJ86R+UnKdknt7WihfTuPIoX10gIrIRWeojG5QFdUQRY/oGb2iN+vJerHerY9J64I1ndlDf2B9/QBjWp9L</latexit>

is
A OA

OA (s) =<latexit sha1_base64="VZXrqEaYzwCZ3Ngh5J9XiNe4kzc=">AAACJnicbVDLSsNAFJ34rPUVdelmsAh1YUlE0E2hVRfurGAf0MQwmU7aoZMHMxOhhHyNG3/FjYuKiDs/xUmbRR8eGDiccy5z73EjRoU0jB9tZXVtfWOzsFXc3tnd29cPDlsijDkmTRyykHdcJAijAWlKKhnpRJwg32Wk7Q5vM7/9QrigYfAkRxGxfdQPqEcxkkpy9KrlIznAiCUPqVMvizNYhdYdYRI59eeEihTOBWa8c2U6esmoGBPAZWLmpARyNBx9bPVCHPskkJghIbqmEUk7QVxSzEhatGJBIoSHqE+6igbIJ8JOJmem8FQpPeiFXL1Awok6O5EgX4iR76pktrNY9DLxP68bS+/aTmgQxZIEePqRFzMoQ5h1BnuUEyzZSBGEOVW7QjxAHGGpmi2qEszFk5dJ66JiGhXz8bJUu8nrKIBjcALKwARXoAbuQQM0AQav4B2Mwaf2pn1oX9r3NLqi5TNHYA7a7x9wJqUl</latexit>

isA|<latexit sha1_base64="3KYxvphc7CZnnmxeVJrjm4nuEY8=">AAACIXicbVDJSgNBEO2JW4xb1KOXxiB4CjMimIsSl4PHCGaBTAw9nZqkSc9Cd40QxvyKF3/FiwdFchN/xs5y0JgHBY/3qqiq58VSaLTtLyuztLyyupZdz21sbm3v5Hf3ajpKFIcqj2SkGh7TIEUIVRQooRErYIEnoe71r8d+/RGUFlF4j4MYWgHrhsIXnKGR2vmSewMSWfvyIRVUD6krwccn6sZaUFeJbg8v6PkitZ0v2EV7AvqfODNSIDNU2vmR24l4EkCIXDKtm44dYytlCgWXMMy5iYaY8T7rQtPQkAWgW+nkwyE9MkqH+pEyFSKdqL8nUhZoPQg80xkw7Ol5bywu8poJ+qVWKsI4QQj5dJGfSIoRHcdFO0IBRzkwhHElzK2U95hiHE2oOROCM//yf1I7KTp20bk7LZSvZnFkyQE5JMfEIWekTG5JhVQJJ8/klbyTD+vFerM+rdG0NWPNZvbJH1jfP0uKovY=</latexit>

Modular conjugation:J

A<latexit sha1_base64="WAFiq2FJJ5VQutp9c5//RmgLF0U=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBahQimJCLqsj4XLCvYBTSiT6bQdOjMJMxOhhGzc+CtuXCji1n9w5984abPQ1gMXDufcy733BBGjSjvOt1VYWl5ZXSuulzY2t7Z37N29lgpjiUkThyyUnQApwqggTU01I51IEsQDRtrB+Drz2w9EKhqKez2JiM/RUNABxUgbqWcfJp7kkIoUelWvCj2O9AgjltyklcuTnl12as4UcJG4OSmDHI2e/eX1QxxzIjRmSKmu60TaT5DUFDOSlrxYkQjhMRqSrqECcaL8ZPpFCo+N0oeDUJoSGk7V3xMJ4kpNeGA6syvVvJeJ/3ndWA8u/ISKKNZE4NmiQcygDmEWCexTSbBmE0MQltTcCvEISYS1Ca5kQnDnX14krdOa69Tcu7Ny/SqPowgOwBGoABecgzq4BQ3QBBg8gmfwCt6sJ+vFerc+Zq0FK5/ZB39gff4AeJyXQQ==</latexit>

J<latexit sha1_base64="sUw7tiBtF+H8EqwBLDH9JL5vMNI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi3hqwX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2Mb2d++wmV5rF8MJME/YgOJQ85o8ZKjft+ueJW3TnIKvFyUoEc9X75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia89jMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14lrYuq51a9xmWldpPHUYQTOIVz8OAKanAHdWgCA4RneIU359F5cd6dj0VrwclnjuEPnM8foIGMzg==</latexit><latexitin D(Ā)<latexit sha1_base64="2UOQplwys+c53w5yUcBFcGwiAzo=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4tQoZREBF3Wx8JlBfuAJpTJdNoOnZmEmYlQQvZu/BU3LhRx6w+482+ctFlo64ELh3Pu5d57gohRpR3n2yqsrK6tbxQ3S1vbO7t79v5BW4WxxKSFQxbKboAUYVSQlqaakW4kCeIBI51gcp35nQciFQ3FvZ5GxOdoJOiQYqSN1LfLiSc5pCKFXs2rQY8jPcaIJTdp1QuQTC7Tk75dcerODHCZuDmpgBzNvv3lDUIccyI0ZkipnutE2k+Q1BQzkpa8WJEI4QkakZ6hAnGi/GT2SwqPjTKAw1CaEhrO1N8TCeJKTXlgOrNb1aKXif95vVgPL/yEiijWROD5omHMoA5hFgwcUEmwZlNDEJbU3ArxGEmEtYmvZEJwF19eJu3TuuvU3buzSuMqj6MIjkAZVIELzkED3IImaAEMHsEzeAVv1pP1Yr1bH/PWgpXPHII/sD5/AHEjmgY=</latexit>

JA | i = | i<latexit sha1_base64="n1hAu67aVLd1GR79sLsHz+BHpYk=">AAACFnicbVDLSsNAFJ3UV62vqEs3g0VwY0lE0I1SdSOuKtgHNCFMppN26OTBzI1QYr/Cjb/ixoUibsWdf+O0zULbHrhwOOde7r3HTwRXYFk/RmFhcWl5pbhaWlvf2Nwyt3caKk4lZXUai1i2fKKY4BGrAwfBWolkJPQFa/r965HffGBS8Ti6h0HC3JB0Ix5wSkBLnnl0611iR7AAHrGTKI4dybs9uMDn81TPLFsVaww8S+yclFGOmmd+O52YpiGLgAqiVNu2EnAzIoFTwYYlJ1UsIbRPuqytaURCptxs/NYQH2ilg4NY6ooAj9W/ExkJlRqEvu4MCfTUtDcS53ntFIIzN+NRkgKL6GRRkAoMMR5lhDtcMgpioAmhkutbMe0RSSjoJEs6BHv65VnSOK7YVsW+OylXr/I4imgP7aNDZKNTVEU3qIbqiKIn9ILe0LvxbLwaH8bnpLVg5DO76B+Mr19URZ46</latexit>in D(A)i=| i

O ⌘ JOA J<latexit sha1_base64="tPqw7IDU9ONvY/lPsYQUJiEIAwM=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBRXJVEBF1W3Ug3VrAXaGKYTCft0MkkzkwKJfQN3Pgqblwo4tatO9/GSRtQW38Y+PjPOcw5vx8zKpVlfRmFhcWl5ZXiamltfWNzy9zeacooEZg0cMQi0faRJIxy0lBUMdKOBUGhz0jLH1xm9daQCEkjfqtGMXFD1OM0oBgpbXnmoRMi1ceIpdfjuxp0yH1Ch1DDj+2dw5pnlq2KNRGcBzuHMshV98xPpxvhJCRcYYak7NhWrNwUCUUxI+OSk0gSIzxAPdLRyFFIpJtO7hnDA+10YRAJ/biCE/f3RIpCKUehrzuzNeVsLTP/q3USFZy5KeVxogjH04+ChEEVwSwc2KWCYMVGGhAWVO8KcR8JhJWOsKRDsGdPnofmccW2KvbNSbl6kcdRBHtgHxwBG5yCKrgCddAAGDyAJ/ACXo1H49l4M96nrQUjn9kFf2R8fAMQ55wD</latexit>

is(Tomita-Takesaki)

In QFT(Bisognano-Wichmann)• For example for a half space cut (Rindler):

is

=

Boost

J=

CPT

<latexit sha1_base64="KAsxAtG3+Jobx17pUDPIp9n5tdA=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoBchqAePEcwDkk2YnXSSIbOzy8ysEpb8hxcPinj1X7z5N06SPWhiQUNR1U13VxALro3rfju5ldW19Y38ZmFre2d3r7h/UNdRohjWWCQi1QyoRsEl1gw3ApuxQhoGAhvB6GbqNx5RaR7JBzOO0Q/pQPI+Z9RYqdO+RWFoJ+V6Qq5It1hyy+4MZJl4GSlBhmq3+NXuRSwJURomqNYtz42Nn1JlOBM4KbQTjTFlIzrAlqWShqj9dHb1hJxYpUf6kbIlDZmpvydSGmo9DgPbGVIz1IveVPzPayWmf+mnXMaJQcnmi/qJICYi0whIjytkRowtoUxxeythQ6ooMzaogg3BW3x5mdTPyp5b9u7PS5XrLI48HMExnIIHF1CBO6hCDRgoeIZXeHOenBfn3fmYt+acbOYQ/sD5/AG3LJH7</latexit><latexit sha1_base64="XT67KhTl9THw7KAfXoeZHwZI6LI=">AAAB9XicbVDLSgMxFL3xWeur6tJNsAh1U2ZEUFxVdOGygn1AO5ZMmmlDM5khyShl6H+4caGIW//FnX9jpp2Fth4IHM65l3ty/FhwbRznGy0tr6yurRc2iptb2zu7pb39po4SRVmDRiJSbZ9oJrhkDcONYO1YMRL6grX80XXmtx6Z0jyS92YcMy8kA8kDTomx0kM3JGZIiUhvJpWrk16p7FSdKfAicXNShhz1Xumr249oEjJpqCBad1wnNl5KlOFUsEmxm2gWEzoiA9axVJKQaS+dpp7gY6v0cRAp+6TBU/X3RkpCrcehbyezlHrey8T/vE5iggsv5TJODJN0dihIBDYRzirAfa4YNWJsCaGK26yYDoki1NiiirYEd/7Li6R5WnWdqnt3Vq5d5nUU4BCOoAIunEMNbqEODaCg4Ble4Q09oRf0jj5mo0so3zmAP0CfP8yhkgI=</latexit>

D(A)s<latexit sha1_base64="N1eK0lTaQQAFA6yHzcECkl4oWJk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipqQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/epYz3</latexit><latexit

OJ<latexit sha1_base64="+Gzb95UIkkioO9xbL4v8MVIrnsg=">AAAB9HicbVDLSgMxFL3js9ZX1aWbYBFclRkRdFl0I26sYB/QDiWTpm1oJjMmdwpl6He4caGIWz/GnX9jpp2Fth4IHM65l3tyglgKg6777aysrq1vbBa2its7u3v7pYPDhokSzXidRTLSrYAaLoXidRQoeSvWnIaB5M1gdJP5zTHXRkTqEScx90M6UKIvGEUr+Z2Q4pBRmd5Pu3fdUtmtuDOQZeLlpAw5at3SV6cXsSTkCpmkxrQ9N0Y/pRoFk3xa7CSGx5SN6IC3LVU05MZPZ6Gn5NQqPdKPtH0KyUz9vZHS0JhJGNjJLKRZ9DLxP6+dYP/KT4WKE+SKzQ/1E0kwIlkDpCc0ZygnllCmhc1K2JBqytD2VLQleItfXiaN84rnVryHi3L1Oq+jAMdwAmfgwSVU4RZqUAcGT/AMr/DmjJ0X5935mI+uOPnOEfyB8/kD2RKSIg==</latexit><latexit sha1_base64="pww6Xbpw4NTqcGSOGiFQAj7aiuY=">AAAB63icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoBch6EU8RTAmkCxhdjKbDJnHMjMrhCW/4MWDIl79IW/+jbPJHjSxoKGo6qa7K0o4M9b3v73Syura+kZ5s7K1vbO7V90/eDQq1YS2iOJKdyJsKGeStiyznHYSTbGIOG1H45vcbz9RbZiSD3aS0FDgoWQxI9jm0h26Qv1qza/7M6BlEhSkBgWa/epXb6BIKqi0hGNjuoGf2DDD2jLC6bTSSw1NMBnjIe06KrGgJsxmt07RiVMGKFbalbRopv6eyLAwZiIi1ymwHZlFLxf/87qpjS/DjMkktVSS+aI45cgqlD+OBkxTYvnEEUw0c7ciMsIaE+viqbgQgsWXl8njWT3w68H9ea1xXcRRhiM4hlMI4AIacAtNaAGBETzDK7x5wnvx3r2PeWvJK2YO4Q+8zx/OEY1p</latexit>

In QFT(Bisognano-Wichmann)• For example for a half space cut (Rindler):

is

=

Boost

J=

CPT

<latexit sha1_base64="KAsxAtG3+Jobx17pUDPIp9n5tdA=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoBchqAePEcwDkk2YnXSSIbOzy8ysEpb8hxcPinj1X7z5N06SPWhiQUNR1U13VxALro3rfju5ldW19Y38ZmFre2d3r7h/UNdRohjWWCQi1QyoRsEl1gw3ApuxQhoGAhvB6GbqNx5RaR7JBzOO0Q/pQPI+Z9RYqdO+RWFoJ+V6Qq5It1hyy+4MZJl4GSlBhmq3+NXuRSwJURomqNYtz42Nn1JlOBM4KbQTjTFlIzrAlqWShqj9dHb1hJxYpUf6kbIlDZmpvydSGmo9DgPbGVIz1IveVPzPayWmf+mnXMaJQcnmi/qJICYi0whIjytkRowtoUxxeythQ6ooMzaogg3BW3x5mdTPyp5b9u7PS5XrLI48HMExnIIHF1CBO6hCDRgoeIZXeHOenBfn3fmYt+acbOYQ/sD5/AG3LJH7</latexit><latexit sha1_base64="XT67KhTl9THw7KAfXoeZHwZI6LI=">AAAB9XicbVDLSgMxFL3xWeur6tJNsAh1U2ZEUFxVdOGygn1AO5ZMmmlDM5khyShl6H+4caGIW//FnX9jpp2Fth4IHM65l3ty/FhwbRznGy0tr6yurRc2iptb2zu7pb39po4SRVmDRiJSbZ9oJrhkDcONYO1YMRL6grX80XXmtx6Z0jyS92YcMy8kA8kDTomx0kM3JGZIiUhvJpWrk16p7FSdKfAicXNShhz1Xumr249oEjJpqCBad1wnNl5KlOFUsEmxm2gWEzoiA9axVJKQaS+dpp7gY6v0cRAp+6TBU/X3RkpCrcehbyezlHrey8T/vE5iggsv5TJODJN0dihIBDYRzirAfa4YNWJsCaGK26yYDoki1NiiirYEd/7Li6R5WnWdqnt3Vq5d5nUU4BCOoAIunEMNbqEODaCg4Ble4Q09oRf0jj5mo0so3zmAP0CfP8yhkgI=</latexit>

D(A)s<latexit sha1_base64="N1eK0lTaQQAFA6yHzcECkl4oWJk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipqQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/epYz3</latexit><latexit

OJ<latexit sha1_base64="pww6Xbpw4NTqcGSOGiFQAj7aiuY=">AAAB63icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoBch6EU8RTAmkCxhdjKbDJnHMjMrhCW/4MWDIl79IW/+jbPJHjSxoKGo6qa7K0o4M9b3v73Syura+kZ5s7K1vbO7V90/eDQq1YS2iOJKdyJsKGeStiyznHYSTbGIOG1H45vcbz9RbZiSD3aS0FDgoWQxI9jm0h26Qv1qza/7M6BlEhSkBgWa/epXb6BIKqi0hGNjuoGf2DDD2jLC6bTSSw1NMBnjIe06KrGgJsxmt07RiVMGKFbalbRopv6eyLAwZiIi1ymwHZlFLxf/87qpjS/DjMkktVSS+aI45cgqlD+OBkxTYvnEEUw0c7ciMsIaE+viqbgQgsWXl8njWT3w68H9ea1xXcRRhiM4hlMI4AIacAtNaAGBETzDK7x5wnvx3r2PeWvJK2YO4Q+8zx/OEY1p</latexit><latexit sha1_base64="+Gzb95UIkkioO9xbL4v8MVIrnsg=">AAAB9HicbVDLSgMxFL3js9ZX1aWbYBFclRkRdFl0I26sYB/QDiWTpm1oJjMmdwpl6He4caGIWz/GnX9jpp2Fth4IHM65l3tyglgKg6777aysrq1vbBa2its7u3v7pYPDhokSzXidRTLSrYAaLoXidRQoeSvWnIaB5M1gdJP5zTHXRkTqEScx90M6UKIvGEUr+Z2Q4pBRmd5Pu3fdUtmtuDOQZeLlpAw5at3SV6cXsSTkCpmkxrQ9N0Y/pRoFk3xa7CSGx5SN6IC3LVU05MZPZ6Gn5NQqPdKPtH0KyUz9vZHS0JhJGNjJLKRZ9DLxP6+dYP/KT4WKE+SKzQ/1E0kwIlkDpCc0ZygnllCmhc1K2JBqytD2VLQleItfXiaN84rnVryHi3L1Oq+jAMdwAmfgwSVU4RZqUAcGT/AMr/DmjJ0X5935mI+uOPnOEfyB8/kD2RKSIg==</latexit>

Important relation:J

1/2<latexit sha1_base64="95vEqAgjzdcptw8Eh0t5Fp7z92E=">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</latexit>†
OA | i = O A| i⇡ Euclidean Rotation<latexit sha1_base64="7YlkrurOAYOv+Ch30L7WpchFVys=">AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8eK9gPaUDbbSbt0swm7G6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTAXXxvO+ndLa+sbmVnm7srO7t3/gHh61dJIphk2WiER1QqpRcIlNw43ATqqQxqHAdji+nfntJ1SaJ/LRTFIMYjqUPOKMGis99FLed6tezZuDrBK/IFUo0Oi7X71BwrIYpWGCat31vdQEOVWGM4HTSi/TmFI2pkPsWippjDrI56dOyZlVBiRKlC1pyFz9PZHTWOtJHNrOmJqRXvZm4n9eNzPRdZBzmWYGJVssijJBTEJmf5MBV8iMmFhCmeL2VsJGVFFmbDoVG4K//PIqaV3UfK/m319W6zdFHGU4gVM4Bx+uoA530IAmMBjCM7zCmyOcF+fd+Vi0lpxi5hj+wPn8AU+9jc0=</latexit><latexit

