Aspects of the AdS/CFT dictionary, II
David Berenstein, UCSB,Durban, 1-15-19
Research supported by
AdS/CFT correspondence
Quantum gravity can be equivalent to quantum field theory in fewer dimensions.
Simplest (original) example
N = 4SYM on S3 ⇥ R$Type IIB superstring on (global) AdS5 ⇥ S5
<latexit sha1_base64="dwjOTb3RbH0JYMRdX3tlywlkgXQ=">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</latexit><latexit sha1_base64="dwjOTb3RbH0JYMRdX3tlywlkgXQ=">AAACbnicdVBNbxMxEPUuXyV8pSBxoEKMiKjSS7SbbptyQCrlQg+gQpq2KE4j2/FurHrtle0FotUe+YPc+A1c+Al4myABgpEsPb2Zec/zaCGFdVH0LQivXL12/cbazdat23fu3muv3z+xujSMj5iW2pxRYrkUio+ccJKfFYaTnEp+Si9eNf3Tj9xYodWxWxR8kpNMiVQw4jw1bX/ZrDAjEt7W8AISGH54A3hO9ecKtIIahufb2ImcW6hwTtycUnhfYwybWPLUGZHNHTFGfwKMW55cbh57Hzg8PABbFt7aj6msketmUlMit7zuy9lwurNSHp7vTNudqPd8b7ef7ELUi6JB3I8b0B8k2wnEnmmqg1Z1NG1/xTPNypwrxySxdhxHhZtUxDjBJK9buLS8IOyCZHzsoSLeaFJdxlXDM8/MINXGP+Xgkv19oyK5tYuc+snmZvt3ryH/1RuXLt2bVEIVpeOKLY3SUoLT0GQPM2E4c3LhAWFG+L8CmxNDmPMxtXwIvy6F/4OTfi/2+F3S2T9YxbGGNtBT1EUxGqB99BodoRFi6HuwHjwKNoIf4cPwcfhkORoGq50H6I8Kuz8Bm4C4Zg==</latexit><latexit sha1_base64="dwjOTb3RbH0JYMRdX3tlywlkgXQ=">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</latexit><latexit sha1_base64="dwjOTb3RbH0JYMRdX3tlywlkgXQ=">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</latexit>
Original Maldacena paper hep-th/9711200
We want to explore CFT at large values of R -large ‘t Hooft
coupling- but with g fixed and small.
AdS/CFT is a strong weak-coupling duality for the
‘t Hooft coupling constant.
Isometries Global symmetries
R4 g2Y MN
Flux = N Gauge group U(N)
State State
AdS CFT
Basic dictionary for simplest case
Standard Table
Field contentZ, Z,�i=1,...4, Aµ
J=1,...4↵ , ↵
J=1,...4<latexit sha1_base64="D02ycfgCBAT+iB3lYvs/3MPN0yM=">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</latexit><latexit sha1_base64="D02ycfgCBAT+iB3lYvs/3MPN0yM=">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</latexit><latexit sha1_base64="D02ycfgCBAT+iB3lYvs/3MPN0yM=">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</latexit><latexit sha1_base64="D02ycfgCBAT+iB3lYvs/3MPN0yM=">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</latexit>
Z is highest weight of SO(6)~SU(4) in 6 rep.Bosons have dim=1, fermions = 3/2
All in adjoint of SU(N) (need to track extra gauge indices)
Single particle state = traceBPS particle maximizes SO(6) R-charge given dimension
This is highest weight state of SO(6) in a symmetric traceless tensor rep.Has a lot of null descendants, so it is massless in 10DCan’t be lifted: does not allow anomalous dimension.
Tr(Zn)|n�2,...<latexit sha1_base64="bSv7JiK7qZzAvesk8k0YvWdQM9E=">AAACA3icdVDLSgMxFM3UV62vqjvdBItQQcrMWNq6K7pxWaEvbMchk6ZtaCYzJhmhjAU3/oobF4q49Sfc+TdmbAUVPRA4nHMPN/d4IaNSmea7kZqbX1hcSi9nVlbX1jeym1tNGUQCkwYOWCDaHpKEUU4aiipG2qEgyPcYaXmj08RvXRMhacDrahwSx0cDTvsUI6UlN7tTF3l4cckPbtyYdwfkCtqHsNsLlJy42ZxZOK6U7GIJmgXTLFu2lRC7XDwqQksrCXJghpqbfdNBHPmEK8yQlB3LDJUTI6EoZmSS6UaShAiP0IB0NOXIJ9KJP2+YwH2t9GA/EPpxBT/V74kY+VKOfU9P+kgN5W8vEf/yOpHqV5yY8jBShOPpon7EoApgUgjsUUGwYmNNEBZU/xXiIRIIK11bRpfwdSn8nzTtgqX5eTFXPZnVkQa7YA/kgQXKoArOQA00AAa34B48gifjzngwno2X6WjKmGW2wQ8Yrx+Wmpbc</latexit><latexit sha1_base64="bSv7JiK7qZzAvesk8k0YvWdQM9E=">AAACA3icdVDLSgMxFM3UV62vqjvdBItQQcrMWNq6K7pxWaEvbMchk6ZtaCYzJhmhjAU3/oobF4q49Sfc+TdmbAUVPRA4nHMPN/d4IaNSmea7kZqbX1hcSi9nVlbX1jeym1tNGUQCkwYOWCDaHpKEUU4aiipG2qEgyPcYaXmj08RvXRMhacDrahwSx0cDTvsUI6UlN7tTF3l4cckPbtyYdwfkCtqHsNsLlJy42ZxZOK6U7GIJmgXTLFu2lRC7XDwqQksrCXJghpqbfdNBHPmEK8yQlB3LDJUTI6EoZmSS6UaShAiP0IB0NOXIJ9KJP2+YwH2t9GA/EPpxBT/V74kY+VKOfU9P+kgN5W8vEf/yOpHqV5yY8jBShOPpon7EoApgUgjsUUGwYmNNEBZU/xXiIRIIK11bRpfwdSn8nzTtgqX5eTFXPZnVkQa7YA/kgQXKoArOQA00AAa34B48gifjzngwno2X6WjKmGW2wQ8Yrx+Wmpbc</latexit><latexit sha1_base64="bSv7JiK7qZzAvesk8k0YvWdQM9E=">AAACA3icdVDLSgMxFM3UV62vqjvdBItQQcrMWNq6K7pxWaEvbMchk6ZtaCYzJhmhjAU3/oobF4q49Sfc+TdmbAUVPRA4nHMPN/d4IaNSmea7kZqbX1hcSi9nVlbX1jeym1tNGUQCkwYOWCDaHpKEUU4aiipG2qEgyPcYaXmj08RvXRMhacDrahwSx0cDTvsUI6UlN7tTF3l4cckPbtyYdwfkCtqHsNsLlJy42ZxZOK6U7GIJmgXTLFu2lRC7XDwqQksrCXJghpqbfdNBHPmEK8yQlB3LDJUTI6EoZmSS6UaShAiP0IB0NOXIJ9KJP2+YwH2t9GA/EPpxBT/V74kY+VKOfU9P+kgN5W8vEf/yOpHqV5yY8jBShOPpon7EoApgUgjsUUGwYmNNEBZU/xXiIRIIK11bRpfwdSn8nzTtgqX5eTFXPZnVkQa7YA/kgQXKoArOQA00AAa34B48gifjzngwno2X6WjKmGW2wQ8Yrx+Wmpbc</latexit><latexit sha1_base64="bSv7JiK7qZzAvesk8k0YvWdQM9E=">AAACA3icdVDLSgMxFM3UV62vqjvdBItQQcrMWNq6K7pxWaEvbMchk6ZtaCYzJhmhjAU3/oobF4q49Sfc+TdmbAUVPRA4nHMPN/d4IaNSmea7kZqbX1hcSi9nVlbX1jeym1tNGUQCkwYOWCDaHpKEUU4aiipG2qEgyPcYaXmj08RvXRMhacDrahwSx0cDTvsUI6UlN7tTF3l4cckPbtyYdwfkCtqHsNsLlJy42ZxZOK6U7GIJmgXTLFu2lRC7XDwqQksrCXJghpqbfdNBHPmEK8yQlB3LDJUTI6EoZmSS6UaShAiP0IB0NOXIJ9KJP2+YwH2t9GA/EPpxBT/V74kY+VKOfU9P+kgN5W8vEf/yOpHqV5yY8jBShOPpon7EoApgUgjsUUGwYmNNEBZU/xXiIRIIK11bRpfwdSn8nzTtgqX5eTFXPZnVkQa7YA/kgQXKoArOQA00AAa34B48gifjzngwno2X6WjKmGW2wQ8Yrx+Wmpbc</latexit>
THE FACT THAT IT STARTS AT n=2 is that the spin of particle in 10D is 2
Convenient to expand Near BPS state
X
k
f(k)Tr(Zk�iZJ�k�i0)
<latexit sha1_base64="1IEKM8ggKUesZ+JeCujQkJhJBHQ=">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</latexit><latexit sha1_base64="1IEKM8ggKUesZ+JeCujQkJhJBHQ=">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</latexit><latexit sha1_base64="1IEKM8ggKUesZ+JeCujQkJhJBHQ=">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</latexit><latexit sha1_base64="1IEKM8ggKUesZ+JeCujQkJhJBHQ=">AAACF3icdZDLSgMxFIYzXmu9VV26CRaxXVhm6tDWXdGNuKrQG3bqkEkzbZjMhSQjlKFv4cZXceNCEbe6823MtBVU9EDgy/+fQ3J+J2JUSF3/0BYWl5ZXVjNr2fWNza3t3M5uW4Qxx6SFQxbyroMEYTQgLUklI92IE+Q7jHQc7zz1O7eECxoGTTmOSN9Hw4C6FCOpJDtXskTs24k3gW7BKzZ5AV7feNCKRtSmCpPLY2VNrwk9mhTtXF4vndYqZbMC9ZKuV42ykUK5ap6Y0FBKWnkwr4ade7cGIY59EkjMkBA9Q49kP0FcUszIJGvFgkQIe2hIegoD5BPRT6Z7TeChUgbQDbk6gYRT9ftEgnwhxr6jOn0kR+K3l4p/eb1YurV+QoMoliTAs4fcmEEZwjQkOKCcYMnGChDmVP0V4hHiCEsVZVaF8LUp/B/a5ZKh+MrM18/mcWTAPjgABWCAKqiDC9AALYDBHXgAT+BZu9cetRftdda6oM1n9sCP0t4+ARnpnqM=</latexit>
Interpreted as defects moving in a sea of Z
Anomalous dimensions can be computed perturbatively and thegap scales as powers of 1/J
The interaction term
∼g2Y MTr([Z,φj][Z,φj])
will make the defects hop to the left or right in theoperator.
