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

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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)

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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)

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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

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sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yu</latexit><latexit 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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

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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

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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.