In QFT(Bisognano-Wichmann)• For example for a half space cut (Rindler):

is

=

Boost

J=

CPT

<latexit sha1_base64="KAsxAtG3+Jobx17pUDPIp9n5tdA=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoBchqAePEcwDkk2YnXSSIbOzy8ysEpb8hxcPinj1X7z5N06SPWhiQUNR1U13VxALro3rfju5ldW19Y38ZmFre2d3r7h/UNdRohjWWCQi1QyoRsEl1gw3ApuxQhoGAhvB6GbqNx5RaR7JBzOO0Q/pQPI+Z9RYqdO+RWFoJ+V6Qq5It1hyy+4MZJl4GSlBhmq3+NXuRSwJURomqNYtz42Nn1JlOBM4KbQTjTFlIzrAlqWShqj9dHb1hJxYpUf6kbIlDZmpvydSGmo9DgPbGVIz1IveVPzPayWmf+mnXMaJQcnmi/qJICYi0whIjytkRowtoUxxeythQ6ooMzaogg3BW3x5mdTPyp5b9u7PS5XrLI48HMExnIIHF1CBO6hCDRgoeIZXeHOenBfn3fmYt+acbOYQ/sD5/AG3LJH7</latexit><latexit sha1_base64="XT67KhTl9THw7KAfXoeZHwZI6LI=">AAAB9XicbVDLSgMxFL3xWeur6tJNsAh1U2ZEUFxVdOGygn1AO5ZMmmlDM5khyShl6H+4caGIW//FnX9jpp2Fth4IHM65l3ty/FhwbRznGy0tr6yurRc2iptb2zu7pb39po4SRVmDRiJSbZ9oJrhkDcONYO1YMRL6grX80XXmtx6Z0jyS92YcMy8kA8kDTomx0kM3JGZIiUhvJpWrk16p7FSdKfAicXNShhz1Xumr249oEjJpqCBad1wnNl5KlOFUsEmxm2gWEzoiA9axVJKQaS+dpp7gY6v0cRAp+6TBU/X3RkpCrcehbyezlHrey8T/vE5iggsv5TJODJN0dihIBDYRzirAfa4YNWJsCaGK26yYDoki1NiiirYEd/7Li6R5WnWdqnt3Vq5d5nUU4BCOoAIunEMNbqEODaCg4Ble4Q09oRf0jj5mo0so3zmAP0CfP8yhkgI=</latexit>

D(A)s<latexit sha1_base64="N1eK0lTaQQAFA6yHzcECkl4oWJk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipqQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/epYz3</latexit><latexit

OJ<latexit sha1_base64="pww6Xbpw4NTqcGSOGiFQAj7aiuY=">AAAB63icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoBch6EU8RTAmkCxhdjKbDJnHMjMrhCW/4MWDIl79IW/+jbPJHjSxoKGo6qa7K0o4M9b3v73Syura+kZ5s7K1vbO7V90/eDQq1YS2iOJKdyJsKGeStiyznHYSTbGIOG1H45vcbz9RbZiSD3aS0FDgoWQxI9jm0h26Qv1qza/7M6BlEhSkBgWa/epXb6BIKqi0hGNjuoGf2DDD2jLC6bTSSw1NMBnjIe06KrGgJsxmt07RiVMGKFbalbRopv6eyLAwZiIi1ymwHZlFLxf/87qpjS/DjMkktVSS+aI45cgqlD+OBkxTYvnEEUw0c7ciMsIaE+viqbgQgsWXl8njWT3w68H9ea1xXcRRhiM4hlMI4AIacAtNaAGBETzDK7x5wnvx3r2PeWvJK2YO4Q+8zx/OEY1p</latexit><latexit sha1_base64="+Gzb95UIkkioO9xbL4v8MVIrnsg=">AAAB9HicbVDLSgMxFL3js9ZX1aWbYBFclRkRdFl0I26sYB/QDiWTpm1oJjMmdwpl6He4caGIWz/GnX9jpp2Fth4IHM65l3tyglgKg6777aysrq1vbBa2its7u3v7pYPDhokSzXidRTLSrYAaLoXidRQoeSvWnIaB5M1gdJP5zTHXRkTqEScx90M6UKIvGEUr+Z2Q4pBRmd5Pu3fdUtmtuDOQZeLlpAw5at3SV6cXsSTkCpmkxrQ9N0Y/pRoFk3xa7CSGx5SN6IC3LVU05MZPZ6Gn5NQqPdKPtH0KyUz9vZHS0JhJGNjJLKRZ9DLxP6+dYP/KT4WKE+SKzQ/1E0kwIlkDpCc0ZygnllCmhc1K2JBqytD2VLQleItfXiaN84rnVryHi3L1Oq+jAMdwAmfgwSVU4RZqUAcGT/AMr/DmjJ0X5935mI+uOPnOEfyB8/kD2RKSIg==</latexit>

Important relation:J

1/2<latexit sha1_base64="95vEqAgjzdcptw8Eh0t5Fp7z92E=">AAACRXicbVBNSyNBFOzRXVfjqlGPXpoNgqc4I4JeVlz1IF5UMCqkY3jTeTNp7Pmg+81CGPPnvHj35j/w4kERr7udmINfBQ1FVT1evwpzrSz5/p03Nv7t+8SPyanK9M+Z2bnq/MKpzQojsSEznZnzECxqlWKDFGk8zw1CEmo8Cy93B/7ZXzRWZekJ9XJsJRCnKlISyEntqjjgYg81wUUZrK71uUiAuhJ0edhv/+FCY0RXXORWcWFU3KUt/vt95kJ0II7RfJVtV2t+3R+CfybBiNTYCEft6q3oZLJIMCWpwdpm4OfUKsGQkhr7FVFYzEFeQoxNR1NI0LbKYQt9vuyUDo8y415KfKi+nSghsbaXhC45OMB+9AbiV16zoGizVao0LwhT+booKjSnjA8q5R1lUJLuOQLSKPdXLrtgQJIrvuJKCD6e/JmcrtUDvx4cr9e2d0Z1TLIl9outsIBtsG22z45Yg0l2ze7ZI3vybrwH79l7eY2OeaOZRfYO3r//4wSxXA==</latexit>†
OA | i = O A| i⇡ Euclidean Rotation<latexit sha1_base64="7YlkrurOAYOv+Ch30L7WpchFVys=">AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8eK9gPaUDbbSbt0swm7G6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTAXXxvO+ndLa+sbmVnm7srO7t3/gHh61dJIphk2WiER1QqpRcIlNw43ATqqQxqHAdji+nfntJ1SaJ/LRTFIMYjqUPOKMGis99FLed6tezZuDrBK/IFUo0Oi7X71BwrIYpWGCat31vdQEOVWGM4HTSi/TmFI2pkPsWippjDrI56dOyZlVBiRKlC1pyFz9PZHTWOtJHNrOmJqRXvZm4n9eNzPRdZBzmWYGJVssijJBTEJmf5MBV8iMmFhCmeL2VsJGVFFmbDoVG4K//PIqaV3UfK/m319W6zdFHGU4gVM4Bx+uoA530IAmMBjCM7zCmyOcF+fd+Vi0lpxi5hj+wPn8AU+9jc0=</latexit><latexit• For more general states, UV structure of entanglement is
the same near the cut. So expect modular flow has
universal geometric description at least acting on
operators close to the cut ….

Powerful constraints …
• Analyticity of correlation functions:s<latexit sha1_base64="N1eK0lTaQQAFA6yHzcECkl4oWJk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipqQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/epYz3</latexit><latexit

Analytic

ish | OA<latexit sha1_base64="Cr5++C4uwquPopMP6EwgVkuxxY0=">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</latexit>}<latexit sha1_base64="1zzMZyHyFfywAeLk1S0cW7o8Dfk=">AAAB+HicbVBNS8NAEN34WetHox69LBbBU0hE0GPRi8cK9gOaUDbbSbt088HuRKihv8SLB0W8+lO8+W/ctjlo64OBx3szzMwLMyk0uu63tba+sbm1Xdmp7u7tH9Tsw6O2TnPFocVTmapuyDRIkUALBUroZgpYHErohOPbmd95BKVFmjzgJIMgZsNERIIzNFLfrvkSInSor8RwhP60b9ddx52DrhKvJHVSotm3v/xByvMYEuSSad3z3AyDgikUXMK06ucaMsbHbAg9QxMWgw6K+eFTemaUAY1SZSpBOld/TxQs1noSh6YzZjjSy95M/M/r5RhdB4VIshwh4YtFUS4ppnSWAh0IBRzlxBDGlTC3Uj5iinE0WVVNCN7yy6ukfeF4ruPdX9YbN2UcFXJCTsk58cgVaZA70iQtwklOnskrebOerBfr3fpYtK5Z5cwx+QPr8wdmnpLp</latexit>

0
OA |<latexit sha1_base64="nVdGhJrJKgedV8EHAk9vZx6H2c8=">AAAB8HicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0EYI2lhGMDGSHGFvM5cs2d07dveEEPIrbCwUsfXn2Plv3CRXaOKDgcd7M8zMi1LBjfX9b6+wsrq2vlHcLG1t7+zulfcPmibJNMMGS0SiWxE1KLjChuVWYCvVSGUk8CEa3kz9hyfUhifq3o5SDCXtKx5zRq2THjsRWkquSNAtV/yqPwNZJkFOKpCj3i1/dXoJyyQqywQ1ph34qQ3HVFvOBE5KncxgStmQ9rHtqKISTTieHTwhJ07pkTjRrpQlM/X3xJhKY0Yycp2S2oFZ9Kbif147s/FlOOYqzSwqNl8UZ4LYhEy/Jz2ukVkxcoQyzd2thA2opsy6jEouhGDx5WXSPKsGfjW4O6/UrvM4inAEx3AKAVxADW6hDg1gIOEZXuHN096L9+59zFsLXj5zCH/gff4AaEKPeg==</latexit>i=1Powerful constraints …
• Analyticity of correlation functions:ish | OA<latexit sha1_base64="Cr5++C4uwquPopMP6EwgVkuxxY0=">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</latexit>

0
OA |

i

0
= h | OA OA(s) | i<latexit sha1_base64="iczFaWItEq6cBijtmi8FM2o0X5c=">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</latexit>s<latexit sha1_base64="N1eK0lTaQQAFA6yHzcECkl4oWJk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipqQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/epYz3</latexit><latexit

Analytic}<latexit sha1_base64="1zzMZyHyFfywAeLk1S0cW7o8Dfk=">AAAB+HicbVBNS8NAEN34WetHox69LBbBU0hE0GPRi8cK9gOaUDbbSbt088HuRKihv8SLB0W8+lO8+W/ctjlo64OBx3szzMwLMyk0uu63tba+sbm1Xdmp7u7tH9Tsw6O2TnPFocVTmapuyDRIkUALBUroZgpYHErohOPbmd95BKVFmjzgJIMgZsNERIIzNFLfrvkSInSor8RwhP60b9ddx52DrhKvJHVSotm3v/xByvMYEuSSad3z3AyDgikUXMK06ucaMsbHbAg9QxMWgw6K+eFTemaUAY1SZSpBOld/TxQs1noSh6YzZjjSy95M/M/r5RhdB4VIshwh4YtFUS4ppnSWAh0IBRzlxBDGlTC3Uj5iinE0WVVNCN7yy6ukfeF4ruPdX9YbN2UcFXJCTsk58cgVaZA70iQtwklOnskrebOerBfr3fpYtK5Z5cwx+QPr8wdmnpLp</latexit><latexit sha1_base64="nVdGhJrJKgedV8EHAk9vZx6H2c8=">AAAB8HicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0EYI2lhGMDGSHGFvM5cs2d07dveEEPIrbCwUsfXn2Plv3CRXaOKDgcd7M8zMi1LBjfX9b6+wsrq2vlHcLG1t7+zulfcPmibJNMMGS0SiWxE1KLjChuVWYCvVSGUk8CEa3kz9hyfUhifq3o5SDCXtKx5zRq2THjsRWkquSNAtV/yqPwNZJkFOKpCj3i1/dXoJyyQqywQ1ph34qQ3HVFvOBE5KncxgStmQ9rHtqKISTTieHTwhJ07pkTjRrpQlM/X3xJhKY0Yycp2S2oFZ9Kbif147s/FlOOYqzSwqNl8UZ4LYhEy/Jz2ukVkxcoQyzd2thA2opsy6jEouhGDx5WXSPKsGfjW4O6/UrvM4inAEx3AKAVxADW6hDg1gIOEZXuHN096L9+59zFsLXj5zCH/gff4AaEKPeg==</latexit>=1

Powerful constraints …
• Analyticity of correlation functions:ish | OA<latexit sha1_base64="Cr5++C4uwquPopMP6EwgVkuxxY0=">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</latexit>

0
OA |

i

0
= h | OA OA(s) | i<latexit sha1_base64="iczFaWItEq6cBijtmi8FM2o0X5c=">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</latexit>s}<latexit sha1_base64="N1eK0lTaQQAFA6yHzcECkl4oWJk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipqQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/epYz3</latexit><latexit

Analytic<latexit sha1_base64="nVdGhJrJKgedV8EHAk9vZx6H2c8=">AAAB8HicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0EYI2lhGMDGSHGFvM5cs2d07dveEEPIrbCwUsfXn2Plv3CRXaOKDgcd7M8zMi1LBjfX9b6+wsrq2vlHcLG1t7+zulfcPmibJNMMGS0SiWxE1KLjChuVWYCvVSGUk8CEa3kz9hyfUhifq3o5SDCXtKx5zRq2THjsRWkquSNAtV/yqPwNZJkFOKpCj3i1/dXoJyyQqywQ1ph34qQ3HVFvOBE5KncxgStmQ9rHtqKISTTieHTwhJ07pkTjRrpQlM/X3xJhKY0Yycp2S2oFZ9Kbif147s/FlOOYqzSwqNl8UZ4LYhEy/Jz2ukVkxcoQyzd2thA2opsy6jEouhGDx5WXSPKsGfjW4O6/UrvM4inAEx3AKAVxADW6hDg1gIOEZXuHN096L9+59zFsLXj5zCH/gff4AaEKPeg==</latexit><latexit sha1_base64="1zzMZyHyFfywAeLk1S0cW7o8Dfk=">AAAB+HicbVBNS8NAEN34WetHox69LBbBU0hE0GPRi8cK9gOaUDbbSbt088HuRKihv8SLB0W8+lO8+W/ctjlo64OBx3szzMwLMyk0uu63tba+sbm1Xdmp7u7tH9Tsw6O2TnPFocVTmapuyDRIkUALBUroZgpYHErohOPbmd95BKVFmjzgJIMgZsNERIIzNFLfrvkSInSor8RwhP60b9ddx52DrhKvJHVSotm3v/xByvMYEuSSad3z3AyDgikUXMK06ucaMsbHbAg9QxMWgw6K+eFTemaUAY1SZSpBOld/TxQs1noSh6YzZjjSy95M/M/r5RhdB4VIshwh4YtFUS4ppnSWAh0IBRzlxBDGlTC3Uj5iinE0WVVNCN7yy6ukfeF4ruPdX9YbN2UcFXJCTsk58cgVaZA70iQtwklOnskrebOerBfr3fpYtK5Z5cwx+QPr8wdmnpLp</latexit>• Unitarity. e.g bounds:
h | OA<latexit sha1_base64="LW+F7hY8VQ34dWLWcxzTu1JA/G4=">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</latexit>

1/2

†
OA |

J
i = h | OA OA
| i
etc.

0

=1

Constraints on Emergenceof gravity in AdS/CFT?
• Modular flow in AdS/CFT ~ boosts near the RyuTakayanagi surface

Constraints on Emergenceof gravity in AdS/CFT?
• Modular flow in AdS/CFT ~ boosts near the RyuTakayanagi surface• Bulk locality likely necessary ingredientConstraints on Emergenceof gravity in AdS/CFT?
• Modular flow in AdS/CFT ~ boosts near the RyuTakayanagi surface• Bulk locality likely necessary ingredient• Look for signatures of such emergenceConstraints on Emergenceof gravity in AdS/CFT?
• Modular flow in AdS/CFT ~ boosts near the RyuTakayanagi surface• Bulk locality likely necessary ingredient• Look for signatures of such emergence• Put constraints on emergenceIn AdS/CFT …
• We start with the holographic entanglement entropy
formula (RT/HRT/LM):A<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit

A(mA )SEE =4GN

a<latexit sha1_base64="b7/vCs5ze5KtVd66W3yyALYBfbk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipSQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/DXYzl</latexit><latexit<latexit sha1_base64="NwDMj53TPLOEpth3EkaMHYPtWRM=">AAACCHicbVDLSsNAFJ34rPUVdenCwSLUTUmkoBuhVYqupKJ9QBvCZDpph04mYWYilJClG3/FjQtF3PoJ7vwbp20W2nrgwuGce7n3Hi9iVCrL+jYWFpeWV1Zza/n1jc2tbXNntynDWGDSwCELRdtDkjDKSUNRxUg7EgQFHiMtb3g59lsPREga8ns1iogToD6nPsVIack1D+7cpFZL4Tns+gLhBFaLgVs9TpMyvHJvUuiaBatkTQDniZ2RAshQd82vbi/EcUC4wgxJ2bGtSDkJEopiRtJ8N5YkQniI+qSjKUcBkU4yeSSFR1rpQT8UuriCE/X3RIICKUeBpzsDpAZy1huL/3mdWPlnTkJ5FCvC8XSRHzOoQjhOBfaoIFixkSYIC6pvhXiAdCBKZ5fXIdizL8+T5knJtkr2bblQucjiyIF9cAiKwAanoAKuQR00AAaP4Bm8gjfjyXgx3o2PaeuCkc3sgT8wPn8AXhKXoQ==</latexit>

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>

In AdS/CFT …
• We start with the holographic entanglement entropy
formula (RT/HRT/LM):A<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit

A(mA )SEE =4GN

a<latexit sha1_base64="b7/vCs5ze5KtVd66W3yyALYBfbk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipSQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/DXYzl</latexit><latexit<latexit sha1_base64="NwDMj53TPLOEpth3EkaMHYPtWRM=">AAACCHicbVDLSsNAFJ34rPUVdenCwSLUTUmkoBuhVYqupKJ9QBvCZDpph04mYWYilJClG3/FjQtF3PoJ7vwbp20W2nrgwuGce7n3Hi9iVCrL+jYWFpeWV1Zza/n1jc2tbXNntynDWGDSwCELRdtDkjDKSUNRxUg7EgQFHiMtb3g59lsPREga8ns1iogToD6nPsVIack1D+7cpFZL4Tns+gLhBFaLgVs9TpMyvHJvUuiaBatkTQDniZ2RAshQd82vbi/EcUC4wgxJ2bGtSDkJEopiRtJ8N5YkQniI+qSjKUcBkU4yeSSFR1rpQT8UuriCE/X3RIICKUeBpzsDpAZy1huL/3mdWPlnTkJ5FCvC8XSRHzOoQjhOBfaoIFixkSYIC6pvhXiAdCBKZ5fXIdizL8+T5knJtkr2bblQucjiyIF9cAiKwAanoAKuQR00AAaP4Bm8gjfjyXgx3o2PaeuCkc3sgT8wPn8AXhKXoQ==</latexit>

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>• Various improvements lead to the JLMS result:<latexit sha1_base64="rIcL3H1G+2v4pKA7bdMaDY+CT3Q=">AAACDXicbVDLSsNAFJ3UV62vqEs3g1VwVRIRdCNUK+KyQl/QpGEynbRDJ5MwMxFKyA+48VfcuFDErXt3/o3TNgttPXDhcM693HuPHzMqlWV9G4Wl5ZXVteJ6aWNza3vH3N1rySgRmDRxxCLR8ZEkjHLSVFQx0okFQaHPSNsf1SZ++4EISSPeUOOYuCEacBpQjJSWPPPIuSFMIe+ql9ZuGxm8hLmAeqkjQugnbJR5ZtmqWFPARWLnpAxy1D3zy+lHOAkJV5ghKbu2FSs3RUJRzEhWchJJYoRHaEC6mnIUEumm028yeKyVPgwioYsrOFV/T6QolHIc+rozRGoo572J+J/XTVRw4aaUx4kiHM8WBQmDKoKTaGCfCoIVG2uCsKD6VoiHSCCsdIAlHYI9//IiaZ1WbKti35+Vq9d5HEVwAA7BCbDBOaiCO1AHTYDBI3gGr+DNeDJejHfjY9ZaMPKZffAHxucPRDGbDg==</latexit>