The idea now is to interpret the position of the defectsin between the Z as a lattice. The number of sites isof order J ∼
√N , so in the large N limit we obtain a
continuum theory.
We need to rewrite the states in a momentum basis,so we apply position dependent phases for inserting thesame defect at different positions.
20
The interaction term + planarity
Defects hop in between Z:nearest neighbor hopping.
Can be mapped to spin chains.
Berenstein, Maldacena, Nastase hep-th/0202021 (BMN)
Minahan, Zarembo hep-th/0212208 (Bethe Ansatz)
Beisert, Staudacher, hep-th/0307042 (USp(2,2|4) integrability)(Used all the Lipatov SL(2) anomalous dimensions from QCD)
Spin chain program
Gave rise to integrability program for strings in AdS x S
Interpolation between weak and strong coupling *Solved (Y system) *.
* Integrability not proved: it is a working assumption. Checked to 5 (or 6) loop order
of some anomalous dimensions
Covers energies up to
o(pN)
<latexit sha1_base64="rLtLN6y4Y0N14fVQ1LCCmhcXVdw=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BItQL2V3Xdp6K3rxJBVsK7ZLyabZNjSbrElWKEv/hRcPinj133jz35htK6jog4HHezPMzAtiRpW27Q8rt7S8srqWXy9sbG5t7xR399pKJBKTFhZMyJsAKcIoJy1NNSM3sSQoChjpBOPzzO/cE6mo4Nd6EhM/QkNOQ4qRNtKtKPfUndTw8rhfLNmV03rV9arQrth2zXGdjLg178SDjlEylMACzX7xvTcQOIkI15ghpbqOHWs/RVJTzMi00EsUiREeoyHpGspRRJSfzi6ewiOjDGAopCmu4Uz9PpGiSKlJFJjOCOmR+u1l4l9eN9Fh3U8pjxNNOJ4vChMGtYDZ+3BAJcGaTQxBWFJzK8QjJBHWJqSCCeHrU/g/absVx/Arr9Q4W8SRBwfgEJSBA2qgAS5AE7QABhw8gCfwbCnr0XqxXuetOWsxsw9+wHr7BPikkHQ=</latexit><latexit sha1_base64="rLtLN6y4Y0N14fVQ1LCCmhcXVdw=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BItQL2V3Xdp6K3rxJBVsK7ZLyabZNjSbrElWKEv/hRcPinj133jz35htK6jog4HHezPMzAtiRpW27Q8rt7S8srqWXy9sbG5t7xR399pKJBKTFhZMyJsAKcIoJy1NNSM3sSQoChjpBOPzzO/cE6mo4Nd6EhM/QkNOQ4qRNtKtKPfUndTw8rhfLNmV03rV9arQrth2zXGdjLg178SDjlEylMACzX7xvTcQOIkI15ghpbqOHWs/RVJTzMi00EsUiREeoyHpGspRRJSfzi6ewiOjDGAopCmu4Uz9PpGiSKlJFJjOCOmR+u1l4l9eN9Fh3U8pjxNNOJ4vChMGtYDZ+3BAJcGaTQxBWFJzK8QjJBHWJqSCCeHrU/g/absVx/Arr9Q4W8SRBwfgEJSBA2qgAS5AE7QABhw8gCfwbCnr0XqxXuetOWsxsw9+wHr7BPikkHQ=</latexit><latexit sha1_base64="rLtLN6y4Y0N14fVQ1LCCmhcXVdw=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BItQL2V3Xdp6K3rxJBVsK7ZLyabZNjSbrElWKEv/hRcPinj133jz35htK6jog4HHezPMzAtiRpW27Q8rt7S8srqWXy9sbG5t7xR399pKJBKTFhZMyJsAKcIoJy1NNSM3sSQoChjpBOPzzO/cE6mo4Nd6EhM/QkNOQ4qRNtKtKPfUndTw8rhfLNmV03rV9arQrth2zXGdjLg178SDjlEylMACzX7xvTcQOIkI15ghpbqOHWs/RVJTzMi00EsUiREeoyHpGspRRJSfzi6ewiOjDGAopCmu4Uz9PpGiSKlJFJjOCOmR+u1l4l9eN9Fh3U8pjxNNOJ4vChMGtYDZ+3BAJcGaTQxBWFJzK8QjJBHWJqSCCeHrU/g/absVx/Arr9Q4W8SRBwfgEJSBA2qgAS5AE7QABhw8gCfwbCnr0XqxXuetOWsxsw9+wHr7BPikkHQ=</latexit><latexit sha1_base64="rLtLN6y4Y0N14fVQ1LCCmhcXVdw=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BItQL2V3Xdp6K3rxJBVsK7ZLyabZNjSbrElWKEv/hRcPinj133jz35htK6jog4HHezPMzAtiRpW27Q8rt7S8srqWXy9sbG5t7xR399pKJBKTFhZMyJsAKcIoJy1NNSM3sSQoChjpBOPzzO/cE6mo4Nd6EhM/QkNOQ4qRNtKtKPfUndTw8rhfLNmV03rV9arQrth2zXGdjLg178SDjlEylMACzX7xvTcQOIkI15ghpbqOHWs/RVJTzMi00EsUiREeoyHpGspRRJSfzi6ewiOjDGAopCmu4Uz9PpGiSKlJFJjOCOmR+u1l4l9eN9Fh3U8pjxNNOJ4vChMGtYDZ+3BAJcGaTQxBWFJzK8QjJBHWJqSCCeHrU/g/absVx/Arr9Q4W8SRBwfgEJSBA2qgAS5AE7QABhw8gCfwbCnr0XqxXuetOWsxsw9+wHr7BPikkHQ=</latexit>
Extended objects: D-branes
There exist half BPS states that are D-brane like extended objects.
These are called giant and dual giant gravitons.
McGreevy, Susskind, Toumbas hep-th/0003075Grisaru, Myers, Tajford hep-th/0008015
Hashimoto, Hirano, Itzhaki hep-th/0008016
Simple description. Write the 5 sphere in cartesian coordinates.
zz +4X
i=1
y2i = 1<latexit sha1_base64="2ZQovfQXXK9s8Cn/AFbm3OG8DDI=">AAACCHicdZDLSsNAFIYnXmu9RV26cLAIglCSUtGNUHTjsoK9QJuGyXTSDp1MwsxESEOXbnwVNy4UcesjuPNtnLQpqOgPAz/fOYcz5/ciRqWyrE9jYXFpeWW1sFZc39jc2jZ3dpsyjAUmDRyyULQ9JAmjnDQUVYy0I0FQ4DHS8kZXWb11R4SkIb9VSUScAA049SlGSiPXPBjDrocEHMMT2JVx4Kb0wp70qjDpVVxtXbNklU+tTNAqW3OTEzsnJZCr7pof3X6I44BwhRmSsmNbkXJSJBTFjEyK3ViSCOERGpCOthwFRDrp9JAJPNKkD/1Q6McVnNLvEykKpEwCT3cGSA3l71oG/6p1YuWfOynlUawIx7NFfsygCmGWCuxTQbBiiTYIC6r/CvEQCYSVzq6oQ5hfCv83zUrZ1v6mWqpd5nEUwD44BMfABmegBq5BHTQABvfgETyDF+PBeDJejbdZ64KRz+yBHzLevwDQTZfr</latexit><latexit sha1_base64="2ZQovfQXXK9s8Cn/AFbm3OG8DDI=">AAACCHicdZDLSsNAFIYnXmu9RV26cLAIglCSUtGNUHTjsoK9QJuGyXTSDp1MwsxESEOXbnwVNy4UcesjuPNtnLQpqOgPAz/fOYcz5/ciRqWyrE9jYXFpeWW1sFZc39jc2jZ3dpsyjAUmDRyyULQ9JAmjnDQUVYy0I0FQ4DHS8kZXWb11R4SkIb9VSUScAA049SlGSiPXPBjDrocEHMMT2JVx4Kb0wp70qjDpVVxtXbNklU+tTNAqW3OTEzsnJZCr7pof3X6I44BwhRmSsmNbkXJSJBTFjEyK3ViSCOERGpCOthwFRDrp9JAJPNKkD/1Q6McVnNLvEykKpEwCT3cGSA3l71oG/6p1YuWfOynlUawIx7NFfsygCmGWCuxTQbBiiTYIC6r/CvEQCYSVzq6oQ5hfCv83zUrZ1v6mWqpd5nEUwD44BMfABmegBq5BHTQABvfgETyDF+PBeDJejbdZ64KRz+yBHzLevwDQTZfr</latexit><latexit sha1_base64="2ZQovfQXXK9s8Cn/AFbm3OG8DDI=">AAACCHicdZDLSsNAFIYnXmu9RV26cLAIglCSUtGNUHTjsoK9QJuGyXTSDp1MwsxESEOXbnwVNy4UcesjuPNtnLQpqOgPAz/fOYcz5/ciRqWyrE9jYXFpeWW1sFZc39jc2jZ3dpsyjAUmDRyyULQ9JAmjnDQUVYy0I0FQ4DHS8kZXWb11R4SkIb9VSUScAA049SlGSiPXPBjDrocEHMMT2JVx4Kb0wp70qjDpVVxtXbNklU+tTNAqW3OTEzsnJZCr7pof3X6I44BwhRmSsmNbkXJSJBTFjEyK3ViSCOERGpCOthwFRDrp9JAJPNKkD/1Q6McVnNLvEykKpEwCT3cGSA3l71oG/6p1YuWfOynlUawIx7NFfsygCmGWCuxTQbBiiTYIC6r/CvEQCYSVzq6oQ5hfCv83zUrZ1v6mWqpd5nEUwD44BMfABmegBq5BHTQABvfgETyDF+PBeDJejbdZ64KRz+yBHzLevwDQTZfr</latexit><latexit sha1_base64="2ZQovfQXXK9s8Cn/AFbm3OG8DDI=">AAACCHicdZDLSsNAFIYnXmu9RV26cLAIglCSUtGNUHTjsoK9QJuGyXTSDp1MwsxESEOXbnwVNy4UcesjuPNtnLQpqOgPAz/fOYcz5/ciRqWyrE9jYXFpeWW1sFZc39jc2jZ3dpsyjAUmDRyyULQ9JAmjnDQUVYy0I0FQ4DHS8kZXWb11R4SkIb9VSUScAA049SlGSiPXPBjDrocEHMMT2JVx4Kb0wp70qjDpVVxtXbNklU+tTNAqW3OTEzsnJZCr7pof3X6I44BwhRmSsmNbkXJSJBTFjEyK3ViSCOERGpCOthwFRDrp9JAJPNKkD/1Q6McVnNLvEykKpEwCT3cGSA3l71oG/6p1YuWfOynlUawIx7NFfsygCmGWCuxTQbBiiTYIC6r/CvEQCYSVzq6oQ5hfCv83zUrZ1v6mWqpd5nEUwD44BMfABmegBq5BHTQABvfgETyDF+PBeDJejbdZ64KRz+yBHzLevwDQTZfr</latexit>
z(t) = z0 exp(it)<latexit sha1_base64="A/UxrIlG2UqeY/zKDZBlrRddy0o=">AAAB/XicdVBNSwJRFH1jX2Zf9rFr80gC3cjMJGqLQGrT0iA10GF483zqwzcfvHcnUpH+SpsWRbTtf7Tr3/RGDSrqwIXDOfdy7z1eJLgC0/wwUkvLK6tr6fXMxubW9k52d6+pwlhS1qChCOWNRxQTPGAN4CDYTSQZ8T3BWt7wIvFbt0wqHgbXMIqY45N+wHucEtCSmz0Y56GAz/DYNXGH3UV5jqHgZnNm8bRatktlbBZNs2LZVkLsSumkhC2tJMihBepu9r3TDWnsswCoIEq1LTMCZ0IkcCrYNNOJFYsIHZI+a2saEJ8pZzK7foqPtdLFvVDqCgDP1O8TE+IrNfI93ekTGKjfXiL+5bVj6FWdCQ+iGFhA54t6scAQ4iQK3OWSURAjTQiVXN+K6YBIQkEHltEhfH2K/ydNu2hpflXK1c4XcaTRITpCeWShCqqhS1RHDUTRGD2gJ/Rs3BuPxovxOm9NGYuZffQDxtsn4BeTkQ==</latexit><latexit sha1_base64="A/UxrIlG2UqeY/zKDZBlrRddy0o=">AAAB/XicdVBNSwJRFH1jX2Zf9rFr80gC3cjMJGqLQGrT0iA10GF483zqwzcfvHcnUpH+SpsWRbTtf7Tr3/RGDSrqwIXDOfdy7z1eJLgC0/wwUkvLK6tr6fXMxubW9k52d6+pwlhS1qChCOWNRxQTPGAN4CDYTSQZ8T3BWt7wIvFbt0wqHgbXMIqY45N+wHucEtCSmz0Y56GAz/DYNXGH3UV5jqHgZnNm8bRatktlbBZNs2LZVkLsSumkhC2tJMihBepu9r3TDWnsswCoIEq1LTMCZ0IkcCrYNNOJFYsIHZI+a2saEJ8pZzK7foqPtdLFvVDqCgDP1O8TE+IrNfI93ekTGKjfXiL+5bVj6FWdCQ+iGFhA54t6scAQ4iQK3OWSURAjTQiVXN+K6YBIQkEHltEhfH2K/ydNu2hpflXK1c4XcaTRITpCeWShCqqhS1RHDUTRGD2gJ/Rs3BuPxovxOm9NGYuZffQDxtsn4BeTkQ==</latexit><latexit sha1_base64="A/UxrIlG2UqeY/zKDZBlrRddy0o=">AAAB/XicdVBNSwJRFH1jX2Zf9rFr80gC3cjMJGqLQGrT0iA10GF483zqwzcfvHcnUpH+SpsWRbTtf7Tr3/RGDSrqwIXDOfdy7z1eJLgC0/wwUkvLK6tr6fXMxubW9k52d6+pwlhS1qChCOWNRxQTPGAN4CDYTSQZ8T3BWt7wIvFbt0wqHgbXMIqY45N+wHucEtCSmz0Y56GAz/DYNXGH3UV5jqHgZnNm8bRatktlbBZNs2LZVkLsSumkhC2tJMihBepu9r3TDWnsswCoIEq1LTMCZ0IkcCrYNNOJFYsIHZI+a2saEJ8pZzK7foqPtdLFvVDqCgDP1O8TE+IrNfI93ekTGKjfXiL+5bVj6FWdCQ+iGFhA54t6scAQ4iQK3OWSURAjTQiVXN+K6YBIQkEHltEhfH2K/ydNu2hpflXK1c4XcaTRITpCeWShCqqhS1RHDUTRGD2gJ/Rs3BuPxovxOm9NGYuZffQDxtsn4BeTkQ==</latexit><latexit sha1_base64="A/UxrIlG2UqeY/zKDZBlrRddy0o=">AAAB/XicdVBNSwJRFH1jX2Zf9rFr80gC3cjMJGqLQGrT0iA10GF483zqwzcfvHcnUpH+SpsWRbTtf7Tr3/RGDSrqwIXDOfdy7z1eJLgC0/wwUkvLK6tr6fXMxubW9k52d6+pwlhS1qChCOWNRxQTPGAN4CDYTSQZ8T3BWt7wIvFbt0wqHgbXMIqY45N+wHucEtCSmz0Y56GAz/DYNXGH3UV5jqHgZnNm8bRatktlbBZNs2LZVkLsSumkhC2tJMihBepu9r3TDWnsswCoIEq1LTMCZ0IkcCrYNNOJFYsIHZI+a2saEJ8pZzK7foqPtdLFvVDqCgDP1O8TE+IrNfI93ekTGKjfXiL+5bVj6FWdCQ+iGFhA54t6scAQ4iQK3OWSURAjTQiVXN+K6YBIQkEHltEhfH2K/ydNu2hpflXK1c4XcaTRITpCeWShCqqhS1RHDUTRGD2gJ/Rs3BuPxovxOm9NGYuZffQDxtsn4BeTkQ==</latexit>
Consider a 3-sphere at fixed z and move it as follows in time
Sits at origin of global AdS coordinates.
z = cos(✓) exp(i�)<latexit sha1_base64="ErrbacEToOr+YriCTWQOgIoLKoI=">AAACBXicdVA9SwNBEN3zM8avqKUWi0GITbg7Q6KFELSxVDAmkAthbzMxS/Y+2J0T42Fj41+xsVDE1v9g579xTyOo6IOBx3szzMzzYyk02vabNTE5NT0zm5vLzy8sLi0XVlbPdJQoDg0eyUi1fKZBihAaKFBCK1bAAl9C0x8eZn7zApQWUXiKoxg6ATsPRV9whkbqFjau6D71eKRLHg4A2Tb14DIuCS8eiO1uoWiX93arbqVK7bJt1xzXyYhbq+xUqGOUDEUyxnG38Or1Ip4EECKXTOu2Y8fYSZlCwSVc571EQ8z4kJ1D29CQBaA76ccX13TLKD3aj5SpEOmH+n0iZYHWo8A3nQHDgf7tZeJfXjvB/m4nFWGcIIT8c1E/kRQjmkVCe0IBRzkyhHElzK2UD5hiHE1weRPC16f0f3Lmlh3DTyrF+sE4jhxZJ5ukRBxSI3VyRI5Jg3ByQ+7IA3m0bq1768l6/mydsMYza+QHrJd3zQCXgA==</latexit><latexit sha1_base64="ErrbacEToOr+YriCTWQOgIoLKoI=">AAACBXicdVA9SwNBEN3zM8avqKUWi0GITbg7Q6KFELSxVDAmkAthbzMxS/Y+2J0T42Fj41+xsVDE1v9g579xTyOo6IOBx3szzMzzYyk02vabNTE5NT0zm5vLzy8sLi0XVlbPdJQoDg0eyUi1fKZBihAaKFBCK1bAAl9C0x8eZn7zApQWUXiKoxg6ATsPRV9whkbqFjau6D71eKRLHg4A2Tb14DIuCS8eiO1uoWiX93arbqVK7bJt1xzXyYhbq+xUqGOUDEUyxnG38Or1Ip4EECKXTOu2Y8fYSZlCwSVc571EQ8z4kJ1D29CQBaA76ccX13TLKD3aj5SpEOmH+n0iZYHWo8A3nQHDgf7tZeJfXjvB/m4nFWGcIIT8c1E/kRQjmkVCe0IBRzkyhHElzK2UD5hiHE1weRPC16f0f3Lmlh3DTyrF+sE4jhxZJ5ukRBxSI3VyRI5Jg3ByQ+7IA3m0bq1768l6/mydsMYza+QHrJd3zQCXgA==</latexit><latexit sha1_base64="ErrbacEToOr+YriCTWQOgIoLKoI=">AAACBXicdVA9SwNBEN3zM8avqKUWi0GITbg7Q6KFELSxVDAmkAthbzMxS/Y+2J0T42Fj41+xsVDE1v9g579xTyOo6IOBx3szzMzzYyk02vabNTE5NT0zm5vLzy8sLi0XVlbPdJQoDg0eyUi1fKZBihAaKFBCK1bAAl9C0x8eZn7zApQWUXiKoxg6ATsPRV9whkbqFjau6D71eKRLHg4A2Tb14DIuCS8eiO1uoWiX93arbqVK7bJt1xzXyYhbq+xUqGOUDEUyxnG38Or1Ip4EECKXTOu2Y8fYSZlCwSVc571EQ8z4kJ1D29CQBaA76ccX13TLKD3aj5SpEOmH+n0iZYHWo8A3nQHDgf7tZeJfXjvB/m4nFWGcIIT8c1E/kRQjmkVCe0IBRzkyhHElzK2UD5hiHE1weRPC16f0f3Lmlh3DTyrF+sE4jhxZJ5ukRBxSI3VyRI5Jg3ByQ+7IA3m0bq1768l6/mydsMYza+QHrJd3zQCXgA==</latexit><latexit sha1_base64="ErrbacEToOr+YriCTWQOgIoLKoI=">AAACBXicdVA9SwNBEN3zM8avqKUWi0GITbg7Q6KFELSxVDAmkAthbzMxS/Y+2J0T42Fj41+xsVDE1v9g579xTyOo6IOBx3szzMzzYyk02vabNTE5NT0zm5vLzy8sLi0XVlbPdJQoDg0eyUi1fKZBihAaKFBCK1bAAl9C0x8eZn7zApQWUXiKoxg6ATsPRV9whkbqFjau6D71eKRLHg4A2Tb14DIuCS8eiO1uoWiX93arbqVK7bJt1xzXyYhbq+xUqGOUDEUyxnG38Or1Ip4EECKXTOu2Y8fYSZlCwSVc571EQ8z4kJ1D29CQBaA76ccX13TLKD3aj5SpEOmH+n0iZYHWo8A3nQHDgf7tZeJfXjvB/m4nFWGcIIT8c1E/kRQjmkVCe0IBRzkyhHElzK2UD5hiHE1weRPC16f0f3Lmlh3DTyrF+sE4jhxZJ5ukRBxSI3VyRI5Jg3ByQ+7IA3m0bq1768l6/mydsMYza+QHrJd3zQCXgA==</latexit>