CF T
=
A

bulk
a

Modular flow for region A
dual to flow in the bulk for region a

In AdS/CFT …
• We start with the holographic entanglement entropy
formula (RT/HRT/LM):A<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit

A(mA )SEE =4GN

a<latexit sha1_base64="b7/vCs5ze5KtVd66W3yyALYBfbk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipSQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/DXYzl</latexit><latexit<latexit sha1_base64="NwDMj53TPLOEpth3EkaMHYPtWRM=">AAACCHicbVDLSsNAFJ34rPUVdenCwSLUTUmkoBuhVYqupKJ9QBvCZDpph04mYWYilJClG3/FjQtF3PoJ7vwbp20W2nrgwuGce7n3Hi9iVCrL+jYWFpeWV1Zza/n1jc2tbXNntynDWGDSwCELRdtDkjDKSUNRxUg7EgQFHiMtb3g59lsPREga8ns1iogToD6nPsVIack1D+7cpFZL4Tns+gLhBFaLgVs9TpMyvHJvUuiaBatkTQDniZ2RAshQd82vbi/EcUC4wgxJ2bGtSDkJEopiRtJ8N5YkQniI+qSjKUcBkU4yeSSFR1rpQT8UuriCE/X3RIICKUeBpzsDpAZy1huL/3mdWPlnTkJ5FCvC8XSRHzOoQjhOBfaoIFixkSYIC6pvhXiAdCBKZ5fXIdizL8+T5knJtkr2bblQucjiyIF9cAiKwAanoAKuQR00AAaP4Bm8gjfjyXgx3o2PaeuCkc3sgT8wPn8AXhKXoQ==</latexit>

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>• Various improvements lead to the JLMS result:<latexit sha1_base64="rIcL3H1G+2v4pKA7bdMaDY+CT3Q=">AAACDXicbVDLSsNAFJ3UV62vqEs3g1VwVRIRdCNUK+KyQl/QpGEynbRDJ5MwMxFKyA+48VfcuFDErXt3/o3TNgttPXDhcM693HuPHzMqlWV9G4Wl5ZXVteJ6aWNza3vH3N1rySgRmDRxxCLR8ZEkjHLSVFQx0okFQaHPSNsf1SZ++4EISSPeUOOYuCEacBpQjJSWPPPIuSFMIe+ql9ZuGxm8hLmAeqkjQugnbJR5ZtmqWFPARWLnpAxy1D3zy+lHOAkJV5ghKbu2FSs3RUJRzEhWchJJYoRHaEC6mnIUEumm028yeKyVPgwioYsrOFV/T6QolHIc+rozRGoo572J+J/XTVRw4aaUx4kiHM8WBQmDKoKTaGCfCoIVG2uCsKD6VoiHSCCsdIAlHYI9//IiaZ1WbKti35+Vq9d5HEVwAA7BCbDBOaiCO1AHTYDBI3gGr+DNeDJejHfjY9ZaMPKZffAHxucPRDGbDg==</latexit>

CF T
=
A

Modular flow for region A
dual to flow in the bulk for region abulkaEntanglement wedge:Ea ⌘ D(a)<latexit sha1_base64="xvgJpmHisvTXq/zkHd7SX0fOTHU=">AAACDHicbVDLSsNAFJ3UV62vqks3g0Wom5KIoMviA1xWsA9oQrmZTtqhk4czk0IJ+QA3/oobF4q49QPc+TdO2iDaemDgcM65zL3HjTiTyjS/jMLS8srqWnG9tLG5tb1T3t1ryTAWhDZJyEPRcUFSzgLaVExx2okEBd/ltO2OLjO/PaZCsjC4U5OIOj4MAuYxAkpLvXLF9kENCfDkOu0Btul9zMb4R7xKq3CsU2bNnAIvEisnFZSj0St/2v2QxD4NFOEgZdcyI+UkIBQjnKYlO5Y0AjKCAe1qGoBPpZNMj0nxkVb62AuFfoHCU/X3RAK+lBPf1clsSznvZeJ/XjdW3rmTsCCKFQ3I7CMv5liFOGsG95mgRPGJJkAE07tiMgQBROn+SroEa/7kRdI6qVlmzbo9rdQv8jqK6AAdoiqy0BmqoxvUQE1E0AN6Qi/o1Xg0no03430WLRj5zD76A+PjG1yRmyY=</latexit>

Heavy probe operators• To get some handle on this we will study correlation

functions of heavy probe operators. These are usually
determined by spacelike geodesics:
O(x)

O(y)

hO(x)O(y)i ⇠ e m`(x,y)<latexit sha1_base64="nfS387eJT+cOEH3dQPVyjCeJ3rY=">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</latexit>

Heavy probe operators• To get some handle on this we will study correlation

functions of heavy probe operators. These are usually
determined by spacelike geodesics:
O(x)

O(y)

hO(x)O(y)i ⇠ e m`(x,y)<latexit sha1_base64="nfS387eJT+cOEH3dQPVyjCeJ3rY=">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</latexit>• In the presence of modular flow expect still semiclassicalanswer:⌦<latexit sha1_base64="zV77cV37hYB+uBZUNouxqQCGOao=">AAACI3icbVDLSgNBEJz1GeMr6tHLYBD0EnZFUEQkqAdvRjAqZNcwO+lNhsw+mOkVw5J/8eKvePGgBC8e/BcnyR6MWtBQVHXT3eUnUmi07U9ranpmdm6+sFBcXFpeWS2trd/oOFUc6jyWsbrzmQYpIqijQAl3iQIW+hJu/e7Z0L99AKVFHF1jLwEvZO1IBIIzNFKzdORKCPCYuiHDDmcyu+zvPO5S9xwksvtM6P6E1TOWEu0OnlDaLJXtij0C/UucnJRJjlqzNHBbMU9DiJBLpnXDsRP0MqZQcAn9optqSBjvsjY0DI1YCNrLRj/26bZRWjSIlakI6Uj9OZGxUOte6JvO4b36tzcU//MaKQaHXiaiJEWI+HhRkEqKMR0GRltCAUfZM4RxJcytlHeYYhxNrEUTgvP75b/kZq/i2BXnar9cPc3jKJBNskV2iEMOSJVckBqpE06eyAt5I+/Ws/VqDayPceuUlc9skAlYX9/Do6Ov</latexit>

O(x)

is

↵

O(y) ⇠ e m`s (x,y) ??
<latexit sha1_base64="YJgUmqujUb57B1hdNXcjlB8bGOw=">AAACBnicbVBNS8NAEN3Ur1q/oh5FWCxCBS2JCHpr0YvHCvYDmlo222m7dDcJuxuxhJ68+Fe8eFDEq7/Bm//GbZuDtj4YeLw3w8w8P+JMacf5tjILi0vLK9nV3Nr6xuaWvb1TU2EsKVRpyEPZ8IkCzgKoaqY5NCIJRPgc6v7gauzX70EqFga3ehhBS5BewLqMEm2ktr3vKSYw3CX4BAvsAedtVXg4Hh7hES6V2nbeKToT4HnipiSPUlTa9pfXCWksINCUE6WarhPpVkKkZpTDKOfFCiJCB6QHTUMDIkC1kskbI3xolA7uhtJUoPFE/T2REKHUUPimUxDdV7PeWPzPa8a6e9FKWBDFGgI6XdSNOdYhHmeCO0wC1XxoCKGSmVsx7RNJqDbJ5UwI7uzL86R2WnSdontzli9fpnFk0R46QAXkonNURteogqqIokf0jF7Rm/VkvVjv1se0NWOlM7voD6zPH/oGlt4=</latexit>

Rules for geodesics(TF, Li, Wang )See also: (Chen, Dong, Lewkowycz,Qi )• We give some rules for when such correlators can becomputed:⌦<latexit sha1_base64="zV77cV37hYB+uBZUNouxqQCGOao=">AAACI3icbVDLSgNBEJz1GeMr6tHLYBD0EnZFUEQkqAdvRjAqZNcwO+lNhsw+mOkVw5J/8eKvePGgBC8e/BcnyR6MWtBQVHXT3eUnUmi07U9ranpmdm6+sFBcXFpeWS2trd/oOFUc6jyWsbrzmQYpIqijQAl3iQIW+hJu/e7Z0L99AKVFHF1jLwEvZO1IBIIzNFKzdORKCPCYuiHDDmcyu+zvPO5S9xwksvtM6P6E1TOWEu0OnlDaLJXtij0C/UucnJRJjlqzNHBbMU9DiJBLpnXDsRP0MqZQcAn9optqSBjvsjY0DI1YCNrLRj/26bZRWjSIlakI6Uj9OZGxUOte6JvO4b36tzcU//MaKQaHXiaiJEWI+HhRkEqKMR0GRltCAUfZM4RxJcytlHeYYhxNrEUTgvP75b/kZq/i2BXnar9cPc3jKJBNskV2iEMOSJVckBqpE06eyAt5I+/Ws/VqDayPceuUlc9skAlYX9/Do6Ov</latexit>

O(x)

A

Ā

<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit<latexit sha1_base64="PjTI1X3daC/ZwUfCQ052fYaiOHo=">AAAB7nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPVi8cK9gPaUDbbTbt0swm7E6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px3cxvP3FtRKIecZLyIKZDJSLBKFqp3Qupzm+mfbfq1bw5yCrxC1KFAo2++9UbJCyLuUImqTFd30sxyKlGwSSfVnqZ4SllYzrkXUsVjbkJ8vm5U3JmlQGJEm1LIZmrvydyGhsziUPbGVMcmWVvJv7ndTOMroNcqDRDrthiUZRJggmZ/U4GQnOGcmIJZVrYWwkbUU0Z2oQqNgR/+eVV0rqo+V7Nf7is1m+LOMpwAqdwDj5cQR3uoQFNYDCGZ3iFNyd1Xpx352PRWnKKmWP4A+fzB1Blj4o=</latexit>

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>

⇠

O(y)

O(x)

is

O(y)
s

↵

Rules for geodesics(TF, Li, Wang )See also: (Chen, Dong, Lewkowycz,Qi )• We give some rules for when such correlators can becomputed:⌦<latexit sha1_base64="zV77cV37hYB+uBZUNouxqQCGOao=">AAACI3icbVDLSgNBEJz1GeMr6tHLYBD0EnZFUEQkqAdvRjAqZNcwO+lNhsw+mOkVw5J/8eKvePGgBC8e/BcnyR6MWtBQVHXT3eUnUmi07U9ranpmdm6+sFBcXFpeWS2trd/oOFUc6jyWsbrzmQYpIqijQAl3iQIW+hJu/e7Z0L99AKVFHF1jLwEvZO1IBIIzNFKzdORKCPCYuiHDDmcyu+zvPO5S9xwksvtM6P6E1TOWEu0OnlDaLJXtij0C/UucnJRJjlqzNHBbMU9DiJBLpnXDsRP0MqZQcAn9optqSBjvsjY0DI1YCNrLRj/26bZRWjSIlakI6Uj9OZGxUOte6JvO4b36tzcU//MaKQaHXiaiJEWI+HhRkEqKMR0GRltCAUfZM4RxJcytlHeYYhxNrEUTgvP75b/kZq/i2BXnar9cPc3jKJBNskV2iEMOSJVckBqpE06eyAt5I+/Ws/VqDayPceuUlc9skAlYX9/Do6Ov</latexit>

O(x)

A

Ā

<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit<latexit sha1_base64="PjTI1X3daC/ZwUfCQ052fYaiOHo=">AAAB7nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPVi8cK9gPaUDbbTbt0swm7E6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px3cxvP3FtRKIecZLyIKZDJSLBKFqp3Qupzm+mfbfq1bw5yCrxC1KFAo2++9UbJCyLuUImqTFd30sxyKlGwSSfVnqZ4SllYzrkXUsVjbkJ8vm5U3JmlQGJEm1LIZmrvydyGhsziUPbGVMcmWVvJv7ndTOMroNcqDRDrthiUZRJggmZ/U4GQnOGcmIJZVrYWwkbUU0Z2oQqNgR/+eVV0rqo+V7Nf7is1m+LOMpwAqdwDj5cQR3uoQFNYDCGZ3iFNyd1Xpx352PRWnKKmWP4A+fzB1Blj4o=</latexit>

O(y)

O(x)

is

O(y)

mA

↵

s

<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>⇠• Consider two geodesics meeting at mA such that their<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>

tangents are related by a local boost about mA with
rapidity 2⇡s<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit><latexit sha1_base64="7HhV+JtmnTy3I6KbvZjQkUFPCf0=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT3mTM7MwyMyuEkH/w4kERr/6PN//GSbIHTSxoKKq66e6KUsGN9f1vb219Y3Nru7BT3N3bPzgsHR03jco0wwZTQul2RA0KLrFhuRXYTjXSJBLYika3M7/1hNpwJR/sOMUwoQPJY86odVKz2k05Mb1S2a/4c5BVEuSkDDnqvdJXt69YlqC0TFBjOoGf2nBCteVM4LTYzQymlI3oADuOSpqgCSfza6fk3Cl9EivtSloyV39PTGhizDiJXGdC7dAsezPxP6+T2fg6nHCZZhYlWyyKM0GsIrPXSZ9rZFaMHaFMc3crYUOqKbMuoKILIVh+eZU0q5XArwT3l+XaTR5HAU7hDC4ggCuowR3UoQEMHuEZXuHNU96L9+59LFrXvHzmBP7A+/wB7KKOsA==</latexit><latexit sha1_base64="FM2A8RIX0oVI5v5/51xH49v4Qus=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1iEuimJCLoRim5cuKhgH9CGMJlM2qGTBzM3YggFf8WNC0Xc+h3u/BunbRbaemDgcM493DvHSwRXYFnfRmlpeWV1rbxe2djc2t4xd/faKk4lZS0ai1h2PaKY4BFrAQfBuolkJPQE63ij64nfeWBS8Ti6hyxhTkgGEQ84JaAl1zwA9xFf4v6tjvjEVbgGbnbimlWrbk2BF4ldkCoq0HTNr74f0zRkEVBBlOrZVgJOTiRwKti40k8VSwgdkQHraRqRkCknn54/xsda8XEQS/0iwFP1dyInoVJZ6OnJkMBQzXsT8T+vl0Jw4eQ8SlJgEZ0tClKBIcaTLrDPJaMgMk0IlVzfiumQSEJBN1bRJdjzX14k7dO6bdXtu7Nq46qoo4wO0RGqIRudowa6QU3UQhTl6Bm9ojfjyXgx3o2P2WjJKDL76A+Mzx9csJRx</latexit>

tx = ⇤s (ty )Rules for geodesics(TF, Li, Wang )See also: (Chen, Dong, Lewkowycz,Qi )• We give some rules for when such correlators can becomputed:⌦<latexit sha1_base64="zV77cV37hYB+uBZUNouxqQCGOao=">AAACI3icbVDLSgNBEJz1GeMr6tHLYBD0EnZFUEQkqAdvRjAqZNcwO+lNhsw+mOkVw5J/8eKvePGgBC8e/BcnyR6MWtBQVHXT3eUnUmi07U9ranpmdm6+sFBcXFpeWS2trd/oOFUc6jyWsbrzmQYpIqijQAl3iQIW+hJu/e7Z0L99AKVFHF1jLwEvZO1IBIIzNFKzdORKCPCYuiHDDmcyu+zvPO5S9xwksvtM6P6E1TOWEu0OnlDaLJXtij0C/UucnJRJjlqzNHBbMU9DiJBLpnXDsRP0MqZQcAn9optqSBjvsjY0DI1YCNrLRj/26bZRWjSIlakI6Uj9OZGxUOte6JvO4b36tzcU//MaKQaHXiaiJEWI+HhRkEqKMR0GRltCAUfZM4RxJcytlHeYYhxNrEUTgvP75b/kZq/i2BXnar9cPc3jKJBNskV2iEMOSJVckBqpE06eyAt5I+/Ws/VqDayPceuUlc9skAlYX9/Do6Ov</latexit>

O(x)

A

Ā

<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit<latexit sha1_base64="PjTI1X3daC/ZwUfCQ052fYaiOHo=">AAAB7nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPVi8cK9gPaUDbbTbt0swm7E6GE/ggvHhTx6u/x5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px3cxvP3FtRKIecZLyIKZDJSLBKFqp3Qupzm+mfbfq1bw5yCrxC1KFAo2++9UbJCyLuUImqTFd30sxyKlGwSSfVnqZ4SllYzrkXUsVjbkJ8vm5U3JmlQGJEm1LIZmrvydyGhsziUPbGVMcmWVvJv7ndTOMroNcqDRDrthiUZRJggmZ/U4GQnOGcmIJZVrYWwkbUU0Z2oQqNgR/+eVV0rqo+V7Nf7is1m+LOMpwAqdwDj5cQR3uoQFNYDCGZ3iFNyd1Xpx352PRWnKKmWP4A+fzB1Blj4o=</latexit>

O(y)

O(x)

is

O(y)

mA

↵

s

<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>⇠• Consider two geodesics meeting at mA such that their<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>

tangents are related by a local boost about mA with
rapidity 2⇡s<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit><latexit sha1_base64="7HhV+JtmnTy3I6KbvZjQkUFPCf0=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6DHoxWME84BkCbOT3mTM7MwyMyuEkH/w4kERr/6PN//GSbIHTSxoKKq66e6KUsGN9f1vb219Y3Nru7BT3N3bPzgsHR03jco0wwZTQul2RA0KLrFhuRXYTjXSJBLYika3M7/1hNpwJR/sOMUwoQPJY86odVKz2k05Mb1S2a/4c5BVEuSkDDnqvdJXt69YlqC0TFBjOoGf2nBCteVM4LTYzQymlI3oADuOSpqgCSfza6fk3Cl9EivtSloyV39PTGhizDiJXGdC7dAsezPxP6+T2fg6nHCZZhYlWyyKM0GsIrPXSZ9rZFaMHaFMc3crYUOqKbMuoKILIVh+eZU0q5XArwT3l+XaTR5HAU7hDC4ggCuowR3UoQEMHuEZXuHNU96L9+59LFrXvHzmBP7A+/wB7KKOsA==</latexit><latexit sha1_base64="FM2A8RIX0oVI5v5/51xH49v4Qus=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1iEuimJCLoRim5cuKhgH9CGMJlM2qGTBzM3YggFf8WNC0Xc+h3u/BunbRbaemDgcM493DvHSwRXYFnfRmlpeWV1rbxe2djc2t4xd/faKk4lZS0ai1h2PaKY4BFrAQfBuolkJPQE63ij64nfeWBS8Ti6hyxhTkgGEQ84JaAl1zwA9xFf4v6tjvjEVbgGbnbimlWrbk2BF4ldkCoq0HTNr74f0zRkEVBBlOrZVgJOTiRwKti40k8VSwgdkQHraRqRkCknn54/xsda8XEQS/0iwFP1dyInoVJZ6OnJkMBQzXsT8T+vl0Jw4eQ8SlJgEZ0tClKBIcaTLrDPJaMgMk0IlVzfiumQSEJBN1bRJdjzX14k7dO6bdXtu7Nq46qoo4wO0RGqIRudowa6QU3UQhTl6Bm9ojfjyXgx3o2P2WjJKDL76A+Mzx9csJRx</latexit>

tx = ⇤s (ty )= exp( m`(x, ⇠)<latexit sha1_base64="tDt5LeKJMTgqYazd+e12ZATaL10=">AAACEnicbVDLSgMxFM3UV62vUZdugkVoQcuMCLoRim5cVrAP6Awlk6ZtaJIZkox0GPoNbvwVNy4UcevKnX9j2g6orQcC555zLzf3BBGjSjvOl5VbWl5ZXcuvFzY2t7Z37N29hgpjiUkdhyyUrQApwqggdU01I61IEsQDRprB8HriN++JVDQUdzqJiM9RX9AexUgbqWOXL6FHRlEJnkBuGGOl0bE3ouWfOpnV5Y5ddCrOFHCRuBkpggy1jv3pdUMccyI0ZkiptutE2k+R1BQzMi54sSIRwkPUJ21DBeJE+en0pDE8MkoX9kJpntBwqv6eSBFXKuGB6eRID9S8NxH/89qx7l34KRVRrInAs0W9mEEdwkk+sEslwZolhiAsqfkrxAMkEdYmxYIJwZ0/eZE0TiuuU3Fvz4rVqyyOPDgAh6AEXHAOquAG1EAdYPAAnsALeLUerWfrzXqfteasbGYf/IH18Q2Zv5pH</latexit>m`(y, ⇠))

Rules for geodesics• Not always true! So we have not “solved” modular flow …

Rules for geodesics• Not always true! So we have not “solved” modular flow …
• Minimize over meeting point ⇠ calculate<latexit sha1_base64="Q3ckis8sxyt1zJ/hVAHRYf+CVtM=">AAAB63icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOZpMh81hmZsWw5Be8eFDEqz/kzb9xNtmDJhY0FFXddHdFCWfG+v63V1pb39jcKm9Xdnb39g+qh0dto1JNaIsornQ3woZyJmnLMstpN9EUi4jTTjS5zf3OI9WGKflgpwkNBR5JFjOCbS71nxgaVGt+3Z8DrZKgIDUo0BxUv/pDRVJBpSUcG9ML/MSGGdaWEU5nlX5qaILJBI9oz1GJBTVhNr91hs6cMkSx0q6kRXP190SGhTFTEblOge3YLHu5+J/XS218HWZMJqmlkiwWxSlHVqH8cTRkmhLLp45gopm7FZEx1phYF0/FhRAsv7xK2hf1wK8H95e1xk0RRxlO4BTOIYAraMAdNKEFBMbwDK/w5gnvxXv3PhatJa+YOYY/8D5/ALKKjf8=</latexit>

s(x, y)<latexit sha1_base64="1gAZm8FK7q4DFFQmB/dbuSvrDn4=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahgpRdEfRY9OKxgv2AdinZNNvGZpMlyYrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmBTFn2rjut1NYWV1b3yhulra2d3b3yvsHLS0TRWiTSC5VJ8CaciZo0zDDaSdWFEcBp+1gfDP1249UaSbFvUlj6kd4KFjICDZWaunq01l62i9X3Jo7A1omXk4qkKPRL3/1BpIkERWGcKx113Nj42dYGUY4nZR6iaYxJmM8pF1LBY6o9rPZtRN0YpUBCqWyJQyaqb8nMhxpnUaB7YywGelFbyr+53UTE175GRNxYqgg80VhwpGRaPo6GjBFieGpJZgoZm9FZIQVJsYGVLIheIsvL5PWec1za97dRaV+ncdRhCM4hip4cAl1uIUGNIHAAzzDK7w50nlx3p2PeWvByWcO4Q+czx/HKo6X</latexit>

Rules for geodesics• Not always true! So we have not “solved” modular flow …
• Minimize over meeting point ⇠ calculates(x, y)<latexit sha1_base64="1gAZm8FK7q4DFFQmB/dbuSvrDn4=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahgpRdEfRY9OKxgv2AdinZNNvGZpMlyYrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmBTFn2rjut1NYWV1b3yhulra2d3b3yvsHLS0TRWiTSC5VJ8CaciZo0zDDaSdWFEcBp+1gfDP1249UaSbFvUlj6kd4KFjICDZWaunq01l62i9X3Jo7A1omXk4qkKPRL3/1BpIkERWGcKx113Nj42dYGUY4nZR6iaYxJmM8pF1LBY6o9rPZtRN0YpUBCqWyJQyaqb8nMhxpnUaB7YywGelFbyr+53UTE175GRNxYqgg80VhwpGRaPo6GjBFieGpJZgoZm9FZIQVJsYGVLIheIsvL5PWec1za97dRaV+ncdRhCM4hip4cAl1uIUGNIHAAzzDK7w50nlx3p2PeWvByWcO4Q+czx/HKo6X</latexit><latexit sha1_base64="Q3ckis8sxyt1zJ/hVAHRYf+CVtM=">AAAB63icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOZpMh81hmZsWw5Be8eFDEqz/kzb9xNtmDJhY0FFXddHdFCWfG+v63V1pb39jcKm9Xdnb39g+qh0dto1JNaIsornQ3woZyJmnLMstpN9EUi4jTTjS5zf3OI9WGKflgpwkNBR5JFjOCbS71nxgaVGt+3Z8DrZKgIDUo0BxUv/pDRVJBpSUcG9ML/MSGGdaWEU5nlX5qaILJBI9oz1GJBTVhNr91hs6cMkSx0q6kRXP190SGhTFTEblOge3YLHu5+J/XS218HWZMJqmlkiwWxSlHVqH8cTRkmhLLp45gopm7FZEx1phYF0/FhRAsv7xK2hf1wK8H95e1xk0RRxlO4BTOIYAraMAdNKEFBMbwDK/w5gnvxXv3PhatJa+YOYY/8D5/ALKKjf8=</latexit>• So co-dimension one slice of parameter space:e.g.

{x, y, s = s(x, y)}<latexit sha1_base64="0zAgR1Ojo/ISeJkR8wuLNyTKnkU=">AAAB/XicbZDLSsNAFIZP6q3WW7zs3AwWoUIpiQi6EYpuXFawF2hCmUwn7dDJhZmJGEPxVdy4UMSt7+HOt3HaZqGtPwx8/OcczpnfizmTyrK+jcLS8srqWnG9tLG5tb1j7u61ZJQIQpsk4pHoeFhSzkLaVExx2okFxYHHadsbXU/q7XsqJIvCO5XG1A3wIGQ+I1hpq2ceOBl6qKZVJNElkhWNJ864Z5atmjUVWgQ7hzLkavTML6cfkSSgoSIcS9m1rVi5GRaKEU7HJSeRNMZkhAe0qzHEAZVuNr1+jI6100d+JPQLFZq6vycyHEiZBp7uDLAayvnaxPyv1k2Uf+FmLIwTRUMyW+QnHKkITaJAfSYoUTzVgIlg+lZEhlhgonRgJR2CPf/lRWid1myrZt+eletXeRxFOIQjqIAN51CHG2hAEwg8wjO8wpvxZLwY78bHrLVg5DP78EfG5w/HW5N9</latexit>

Rules for geodesics• Not always true! So we have not “solved” modular flow …
• Minimize over meeting point ⇠ calculates(x, y)<latexit sha1_base64="1gAZm8FK7q4DFFQmB/dbuSvrDn4=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahgpRdEfRY9OKxgv2AdinZNNvGZpMlyYrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmBTFn2rjut1NYWV1b3yhulra2d3b3yvsHLS0TRWiTSC5VJ8CaciZo0zDDaSdWFEcBp+1gfDP1249UaSbFvUlj6kd4KFjICDZWaunq01l62i9X3Jo7A1omXk4qkKPRL3/1BpIkERWGcKx113Nj42dYGUY4nZR6iaYxJmM8pF1LBY6o9rPZtRN0YpUBCqWyJQyaqb8nMhxpnUaB7YywGelFbyr+53UTE175GRNxYqgg80VhwpGRaPo6GjBFieGpJZgoZm9FZIQVJsYGVLIheIsvL5PWec1za97dRaV+ncdRhCM4hip4cAl1uIUGNIHAAzzDK7w50nlx3p2PeWvByWcO4Q+czx/HKo6X</latexit><latexit sha1_base64="Q3ckis8sxyt1zJ/hVAHRYf+CVtM=">AAAB63icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOZpMh81hmZsWw5Be8eFDEqz/kzb9xNtmDJhY0FFXddHdFCWfG+v63V1pb39jcKm9Xdnb39g+qh0dto1JNaIsornQ3woZyJmnLMstpN9EUi4jTTjS5zf3OI9WGKflgpwkNBR5JFjOCbS71nxgaVGt+3Z8DrZKgIDUo0BxUv/pDRVJBpSUcG9ML/MSGGdaWEU5nlX5qaILJBI9oz1GJBTVhNr91hs6cMkSx0q6kRXP190SGhTFTEblOge3YLHu5+J/XS218HWZMJqmlkiwWxSlHVqH8cTRkmhLLp45gopm7FZEx1phYF0/FhRAsv7xK2hf1wK8H95e1xk0RRxlO4BTOIYAraMAdNKEFBMbwDK/w5gnvxXv3PhatJa+YOYY/8D5/ALKKjf8=</latexit>• So co-dimension one slice of parameter space:e.g.

{x, y, s = s(x, y)}<latexit sha1_base64="0zAgR1Ojo/ISeJkR8wuLNyTKnkU=">AAAB/XicbZDLSsNAFIZP6q3WW7zs3AwWoUIpiQi6EYpuXFawF2hCmUwn7dDJhZmJGEPxVdy4UMSt7+HOt3HaZqGtPwx8/OcczpnfizmTyrK+jcLS8srqWnG9tLG5tb1j7u61ZJQIQpsk4pHoeFhSzkLaVExx2okFxYHHadsbXU/q7XsqJIvCO5XG1A3wIGQ+I1hpq2ceOBl6qKZVJNElkhWNJ864Z5atmjUVWgQ7hzLkavTML6cfkSSgoSIcS9m1rVi5GRaKEU7HJSeRNMZkhAe0qzHEAZVuNr1+jI6100d+JPQLFZq6vycyHEiZBp7uDLAayvnaxPyv1k2Uf+FmLIwTRUMyW+QnHKkITaJAfSYoUTzVgIlg+lZEhlhgonRgJR2CPf/lRWid1myrZt+eletXeRxFOIQjqIAN51CHG2hAEwg8wjO8wpvxZLwY78bHrLVg5DP78EfG5w/HW5N9</latexit>• We will also need for linear deformations thereof:
exp( m(`(x, ⇠) + `(y, ⇠) +<latexit sha1_base64="umD5wFg0i4uLqnWcoV1z7WgeBrE=">AAACG3icbVDLSgMxFM3UV62vqks3wSK0qGWmCLosunFZwT6gM5ZM5k4bmnmQZKRl6H+48VfcuFDEleDCvzF9INp6IHDuOfdyc48bcyaVaX4ZmaXlldW17HpuY3Nreye/u9eQUSIo1GnEI9FyiQTOQqgrpji0YgEkcDk03f7V2G/eg5AsCm/VMAYnIN2Q+YwSpaVOvmLDIC7iUxzgog2cFwcn9oCV8DGeVMOfygOuyF2lhEudfMEsmxPgRWLNSAHNUOvkP2wvokkAoaKcSNm2zFg5KRGKUQ6jnJ1IiAntky60NQ1JANJJJ7eN8JFWPOxHQr9Q4Yn6eyIlgZTDwNWdAVE9Oe+Nxf+8dqL8CydlYZwoCOl0kZ9wrCI8Dgp7TABVfKgJoYLpv2LaI4JQpePM6RCs+ZMXSaNStsyydXNWqF7O4siiA3SIishC56iKrlEN1RFFD+gJvaBX49F4Nt6M92lrxpjN7KM/MD6/AZ38nWc=</latexit><latexit sha1_base64="AU/vq+DAgMm1x9RhUc20aA7BJwY=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK9gPaUDabTbt2kw27E6GE/gcvHhTx6v/x5r9x2+agrQ8GHu/NMDMvSKUw6LrfTmltfWNzq7xd2dnd2z+oHh61jco04y2mpNLdgBouRcJbKFDybqo5jQPJO8H4duZ3nrg2QiUPOEm5H9NhIiLBKFqp3Q+5RDqo1ty6OwdZJV5BalCgOah+9UPFspgnyCQ1pue5Kfo51SiY5NNKPzM8pWxMh7xnaUJjbvx8fu2UnFklJJHSthIkc/X3RE5jYyZxYDtjiiOz7M3E/7xehtG1n4skzZAnbLEoyiRBRWavk1BozlBOLKFMC3srYSOqKUMbUMWG4C2/vEraF3XPrXv3l7XGTRFHGU7gFM7BgytowB00oQUMHuEZXuHNUc6L8+58LFpLTjFzDH/gfP4AkYyPHA==</latexit>2))Some intuition:• Following Jaﬀeris, Suh (2014):|<latexit sha1_base64="ZKjmQxIgvkNXoVQmEQGiIDWbHsY=">AAACIHicbVDLSsNAFJ34rPUVdelmsAiuSiJC3ShVNy4r2Ac0MUymk3bo5MHMjVBiP8WNv+LGhSK6069x2kaorQcGDuecy517/ERwBZb1ZSwsLi2vrBbWiusbm1vb5s5uQ8WppKxOYxHLlk8UEzxideAgWCuRjIS+YE2/fzXym/dMKh5HtzBImBuSbsQDTgloyTMrjmABPGAnUdxT2JG824NzfKZZL/Yu7jKuhngq85vwzJJVtsbA88TOSQnlqHnmp9OJaRqyCKggSrVtKwE3IxI4FWxYdFLFEkL7pMvamkYkZMrNxgcO8aFWOjiIpX4R4LE6PZGRUKlB6OtkSKCnZr2R+J/XTiE4dTMeJSmwiE4WBanAEONRW7jDJaMgBpoQKrn+K6Y9IgkF3WlRl2DPnjxPGsdl2yrbNyel6mVeRwHtowN0hGxUQVV0jWqojih6RM/oFb0ZT8aL8W58TKILRj6zh/7A+P4BNl6i/Q==</latexit>iss i = ⇢A |Using ``boost invariance’’:<latexit sha1_base64="tjtCTrfiScJEiVpPbbZNcVJYKNg=">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</latexit>

h s | O(x)O(y) | s i = h | O(x)<latexit sha1_base64="rcTebQ7sk7ciXbSKjNGwm1mpDLM=">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</latexit>

is

A|

is
i = ⇢Ā |<latexit sha1_base64="pXkoh1C7yPxcLM3tgf9mVffCq2A=">AAACEnicbVDLSsNAFJ34rPUVdelmsAi6KYkIulGqblxWsA9oYphMJ+3QyYOZG6HEfIMbf8WNC0XcunLn3zhts9DWAxcO59zLvff4ieAKLOvbmJtfWFxaLq2UV9fWNzbNre2milNJWYPGIpZtnygmeMQawEGwdiIZCX3BWv7gauS37plUPI5uYZgwNyS9iAecEtCSZx6eYUf2Yy9zfCKzizy/y7jKsSNYAA/YSRTXPu/14dwzK1bVGgPPErsgFVSg7plfTjemacgioIIo1bGtBNyMSOBUsLzspIolhA5Ij3U0jUjIlJuNX8rxvla6OIilrgjwWP09kZFQqWHo686QQF9NeyPxP6+TQnDqZjxKUmARnSwKUoEhxqN8cJdLRkEMNSFUcn0rpn0iCQWdYlmHYE+/PEuaR1Xbqto3x5XaZRFHCe2iPXSAbHSCauga1VEDUfSIntErejOejBfj3fiYtM4ZxcwO+gPj8wdQV53d</latexit>ii = ⇢A ⌦ ⇢Ā1 | i = | iO(y) | iSome intuition:• Following Jaﬀeris, Suh (2014):|<latexit sha1_base64="ZKjmQxIgvkNXoVQmEQGiIDWbHsY=">AAACIHicbVDLSsNAFJ34rPUVdelmsAiuSiJC3ShVNy4r2Ac0MUymk3bo5MHMjVBiP8WNv+LGhSK6069x2kaorQcGDuecy517/ERwBZb1ZSwsLi2vrBbWiusbm1vb5s5uQ8WppKxOYxHLlk8UEzxideAgWCuRjIS+YE2/fzXym/dMKh5HtzBImBuSbsQDTgloyTMrjmABPGAnUdxT2JG824NzfKZZL/Yu7jKuhngq85vwzJJVtsbA88TOSQnlqHnmp9OJaRqyCKggSrVtKwE3IxI4FWxYdFLFEkL7pMvamkYkZMrNxgcO8aFWOjiIpX4R4LE6PZGRUKlB6OtkSKCnZr2R+J/XTiE4dTMeJSmwiE4WBanAEONRW7jDJaMgBpoQKrn+K6Y9IgkF3WlRl2DPnjxPGsdl2yrbNyel6mVeRwHtowN0hGxUQVV0jWqojih6RM/oFb0ZT8aL8W58TKILRj6zh/7A+P4BNl6i/Q==</latexit>iss i = ⇢A |Using ``boost invariance’’:<latexit sha1_base64="tjtCTrfiScJEiVpPbbZNcVJYKNg=">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</latexit>

h s | O(x)O(y) | s i = h | O(x)<latexit sha1_base64="rcTebQ7sk7ciXbSKjNGwm1mpDLM=">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</latexit>

is

A|

is
i = ⇢Ā |<latexit sha1_base64="pXkoh1C7yPxcLM3tgf9mVffCq2A=">AAACEnicbVDLSsNAFJ34rPUVdelmsAi6KYkIulGqblxWsA9oYphMJ+3QyYOZG6HEfIMbf8WNC0XcunLn3zhts9DWAxcO59zLvff4ieAKLOvbmJtfWFxaLq2UV9fWNzbNre2milNJWYPGIpZtnygmeMQawEGwdiIZCX3BWv7gauS37plUPI5uYZgwNyS9iAecEtCSZx6eYUf2Yy9zfCKzizy/y7jKsSNYAA/YSRTXPu/14dwzK1bVGgPPErsgFVSg7plfTjemacgioIIo1bGtBNyMSOBUsLzspIolhA5Ij3U0jUjIlJuNX8rxvla6OIilrgjwWP09kZFQqWHo686QQF9NeyPxP6+TQnDqZjxKUmARnSwKUoEhxqN8cJdLRkEMNSFUcn0rpn0iCQWdYlmHYE+/PEuaR1Xbqto3x5XaZRFHCe2iPXSAbHSCauga1VEDUfSIntErejOejBfj3fiYtM4ZxcwO+gPj8wdQV53d</latexit>ii = ⇢A ⌦ ⇢Ā1 | i = | iO(y) | i• In this state, observables restricted to D(A) or D(Ā)
unaﬀected: Entanglement wedges unaﬀected!!
<latexit sha1_base64="f1yVLQiq0VmWHDotiuP9ysW02eI=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1iEuimJFHRZHwuXFewD2lAm00k7dDIJMxOhhn6JGxeKuPVT3Pk3TtostPXAwOGce7lnjh9zprTjfFuFtfWNza3idmlnd2+/bB8ctlWUSEJbJOKR7PpYUc4EbWmmOe3GkuLQ57TjT24yv/NIpWKReNDTmHohHgkWMIK1kQZ2uR9iPSaYp7ez6tUZGtgVp+bMgVaJm5MK5GgO7K/+MCJJSIUmHCvVc51YeymWmhFOZ6V+omiMyQSPaM9QgUOqvHQefIZOjTJEQSTNExrN1d8bKQ6Vmoa+mcxiqmUvE//zeokOLr2UiTjRVJDFoSDhSEcoawENmaRE86khmEhmsiIyxhITbboqmRLc5S+vkvZ5zXVq7n290rjO6yjCMZxAFVy4gAbcQRNaQCCBZ3iFN+vJerHerY/FaMHKd47gD6zPH6FPkmU=</latexit><latexit sha1_base64="RpdS0WcLWDvUx9m2UekJPAFP46c=">AAAB/nicbVDLSsNAFJ34rPUVFVduBotQNyURQZf1sXBZwT6gKeVmOmmHTiZhZiKUEPBX3LhQxK3f4c6/cdJmoa0HBg7n3Ms9c/yYM6Ud59taWl5ZXVsvbZQ3t7Z3du29/ZaKEklok0Q8kh0fFOVM0KZmmtNOLCmEPqdtf3yT++1HKhWLxIOexLQXwlCwgBHQRurbh14IekSAp7dZ1fNBplfZKe7bFafmTIEXiVuQCirQ6Ntf3iAiSUiFJhyU6rpOrHspSM0Ip1nZSxSNgYxhSLuGCgip6qXT+Bk+McoAB5E0T2g8VX9vpBAqNQl9M5mHVfNeLv7ndRMdXPZSJuJEU0Fmh4KEYx3hvAs8YJISzSeGAJHMZMVkBBKINo2VTQnu/JcXSeus5jo19/68Ur8u6iihI3SMqshFF6iO7lADNRFBKXpGr+jNerJerHfrYza6ZBU7B+gPrM8fgOeVKg==</latexit>