� = 1<latexit sha1_base64="Qw7ebSUcFqFlDjRe39TChl9xSuw=">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg==</latexit><latexit sha1_base64="Qw7ebSUcFqFlDjRe39TChl9xSuw=">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg==</latexit><latexit sha1_base64="Qw7ebSUcFqFlDjRe39TChl9xSuw=">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg==</latexit><latexit sha1_base64="Qw7ebSUcFqFlDjRe39TChl9xSuw=">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg==</latexit>
⇢ = 0<latexit sha1_base64="3UAsdzuJ9FQ9O/8QIOSp6bI+u4E=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4Kpk6tHUhFN24rGAf0A4lk2ba2EwyJBmhDP0HNy4Ucev/uPNvzLQVVPTAhcM593LvPUHMmTYIfTi5ldW19Y38ZmFre2d3r7h/0NYyUYS2iORSdQOsKWeCtgwznHZjRXEUcNoJJleZ37mnSjMpbs00pn6ER4KFjGBjpXZfjeUFGhRLqHxer1a8KkRlhGpuxc1IpeadedC1SoYSWKI5KL73h5IkERWGcKx1z0Wx8VOsDCOczgr9RNMYkwke0Z6lAkdU++n82hk8scoQhlLZEgbO1e8TKY60nkaB7YywGevfXib+5fUSE9b9lIk4MVSQxaIw4dBImL0Oh0xRYvjUEkwUs7dCMsYKE2MDKtgQvj6F/5N2pexafuOVGpfLOPLgCByDU+CCGmiAa9AELUDAHXgAT+DZkc6j8+K8LlpzznLmEPyA8/YJcl2PCg==</latexit><latexit sha1_base64="3UAsdzuJ9FQ9O/8QIOSp6bI+u4E=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4Kpk6tHUhFN24rGAf0A4lk2ba2EwyJBmhDP0HNy4Ucev/uPNvzLQVVPTAhcM593LvPUHMmTYIfTi5ldW19Y38ZmFre2d3r7h/0NYyUYS2iORSdQOsKWeCtgwznHZjRXEUcNoJJleZ37mnSjMpbs00pn6ER4KFjGBjpXZfjeUFGhRLqHxer1a8KkRlhGpuxc1IpeadedC1SoYSWKI5KL73h5IkERWGcKx1z0Wx8VOsDCOczgr9RNMYkwke0Z6lAkdU++n82hk8scoQhlLZEgbO1e8TKY60nkaB7YywGevfXib+5fUSE9b9lIk4MVSQxaIw4dBImL0Oh0xRYvjUEkwUs7dCMsYKE2MDKtgQvj6F/5N2pexafuOVGpfLOPLgCByDU+CCGmiAa9AELUDAHXgAT+DZkc6j8+K8LlpzznLmEPyA8/YJcl2PCg==</latexit><latexit sha1_base64="3UAsdzuJ9FQ9O/8QIOSp6bI+u4E=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4Kpk6tHUhFN24rGAf0A4lk2ba2EwyJBmhDP0HNy4Ucev/uPNvzLQVVPTAhcM593LvPUHMmTYIfTi5ldW19Y38ZmFre2d3r7h/0NYyUYS2iORSdQOsKWeCtgwznHZjRXEUcNoJJleZ37mnSjMpbs00pn6ER4KFjGBjpXZfjeUFGhRLqHxer1a8KkRlhGpuxc1IpeadedC1SoYSWKI5KL73h5IkERWGcKx1z0Wx8VOsDCOczgr9RNMYkwke0Z6lAkdU++n82hk8scoQhlLZEgbO1e8TKY60nkaB7YywGevfXib+5fUSE9b9lIk4MVSQxaIw4dBImL0Oh0xRYvjUEkwUs7dCMsYKE2MDKtgQvj6F/5N2pexafuOVGpfLOPLgCByDU+CCGmiAa9AELUDAHXgAT+DZkc6j8+K8LlpzznLmEPyA8/YJcl2PCg==</latexit><latexit sha1_base64="3UAsdzuJ9FQ9O/8QIOSp6bI+u4E=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4Kpk6tHUhFN24rGAf0A4lk2ba2EwyJBmhDP0HNy4Ucev/uPNvzLQVVPTAhcM593LvPUHMmTYIfTi5ldW19Y38ZmFre2d3r7h/0NYyUYS2iORSdQOsKWeCtgwznHZjRXEUcNoJJleZ37mnSjMpbs00pn6ER4KFjGBjpXZfjeUFGhRLqHxer1a8KkRlhGpuxc1IpeadedC1SoYSWKI5KL73h5IkERWGcKx1z0Wx8VOsDCOczgr9RNMYkwke0Z6lAkdU++n82hk8scoQhlLZEgbO1e8TKY60nkaB7YywGevfXib+5fUSE9b9lIk4MVSQxaIw4dBImL0Oh0xRYvjUEkwUs7dCMsYKE2MDKtgQvj6F/5N2pexafuOVGpfLOPLgCByDU+CCGmiAa9AELUDAHXgAT+DZkc6j8+K8LlpzznLmEPyA8/YJcl2PCg==</latexit>
Dual giant: analytic continuation
⇠⇠ �X
i
y2i = �1<latexit sha1_base64="T75+gMRpB8rOBI5nXmmMpD5gIus=">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</latexit><latexit sha1_base64="T75+gMRpB8rOBI5nXmmMpD5gIus=">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</latexit><latexit sha1_base64="T75+gMRpB8rOBI5nXmmMpD5gIus=">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</latexit><latexit sha1_base64="T75+gMRpB8rOBI5nXmmMpD5gIus=">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</latexit>
✓ = 0<latexit sha1_base64="VtjVYMSbym7RPVXEZKG2onDKsuY=">AAAB73icbZA9SwNBEIbn4leMX1FLm8UgWIU7EbQRgjaWEUwMJEfY28wlS/Y+3J0TQsifsLFQxNa/Y+e/cZNcoYkvLDy8M8POvEGqpCHX/XYKK6tr6xvFzdLW9s7uXnn/oGmSTAtsiEQluhVwg0rG2CBJClupRh4FCh+C4c20/vCE2sgkvqdRin7E+7EMpeBkrVaHBkj8yu2WK27VnYktg5dDBXLVu+WvTi8RWYQxCcWNaXtuSv6Ya5JC4aTUyQymXAx5H9sWYx6h8cezfSfsxDo9FibavpjYzP09MeaRMaMosJ0Rp4FZrE3N/2rtjMJLfyzjNCOMxfyjMFOMEjY9nvWkRkFqZIELLe2uTAy45oJsRCUbgrd48jI0z6qe5bvzSu06j6MIR3AMp+DBBdTgFurQAAEKnuEV3pxH58V5dz7mrQUnnzmEP3I+fwCYO4+p</latexit><latexit sha1_base64="VtjVYMSbym7RPVXEZKG2onDKsuY=">AAAB73icbZA9SwNBEIbn4leMX1FLm8UgWIU7EbQRgjaWEUwMJEfY28wlS/Y+3J0TQsifsLFQxNa/Y+e/cZNcoYkvLDy8M8POvEGqpCHX/XYKK6tr6xvFzdLW9s7uXnn/oGmSTAtsiEQluhVwg0rG2CBJClupRh4FCh+C4c20/vCE2sgkvqdRin7E+7EMpeBkrVaHBkj8yu2WK27VnYktg5dDBXLVu+WvTi8RWYQxCcWNaXtuSv6Ya5JC4aTUyQymXAx5H9sWYx6h8cezfSfsxDo9FibavpjYzP09MeaRMaMosJ0Rp4FZrE3N/2rtjMJLfyzjNCOMxfyjMFOMEjY9nvWkRkFqZIELLe2uTAy45oJsRCUbgrd48jI0z6qe5bvzSu06j6MIR3AMp+DBBdTgFurQAAEKnuEV3pxH58V5dz7mrQUnnzmEP3I+fwCYO4+p</latexit><latexit sha1_base64="VtjVYMSbym7RPVXEZKG2onDKsuY=">AAAB73icbZA9SwNBEIbn4leMX1FLm8UgWIU7EbQRgjaWEUwMJEfY28wlS/Y+3J0TQsifsLFQxNa/Y+e/cZNcoYkvLDy8M8POvEGqpCHX/XYKK6tr6xvFzdLW9s7uXnn/oGmSTAtsiEQluhVwg0rG2CBJClupRh4FCh+C4c20/vCE2sgkvqdRin7E+7EMpeBkrVaHBkj8yu2WK27VnYktg5dDBXLVu+WvTi8RWYQxCcWNaXtuSv6Ya5JC4aTUyQymXAx5H9sWYx6h8cezfSfsxDo9FibavpjYzP09MeaRMaMosJ0Rp4FZrE3N/2rtjMJLfyzjNCOMxfyjMFOMEjY9nvWkRkFqZIELLe2uTAy45oJsRCUbgrd48jI0z6qe5bvzSu06j6MIR3AMp+DBBdTgFurQAAEKnuEV3pxH58V5dz7mrQUnnzmEP3I+fwCYO4+p</latexit><latexit sha1_base64="VtjVYMSbym7RPVXEZKG2onDKsuY=">AAAB73icbZA9SwNBEIbn4leMX1FLm8UgWIU7EbQRgjaWEUwMJEfY28wlS/Y+3J0TQsifsLFQxNa/Y+e/cZNcoYkvLDy8M8POvEGqpCHX/XYKK6tr6xvFzdLW9s7uXnn/oGmSTAtsiEQluhVwg0rG2CBJClupRh4FCh+C4c20/vCE2sgkvqdRin7E+7EMpeBkrVaHBkj8yu2WK27VnYktg5dDBXLVu+WvTi8RWYQxCcWNaXtuSv6Ya5JC4aTUyQymXAx5H9sWYx6h8cezfSfsxDo9FibavpjYzP09MeaRMaMosJ0Rp4FZrE3N/2rtjMJLfyzjNCOMxfyjMFOMEjY9nvWkRkFqZIELLe2uTAy45oJsRCUbgrd48jI0z6qe5bvzSu06j6MIR3AMp+DBBdTgFurQAAEKnuEV3pxH58V5dz7mrQUnnzmEP3I+fwCYO4+p</latexit>
Sits at “origin” in z
⇠ = cosh ⇢ exp(it)<latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit><latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit><latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit><latexit sha1_base64="ck8pdC+ekZH4nUmSP+ZG7r8lEyk=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odOn4MoA7ncAFXEMIN3MEDdKALAhJ4hXdv4r15H6uuat66tDP4I+/zBzjGijg=</latexit><latexit sha1_base64="32Jo/CzqMvY+HKRgTOn0PWRR5wo=">AAAB93icbZBLSwMxFIXv+Ky16uhO3ASLUDdlxo1uBMGNywr2AZ2hZNJMJzSTDElGWobixr/ixoUi/hN3/hvTx0JbDwQ+zkm4uSfKONPG876dtfWNza3t0k55t7K3f+AeVlpa5orQJpFcqk6ENeVM0KZhhtNOpihOI07b0fB2mrcfqdJMigczzmiY4oFgMSPYWKvnHgcjdo0CInWCApVIFNBRVmPInPfcqlf3ZkKr4C+gCgs1eu5X0JckT6kwhGOtu76XmbDAyjDC6aQc5JpmmAzxgHYtCpxSHRazFSbozDp9FEtljzBo5v5+UeBU63Ea2ZspNolezqbmf1k3N/FVWDCR5YYKMh8U5xwZiaZ9oD5TlBg+toCJYvaviCRYYWJsa2Vbgr+88iq0Luq+5XsPSnACp1ADHy7hBu6gAU0g8AQv8AbvzrPz6nzM61pzFr0dwR85nz/2zJS3</latexit><latexit sha1_base64="ZdZpD4kWHpDg34n7mB1VVQwx4kU=">AAAB93icdZBPS8MwGMbfzn9zTq3exEtwCPMyOkH0IghePE5wf2AtI83SNSxtSpLKRhle/CpePCjiN/HmtzHdOlDRBwIPvyfhzfv4CWdKO86nVVpZXVvfKG9WtqrbO7v2XrWjRCoJbRPBhez5WFHOYtrWTHPaSyTFkc9p1x9f53n3nkrFRHynpwn1IjyKWcAI1gYN7AN3wi6RS4QKkStDgVw6SeoM6ZOBXXMaZ04u5DScpSlIsyA1KNQa2B/uUJA0orEmHCvVbzqJ9jIsNSOczipuqmiCyRiPaN/YGEdUedl8hRk6NmSIAiHNiTWa0+8vMhwpNY18czPCOlS/sxz+lfVTHVx4GYuTVNOYLAYFKUdaoLwPNGSSEs2nxmAimfkrIiGWmGjTWsWUsNwU/W86p42m8bcOlOEQjqAOTTiHK7iBFrSBwAM8wQu8Wo/Ws/W2qKtkFb3tww9Z718E/5TB</latexit><latexit sha1_base64="CTa2+xbyceepiJsiPBofYdZTmaM=">AAACAnicdZDLSsNAFIYn9VbrLepK3AwWoW7KRBDdCEU3LivYCzShTKaTZugkE2Ym0hKKG1/FjQtF3PoU7nwbJ20EFf1h4Oc753Dm/H7CmdIIfVilhcWl5ZXyamVtfWNzy97eaSuRSkJbRHAhuz5WlLOYtjTTnHYTSXHkc9rxR5d5vXNLpWIivtGThHoRHsYsYARrg/r2njtm59AlQoXQlaGALh0nNQb1Ud+uovoJygVRHX2ZgjgFqYJCzb797g4ESSMaa8KxUj0HJdrLsNSMcDqtuKmiCSYjPKQ9Y2McUeVlsxOm8NCQAQyENC/WcEa/T2Q4UmoS+aYzwjpUv2s5/KvWS3Vw5mUsTlJNYzJfFKQcagHzPOCASUo0nxiDiWTmr5CEWGKiTWoVE8LXpfB/0z6uO8Zfo2rjooijDPbBAagBB5yCBrgCTdACBNyBB/AEnq1769F6sV7nrSWrmNkFP2S9fQJ435Yq</latexit><latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit><latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit><latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit><latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit><latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit><latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit>
� = 1<latexit sha1_base64="Qw7ebSUcFqFlDjRe39TChl9xSuw=">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg==</latexit><latexit sha1_base64="Qw7ebSUcFqFlDjRe39TChl9xSuw=">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg==</latexit><latexit sha1_base64="Qw7ebSUcFqFlDjRe39TChl9xSuw=">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg==</latexit><latexit sha1_base64="Qw7ebSUcFqFlDjRe39TChl9xSuw=">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg==</latexit>
R-charge = Energy
E = N(1� |z|2) = N(1� cos2(✓))<latexit sha1_base64="1vxO4la17+xHCLxR9iNWtuAxabI=">AAACDHicdVDLSgMxFM34rPVVdekmWIR2YZkpim6EogiupIJ9QKctmTRtQzMPkjtCnfYD3Pgrblwo4tYPcOffmGmnoKIHAifnnEtyjxMIrsA0P425+YXFpeXUSnp1bX1jM7O1XVV+KCmrUF/4su4QxQT3WAU4CFYPJCOuI1jNGZzHfu2WScV97waGAWu6pOfxLqcEtNTOZC9O8VXOOhjdjVrFPJ5ebOqrVjFnQ58Byed1yiwcmTGwWTBnJFGsRMmiBOV25sPu+DR0mQdUEKUalhlAMyISOBVsnLZDxQJCB6THGpp6xGWqGU2WGeN9rXRw15f6eIAn6veJiLhKDV1HJ10CffXbi8W/vEYI3ZNmxL0gBObR6UPdUGDwcdwM7nDJKIihJoRKrv+KaZ9IQkH3l9YlzDbF/5NqsWBpfn2YLZ0ldaTQLtpDOWShY1RCl6iMKoiie/SIntGL8WA8Ga/G2zQ6ZyQzO+gHjPcvhVWYIg==</latexit><latexit sha1_base64="1vxO4la17+xHCLxR9iNWtuAxabI=">AAACDHicdVDLSgMxFM34rPVVdekmWIR2YZkpim6EogiupIJ9QKctmTRtQzMPkjtCnfYD3Pgrblwo4tYPcOffmGmnoKIHAifnnEtyjxMIrsA0P425+YXFpeXUSnp1bX1jM7O1XVV+KCmrUF/4su4QxQT3WAU4CFYPJCOuI1jNGZzHfu2WScV97waGAWu6pOfxLqcEtNTOZC9O8VXOOhjdjVrFPJ5ebOqrVjFnQ58Byed1yiwcmTGwWTBnJFGsRMmiBOV25sPu+DR0mQdUEKUalhlAMyISOBVsnLZDxQJCB6THGpp6xGWqGU2WGeN9rXRw15f6eIAn6veJiLhKDV1HJ10CffXbi8W/vEYI3ZNmxL0gBObR6UPdUGDwcdwM7nDJKIihJoRKrv+KaZ9IQkH3l9YlzDbF/5NqsWBpfn2YLZ0ldaTQLtpDOWShY1RCl6iMKoiie/SIntGL8WA8Ga/G2zQ6ZyQzO+gHjPcvhVWYIg==</latexit><latexit sha1_base64="1vxO4la17+xHCLxR9iNWtuAxabI=">AAACDHicdVDLSgMxFM34rPVVdekmWIR2YZkpim6EogiupIJ9QKctmTRtQzMPkjtCnfYD3Pgrblwo4tYPcOffmGmnoKIHAifnnEtyjxMIrsA0P425+YXFpeXUSnp1bX1jM7O1XVV+KCmrUF/4su4QxQT3WAU4CFYPJCOuI1jNGZzHfu2WScV97waGAWu6pOfxLqcEtNTOZC9O8VXOOhjdjVrFPJ5ebOqrVjFnQ58Byed1yiwcmTGwWTBnJFGsRMmiBOV25sPu+DR0mQdUEKUalhlAMyISOBVsnLZDxQJCB6THGpp6xGWqGU2WGeN9rXRw15f6eIAn6veJiLhKDV1HJ10CffXbi8W/vEYI3ZNmxL0gBObR6UPdUGDwcdwM7nDJKIihJoRKrv+KaZ9IQkH3l9YlzDbF/5NqsWBpfn2YLZ0ldaTQLtpDOWShY1RCl6iMKoiie/SIntGL8WA8Ga/G2zQ6ZyQzO+gHjPcvhVWYIg==</latexit><latexit sha1_base64="1vxO4la17+xHCLxR9iNWtuAxabI=">AAACDHicdVDLSgMxFM34rPVVdekmWIR2YZkpim6EogiupIJ9QKctmTRtQzMPkjtCnfYD3Pgrblwo4tYPcOffmGmnoKIHAifnnEtyjxMIrsA0P425+YXFpeXUSnp1bX1jM7O1XVV+KCmrUF/4su4QxQT3WAU4CFYPJCOuI1jNGZzHfu2WScV97waGAWu6pOfxLqcEtNTOZC9O8VXOOhjdjVrFPJ5ebOqrVjFnQ58Byed1yiwcmTGwWTBnJFGsRMmiBOV25sPu+DR0mQdUEKUalhlAMyISOBVsnLZDxQJCB6THGpp6xGWqGU2WGeN9rXRw15f6eIAn6veJiLhKDV1HJ10CffXbi8W/vEYI3ZNmxL0gBObR6UPdUGDwcdwM7nDJKIihJoRKrv+KaZ9IQkH3l9YlzDbF/5NqsWBpfn2YLZ0ldaTQLtpDOWShY1RCl6iMKoiie/SIntGL8WA8Ga/G2zQ6ZyQzO+gHjPcvhVWYIg==</latexit>