Some intuition:• Following Jaﬀeris, Suh (2014):|<latexit sha1_base64="ZKjmQxIgvkNXoVQmEQGiIDWbHsY=">AAACIHicbVDLSsNAFJ34rPUVdelmsAiuSiJC3ShVNy4r2Ac0MUymk3bo5MHMjVBiP8WNv+LGhSK6069x2kaorQcGDuecy517/ERwBZb1ZSwsLi2vrBbWiusbm1vb5s5uQ8WppKxOYxHLlk8UEzxideAgWCuRjIS+YE2/fzXym/dMKh5HtzBImBuSbsQDTgloyTMrjmABPGAnUdxT2JG824NzfKZZL/Yu7jKuhngq85vwzJJVtsbA88TOSQnlqHnmp9OJaRqyCKggSrVtKwE3IxI4FWxYdFLFEkL7pMvamkYkZMrNxgcO8aFWOjiIpX4R4LE6PZGRUKlB6OtkSKCnZr2R+J/XTiE4dTMeJSmwiE4WBanAEONRW7jDJaMgBpoQKrn+K6Y9IgkF3WlRl2DPnjxPGsdl2yrbNyel6mVeRwHtowN0hGxUQVV0jWqojih6RM/oFb0ZT8aL8W58TKILRj6zh/7A+P4BNl6i/Q==</latexit>iss i = ⇢A |Using ``boost invariance’’:A|<latexit sha1_base64="tjtCTrfiScJEiVpPbbZNcVJYKNg=">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</latexit>ish s | O(x)O(y) | s i = h | O(x)<latexit sha1_base64="rcTebQ7sk7ciXbSKjNGwm1mpDLM=">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</latexit>isi = ⇢Ā |i<latexit sha1_base64="pXkoh1C7yPxcLM3tgf9mVffCq2A=">AAACEnicbVDLSsNAFJ34rPUVdelmsAi6KYkIulGqblxWsA9oYphMJ+3QyYOZG6HEfIMbf8WNC0XcunLn3zhts9DWAxcO59zLvff4ieAKLOvbmJtfWFxaLq2UV9fWNzbNre2milNJWYPGIpZtnygmeMQawEGwdiIZCX3BWv7gauS37plUPI5uYZgwNyS9iAecEtCSZx6eYUf2Yy9zfCKzizy/y7jKsSNYAA/YSRTXPu/14dwzK1bVGgPPErsgFVSg7plfTjemacgioIIo1bGtBNyMSOBUsLzspIolhA5Ij3U0jUjIlJuNX8rxvla6OIilrgjwWP09kZFQqWHo686QQF9NeyPxP6+TQnDqZjxKUmARnSwKUoEhxqN8cJdLRkEMNSFUcn0rpn0iCQWdYlmHYE+/PEuaR1Xbqto3x5XaZRFHCe2iPXSAbHSCauga1VEDUfSIntErejOejBfj3fiYtM4ZxcwO+gPj8wdQV53d</latexit>i = ⇢A ⌦ ⇢Ā1 | i = | iO(y) | i• In this state, observables restricted to D(A) or D(Ā)
unaﬀected: Entanglement wedges unaﬀected!!
<latexit sha1_base64="f1yVLQiq0VmWHDotiuP9ysW02eI=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1iEuimJFHRZHwuXFewD2lAm00k7dDIJMxOhhn6JGxeKuPVT3Pk3TtostPXAwOGce7lnjh9zprTjfFuFtfWNza3idmlnd2+/bB8ctlWUSEJbJOKR7PpYUc4EbWmmOe3GkuLQ57TjT24yv/NIpWKReNDTmHohHgkWMIK1kQZ2uR9iPSaYp7ez6tUZGtgVp+bMgVaJm5MK5GgO7K/+MCJJSIUmHCvVc51YeymWmhFOZ6V+omiMyQSPaM9QgUOqvHQefIZOjTJEQSTNExrN1d8bKQ6Vmoa+mcxiqmUvE//zeokOLr2UiTjRVJDFoSDhSEcoawENmaRE86khmEhmsiIyxhITbboqmRLc5S+vkvZ5zXVq7n290rjO6yjCMZxAFVy4gAbcQRNaQCCBZ3iFN+vJerHerY/FaMHKd47gD6zPH6FPkmU=</latexit><latexit sha1_base64="k3Ghl1MhHA/5AwXp6k1vYiifIjw=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8eK9gPaUDbbTbp0swm7E6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvSKUw6LrfTmltfWNzq7xd2dnd2z+oHh61TZJpxlsskYnuBtRwKRRvoUDJu6nmNA4k7wTj25nfeeLaiEQ94iTlfkwjJULBKFrpIRqYQbXm1t05yCrxClKDAs1B9as/TFgWc4VMUmN6npuin1ONgkk+rfQzw1PKxjTiPUsVjbnx8/mpU3JmlSEJE21LIZmrvydyGhsziQPbGVMcmWVvJv7n9TIMr/1cqDRDrthiUZhJggmZ/U2GQnOGcmIJZVrYWwkbUU0Z2nQqNgRv+eVV0r6oe27du7+sNW6KOMpwAqdwDh5cQQPuoAktYBDBM7zCmyOdF+fd+Vi0lpxi5hj+wPn8AVXSjdE=</latexit><latexit sha1_base64="RpdS0WcLWDvUx9m2UekJPAFP46c=">AAAB/nicbVDLSsNAFJ34rPUVFVduBotQNyURQZf1sXBZwT6gKeVmOmmHTiZhZiKUEPBX3LhQxK3f4c6/cdJmoa0HBg7n3Ms9c/yYM6Ud59taWl5ZXVsvbZQ3t7Z3du29/ZaKEklok0Q8kh0fFOVM0KZmmtNOLCmEPqdtf3yT++1HKhWLxIOexLQXwlCwgBHQRurbh14IekSAp7dZ1fNBplfZKe7bFafmTIEXiVuQCirQ6Ntf3iAiSUiFJhyU6rpOrHspSM0Ip1nZSxSNgYxhSLuGCgip6qXT+Bk+McoAB5E0T2g8VX9vpBAqNQl9M5mHVfNeLv7ndRMdXPZSJuJEU0Fmh4KEYx3hvAs8YJISzSeGAJHMZMVkBBKINo2VTQnu/JcXSeus5jo19/68Ur8u6iihI3SMqshFF6iO7lADNRFBKXpGr+jNerJerHfrYza6ZBU7B+gPrM8fgOeVKg==</latexit>

gs

??

x
<latexit sha1_base64="f2yzimwbR/Dgjzp6tZ360fHRqNI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N2IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHst7M0nQj+hQ8pAzaqzUeOqXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqN3kcRTiBUzgHD66gBndQhyYwQHiGV3hzHpwX5935WLQWnHzmGP7A+fwB5jmM/A==</latexit><latexit<latexit sha1_base64="g52mBfVlfrpPbt9ADvc/q3MAFxY=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0FuLXjxWsR/QhrLZbtqlm03YnQgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsY38789hPXRsTqEScJ9yM6VCIUjKKVHmq1frniVt05yCrxclKBHI1++as3iFkacYVMUmO6npugn1GNgkk+LfVSwxPKxnTIu5YqGnHjZ/NLp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP66YYXvuZUEmKXLHFojCVBGMye5sMhOYM5cQSyrSwtxI2opoytOGUbAje8surpHVR9dyqd39Zqd/kcRThBE7hHDy4gjrcQQOawCCEZ3iFN2fsvDjvzseiteDkM8fwB87nDxSDjQw=</latexit>

Ea

Eā

<latexit sha1_base64="3HsdEBrZMYop2xl3X7ojJScomjA=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiCC4r2Ae0Q7mTpm1oJjMmmUIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z6cIBZcG9f9dgpr6xubW8Xt0s7u3v5B+fCoqaNEUdagkYhUO0DNBJesYbgRrB0rhmEgWCsY32Z+a8KU5pF8NNOY+SEOJR9wisZKfjdEM6Io0rtZD3vlilt15yCrxMtJBXLUe+Wvbj+iScikoQK17nhubPwUleFUsFmpm2gWIx3jkHUslRgy7afz0DNyZpU+GUTKPmnIXP29kWKo9TQM7GQWUi97mfif10nM4NpPuYwTwyRdHBokgpiIZA2QPleMGjG1BKniNiuhI1RIje2pZEvwlr+8SpoXVc+teg+XldpNXkcRTuAUzsGDK6jBPdShARSe4Ble4c2ZOC/Ou/OxGC04+c4x/IHz+QPsqJIv</latexit>s<latexit sha1_base64="o4RaBwsfV0QeFAQKiOVzFLnI9lo=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCq5KIoMuiCC4r2FpoQriZTtuhk0mYmQg1BH/FjQtF3Pof7vwbJ20W2npg4HDOvdwzJ0w4U9pxvq3K0vLK6lp1vbaxubW9Y+/udVScSkLbJOax7IagKGeCtjXTnHYTSSEKOb0Px1eFf/9ApWKxuNOThPoRDAUbMALaSIF94EWgRwR4dp0HmReCxJAHdt1pOFPgReKWpI5KtAL7y+vHJI2o0ISDUj3XSbSfgdSMcJrXvFTRBMgYhrRnqICIKj+bps/xsVH6eBBL84TGU/X3RgaRUpMoNJNFVjXvFeJ/Xi/Vgws/YyJJNRVkdmiQcqxjXFSB+0xSovnEECCSmayYjEAC0aawminBnf/yIumcNlyn4d6e1ZuXZR1VdIiO0Aly0TlqohvUQm1E0CN6Rq/ozXqyXqx362M2WrHKnX30B9bnD50slU8=</latexit>??
<latexit sha1_base64="g52mBfVlfrpPbt9ADvc/q3MAFxY=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0FuLXjxWsR/QhrLZbtqlm03YnQgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsY38789hPXRsTqEScJ9yM6VCIUjKKVHmq1frniVt05yCrxclKBHI1++as3iFkacYVMUmO6npugn1GNgkk+LfVSwxPKxnTIu5YqGnHjZ/NLp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP66YYXvuZUEmKXLHFojCVBGMye5sMhOYM5cQSyrSwtxI2opoytOGUbAje8surpHVR9dyqd39Zqd/kcRThBE7hHDy4gjrcQQOawCCEZ3iFN2fsvDjvzseiteDkM8fwB87nDxSDjQw=</latexit>

tune

??
<latexit sha1_base64="g52mBfVlfrpPbt9ADvc/q3MAFxY=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0FuLXjxWsR/QhrLZbtqlm03YnQgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsY38789hPXRsTqEScJ9yM6VCIUjKKVHmq1frniVt05yCrxclKBHI1++as3iFkacYVMUmO6npugn1GNgkk+LfVSwxPKxnTIu5YqGnHjZ/NLp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP66YYXvuZUEmKXLHFojCVBGMye5sMhOYM5cQSyrSwtxI2opoytOGUbAje8surpHVR9dyqd39Zqd/kcRThBE7hHDy4gjrcQQOawCCEZ3iFN2fsvDjvzseiteDkM8fwB87nDxSDjQw=</latexit>

Also: replica trick argumenty<latexit sha1_base64="l29WxoUb9DEbvmhLG7jHtZ0OU24=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokIeix68diC/YA2lM120q7dbMLuRgihv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZwqhi0Wi1h1A6pRcIktw43AbqKQRoHATjC5m/mdJ1Sax/LBZAn6ER1JHnJGjZWa2aBSdWvuHGSVeAWpQoHGoPLVH8YsjVAaJqjWPc9NjJ9TZTgTOC33U40JZRM6wp6lkkao/Xx+6JScW2VIwljZkobM1d8TOY20zqLAdkbUjPWyNxP/83qpCW/8nMskNSjZYlGYCmJiMvuaDLlCZkRmCWWK21sJG1NFmbHZlG0I3vLLq6R9WfPcmte8qtZvizhKcApncAEeXEMd7qEBLWCA8Ayv8OY8Oi/Ou/OxaF1zipkT+APn8wfnvYz9</latexit>Mirrors for mirror ops• ComplexDboosts: EuclideanErotations - ⇡ rotation: mirror!
<latexit sha1_base64="7YlkrurOAYOv+Ch30L7WpchFVys=">AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8eK9gPaUDbbSbt0swm7G6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTAXXxvO+ndLa+sbmVnm7srO7t3/gHh61dJIphk2WiER1QqpRcIlNw43ATqqQxqHAdji+nfntJ1SaJ/LRTFIMYjqUPOKMGis99FLed6tezZuDrBK/IFUo0Oi7X71BwrIYpWGCat31vdQEOVWGM4HTSi/TmFI2pkPsWippjDrI56dOyZlVBiRKlC1pyFz9PZHTWOtJHNrOmJqRXvZm4n9eNzPRdZBzmWYGJVssijJBTEJmf5MBV8iMmFhCmeL2VsJGVFFmbDoVG4K//PIqaV3UfK/m319W6zdFHGU4gVM4Bx+uoA530IAmMBjCM7zCmyOcF+fd+Vi0lpxi5hj+wPn8AU+9jc0=</latexit><latexit<latexit sha1_base64="AWN2vRavkxBkeMhezqPgRlD7R7U=">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</latexit>

O(x)

O(x)

1/2

O(y) = hO(x)OJ (y)iA<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>• Reflected back on itself so: y = x<latexit sha1_base64="QUt5gvxrbVMJ8H8oVIVZ0DrvFb4=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ItQ9OKxgq2FNpTNdtMu3WzC7kQMob/BiwdFvPqDvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NaWV1b3yhvVra2d3b3qvsHbROnmvEWi2WsOwE1XArFWyhQ8k6iOY0CyR+C8c3Uf3jk2ohY3WOWcD+iQyVCwShaqZWRK/LUr9bcujsDWSZeQWpQoNmvfvUGMUsjrpBJakzXcxP0c6pRMMknlV5qeELZmA5511JFI278fHbshJxYZUDCWNtSSGbq74mcRsZkUWA7I4ojs+hNxf+8borhpZ8LlaTIFZsvClNJMCbTz8lAaM5QZpZQpoW9lbAR1ZShzadiQ/AWX14m7bO659a9u/Na47qIowxHcAyn4MEFNOAWmtACBgKe4RXeHOW8OO/Ox7y15BQzh/AHzucP8amOGg==</latexit>Mirrors for mirror ops• ComplexDboosts: EuclideanErotations - ⇡ rotation: mirror!
<latexit sha1_base64="7YlkrurOAYOv+Ch30L7WpchFVys=">AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8eK9gPaUDbbSbt0swm7G6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTAXXxvO+ndLa+sbmVnm7srO7t3/gHh61dJIphk2WiER1QqpRcIlNw43ATqqQxqHAdji+nfntJ1SaJ/LRTFIMYjqUPOKMGis99FLed6tezZuDrBK/IFUo0Oi7X71BwrIYpWGCat31vdQEOVWGM4HTSi/TmFI2pkPsWippjDrI56dOyZlVBiRKlC1pyFz9PZHTWOtJHNrOmJqRXvZm4n9eNzPRdZBzmWYGJVssijJBTEJmf5MBV8iMmFhCmeL2VsJGVFFmbDoVG4K//PIqaV3UfK/m319W6zdFHGU4gVM4Bx+uoA530IAmMBjCM7zCmyOcF+fd+Vi0lpxi5hj+wPn8AU+9jc0=</latexit><latexit<latexit sha1_base64="AWN2vRavkxBkeMhezqPgRlD7R7U=">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</latexit>

O(x)

O(x)

1/2

O(y) = hO(x)OJ (y)iA<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>• Reflected back on itself so: y = x<latexit sha1_base64="QUt5gvxrbVMJ8H8oVIVZ0DrvFb4=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ItQ9OKxgq2FNpTNdtMu3WzC7kQMob/BiwdFvPqDvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NaWV1b3yhvVra2d3b3qvsHbROnmvEWi2WsOwE1XArFWyhQ8k6iOY0CyR+C8c3Uf3jk2ohY3WOWcD+iQyVCwShaqZWRK/LUr9bcujsDWSZeQWpQoNmvfvUGMUsjrpBJakzXcxP0c6pRMMknlV5qeELZmA5511JFI278fHbshJxYZUDCWNtSSGbq74mcRsZkUWA7I4ojs+hNxf+8borhpZ8LlaTIFZsvClNJMCbTz8lAaM5QZpZQpoW9lbAR1ZShzadiQ/AWX14m7bO659a9u/Na47qIowxHcAyn4MEFNOAWmtACBgKe4RXeHOW8OO/Ox7y15BQzh/AHzucP8amOGg==</latexit>• Use to map out RT surface:⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>Mirrors for mirror ops• ComplexDboosts: EuclideanErotations - ⇡ rotation: mirror!
<latexit sha1_base64="7YlkrurOAYOv+Ch30L7WpchFVys=">AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8eK9gPaUDbbSbt0swm7G6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTAXXxvO+ndLa+sbmVnm7srO7t3/gHh61dJIphk2WiER1QqpRcIlNw43ATqqQxqHAdji+nfntJ1SaJ/LRTFIMYjqUPOKMGis99FLed6tezZuDrBK/IFUo0Oi7X71BwrIYpWGCat31vdQEOVWGM4HTSi/TmFI2pkPsWippjDrI56dOyZlVBiRKlC1pyFz9PZHTWOtJHNrOmJqRXvZm4n9eNzPRdZBzmWYGJVssijJBTEJmf5MBV8iMmFhCmeL2VsJGVFFmbDoVG4K//PIqaV3UfK/m319W6zdFHGU4gVM4Bx+uoA530IAmMBjCM7zCmyOcF+fd+Vi0lpxi5hj+wPn8AU+9jc0=</latexit><latexit<latexit sha1_base64="AWN2vRavkxBkeMhezqPgRlD7R7U=">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</latexit>