For giants:
For dual giants:
E = N(|⇠|2 � 1) = N(cosh2(⇢)� 1)<latexit sha1_base64="HRU5xhSCAJINAWy9L9m2JDMIQX8=">AAACDXicdZDLSgMxFIYz9VbrbdSlm2AV2oVlpii6EYoiuJIK9gKdtmTStBOamQxJRixtX8CNr+LGhSJu3bvzbcy0U1DRHwJ/vnMOyfndkFGpLOvTSM3NLywupZczK6tr6xvm5lZV8khgUsGccVF3kSSMBqSiqGKkHgqCfJeRmts/j+u1WyIk5cGNGoSk6aNeQLsUI6VR29y7OIVXuZFzR0et4oGdh/HVwVx6rWLOER7Pa9g2s1bhyIoFrYI1MwmxE5IFicpt88PpcBz5JFCYISkbthWq5hAJRTEj44wTSRIi3Ec90tA2QD6RzeFkmzHc16QDu1zoEyg4od8nhsiXcuC7utNHypO/azH8q9aIVPekOaRBGCkS4OlD3YhBxWEcDexQQbBiA20QFlT/FWIPCYSVDjCjQ5htCv831WLB1v76MFs6S+JIgx2wC3LABsegBC5BGVQABvfgETyDF+PBeDJejbdpa8pIZrbBDxnvX07jmIo=</latexit><latexit sha1_base64="HRU5xhSCAJINAWy9L9m2JDMIQX8=">AAACDXicdZDLSgMxFIYz9VbrbdSlm2AV2oVlpii6EYoiuJIK9gKdtmTStBOamQxJRixtX8CNr+LGhSJu3bvzbcy0U1DRHwJ/vnMOyfndkFGpLOvTSM3NLywupZczK6tr6xvm5lZV8khgUsGccVF3kSSMBqSiqGKkHgqCfJeRmts/j+u1WyIk5cGNGoSk6aNeQLsUI6VR29y7OIVXuZFzR0et4oGdh/HVwVx6rWLOER7Pa9g2s1bhyIoFrYI1MwmxE5IFicpt88PpcBz5JFCYISkbthWq5hAJRTEj44wTSRIi3Ec90tA2QD6RzeFkmzHc16QDu1zoEyg4od8nhsiXcuC7utNHypO/azH8q9aIVPekOaRBGCkS4OlD3YhBxWEcDexQQbBiA20QFlT/FWIPCYSVDjCjQ5htCv831WLB1v76MFs6S+JIgx2wC3LABsegBC5BGVQABvfgETyDF+PBeDJejbdpa8pIZrbBDxnvX07jmIo=</latexit><latexit sha1_base64="HRU5xhSCAJINAWy9L9m2JDMIQX8=">AAACDXicdZDLSgMxFIYz9VbrbdSlm2AV2oVlpii6EYoiuJIK9gKdtmTStBOamQxJRixtX8CNr+LGhSJu3bvzbcy0U1DRHwJ/vnMOyfndkFGpLOvTSM3NLywupZczK6tr6xvm5lZV8khgUsGccVF3kSSMBqSiqGKkHgqCfJeRmts/j+u1WyIk5cGNGoSk6aNeQLsUI6VR29y7OIVXuZFzR0et4oGdh/HVwVx6rWLOER7Pa9g2s1bhyIoFrYI1MwmxE5IFicpt88PpcBz5JFCYISkbthWq5hAJRTEj44wTSRIi3Ec90tA2QD6RzeFkmzHc16QDu1zoEyg4od8nhsiXcuC7utNHypO/azH8q9aIVPekOaRBGCkS4OlD3YhBxWEcDexQQbBiA20QFlT/FWIPCYSVDjCjQ5htCv831WLB1v76MFs6S+JIgx2wC3LABsegBC5BGVQABvfgETyDF+PBeDJejbdpa8pIZrbBDxnvX07jmIo=</latexit><latexit sha1_base64="HRU5xhSCAJINAWy9L9m2JDMIQX8=">AAACDXicdZDLSgMxFIYz9VbrbdSlm2AV2oVlpii6EYoiuJIK9gKdtmTStBOamQxJRixtX8CNr+LGhSJu3bvzbcy0U1DRHwJ/vnMOyfndkFGpLOvTSM3NLywupZczK6tr6xvm5lZV8khgUsGccVF3kSSMBqSiqGKkHgqCfJeRmts/j+u1WyIk5cGNGoSk6aNeQLsUI6VR29y7OIVXuZFzR0et4oGdh/HVwVx6rWLOER7Pa9g2s1bhyIoFrYI1MwmxE5IFicpt88PpcBz5JFCYISkbthWq5hAJRTEj44wTSRIi3Ec90tA2QD6RzeFkmzHc16QDu1zoEyg4od8nhsiXcuC7utNHypO/azH8q9aIVPekOaRBGCkS4OlD3YhBxWEcDexQQbBiA20QFlT/FWIPCYSVDjCjQ5htCv831WLB1v76MFs6S+JIgx2wC3LABsegBC5BGVQABvfgETyDF+PBeDJejbdpa8pIZrbBDxnvX07jmIo=</latexit>
Energies up to
O(N)<latexit sha1_base64="pHGyu6r4U0O4KCziVrGUXTuKBqc=">AAAB7HicdVBdSwJBFL1rX2ZfVo+9DElgL7Jri9qb1EtPZdCqoCKz46wOzs4uM7OBiL+hlx6K6LUf1Fv/plk1qKgDFw7n3Mu99/gxZ0rb9oeVWVldW9/Ibua2tnd29/L7B00VJZJQj0Q8km0fK8qZoJ5mmtN2LCkOfU5b/vgy9Vv3VCoWiTs9iWkvxEPBAkawNpJ3U0TXp/18wS6d1yplt4Lskm1XnbKTknLVPXORY5QUBVii0c+/dwcRSUIqNOFYqY5jx7o3xVIzwuks100UjTEZ4yHtGCpwSFVvOj92hk6MMkBBJE0Jjebq94kpDpWahL7pDLEeqd9eKv7ldRId1HpTJuJEU0EWi4KEIx2h9HM0YJISzSeGYCKZuRWREZaYaJNPzoTw9Sn6nzTLJcfwW7dQv1jGkYUjOIYiOFCFOlxBAzwgwOABnuDZEtaj9WK9Lloz1nLmEH7AevsEv56N/A==</latexit><latexit sha1_base64="pHGyu6r4U0O4KCziVrGUXTuKBqc=">AAAB7HicdVBdSwJBFL1rX2ZfVo+9DElgL7Jri9qb1EtPZdCqoCKz46wOzs4uM7OBiL+hlx6K6LUf1Fv/plk1qKgDFw7n3Mu99/gxZ0rb9oeVWVldW9/Ibua2tnd29/L7B00VJZJQj0Q8km0fK8qZoJ5mmtN2LCkOfU5b/vgy9Vv3VCoWiTs9iWkvxEPBAkawNpJ3U0TXp/18wS6d1yplt4Lskm1XnbKTknLVPXORY5QUBVii0c+/dwcRSUIqNOFYqY5jx7o3xVIzwuks100UjTEZ4yHtGCpwSFVvOj92hk6MMkBBJE0Jjebq94kpDpWahL7pDLEeqd9eKv7ldRId1HpTJuJEU0EWi4KEIx2h9HM0YJISzSeGYCKZuRWREZaYaJNPzoTw9Sn6nzTLJcfwW7dQv1jGkYUjOIYiOFCFOlxBAzwgwOABnuDZEtaj9WK9Lloz1nLmEH7AevsEv56N/A==</latexit><latexit sha1_base64="pHGyu6r4U0O4KCziVrGUXTuKBqc=">AAAB7HicdVBdSwJBFL1rX2ZfVo+9DElgL7Jri9qb1EtPZdCqoCKz46wOzs4uM7OBiL+hlx6K6LUf1Fv/plk1qKgDFw7n3Mu99/gxZ0rb9oeVWVldW9/Ibua2tnd29/L7B00VJZJQj0Q8km0fK8qZoJ5mmtN2LCkOfU5b/vgy9Vv3VCoWiTs9iWkvxEPBAkawNpJ3U0TXp/18wS6d1yplt4Lskm1XnbKTknLVPXORY5QUBVii0c+/dwcRSUIqNOFYqY5jx7o3xVIzwuks100UjTEZ4yHtGCpwSFVvOj92hk6MMkBBJE0Jjebq94kpDpWahL7pDLEeqd9eKv7ldRId1HpTJuJEU0EWi4KEIx2h9HM0YJISzSeGYCKZuRWREZaYaJNPzoTw9Sn6nzTLJcfwW7dQv1jGkYUjOIYiOFCFOlxBAzwgwOABnuDZEtaj9WK9Lloz1nLmEH7AevsEv56N/A==</latexit><latexit sha1_base64="pHGyu6r4U0O4KCziVrGUXTuKBqc=">AAAB7HicdVBdSwJBFL1rX2ZfVo+9DElgL7Jri9qb1EtPZdCqoCKz46wOzs4uM7OBiL+hlx6K6LUf1Fv/plk1qKgDFw7n3Mu99/gxZ0rb9oeVWVldW9/Ibua2tnd29/L7B00VJZJQj0Q8km0fK8qZoJ5mmtN2LCkOfU5b/vgy9Vv3VCoWiTs9iWkvxEPBAkawNpJ3U0TXp/18wS6d1yplt4Lskm1XnbKTknLVPXORY5QUBVii0c+/dwcRSUIqNOFYqY5jx7o3xVIzwuks100UjTEZ4yHtGCpwSFVvOj92hk6MMkBBJE0Jjebq94kpDpWahL7pDLEeqd9eKv7ldRId1HpTJuJEU0EWi4KEIx2h9HM0YJISzSeGYCKZuRWREZaYaJNPzoTw9Sn6nzTLJcfwW7dQv1jGkYUjOIYiOFCFOlxBAzwgwOABnuDZEtaj9WK9Lloz1nLmEH7AevsEv56N/A==</latexit>
Multi-traces, although BPS, are not orthogonal at finite N.