O(x)

O(x)

1/2

O(y) = hO(x)OJ (y)iA<latexit sha1_base64="BAeVOBC5ObWqGCFk52KlP7hcwRg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPVi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUuOmXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqt3kcRTiBUzgHD66gBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBkt2MxQ==</latexit><latexit

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>

Shadow⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>• Reflected back on itself so: y = x<latexit sha1_base64="QUt5gvxrbVMJ8H8oVIVZ0DrvFb4=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ItQ9OKxgq2FNpTNdtMu3WzC7kQMob/BiwdFvPqDvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NaWV1b3yhvVra2d3b3qvsHbROnmvEWi2WsOwE1XArFWyhQ8k6iOY0CyR+C8c3Uf3jk2ohY3WOWcD+iQyVCwShaqZWRK/LUr9bcujsDWSZeQWpQoNmvfvUGMUsjrpBJakzXcxP0c6pRMMknlV5qeELZmA5511JFI278fHbshJxYZUDCWNtSSGbq74mcRsZkUWA7I4ojs+hNxf+8borhpZ8LlaTIFZsvClNJMCbTz8lAaM5QZpZQpoW9lbAR1ZShzadiQ/AWX14m7bO659a9u/Na47qIowxHcAyn4MEFNOAWmtACBgKe4RXeHOW8OO/Ox7y15BQzh/AHzucP8amOGg==</latexit>• Use to map out RT surface:⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>

Nested flows• We can use these rules for more complicated correlators:
is
is
is
is
O(sAB ) ⌘ ⇢A ⇢B O⇢B ⇢A<latexit sha1_base64="BLZc476t64FKDi2hbx2TAWA72KM=">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</latexit>

D(B) ⇢ D(A)<latexit sha1_base64="1Hl8vgUrolP2rnxIaOTHpcs1gtA=">AAACDnicbVDLSsNAFJ34rPUVdelmsBTaTUlE0GWtLlxWsA9oQplMb9qhkwczE6GEfoEbf8WNC0Xcunbn3zhps6itBy4czrmXe+/xYs6ksqwfY219Y3Nru7BT3N3bPzg0j47bMkoEhRaNeCS6HpHAWQgtxRSHbiyABB6Hjje+yfzOIwjJovBBTWJwAzIMmc8oUVrqm2UnIGpECU9vp5VGFTsy8SQovChfV/tmyapZM+BVYuekhHI0++a3M4hoEkCoKCdS9mwrVm5KhGKUw7ToJBJiQsdkCD1NQxKAdNPZO1Nc1soA+5HQFSo8UxcnUhJIOQk83ZldKZe9TPzP6yXKv3JTFsaJgpDOF/kJxyrCWTZ4wARQxSeaECqYvhXTERGEKp1gUYdgL7+8StrnNduq2fcXpXojj6OATtEZqiAbXaI6ukNN1EIUPaEX9IbejWfj1fgwPueta0Y+c4L+wPj6BeNGm1g=</latexit>

x

??
s

y
<latexit sha1_base64="l29WxoUb9DEbvmhLG7jHtZ0OU24=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokIeix68diC/YA2lM120q7dbMLuRgihv8CLB0W8+pO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZwqhi0Wi1h1A6pRcIktw43AbqKQRoHATjC5m/mdJ1Sax/LBZAn6ER1JHnJGjZWa2aBSdWvuHGSVeAWpQoHGoPLVH8YsjVAaJqjWPc9NjJ9TZTgTOC33U40JZRM6wp6lkkao/Xx+6JScW2VIwljZkobM1d8TOY20zqLAdkbUjPWyNxP/83qpCW/8nMskNSjZYlGYCmJiMvuaDLlCZkRmCWWK21sJG1NFmbHZlG0I3vLLq6R9WfPcmte8qtZvizhKcApncAEeXEMd7qEBLWCA8Ayv8OY8Oi/Ou/OxaF1zipkT+APn8wfnvYz9</latexit><latexit sha1_base64="g52mBfVlfrpPbt9ADvc/q3MAFxY=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0FuLXjxWsR/QhrLZbtqlm03YnQgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsY38789hPXRsTqEScJ9yM6VCIUjKKVHmq1frniVt05yCrxclKBHI1++as3iFkacYVMUmO6npugn1GNgkk+LfVSwxPKxnTIu5YqGnHjZ/NLp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP66YYXvuZUEmKXLHFojCVBGMye5sMhOYM5cQSyrSwtxI2opoytOGUbAje8surpHVR9dyqd39Zqd/kcRThBE7hHDy4gjrcQQOawCCEZ3iFN2fsvDjvzseiteDkM8fwB87nDxSDjQw=</latexit><latexit sha1_base64="f2yzimwbR/Dgjzp6tZ360fHRqNI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N2IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHst7M0nQj+hQ8pAzaqzUeOqXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqN3kcRTiBUzgHD66gBndQhyYwQHiGV3hzHpwX5935WLQWnHzmGP7A+fwB5jmM/A==</latexit><latexit

mB<latexit sha1_base64="9qWD6uLLjK8yI1ztd9E3ADWHVc0=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMcQLx4jmgckS5idzCZDZmaXmV4hhHyCFw+KePWLvPk3TpI9aLSgoajqprsrSqWw6PtfXmFtfWNzq7hd2tnd2z8oHx61bJIZxpsskYnpRNRyKTRvokDJO6nhVEWSt6PxzdxvP3JjRaIfcJLyUNGhFrFgFJ10r/r1frniV/0FyF8S5KQCORr98mdvkLBMcY1MUmu7gZ9iOKUGBZN8VupllqeUjemQdx3VVHEbThenzsiZUwYkTowrjWSh/pyYUmXtREWuU1Ec2VVvLv7ndTOMr8Op0GmGXLPlojiTBBMy/5sMhOEM5cQRyoxwtxI2ooYydOmUXAjB6st/SeuiGvjV4O6yUqvncRThBE7hHAK4ghrcQgOawGAIT/ACr570nr03733ZWvDymWP4Be/jGxSyjaY=</latexit>

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mA

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<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit><latexit sha1_base64="g52mBfVlfrpPbt9ADvc/q3MAFxY=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0FuLXjxWsR/QhrLZbtqlm03YnQgl9B948aCIV/+RN/+N2zYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsY38789hPXRsTqEScJ9yM6VCIUjKKVHmq1frniVt05yCrxclKBHI1++as3iFkacYVMUmO6npugn1GNgkk+LfVSwxPKxnTIu5YqGnHjZ/NLp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP66YYXvuZUEmKXLHFojCVBGMye5sMhOYM5cQSyrSwtxI2opoytOGUbAje8surpHVR9dyqd39Zqd/kcRThBE7hHDy4gjrcQQOawCCEZ3iFN2fsvDjvzseiteDkM8fwB87nDxSDjQw=</latexit>• As long as we can thread the geodesic through -satisfying the boost conditions at each RT surface

Nested flows• We would like to combine these two ideas (mirrors anddouble flow) to compute:<latexit sha1_base64="Rz24AJLzhfXRzuSDXsDj0DVXcds=">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</latexit>

double flow<latexit sha1_base64="5GkLUTd2xFY3aqB0Uo4TC/Wxnp4=">AAAB+HicbVBNS8NAEN34WetHox69LBbBU0hE0GPRi8cK9gOaUDbbSbt088HuRKihv8SLB0W8+lO8+W/ctjlo64OBx3szzMwLMyk0uu63tba+sbm1Xdmp7u7tH9Tsw6O2TnPFocVTmapuyDRIkUALBUroZgpYHErohOPbmd95BKVFmjzgJIMgZsNERIIzNFLfrvkSIvSn1FdiOEKnb9ddx52DrhKvJHVSotm3v/xByvMYEuSSad3z3AyDgikUXMK06ucaMsbHbAg9QxMWgw6K+eFTemaUAY1SZSpBOld/TxQs1noSh6YzZjjSy95M/M/r5RhdB4VIshwh4YtFUS4ppnSWAh0IBRzlxBDGlTC3Uj5iinE0WVVNCN7yy6ukfeF4ruPdX9YbN2UcFXJCTsk58cgVaZA70iQtwklOnskrebOerBfr3fpYtK5Z5cwx+QPr8wdpLpLp</latexit>}DmirrorO(x)isBis+1/2A

O(x)ENested flows• We would like to combine these two ideas (mirrors anddouble flow) to compute:<latexit sha1_base64="Rz24AJLzhfXRzuSDXsDj0DVXcds=">AAACNnicbVDLSgNBEJz1bXxFPXoZDIIixl0R9CASowcvooJRIRvD7KQ3GTL7YKZXDMt+lRe/w5sXD4p49ROcxAhqLBioqeqmu8uLpdBo20/W0PDI6Nj4xGRuanpmdi4/v3Cpo0RxqPBIRuraYxqkCKGCAiVcxwpY4Em48tqHXf/qFpQWUXiBnRhqAWuGwhecoZHq+RNXgo971A0YtjiT6Wm2erdG3SOQyOrlm1To7Pt3cJNuCL3ubG5lA/VKNFu4X88X7KLdAx0kTp8USB9n9fyj24h4EkCIXDKtq44dYy1lCgWXkOXcREPMeJs1oWpoyALQtbR3dkZXjNKgfqTMC5H21J8dKQu07gSeqexuq/96XfE/r5qgv1tLRRgnCCH/GuQnkmJEuxnShlDAUXYMYVwJsyvlLaYYR5N0zoTg/D15kFxuFR276JxvF0rlfhwTZIksk1XikB1SIsfkjFQIJ/fkibyQV+vBerberPev0iGr37NIfsH6+AQhuKrv</latexit>

double flow<latexit sha1_base64="5GkLUTd2xFY3aqB0Uo4TC/Wxnp4=">AAAB+HicbVBNS8NAEN34WetHox69LBbBU0hE0GPRi8cK9gOaUDbbSbt088HuRKihv8SLB0W8+lO8+W/ctjlo64OBx3szzMwLMyk0uu63tba+sbm1Xdmp7u7tH9Tsw6O2TnPFocVTmapuyDRIkUALBUroZgpYHErohOPbmd95BKVFmjzgJIMgZsNERIIzNFLfrvkSIvSn1FdiOEKnb9ddx52DrhKvJHVSotm3v/xByvMYEuSSad3z3AyDgikUXMK06ucaMsbHbAg9QxMWgw6K+eFTemaUAY1SZSpBOld/TxQs1noSh6YzZjjSy95M/M/r5RhdB4VIshwh4YtFUS4ppnSWAh0IBRzlxBDGlTC3Uj5iinE0WVVNCN7yy6ukfeF4ruPdX9YbN2UcFXJCTsk58cgVaZA70iQtwklOnskrebOerBfr3fpYtK5Z5cwx+QPr8wdpLpLp</latexit>}DmirrorO(x)isBis+1/2A

O(x)E• Some intuition: double flow in the Rindler case gives

action of two boosts = translation. More generally acting
on the geodesic correlator we will be able to extract
properties of this translation deep in the bulk at:⇠(x) 2 mA<latexit sha1_base64="JlgDdeS3WP2HMmIv6vsYKnCT+ng=">AAAB9XicbVBNSwMxEJ2tX7V+VT16CRahXsquCHqsevFYwX5Au5Zsmm1Dk+ySZLVl6f/w4kERr/4Xb/4b03YP2vpg4PHeDDPzgpgzbVz328mtrK6tb+Q3C1vbO7t7xf2Dho4SRWidRDxSrQBrypmkdcMMp61YUSwCTpvB8GbqNx+p0iyS92YcU1/gvmQhI9hY6aEzYuXRKeowiUT3qlssuRV3BrRMvIyUIEOtW/zq9CKSCCoN4VjrtufGxk+xMoxwOil0Ek1jTIa4T9uWSiyo9tPZ1RN0YpUeCiNlSxo0U39PpFhoPRaB7RTYDPSiNxX/89qJCS/9lMk4MVSS+aIw4chEaBoB6jFFieFjSzBRzN6KyAArTIwNqmBD8BZfXiaNs4rnVry781L1OosjD0dwDGXw4AKqcAs1qAMBBc/wCm/Ok/PivDsf89ack80cwh84nz8KbZGM</latexit>

Nested boosts• Still hard to satisfy the boost condition while varying sO(x)B<latexit sha1_base64="omio59kvEVQ455XgB3RHdtTRWpI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GOpF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipWR+UK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasJbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXldq9TyOIpzBOVyCBzdQg3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4AlGGMxg==</latexit><latexit

mB mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit><latexit sha1_base64="9qWD6uLLjK8yI1ztd9E3ADWHVc0=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMcQLx4jmgckS5idzCZDZmaXmV4hhHyCFw+KePWLvPk3TpI9aLSgoajqprsrSqWw6PtfXmFtfWNzq7hd2tnd2z8oHx61bJIZxpsskYnpRNRyKTRvokDJO6nhVEWSt6PxzdxvP3JjRaIfcJLyUNGhFrFgFJ10r/r1frniV/0FyF8S5KQCORr98mdvkLBMcY1MUmu7gZ9iOKUGBZN8VupllqeUjemQdx3VVHEbThenzsiZUwYkTowrjWSh/pyYUmXtREWuU1Ec2VVvLv7ndTOMr8Op0GmGXLPlojiTBBMy/5sMhOEM5cQRyoxwtxI2ooYydOmUXAjB6st/SeuiGvjV4O6yUqvncRThBE7hHAK4ghrcQgOawGAIT/ACr570nr03733ZWvDymWP4Be/jGxSyjaY=</latexit>⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>

Nested boosts• Still hard to satisfy the boost condition while varying sO(x)B<latexit sha1_base64="omio59kvEVQ455XgB3RHdtTRWpI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GOpF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipWR+UK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasJbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6btVrXldq9TyOIpzBOVyCBzdQg3toQAsYIDzDK7w5j86L8+58LFsLTj5zCn/gfP4AlGGMxg==</latexit><latexit

mB mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>

Note that when:<latexit sha1_base64="9qWD6uLLjK8yI1ztd9E3ADWHVc0=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMcQLx4jmgckS5idzCZDZmaXmV4hhHyCFw+KePWLvPk3TpI9aLSgoajqprsrSqWw6PtfXmFtfWNzq7hd2tnd2z8oHx61bJIZxpsskYnpRNRyKTRvokDJO6nhVEWSt6PxzdxvP3JjRaIfcJLyUNGhFrFgFJ10r/r1frniV/0FyF8S5KQCORr98mdvkLBMcY1MUmu7gZ9iOKUGBZN8VupllqeUjemQdx3VVHEbThenzsiZUwYkTowrjWSh/pyYUmXtREWuU1Ec2VVvLv7ndTOMr8Op0GmGXLPlojiTBBMy/5sMhOEM5cQRyoxwtxI2ooYydOmUXAjB6st/SeuiGvjV4O6yUqvncRThBE7hHAK4ghrcQgOawGAIT/ACr570nr03733ZWvDymWP4Be/jGxSyjaY=</latexit>

A=B<latexit sha1_base64="vktg59PxmUnBvc59akW1nDUw6UE=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ItQ68VjRfsBbSib7aZdutmE3YlQQn+CFw+KePUXefPfuG1z0OqDgcd7M8zMCxIpDLrul1NYWV1b3yhulra2d3b3yvsHLROnmvEmi2WsOwE1XArFmyhQ8k6iOY0CydvB+Gbmtx+5NiJWDzhJuB/RoRKhYBStdH99Ve+XK27VnYP8JV5OKpCj0S9/9gYxSyOukElqTNdzE/QzqlEwyaelXmp4QtmYDnnXUkUjbvxsfuqUnFhlQMJY21JI5urPiYxGxkyiwHZGFEdm2ZuJ/3ndFMNLPxMqSZErtlgUppJgTGZ/k4HQnKGcWEKZFvZWwkZUU4Y2nZINwVt++S9pnVU9t+rdnVdq9TyOIhzBMZyCBxdQg1toQBMYDOEJXuDVkc6z8+a8L1oLTj5zCL/gfHwDnfGNWA==</latexit>D! O(x)<latexit sha1_base64="jNH1zNfhc2njZ97KEpV2nYUELPk=">AAACMHicbVBdSxtBFJ31o41R21QffRkMQnyJu1LQh1KiFto3IxgVsjHcndxNBmc/mLmrhiU/yZf+lPalgkV89Vc4+XjQxAMDh3PuYe49QaqkIde9d+bmFxY/fCwsFZdXVj99Ln1ZOzNJpgU2RKISfRGAQSVjbJAkhRepRogChefB1dHQP79GbWQSn1I/xVYE3ViGUgBZqV366WvZ7RFondxwX2FI37gfAfUEqPx4ULnd5v4PVATtg8vc29kdzLij/Pd2qexW3RH4LPEmpMwmqLdLf/xOIrIIYxIKjGl6bkqtHDRJoXBQ9DODKYgr6GLT0hgiNK18dPCAb1mlw8NE2xcTH6mvEzlExvSjwE4OtzXT3lB8z2tmFO63chmnGWEsxh+FmeKU8GF7vCM1ClJ9S0BoaXflogcaBNmOi7YEb/rkWXK2W/XcqnfytVw7nNRRYBtsk1WYx/ZYjf1iddZggt2xv+yB/Xd+O/+cR+dpPDrnTDLr7A2c5xc8cakL</latexit>

⇠(x)

1/2
A O(x)

E<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>• For small deformations A ! B , reflected geodesics come<latexit sha1_base64="GbpplQApm9EzC2rCgHzQlPYrvMk=">AAAB+HicbVBNSwMxEM3Wr1o/uurRS7AInsquCHqs9eKxgv2AdinZNNuGZpMlmVXq0l/ixYMiXv0p3vw3pu0etPXBwOO9GWbmhYngBjzv2ymsrW9sbhW3Szu7e/tl9+CwZVSqKWtSJZTuhMQwwSVrAgfBOolmJA4Fa4fjm5nffmDacCXvYZKwICZDySNOCVip75avcU/z4QiI1uoR1/tuxat6c+BV4uekgnI0+u5Xb6BoGjMJVBBjur6XQJARDZwKNi31UsMSQsdkyLqWShIzE2Tzw6f41CoDHCltSwKeq78nMhIbM4lD2xkTGJllbyb+53VTiK6CjMskBSbpYlGUCgwKz1LAA64ZBTGxhFDN7a2YjogmFGxWJRuCv/zyKmmdV32v6t9dVGr1PI4iOkYn6Az56BLV0C1qoCaiKEXP6BW9OU/Oi/PufCxaC04+c4T+wPn8AQStkqk=</latexit>

close enough to use the rules to linear order in the deformation

Nested boosts• Consider:D

iM + 1 ⌘ rD
<latexit sha1_base64="QI0z49pG3CE236r5UsVcPKj7A3Y=">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</latexit>

O

is
B

is+1/2AED
1/2
O A O O

O

E

1/2
B O

E

Nested boosts• Consider:D

iM + 1 ⌘ rD
<latexit sha1_base64="QI0z49pG3CE236r5UsVcPKj7A3Y=">AAACynicfVFLTxsxEPYu5RVegR65WI2QKlWE3QgJDghBitQeWkGlBpCyIfI6s4mF94E9GxRZe+MXcuPYf1JvErW8R7L0+ZuZb8b+wkwKjZ734LgzH2bn5hcWK0vLK6tr1fWNc53mikOLpzJVlyHTIEUCLRQo4TJTwOJQwkV4/bXMXwxBaZEmv3GUQSdm/UREgjO0VLf6R9AgZjjgTJqfBf1C/UoAN7kY0iBSjBsaSIjw4H/RaUGDE5DIus0rI/S/2/GV2Ra6FNhpFE/LlegP8LCwWvpG4XuSVuSt9vcXeXMoLbrVmlf3xkFfAn8KamQaZ93qfdBLeR5Dglwyrdu+l2HHMIWCSygqQa4hY/ya9aFtYcJi0B0ztqKgW5bp0ShV9iRIx+zjDsNirUdxaCvLbfXzXEm+lmvnGO13jEiyHCHhk0FRLimmtPSV9oQCjnJkAeNK2F0pHzBrIVr3K/YT/OdPfgnOG3Xfq/u/dmtHzel3LJBN8ol8Jj7ZI0fkOzkjLcKdb07sDJ1b94er3JFrJqWuM+35SJ6Ee/cXYLvc6w==</latexit>

O

is
B

is+1/2AED
1/2
O A O O• Which we calculated using the rules:

O

E

1/2
B O

(somewhere deepin the bulk)z (⇠)<latexit sha1_base64="gqz+KmL3PlOZNE3R507p255tD9w=">AAAB+HicbVBNS8NAEN34WetHox69LBahHiyJCHosevFYwX5AE8tmM2mXbjZhdyO2pb/EiwdFvPpTvPlv3LY5aOuDgcd7M8zMC1LOlHacb2tldW19Y7OwVdze2d0r2fsHTZVkkkKDJjyR7YAo4ExAQzPNoZ1KIHHAoRUMbqZ+6xGkYom418MU/Jj0BIsYJdpIXbvkhcA1waOHs4r3xE67dtmpOjPgZeLmpIxy1Lv2lxcmNItBaMqJUh3XSbU/JlIzymFS9DIFKaED0oOOoYLEoPzx7PAJPjFKiKNEmhIaz9TfE2MSKzWMA9MZE91Xi95U/M/rZDq68sdMpJkGQeeLooxjneBpCjhkEqjmQ0MIlczcimmfSEK1yapoQnAXX14mzfOq61Tdu4ty7TqPo4CO0DGqIBddohq6RXXUQBRl6Bm9ojdrZL1Y79bHvHXFymcO0R9Ynz+Nm5Ja</latexit>

EmB<latexit sha1_base64="9qWD6uLLjK8yI1ztd9E3ADWHVc0=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMcQLx4jmgckS5idzCZDZmaXmV4hhHyCFw+KePWLvPk3TpI9aLSgoajqprsrSqWw6PtfXmFtfWNzq7hd2tnd2z8oHx61bJIZxpsskYnpRNRyKTRvokDJO6nhVEWSt6PxzdxvP3JjRaIfcJLyUNGhFrFgFJ10r/r1frniV/0FyF8S5KQCORr98mdvkLBMcY1MUmu7gZ9iOKUGBZN8VupllqeUjemQdx3VVHEbThenzsiZUwYkTowrjWSh/pyYUmXtREWuU1Ec2VVvLv7ndTOMr8Op0GmGXLPlojiTBBMy/5sMhOEM5cQRyoxwtxI2ooYydOmUXAjB6st/SeuiGvjV4O6yUqvncRThBE7hHAK4ghrcQgOawGAIT/ACr570nr03733ZWvDymWP4Be/jGxSyjaY=</latexit>

M = ie2⇡(s s? ) z + (⇠) + ie 2⇡(s s? ) z (⇠)<latexit sha1_base64="G7AE/9n2FtNcW+0iAmFqStrH/1M=">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</latexit>

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>z + (⇠)<latexit sha1_base64="XCBt6vgRU5hLOkQwWHvnu2U3h5w=">AAAB+HicbVBNS8NAEN34WetHox69LBahIpREBD0WvXisYD+giWWzmbRLN5uwuxHb0l/ixYMiXv0p3vw3btsctPXBwOO9GWbmBSlnSjvOt7Wyura+sVnYKm7v7O6V7P2DpkoySaFBE57IdkAUcCagoZnm0E4lkDjg0AoGN1O/9QhSsUTc62EKfkx6gkWMEm2krl3yQuCa4NHDWcV7Yqddu+xUnRnwMnFzUkY56l37ywsTmsUgNOVEqY7rpNofE6kZ5TApepmClNAB6UHHUEFiUP54dvgEnxglxFEiTQmNZ+rviTGJlRrGgemMie6rRW8q/ud1Mh1d+WMm0kyDoPNFUcaxTvA0BRwyCVTzoSGESmZuxbRPJKHaZFU0IbiLLy+T5nnVdaru3UW5dp3HUUBH6BhVkIsuUQ3dojpqIIoy9Ixe0Zs1sl6sd+tj3rpi5TOH6A+szx+KiZJY</latexit>⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>

Nested boosts• Consider:D

iM + 1 ⌘ rD
<latexit sha1_base64="QI0z49pG3CE236r5UsVcPKj7A3Y=">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</latexit>O

is+1/2

is
B

A

ED
1/2
O A O O

O

E

1/2
B O

• Which we calculated using the rules:(somewhere deepin the bulk)z (⇠)<latexit sha1_base64="gqz+KmL3PlOZNE3R507p255tD9w=">AAAB+HicbVBNS8NAEN34WetHox69LBahHiyJCHosevFYwX5AE8tmM2mXbjZhdyO2pb/EiwdFvPpTvPlv3LY5aOuDgcd7M8zMC1LOlHacb2tldW19Y7OwVdze2d0r2fsHTZVkkkKDJjyR7YAo4ExAQzPNoZ1KIHHAoRUMbqZ+6xGkYom418MU/Jj0BIsYJdpIXbvkhcA1waOHs4r3xE67dtmpOjPgZeLmpIxy1Lv2lxcmNItBaMqJUh3XSbU/JlIzymFS9DIFKaED0oOOoYLEoPzx7PAJPjFKiKNEmhIaz9TfE2MSKzWMA9MZE91Xi95U/M/rZDq68sdMpJkGQeeLooxjneBpCjhkEqjmQ0MIlczcimmfSEK1yapoQnAXX14mzfOq61Tdu4ty7TqPo4CO0DGqIBddohq6RXXUQBRl6Bm9ojdrZL1Y79bHvHXFymcO0R9Ynz+Nm5Ja</latexit>mB

E<latexit sha1_base64="9qWD6uLLjK8yI1ztd9E3ADWHVc0=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMcQLx4jmgckS5idzCZDZmaXmV4hhHyCFw+KePWLvPk3TpI9aLSgoajqprsrSqWw6PtfXmFtfWNzq7hd2tnd2z8oHx61bJIZxpsskYnpRNRyKTRvokDJO6nhVEWSt6PxzdxvP3JjRaIfcJLyUNGhFrFgFJ10r/r1frniV/0FyF8S5KQCORr98mdvkLBMcY1MUmu7gZ9iOKUGBZN8VupllqeUjemQdx3VVHEbThenzsiZUwYkTowrjWSh/pyYUmXtREWuU1Ec2VVvLv7ndTOMr8Op0GmGXLPlojiTBBMy/5sMhOEM5cQRyoxwtxI2ooYydOmUXAjB6st/SeuiGvjV4O6yUqvncRThBE7hHAK4ghrcQgOawGAIT/ACr570nr03733ZWvDymWP4Be/jGxSyjaY=</latexit>

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>z + (⇠)<latexit sha1_base64="XCBt6vgRU5hLOkQwWHvnu2U3h5w=">AAAB+HicbVBNS8NAEN34WetHox69LBahIpREBD0WvXisYD+giWWzmbRLN5uwuxHb0l/ixYMiXv0p3vw3btsctPXBwOO9GWbmBSlnSjvOt7Wyura+sVnYKm7v7O6V7P2DpkoySaFBE57IdkAUcCagoZnm0E4lkDjg0AoGN1O/9QhSsUTc62EKfkx6gkWMEm2krl3yQuCa4NHDWcV7Yqddu+xUnRnwMnFzUkY56l37ywsTmsUgNOVEqY7rpNofE6kZ5TApepmClNAB6UHHUEFiUP54dvgEnxglxFEiTQmNZ+rviTGJlRrGgemMie6rRW8q/ud1Mh1d+WMm0kyDoPNFUcaxTvA0BRwyCVTzoSGESmZuxbRPJKHaZFU0IbiLLy+T5nnVdaru3UW5dp3HUUBH6BhVkIsuUQ3dojpqIIoy9Ixe0Zs1sl6sd+tj3rpi5TOH6A+szx+KiZJY</latexit>⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>

M = ie2⇡(s s? ) z + (⇠) + ie 2⇡(s s? ) z (⇠)<latexit sha1_base64="G7AE/9n2FtNcW+0iAmFqStrH/1M=">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</latexit>• For nested regions, can show that in the “thermal” strip:<latexit sha1_base64="65bKd3LcdIA/nzWFN0Np+GGPh/A=">AAACFnicbVDLSgMxFL1TX7W+Rl26CRbBTcuMFHRZdKO7CvYBnVIyaaYNTWbGJCOUYb7Cjb/ixoUibsWdf2PazkJbDwROzjmX5B4/5kxpx/m2Ciura+sbxc3S1vbO7p69f9BSUSIJbZKIR7LjY0U5C2lTM81pJ5YUC5/Ttj++mvrtByoVi8I7PYlpT+BhyAJGsDZS365UvEBikrpZWsuQx+k9Sj0p0I3IkJrffwX6dtmpOjOgZeLmpAw5Gn37yxtEJBE01IRjpbquE+teiqVmhNOs5CWKxpiM8ZB2DQ2xoKqXztbK0IlRBiiIpDmhRjP190SKhVIT4ZukwHqkFr2p+J/XTXRw0UtZGCeahmT+UJBwpCM07QgNmKRE84khmEhm/orICJsWtGmyZEpwF1deJq2zqutU3dtauX6Z11GEIziGU3DhHOpwDQ1oAoFHeIZXeLOerBfr3fqYRwtWPnMIf2B9/gDaQZ6M</latexit>

11 Ims 
4
4

:

ImM

0

<latexit sha1_base64="bdv9vThVzbcahCfPldh/t0hTc0g=">AAACBHicbZDLSsNAFIYnXmu9RV12M1gEVyURQZdFN7oQKtgLNKFMpift0JkkzkyEErpw46u4caGIWx/CnW/jpM1CW38Y+PjPOcw5f5BwprTjfFtLyyura+uljfLm1vbOrr2331JxKik0acxj2QmIAs4iaGqmOXQSCUQEHNrB6DKvtx9AKhZHd3qcgC/IIGIho0Qbq2dXMk8KfC0m2BNEDynh2Y3hAdxjp2dXnZozFV4Et4AqKtTo2V9eP6apgEhTTpTquk6i/YxIzSiHSdlLFSSEjsgAugYjIkD52fSICT4yTh+HsTQv0njq/p7IiFBqLALTmW+q5mu5+V+tm+rw3M9YlKQaIjr7KEw51jHOE8F9JoFqPjZAqGRmV0yHRBKqTW5lE4I7f/IitE5qrlNzb0+r9YsijhKqoEN0jFx0huroCjVQE1H0iJ7RK3qznqwX6936mLUuWcXMAfoj6/MHmQOXZg==</latexit>

±

z (⇠)
<latexit sha1_base64="G2VochCB1iXN15bIEzcXoOrSDtc=">AAACAXicbVDLSsNAFJ3UV62vqBvBzWAR6qYkIuiy6MZlBfuAJpbJ5KYdOnk4MxFrqBt/xY0LRdz6F+78G6dtFtp64MLhnHu59x4v4Uwqy/o2CguLS8srxdXS2vrG5pa5vdOUcSooNGjMY9H2iATOImgopji0EwEk9Di0vMHF2G/dgZAsjq7VMAE3JL2IBYwSpaWuuef4wBXBDzdOElace3aEnR7cYqtrlq2qNQGeJ3ZOyihHvWt+OX5M0xAiRTmRsmNbiXIzIhSjHEYlJ5WQEDogPehoGpEQpJtNPhjhQ634OIiFrkjhifp7IiOhlMPQ050hUX05643F/7xOqoIzN2NRkiqI6HRRkHKsYjyOA/tMAFV8qAmhgulbMe0TQajSoZV0CPbsy/OkeVy1rap9dVKunedxFNE+OkAVZKNTVEOXqI4aiKJH9Ixe0ZvxZLwY78bHtLVg5DO76A+Mzx/GpJXJ</latexit>0Nested boosts• Consider:D

iM + 1 ⌘ rD
<latexit sha1_base64="QI0z49pG3CE236r5UsVcPKj7A3Y=">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</latexit>O

is+1/2

is
B

A

ED
1/2
O A O O

O

E

1/2
B O

• Which we calculated using the rules:(somewhere deepin the bulk)z (⇠)<latexit sha1_base64="gqz+KmL3PlOZNE3R507p255tD9w=">AAAB+HicbVBNS8NAEN34WetHox69LBahHiyJCHosevFYwX5AE8tmM2mXbjZhdyO2pb/EiwdFvPpTvPlv3LY5aOuDgcd7M8zMC1LOlHacb2tldW19Y7OwVdze2d0r2fsHTZVkkkKDJjyR7YAo4ExAQzPNoZ1KIHHAoRUMbqZ+6xGkYom418MU/Jj0BIsYJdpIXbvkhcA1waOHs4r3xE67dtmpOjPgZeLmpIxy1Lv2lxcmNItBaMqJUh3XSbU/JlIzymFS9DIFKaED0oOOoYLEoPzx7PAJPjFKiKNEmhIaz9TfE2MSKzWMA9MZE91Xi95U/M/rZDq68sdMpJkGQeeLooxjneBpCjhkEqjmQ0MIlczcimmfSEK1yapoQnAXX14mzfOq61Tdu4ty7TqPo4CO0DGqIBddohq6RXXUQBRl6Bm9ojdrZL1Y79bHvHXFymcO0R9Ynz+Nm5Ja</latexit>mB

E<latexit sha1_base64="9qWD6uLLjK8yI1ztd9E3ADWHVc0=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMcQLx4jmgckS5idzCZDZmaXmV4hhHyCFw+KePWLvPk3TpI9aLSgoajqprsrSqWw6PtfXmFtfWNzq7hd2tnd2z8oHx61bJIZxpsskYnpRNRyKTRvokDJO6nhVEWSt6PxzdxvP3JjRaIfcJLyUNGhFrFgFJ10r/r1frniV/0FyF8S5KQCORr98mdvkLBMcY1MUmu7gZ9iOKUGBZN8VupllqeUjemQdx3VVHEbThenzsiZUwYkTowrjWSh/pyYUmXtREWuU1Ec2VVvLv7ndTOMr8Op0GmGXLPlojiTBBMy/5sMhOEM5cQRyoxwtxI2ooYydOmUXAjB6st/SeuiGvjV4O6yUqvncRThBE7hHAK4ghrcQgOawGAIT/ACr570nr03733ZWvDymWP4Be/jGxSyjaY=</latexit>

mA<latexit sha1_base64="6AB0rawL5G8AXBuJsGoczloD6L8=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJqYxnRmEByhL3NXLJkd+/Y3RNCyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgU31ve/vcLK6tr6RnGztLW9s7tX3j94NEmmGTZYIhLdiqhBwRU2LLcCW6lGKiOBzWh4M/WbT6gNT9SDHaUYStpXPOaMWifdy+5Vt1zxq/4MZJkEOalAjnq3/NXpJSyTqCwT1Jh24Kc2HFNtORM4KXUygyllQ9rHtqOKSjTheHbqhJw4pUfiRLtSlszU3xNjKo0Zych1SmoHZtGbiv957czGl+GYqzSzqNh8UZwJYhMy/Zv0uEZmxcgRyjR3txI2oJoy69IpuRCCxZeXyeNZNfCrwd15pXadx1GEIziGUwjgAmpwC3VoAIM+PMMrvHnCe/HevY95a8HLZw7hD7zPHxMujaU=</latexit>z + (⇠)<latexit sha1_base64="XCBt6vgRU5hLOkQwWHvnu2U3h5w=">AAAB+HicbVBNS8NAEN34WetHox69LBahIpREBD0WvXisYD+giWWzmbRLN5uwuxHb0l/ixYMiXv0p3vw3btsctPXBwOO9GWbmBSlnSjvOt7Wyura+sVnYKm7v7O6V7P2DpkoySaFBE57IdkAUcCagoZnm0E4lkDjg0AoGN1O/9QhSsUTc62EKfkx6gkWMEm2krl3yQuCa4NHDWcV7Yqddu+xUnRnwMnFzUkY56l37ywsTmsUgNOVEqY7rpNofE6kZ5TApepmClNAB6UHHUEFiUP54dvgEnxglxFEiTQmNZ+rviTGJlRrGgemMie6rRW8q/ud1Mh1d+WMm0kyDoPNFUcaxTvA0BRwyCVTzoSGESmZuxbRPJKHaZFU0IbiLLy+T5nnVdaru3UW5dp3HUUBH6BhVkIsuUQ3dojpqIIoy9Ixe0Zs1sl6sd+tj3rpi5TOH6A+szx+KiZJY</latexit>⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>

M = ie2⇡(s s? ) z + (⇠) + ie 2⇡(s s? ) z (⇠)<latexit sha1_base64="G7AE/9n2FtNcW+0iAmFqStrH/1M=">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</latexit>

saturates chaos bound (T=1)• For nested regions, can show that in the “thermal” strip:<latexit sha1_base64="65bKd3LcdIA/nzWFN0Np+GGPh/A=">AAACFnicbVDLSgMxFL1TX7W+Rl26CRbBTcuMFHRZdKO7CvYBnVIyaaYNTWbGJCOUYb7Cjb/ixoUibsWdf2PazkJbDwROzjmX5B4/5kxpx/m2Ciura+sbxc3S1vbO7p69f9BSUSIJbZKIR7LjY0U5C2lTM81pJ5YUC5/Ttj++mvrtByoVi8I7PYlpT+BhyAJGsDZS365UvEBikrpZWsuQx+k9Sj0p0I3IkJrffwX6dtmpOjOgZeLmpAw5Gn37yxtEJBE01IRjpbquE+teiqVmhNOs5CWKxpiM8ZB2DQ2xoKqXztbK0IlRBiiIpDmhRjP190SKhVIT4ZukwHqkFr2p+J/XTXRw0UtZGCeahmT+UJBwpCM07QgNmKRE84khmEhm/orICJsWtGmyZEpwF1deJq2zqutU3dtauX6Z11GEIziGU3DhHOpwDQ1oAoFHeIZXeLOerBfr3fqYRwtWPnMIf2B9/gDaQZ6M</latexit>