Need another orthogonal basis to describe finite N physics.
How do these fit? Dual operators
To build gauge invariants, need to contract upper and lower indices in Z
Zi1i01. . . Zik
i0k<latexit sha1_base64="hNRZZTN9JcYEP5Yy7lTfjbKEOm8=">AAACDHicdZC9TsMwEMcvfJbyFWBksagQTFWCQDBWsDAWiX6INkSO67RWHSeyHaQq6gOw8CosDCDEygOw8TY4bSoBgpOs++t3d7bvHyScKe04n9bc/MLi0nJppby6tr6xaW9tN1WcSkIbJOaxbAdYUc4EbWimOW0nkuIo4LQVDC/yeuuOSsVica1HCfUi3BcsZARrg3y7cnObMd8d+xk7MAl1e7FWaAKHU2iSXXGqJ04eyKk6M1EQtyAVKKLu2x/mGpJGVGjCsVId10m0l2GpGeF0XO6miiaYDHGfdowUOKLKyybLjNG+IT0UxtIcodGEfp/IcKTUKApMZ4T1QP2u5fCvWifV4ZmXMZGkmgoyfShMOdIxyp1BPSYp0XxkBCaSmb8iMsASE238KxsTZpui/0XzqOoafXVcqZ0XdpRgF/bgEFw4hRpcQh0aQOAeHuEZXqwH68l6td6mrXNWMbMDP8J6/wJCKJsd</latexit><latexit sha1_base64="hNRZZTN9JcYEP5Yy7lTfjbKEOm8=">AAACDHicdZC9TsMwEMcvfJbyFWBksagQTFWCQDBWsDAWiX6INkSO67RWHSeyHaQq6gOw8CosDCDEygOw8TY4bSoBgpOs++t3d7bvHyScKe04n9bc/MLi0nJppby6tr6xaW9tN1WcSkIbJOaxbAdYUc4EbWimOW0nkuIo4LQVDC/yeuuOSsVica1HCfUi3BcsZARrg3y7cnObMd8d+xk7MAl1e7FWaAKHU2iSXXGqJ04eyKk6M1EQtyAVKKLu2x/mGpJGVGjCsVId10m0l2GpGeF0XO6miiaYDHGfdowUOKLKyybLjNG+IT0UxtIcodGEfp/IcKTUKApMZ4T1QP2u5fCvWifV4ZmXMZGkmgoyfShMOdIxyp1BPSYp0XxkBCaSmb8iMsASE238KxsTZpui/0XzqOoafXVcqZ0XdpRgF/bgEFw4hRpcQh0aQOAeHuEZXqwH68l6td6mrXNWMbMDP8J6/wJCKJsd</latexit><latexit sha1_base64="hNRZZTN9JcYEP5Yy7lTfjbKEOm8=">AAACDHicdZC9TsMwEMcvfJbyFWBksagQTFWCQDBWsDAWiX6INkSO67RWHSeyHaQq6gOw8CosDCDEygOw8TY4bSoBgpOs++t3d7bvHyScKe04n9bc/MLi0nJppby6tr6xaW9tN1WcSkIbJOaxbAdYUc4EbWimOW0nkuIo4LQVDC/yeuuOSsVica1HCfUi3BcsZARrg3y7cnObMd8d+xk7MAl1e7FWaAKHU2iSXXGqJ04eyKk6M1EQtyAVKKLu2x/mGpJGVGjCsVId10m0l2GpGeF0XO6miiaYDHGfdowUOKLKyybLjNG+IT0UxtIcodGEfp/IcKTUKApMZ4T1QP2u5fCvWifV4ZmXMZGkmgoyfShMOdIxyp1BPSYp0XxkBCaSmb8iMsASE238KxsTZpui/0XzqOoafXVcqZ0XdpRgF/bgEFw4hRpcQh0aQOAeHuEZXqwH68l6td6mrXNWMbMDP8J6/wJCKJsd</latexit><latexit sha1_base64="hNRZZTN9JcYEP5Yy7lTfjbKEOm8=">AAACDHicdZC9TsMwEMcvfJbyFWBksagQTFWCQDBWsDAWiX6INkSO67RWHSeyHaQq6gOw8CosDCDEygOw8TY4bSoBgpOs++t3d7bvHyScKe04n9bc/MLi0nJppby6tr6xaW9tN1WcSkIbJOaxbAdYUc4EbWimOW0nkuIo4LQVDC/yeuuOSsVica1HCfUi3BcsZARrg3y7cnObMd8d+xk7MAl1e7FWaAKHU2iSXXGqJ04eyKk6M1EQtyAVKKLu2x/mGpJGVGjCsVId10m0l2GpGeF0XO6miiaYDHGfdowUOKLKyybLjNG+IT0UxtIcodGEfp/IcKTUKApMZ4T1QP2u5fCvWifV4ZmXMZGkmgoyfShMOdIxyp1BPSYp0XxkBCaSmb8iMsASE238KxsTZpui/0XzqOoafXVcqZ0XdpRgF/bgEFw4hRpcQh0aQOAeHuEZXqwH68l6td6mrXNWMbMDP8J6/wJCKJsd</latexit>
Free theory has a symmetry of upper and lower indices
SU(N)U ⇥ SU(N)D<latexit sha1_base64="fk5FTS1iEWuqImgjfRog3vWhpkQ=">AAAB/nicdVDLSsNAFL3xWesrKq7cDBahbkoiii6LunAlFU1baEOYTCft0MmDmYlQQsFfceNCEbd+hzv/xkmbgooeGDj3nHu5d46fcCaVZX0ac/MLi0vLpZXy6tr6xqa5td2UcSoIdUjMY9H2saScRdRRTHHaTgTFoc9pyx9e5H7rngrJ4uhOjRLqhrgfsYARrLTkmbu3TvX60HO6ioVUoml16ZkVq3Zi5UBWzZqRQrELpQIFGp750e3FJA1ppAjHUnZsK1FuhoVihNNxuZtKmmAyxH3a0TTCepmbTc4fowOt9FAQC/0ihSbq94kMh1KOQl93hlgN5G8vF//yOqkKztyMRUmqaESmi4KUIxWjPAvUY4ISxUeaYCKYvhWRARaYKJ1YWYcw+yn6nzSParbmN8eV+nkRRwn2YB+qYMMp1OEKGuAAgQwe4RlejAfjyXg13qatc0YxswM/YLx/AbxDlAo=</latexit><latexit sha1_base64="fk5FTS1iEWuqImgjfRog3vWhpkQ=">AAAB/nicdVDLSsNAFL3xWesrKq7cDBahbkoiii6LunAlFU1baEOYTCft0MmDmYlQQsFfceNCEbd+hzv/xkmbgooeGDj3nHu5d46fcCaVZX0ac/MLi0vLpZXy6tr6xqa5td2UcSoIdUjMY9H2saScRdRRTHHaTgTFoc9pyx9e5H7rngrJ4uhOjRLqhrgfsYARrLTkmbu3TvX60HO6ioVUoml16ZkVq3Zi5UBWzZqRQrELpQIFGp750e3FJA1ppAjHUnZsK1FuhoVihNNxuZtKmmAyxH3a0TTCepmbTc4fowOt9FAQC/0ihSbq94kMh1KOQl93hlgN5G8vF//yOqkKztyMRUmqaESmi4KUIxWjPAvUY4ISxUeaYCKYvhWRARaYKJ1YWYcw+yn6nzSParbmN8eV+nkRRwn2YB+qYMMp1OEKGuAAgQwe4RlejAfjyXg13qatc0YxswM/YLx/AbxDlAo=</latexit><latexit sha1_base64="fk5FTS1iEWuqImgjfRog3vWhpkQ=">AAAB/nicdVDLSsNAFL3xWesrKq7cDBahbkoiii6LunAlFU1baEOYTCft0MmDmYlQQsFfceNCEbd+hzv/xkmbgooeGDj3nHu5d46fcCaVZX0ac/MLi0vLpZXy6tr6xqa5td2UcSoIdUjMY9H2saScRdRRTHHaTgTFoc9pyx9e5H7rngrJ4uhOjRLqhrgfsYARrLTkmbu3TvX60HO6ioVUoml16ZkVq3Zi5UBWzZqRQrELpQIFGp750e3FJA1ppAjHUnZsK1FuhoVihNNxuZtKmmAyxH3a0TTCepmbTc4fowOt9FAQC/0ihSbq94kMh1KOQl93hlgN5G8vF//yOqkKztyMRUmqaESmi4KUIxWjPAvUY4ISxUeaYCKYvhWRARaYKJ1YWYcw+yn6nzSParbmN8eV+nkRRwn2YB+qYMMp1OEKGuAAgQwe4RlejAfjyXg13qatc0YxswM/YLx/AbxDlAo=</latexit><latexit sha1_base64="fk5FTS1iEWuqImgjfRog3vWhpkQ=">AAAB/nicdVDLSsNAFL3xWesrKq7cDBahbkoiii6LunAlFU1baEOYTCft0MmDmYlQQsFfceNCEbd+hzv/xkmbgooeGDj3nHu5d46fcCaVZX0ac/MLi0vLpZXy6tr6xqa5td2UcSoIdUjMY9H2saScRdRRTHHaTgTFoc9pyx9e5H7rngrJ4uhOjRLqhrgfsYARrLTkmbu3TvX60HO6ioVUoml16ZkVq3Zi5UBWzZqRQrELpQIFGp750e3FJA1ppAjHUnZsK1FuhoVihNNxuZtKmmAyxH3a0TTCepmbTc4fowOt9FAQC/0ihSbq94kMh1KOQl93hlgN5G8vF//yOqkKztyMRUmqaESmi4KUIxWjPAvUY4ISxUeaYCKYvhWRARaYKJ1YWYcw+yn6nzSParbmN8eV+nkRRwn2YB+qYMMp1OEKGuAAgQwe4RlejAfjyXg13qatc0YxswM/YLx/AbxDlAo=</latexit>
where a particular diagonal embedding is gauged.
Decompose tensor into irreps of this extra symmetry:Young diagram for upper and lower indices.
This is the Schur polynomial basis.
Unique gauge invariant built from this data.
Corley, Jevicki, Ramgoolam hep-th/0111222
How they match
• Giants are sub-determinants (1-column)
• Balasubramanian, Berkooz, Naqvi, Strassler hep-th/0107119
• Dual giants are 1-row.
• CJR
These are D-branes, so strings must end on them (somehow).
Berenstein, Balasubramanian, Feng, Huang, hep-th/0411205
Idea: intertwine indices of extra letters with the Z in a column or rowto produce open strings spin chains with interesting boundary conditions.
Many years of work by various groups
R. de Mello Koch et al; S. Ramgoolam et al; (constructing restricted Schur basis
and learning how to count states and compute)
Berenstein et al. (understanding boundary conditions on spin chains+collective coordinates+solving for ground state)
What does it look like?
When considering bigger sectors with SUSY,
{Q,Q} / exp(ip)
one can associate a central charge to the spin chain (Beisert).
Y ! [Y, Z]
Which shows effectively as follows
Evaluating one gets
This is evaluated on a single magnon.
Produces a non-trivial dispersion relation for impurity
�� J 'q
1 + h(�) sin2 p/2
Good to “all orders”h(�) = �
This is the strong ’t Hooft coupling constant.
This is the dispersion relation of a giant magnon: a piece of string
that is a straight chord between two end-points on a disk, moving with angular velocity one
S3 ! S5
#D2
(Hofmann -Maldacena)
Same coordinates as before
See also talk by Jaco van Zyl.
D.B., E. Dziekowski arXiv:1408.3620
The D-branes have sourced the central charge of Beisert.
Bubbling geometries
Can one solve the most general half BPS state in
gravity?
Matrix model ideas suggest that the Schurs are well described by free fermions.
Ideal representation is by using a holomorphic representation of this structure: free fermions in phase space (same physics as Quantum Hall
droplets)
DB, hep-th/0403110
In this representation traces are single quanta of edge states, whereas dual giants and giants are fermions (eigenvalues) and holes removed from the droplet
respectively
Suggests that the most general “classical limit” is of incompressible droplets in a 2D plane.
This is realized exactly in gravity!
LLM geometries
ds2 = � yq14 � z2
(dt+ Vidxi)2 +
q14 � z2
y(dy2 + dxidxi)
+y
s12 � z12 + z
!d⌦2
3 + y
s12 + z12 � z
!d⌦2
3
Lin, Lunin, Maldacena, JHEP 0410 (2004) 025
Solutions of type IIB supergravity, that preserve half the SUSY
+ attendant fluxes
depend on a single function z(~x, y)
@i@iz + y@y
✓@yz
y
◆= 0
Satisfies an ordinary linear PDE
Reality of the metric requires that
|z| 1
2
Degeneration at y=0
This provides a two coloring of the plane.
These are all horizon free smooth 10 D geometries.
Absence of conical singularity forces
z(0, ~x) = ±1
2
A typical LLM geometry
Topology of picture encodes topology of spacetime: different picture topologies give rise to different spacetime
topologies.
Have conserved area and an energy function.
N =
Z⇢(~x)d2x
They have a conserved “flux”
An energy function
Area of the droplet
E =
Z⇢(~x)~x2d2x� E�
Complete classification
Has energies of order
O(N2)<latexit sha1_base64="LmlaKfhQyGSZDBv5Fqvma5WdWf8=">AAAB7nicdVDLSgNBEOz1GeMr6tHLYBDiJeyuSxJvQS+eNIJ5QLKG2clsMmT2wcysEJZ8hBcPinj1e7z5N84mEVS0oKGo6qa7y4s5k8o0P4yl5ZXVtfXcRn5za3tnt7C335JRIghtkohHouNhSTkLaVMxxWknFhQHHqdtb3yR+e17KiSLwls1iakb4GHIfEaw0lL7uoSu7uyTfqFols9qFdupILNsmlXLtjJiV51TB1layVCEBRr9wntvEJEkoKEiHEvZtcxYuSkWihFOp/leImmMyRgPaVfTEAdUuuns3Ck61soA+ZHQFSo0U79PpDiQchJ4ujPAaiR/e5n4l9dNlF9zUxbGiaIhmS/yE45UhLLf0YAJShSfaIKJYPpWREZYYKJ0Qnkdwten6H/SssuW5jdOsX6+iCMHh3AEJbCgCnW4hAY0gcAYHuAJno3YeDRejNd565KxmDmAHzDePgHmf46g</latexit><latexit sha1_base64="LmlaKfhQyGSZDBv5Fqvma5WdWf8=">AAAB7nicdVDLSgNBEOz1GeMr6tHLYBDiJeyuSxJvQS+eNIJ5QLKG2clsMmT2wcysEJZ8hBcPinj1e7z5N84mEVS0oKGo6qa7y4s5k8o0P4yl5ZXVtfXcRn5za3tnt7C335JRIghtkohHouNhSTkLaVMxxWknFhQHHqdtb3yR+e17KiSLwls1iakb4GHIfEaw0lL7uoSu7uyTfqFols9qFdupILNsmlXLtjJiV51TB1layVCEBRr9wntvEJEkoKEiHEvZtcxYuSkWihFOp/leImmMyRgPaVfTEAdUuuns3Ck61soA+ZHQFSo0U79PpDiQchJ4ujPAaiR/e5n4l9dNlF9zUxbGiaIhmS/yE45UhLLf0YAJShSfaIKJYPpWREZYYKJ0Qnkdwten6H/SssuW5jdOsX6+iCMHh3AEJbCgCnW4hAY0gcAYHuAJno3YeDRejNd565KxmDmAHzDePgHmf46g</latexit><latexit sha1_base64="LmlaKfhQyGSZDBv5Fqvma5WdWf8=">AAAB7nicdVDLSgNBEOz1GeMr6tHLYBDiJeyuSxJvQS+eNIJ5QLKG2clsMmT2wcysEJZ8hBcPinj1e7z5N84mEVS0oKGo6qa7y4s5k8o0P4yl5ZXVtfXcRn5za3tnt7C335JRIghtkohHouNhSTkLaVMxxWknFhQHHqdtb3yR+e17KiSLwls1iakb4GHIfEaw0lL7uoSu7uyTfqFols9qFdupILNsmlXLtjJiV51TB1layVCEBRr9wntvEJEkoKEiHEvZtcxYuSkWihFOp/leImmMyRgPaVfTEAdUuuns3Ck61soA+ZHQFSo0U79PpDiQchJ4ujPAaiR/e5n4l9dNlF9zUxbGiaIhmS/yE45UhLLf0YAJShSfaIKJYPpWREZYYKJ0Qnkdwten6H/SssuW5jdOsX6+iCMHh3AEJbCgCnW4hAY0gcAYHuAJno3YeDRejNd565KxmDmAHzDePgHmf46g</latexit><latexit sha1_base64="LmlaKfhQyGSZDBv5Fqvma5WdWf8=">AAAB7nicdVDLSgNBEOz1GeMr6tHLYBDiJeyuSxJvQS+eNIJ5QLKG2clsMmT2wcysEJZ8hBcPinj1e7z5N84mEVS0oKGo6qa7y4s5k8o0P4yl5ZXVtfXcRn5za3tnt7C335JRIghtkohHouNhSTkLaVMxxWknFhQHHqdtb3yR+e17KiSLwls1iakb4GHIfEaw0lL7uoSu7uyTfqFols9qFdupILNsmlXLtjJiV51TB1layVCEBRr9wntvEJEkoKEiHEvZtcxYuSkWihFOp/leImmMyRgPaVfTEAdUuuns3Ck61soA+ZHQFSo0U79PpDiQchJ4ujPAaiR/e5n4l9dNlF9zUxbGiaIhmS/yE45UhLLf0YAJShSfaIKJYPpWREZYYKJ0Qnkdwten6H/SssuW5jdOsX6+iCMHh3AEJbCgCnW4hAY0gcAYHuAJno3YeDRejNd565KxmDmAHzDePgHmf46g</latexit>
Beyond
• Same energies, but not BPS= black hole physics
• Thermal field theory at high T
• Much less is known
• Specific heat of free field theory is of the correct order if magnitude to compare to specific heat of black hole
Other results
• Wilson loops and localization (different kind of matrix models)
• Helicity methods (integrability in S-matrix)
• Other CFT’s: ABJM, SUSY flows
• HKLL bulk reconstruction methods.
Conclusion
• We know a lot of pieces of the dictionary
• Still a lot to learn
• Black holes and horizon dynamics are still a mystery.