11 Ims 
4
4

:

ImM

0

<latexit sha1_base64="bdv9vThVzbcahCfPldh/t0hTc0g=">AAACBHicbZDLSsNAFIYnXmu9RV12M1gEVyURQZdFN7oQKtgLNKFMpift0JkkzkyEErpw46u4caGIWx/CnW/jpM1CW38Y+PjPOcw5f5BwprTjfFtLyyura+uljfLm1vbOrr2331JxKik0acxj2QmIAs4iaGqmOXQSCUQEHNrB6DKvtx9AKhZHd3qcgC/IIGIho0Qbq2dXMk8KfC0m2BNEDynh2Y3hAdxjp2dXnZozFV4Et4AqKtTo2V9eP6apgEhTTpTquk6i/YxIzSiHSdlLFSSEjsgAugYjIkD52fSICT4yTh+HsTQv0njq/p7IiFBqLALTmW+q5mu5+V+tm+rw3M9YlKQaIjr7KEw51jHOE8F9JoFqPjZAqGRmV0yHRBKqTW5lE4I7f/IitE5qrlNzb0+r9YsijhKqoEN0jFx0huroCjVQE1H0iJ7RK3qznqwX6936mLUuWcXMAfoj6/MHmQOXZg==</latexit>

±

z (⇠)
<latexit sha1_base64="G2VochCB1iXN15bIEzcXoOrSDtc=">AAACAXicbVDLSsNAFJ3UV62vqBvBzWAR6qYkIuiy6MZlBfuAJpbJ5KYdOnk4MxFrqBt/xY0LRdz6F+78G6dtFtp64MLhnHu59x4v4Uwqy/o2CguLS8srxdXS2vrG5pa5vdOUcSooNGjMY9H2iATOImgopji0EwEk9Di0vMHF2G/dgZAsjq7VMAE3JL2IBYwSpaWuuef4wBXBDzdOElace3aEnR7cYqtrlq2qNQGeJ3ZOyihHvWt+OX5M0xAiRTmRsmNbiXIzIhSjHEYlJ5WQEDogPehoGpEQpJtNPhjhQ634OIiFrkjhifp7IiOhlMPQ050hUX05643F/7xOqoIzN2NRkiqI6HRRkHKsYjyOA/tMAFV8qAmhgulbMe0TQajSoZV0CPbsy/OkeVy1rap9dVKunedxFNE+OkAVZKNTVEOXqI4aiKJH9Ixe0ZvxZLwY78bHtLVg5DO76A+Mzx/GpJXJ</latexit>

0EWN• This is the entanglement wedge nesting property
D(B) ⇢ D(A)<latexit sha1_base64="1Hl8vgUrolP2rnxIaOTHpcs1gtA=">AAACDnicbVDLSsNAFJ34rPUVdelmsBTaTUlE0GWtLlxWsA9oQplMb9qhkwczE6GEfoEbf8WNC0Xcunbn3zhps6itBy4czrmXe+/xYs6ksqwfY219Y3Nru7BT3N3bPzg0j47bMkoEhRaNeCS6HpHAWQgtxRSHbiyABB6Hjje+yfzOIwjJovBBTWJwAzIMmc8oUVrqm2UnIGpECU9vp5VGFTsy8SQovChfV/tmyapZM+BVYuekhHI0++a3M4hoEkCoKCdS9mwrVm5KhGKUw7ToJBJiQsdkCD1NQxKAdNPZO1Nc1soA+5HQFSo8UxcnUhJIOQk83ZldKZe9TPzP6yXKv3JTFsaJgpDOF/kJxyrCWTZ4wARQxSeaECqYvhXTERGEKp1gUYdgL7+8StrnNduq2fcXpXojj6OATtEZqiAbXaI6ukNN1EIUPaEX9IbejWfj1fgwPueta0Y+c4L+wPj6BeNGm1g=</latexit>

E b ⇢ Ea<latexit sha1_base64="ykxNtKFjZ907RHFer2+hceLREUg=">AAACDHicbVDLSsNAFJ3UV62vqks3g0VwVRIRdFkUwWUF+4AmlMn0ph06eTBzI5TQD3Djr7hxoYhbP8Cdf+OkzUJbDwwczjmXuff4iRQabfvbKq2srq1vlDcrW9s7u3vV/YO2jlPFocVjGauuzzRIEUELBUroJgpY6Evo+OPr3O88gNIiju5xkoAXsmEkAsEZGqlfrbkhwxFnMruZ9n3q6tTXgPS3ykzKrtsz0GXiFKRGCjT71S93EPM0hAi5ZFr3HDtBL2MKBZcwrbiphoTxMRtCz9CIhaC9bHbMlJ4YZUCDWJkXIZ2pvycyFmo9CX2TzJfUi14u/uf1UgwuvUxESYoQ8flHQSopxjRvhg6EAo5yYgjjSphdKR8xxTia/iqmBGfx5GXSPqs7dt25O681roo6yuSIHJNT4pAL0iC3pElahJNH8kxeyZv1ZL1Y79bHPFqyiplD8gfW5w8ZzZui</latexit>(spacelike movement of
the RT surface)• This property can be proven directly in the bulk,
assuming: classical GR and bulk NEC(Wall 2012)

EWN• This is the entanglement wedge nesting property
D(B) ⇢ D(A)<latexit sha1_base64="1Hl8vgUrolP2rnxIaOTHpcs1gtA=">AAACDnicbVDLSsNAFJ34rPUVdelmsBTaTUlE0GWtLlxWsA9oQplMb9qhkwczE6GEfoEbf8WNC0Xcunbn3zhps6itBy4czrmXe+/xYs6ksqwfY219Y3Nru7BT3N3bPzg0j47bMkoEhRaNeCS6HpHAWQgtxRSHbiyABB6Hjje+yfzOIwjJovBBTWJwAzIMmc8oUVrqm2UnIGpECU9vp5VGFTsy8SQovChfV/tmyapZM+BVYuekhHI0++a3M4hoEkCoKCdS9mwrVm5KhGKUw7ToJBJiQsdkCD1NQxKAdNPZO1Nc1soA+5HQFSo8UxcnUhJIOQk83ZldKZe9TPzP6yXKv3JTFsaJgpDOF/kJxyrCWTZ4wARQxSeaECqYvhXTERGEKp1gUYdgL7+8StrnNduq2fcXpXojj6OATtEZqiAbXaI6ukNN1EIUPaEX9IbejWfj1fgwPueta0Y+c4L+wPj6BeNGm1g=</latexit>

E b ⇢ Ea<latexit sha1_base64="ykxNtKFjZ907RHFer2+hceLREUg=">AAACDHicbVDLSsNAFJ3UV62vqks3g0VwVRIRdFkUwWUF+4AmlMn0ph06eTBzI5TQD3Djr7hxoYhbP8Cdf+OkzUJbDwwczjmXuff4iRQabfvbKq2srq1vlDcrW9s7u3vV/YO2jlPFocVjGauuzzRIEUELBUroJgpY6Evo+OPr3O88gNIiju5xkoAXsmEkAsEZGqlfrbkhwxFnMruZ9n3q6tTXgPS3ykzKrtsz0GXiFKRGCjT71S93EPM0hAi5ZFr3HDtBL2MKBZcwrbiphoTxMRtCz9CIhaC9bHbMlJ4YZUCDWJkXIZ2pvycyFmo9CX2TzJfUi14u/uf1UgwuvUxESYoQ8flHQSopxjRvhg6EAo5yYgjjSphdKR8xxTia/iqmBGfx5GXSPqs7dt25O681roo6yuSIHJNT4pAL0iC3pElahJNH8kxeyZv1ZL1Y79bHPFqyiplD8gfW5w8ZzZui</latexit>(spacelike movement of
the RT surface)• This property can be proven directly in the bulk,
assuming: classical GR and bulk NEC(Wall 2012)• Here we have a purely boundary argument, based on
analyticity and unitarity

EWN• This is the entanglement wedge nesting property
D(B) ⇢ D(A)<latexit sha1_base64="1Hl8vgUrolP2rnxIaOTHpcs1gtA=">AAACDnicbVDLSsNAFJ34rPUVdelmsBTaTUlE0GWtLlxWsA9oQplMb9qhkwczE6GEfoEbf8WNC0Xcunbn3zhps6itBy4czrmXe+/xYs6ksqwfY219Y3Nru7BT3N3bPzg0j47bMkoEhRaNeCS6HpHAWQgtxRSHbiyABB6Hjje+yfzOIwjJovBBTWJwAzIMmc8oUVrqm2UnIGpECU9vp5VGFTsy8SQovChfV/tmyapZM+BVYuekhHI0++a3M4hoEkCoKCdS9mwrVm5KhGKUw7ToJBJiQsdkCD1NQxKAdNPZO1Nc1soA+5HQFSo8UxcnUhJIOQk83ZldKZe9TPzP6yXKv3JTFsaJgpDOF/kJxyrCWTZ4wARQxSeaECqYvhXTERGEKp1gUYdgL7+8StrnNduq2fcXpXojj6OATtEZqiAbXaI6ukNN1EIUPaEX9IbejWfj1fgwPueta0Y+c4L+wPj6BeNGm1g=</latexit>

E b ⇢ Ea<latexit sha1_base64="ykxNtKFjZ907RHFer2+hceLREUg=">AAACDHicbVDLSsNAFJ3UV62vqks3g0VwVRIRdFkUwWUF+4AmlMn0ph06eTBzI5TQD3Djr7hxoYhbP8Cdf+OkzUJbDwwczjmXuff4iRQabfvbKq2srq1vlDcrW9s7u3vV/YO2jlPFocVjGauuzzRIEUELBUroJgpY6Evo+OPr3O88gNIiju5xkoAXsmEkAsEZGqlfrbkhwxFnMruZ9n3q6tTXgPS3ykzKrtsz0GXiFKRGCjT71S93EPM0hAi5ZFr3HDtBL2MKBZcwrbiphoTxMRtCz9CIhaC9bHbMlJ4YZUCDWJkXIZ2pvycyFmo9CX2TzJfUi14u/uf1UgwuvUxESYoQ8flHQSopxjRvhg6EAo5yYgjjSphdKR8xxTia/iqmBGfx5GXSPqs7dt25O681roo6yuSIHJNT4pAL0iC3pElahJNH8kxeyZv1ZL1Y79bHPFqyiplD8gfW5w8ZzZui</latexit>(spacelike movement of
the RT surface)• This property can be proven directly in the bulk,
assuming: classical GR and bulk NEC(Wall 2012)• Here we have a purely boundary argument, based on
analyticity and unitarity• Imposing this condition near the boundary of AdS gives
the QNEC of the boundary CFT(Leichenhaur, Koeller)

QNEC near boundary(Bousso, Fisher, Koeller, Leichenauer, Wall)

O(x)

@B
<latexit sha1_base64="JiK8pyRKpEHL0MezFzEqPHbo3IY=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4KokIeiz14rGC/cA2lM120y7dbMLuRCih/8KLB0W8+m+8+W/ctDlo64OFx3szszMvSKQw6Lrfztr6xubWdmmnvLu3f3BYOTpumzjVjLdYLGPdDajhUijeQoGSdxPNaRRI3gkmt7nfeeLaiFg94DThfkRHSoSCUbTSYz+hGgWVpDGoVN2aOwdZJV5BqlCgOah89YcxSyOukElqTM9zE/SzfB6TfFbup4YnlE3oiPcsVTTixs/mG8/IuVWGJIy1fQrJXP3dkdHImGkU2MqI4tgse7n4n9dLMbzxM6GSFLlii4/CVBKMSX4+GQrNGcqpJZRpYXclbEw1ZWhDKtsQvOWTV0n7sua5Ne/+qlpvFHGU4BTO4AI8uIY63EETWsBAwTO8wptjnBfn3flYlK45Rc8J/IHz+QMeTJCJ</latexit>hT++ i<latexit sha1_base64="fiuyc/0UnFB9y70km0YSW7Jhl4E=">AAACDXicbVDLSgNBEJz1bXxFPXoZjKIQCLsi6EFEFMGjYl6QDcvspDcZnH040yuEZX/Ai7/ixYMiXr1782+cxBw0WtBQVHXT3eUnUmi07U9rYnJqemZ2br6wsLi0vFJcXavrOFUcajyWsWr6TIMUEdRQoIRmooCFvoSGf3M28Bt3oLSIoyr2E2iHrBuJQHCGRvKKW66EAI9o1cvK5Zy6SnR7eEzdLtxSeu1l5+f5zo5XLNkVewj6lzgjUiIjXHrFD7cT8zSECLlkWrccO8F2xhQKLiEvuKmGhPEb1oWWoRELQbez4Tc53TZKhwaxMhUhHao/JzIWat0PfdMZMuzpcW8g/ue1UgwO25mIkhQh4t+LglRSjOkgGtoRCjjKviGMK2FupbzHFONoAiyYEJzxl/+S+l7FsSvO1X7p5HQUxxzZIJtklzjkgJyQC3JJaoSTe/JInsmL9WA9Wa/W23frhDWaWSe/YL1/AVHPmco=</latexit>⇠(x)<latexit sha1_base64="JlaVYClB/cy3Ka4U+/qreK9mBhI=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EixCvZRdEfRY9OKxgv2AdinZNNvGZpMlyUrL0v/gxYMiXv0/3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YP3MOjppapIrRBJJeqHWJNORO0YZjhtJ0oiuOQ01Y4up35rSeqNJPiwUwSGsR4IFjECDZWanbHrDI+77llr+rNgVaJn5My5Kj33K9uX5I0psIQjrXu+F5iggwrwwin01I31TTBZIQHtGOpwDHVQTa/dorOrNJHkVS2hEFz9fdEhmOtJ3FoO2NshnrZm4n/eZ3URNdBxkSSGirIYlGUcmQkmr2O+kxRYvjEEkwUs7ciMsQKE2MDKtkQ/OWXV0nzoup7Vf/+sly7yeMowgmcQgV8uIIa3EEdGkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w//1Y68</latexit>

Metricfall oﬀ nearbdry of AdS

00
SEE
Shape ofRT surface
near bdry• This connects to a general proof of the QNEC, which usedsuch a modular flow correlator, but calculated it using other
CFT methods(Balakrishnan, TF, Khandker, Wang)• Such methods fail unless O(x) is close to @B<latexit sha1_base64="JiK8pyRKpEHL0MezFzEqPHbo3IY=">AAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4KokIeiz14rGC/cA2lM120y7dbMLuRCih/8KLB0W8+m+8+W/ctDlo64OFx3szszMvSKQw6Lrfztr6xubWdmmnvLu3f3BYOTpumzjVjLdYLGPdDajhUijeQoGSdxPNaRRI3gkmt7nfeeLaiFg94DThfkRHSoSCUbTSYz+hGgWVpDGoVN2aOwdZJV5BqlCgOah89YcxSyOukElqTM9zE/SzfB6TfFbup4YnlE3oiPcsVTTixs/mG8/IuVWGJIy1fQrJXP3dkdHImGkU2MqI4tgse7n4n9dLMbzxM6GSFLlii4/CVBKMSX4+GQrNGcqpJZRpYXclbEw1ZWhDKtsQvOWTV0n7sua5Ne/+qlpvFHGU4BTO4AI8uIY63EETWsBAwTO8wptjnBfn3flYlK45Rc8J/IHz+QMeTJCJ</latexit><latexit sha1_base64="SXJzCIDk724qYqZlpkFBPzq6hPs=">AAAB9XicbVDLSgMxFL3xWeur6tJNsAh1U2ZE0GXRjTsr2Ae0Y8mkmTY0kxmSjFqG/ocbF4q49V/c+Tdm2llo64HA4Zx7uSfHjwXXxnG+0dLyyuraemGjuLm1vbNb2ttv6ihRlDVoJCLV9olmgkvWMNwI1o4VI6EvWMsfXWV+64EpzSN5Z8Yx80IykDzglBgr3XdDYoaUiPRmUnk66ZXKTtWZAi8SNydlyFHvlb66/YgmIZOGCqJ1x3Vi46VEGU4FmxS7iWYxoSMyYB1LJQmZ9tJp6gk+tkofB5GyTxo8VX9vpCTUehz6djJLqee9TPzP6yQmuPBSLuPEMElnh4JEYBPhrALc54pRI8aWEKq4zYrpkChCjS2qaEtw57+8SJqnVdepurdn5dplXkcBDuEIKuDCOdTgGurQAAoKnuEV3tAjekHv6GM2uoTynQP4A/T5AzODkkw=</latexit>• To get away from this, we needed to use holographic CFTs

Other things …
• We have only started to use these new tools …
• Can get more information about the bulk:
hij

can we get:<latexit sha1_base64="RwNvmLyxdU5XNhwYBbCRhDWyiZo=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bj04jGCeUCyhNnJJJlkdmaZ6RXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEiks+v63t7a+sbm1Xdgp7u7tHxyWjo4bVqeG8TrTUptWRC2XQvE6CpS8lRhO40jyZjS+m/nNJ26s0OoRJwkPYzpQoi8YRSc1ht1MjKbdUtmv+HOQVRLkpAw5at3SV6enWRpzhUxSa9uBn2CYUYOCST4tdlLLE8rGdMDbjioacxtm82un5NwpPdLXxpVCMld/T2Q0tnYSR64zpji0y95M/M9rp9i/CTOhkhS5YotF/VQS1GT2OukJwxnKiSOUGeFuJWxIDWXoAiq6EILll1dJ47IS+JXg4apcvc3jKMApnMEFBHANVbiHGtSBwQie4RXePO29eO/ex6J1zctnTuAPvM8f1JaPSA==</latexit>

↵
bulk
Kij
, T++
, . . .??
<latexit sha1_base64="BxDT2nIXZ75Oh0l8TiRDNdg5JK0=">AAACFHicbVDLSsNAFJ3UV62vqEs3g0UQWkoigu5adCO4qdAXNGmYTCbt2MmDmYlQQj7Cjb/ixoUibl2482+ctllo64ELh3Pu5d573JhRIQ3jWyusrK6tbxQ3S1vbO7t7+v5BR0QJx6SNIxbxnosEYTQkbUklI72YExS4jHTd8fXU7z4QLmgUtuQkJnaAhiH1KUZSSY5euXVSep8NLMTiEarClpNWKtkgtXgA3YSNsyq0mBdJAet1Ry8bNWMGuEzMnJRBjqajf1lehJOAhBIzJETfNGJpp4hLihnJSlYiSIzwGA1JX9EQBUTY6eypDJ4oxYN+xFWFEs7U3xMpCoSYBK7qDJAciUVvKv7n9RPpX9opDeNEkhDPF/kJgzKC04SgRznBkk0UQZhTdSvEI8QRlirHkgrBXHx5mXTOaqZRM+/Oy42rPI4iOALH4BSY4AI0wA1ogjbA4BE8g1fwpj1pL9q79jFvLWj5zCH4A+3zBz0XnaY=</latexit>• What about the bulk NEC? Quantum Focusing Condition?
Einstein’s equations?

• Stringy corrections? Non saturation of the chaos bound?

ありがとう