David Berenstein, UCSB, Durban, 1-15-19

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

3 =4SY M on S R N ⇥ $ Type IIB superstring on (global) AdS S5 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 5 ⇥

Original Maldacena paper hep-th/9711200 Basic dictionary for simplest case AdS/CFT is a strStandardong weak-coupling Table duality for the ‘t Hooft coupling constant.

AdS CFT Isometries Global symmetries 4 2 R gY M N Flux = N Gauge group U(N) State State

We want to explore CFT at large values of R -large ‘t Hooft coupling- but with g ﬁxed and small. Field content

Z, Z,¯ i=1,...4,Aµ J=1,...4, ¯↵˙

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 ↵ J=1,...4

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 = trace

BPS particle maximizes SO(6) R-charge given dimension

Tr(Zn)

AAACA3icdVDLSgMxFM3UV62vqjvdBItQQcrMWNq6K7pxWaEvbMchk6ZtaCYzJhmhjAU3/oobF4q49Sfc+TdmbAUVPRA4nHMPN/d4IaNSmea7kZqbX1hcSi9nVlbX1jeym1tNGUQCkwYOWCDaHpKEUU4aiipG2qEgyPcYaXmj08RvXRMhacDrahwSx0cDTvsUI6UlN7tTF3l4cckPbtyYdwfkCtqHsNsLlJy42ZxZOK6U7GIJmgXTLFu2lRC7XDwqQksrCXJghpqbfdNBHPmEK8yQlB3LDJUTI6EoZmSS6UaShAiP0IB0NOXIJ9KJP2+YwH2t9GA/EPpxBT/V74kY+VKOfU9P+kgN5W8vEf/yOpHqV5yY8jBShOPpon7EoApgUgjsUUGwYmNNEBZU/xXiIRIIK11bRpfwdSn8nzTtgqX5eTFXPZnVkQa7YA/kgQXKoArOQA00AAa34B48gifjzngwno2X6WjKmGW2wQ8Yrx+Wmpbc n 2,... | 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 10D Can’t be lifted: does not allow anomalous dimension.

THE FACT THAT IT STARTS AT n=2 is that the spin of particle in 10D is 2 Convenient to expand Near BPS state

k J k f(k)Tr(Z iZ i0 ) 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 k

Interpreted as defects moving in a sea of Z

Anomalous dimensions can be computed perturbatively and the gap scales as powers of 1/J The interaction termThe interaction term + planarity g2 Tr([Z, φj][Z,¯ φj]) ∼ YM will make the defects hop to the left or right in the Defects hop in between Z: operator. nearest neighbor hopping.

Can be mapped to spin chains.

The idea now is to interpret the position of the defects in between the Z as a lattice. The number of sites is of 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 the same defect at diﬀerent positions.

20 Spin chain program

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) 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)

AAAB8XicdVBNSwMxEM3Wr1q/qh69BItQL2V3Xdp6K3rxJBVsK7ZLyabZNjSbrElWKEv/hRcPinj133jz35htK6jog4HHezPMzAtiRpW27Q8rt7S8srqWXy9sbG5t7xR399pKJBKTFhZMyJsAKcIoJy1NNSM3sSQoChjpBOPzzO/cE6mo4Nd6EhM/QkNOQ4qRNtKtKPfUndTw8rhfLNmV03rV9arQrth2zXGdjLg178SDjlEylMACzX7xvTcQOIkI15ghpbqOHWs/RVJTzMi00EsUiREeoyHpGspRRJSfzi6ewiOjDGAopCmu4Uz9PpGiSKlJFJjOCOmR+u1l4l9eN9Fh3U8pjxNNOJ4vChMGtYDZ+3BAJcGaTQxBWFJzK8QjJBHWJqSCCeHrU/g/absVx/Arr9Q4W8SRBwfgEJSBA2qgAS5AE7QABhw8gCfwbCnr0XqxXuetOWsxsw9+wHr7BPikkHQ= 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/0003075 Grisaru, Myers, Tajford hep-th/0008015 Hashimoto, Hirano, Itzhaki hep-th/0008016 Simple description. Write the 5 sphere in cartesian coordinates.

4 2 zz¯ + yi =1

AAACCHicdZDLSsNAFIYnXmu9RV26cLAIglCSUtGNUHTjsoK9QJuGyXTSDp1MwsxESEOXbnwVNy4UcesjuPNtnLQpqOgPAz/fOYcz5/ciRqWyrE9jYXFpeWW1sFZc39jc2jZ3dpsyjAUmDRyyULQ9JAmjnDQUVYy0I0FQ4DHS8kZXWb11R4SkIb9VSUScAA049SlGSiPXPBjDrocEHMMT2JVx4Kb0wp70qjDpVVxtXbNklU+tTNAqW3OTEzsnJZCr7pof3X6I44BwhRmSsmNbkXJSJBTFjEyK3ViSCOERGpCOthwFRDrp9JAJPNKkD/1Q6McVnNLvEykKpEwCT3cGSA3l71oG/6p1YuWfOynlUawIx7NFfsygCmGWCuxTQbBiiTYIC6r/CvEQCYSVzq6oQ5hfCv83zUrZ1v6mWqpd5nEUwD44BMfABmegBq5BHTQABvfgETyDF+PBeDJejbdZ64KRz+yBHzLevwDQTZfr i=1 X Consider a 3-sphere at ﬁxed z and move it as follows in time z(t)=z exp(it) AAAB/XicdVBNSwJRFH1jX2Zf9rFr80gC3cjMJGqLQGrT0iA10GF483zqwzcfvHcnUpH+SpsWRbTtf7Tr3/RGDSrqwIXDOfdy7z1eJLgC0/wwUkvLK6tr6fXMxubW9k52d6+pwlhS1qChCOWNRxQTPGAN4CDYTSQZ8T3BWt7wIvFbt0wqHgbXMIqY45N+wHucEtCSmz0Y56GAz/DYNXGH3UV5jqHgZnNm8bRatktlbBZNs2LZVkLsSumkhC2tJMihBepu9r3TDWnsswCoIEq1LTMCZ0IkcCrYNNOJFYsIHZI+a2saEJ8pZzK7foqPtdLFvVDqCgDP1O8TE+IrNfI93ekTGKjfXiL+5bVj6FWdCQ+iGFhA54t6scAQ4iQK3OWSURAjTQiVXN+K6YBIQkEHltEhfH2K/ydNu2hpflXK1c4XcaTRITpCeWShCqqhS1RHDUTRGD2gJ/Rs3BuPxovxOm9NGYuZffQDxtsn4BeTkQ== 0 z = cos(✓)exp(i)

AAACBXicdVA9SwNBEN3zM8avqKUWi0GITbg7Q6KFELSxVDAmkAthbzMxS/Y+2J0T42Fj41+xsVDE1v9g579xTyOo6IOBx3szzMzzYyk02vabNTE5NT0zm5vLzy8sLi0XVlbPdJQoDg0eyUi1fKZBihAaKFBCK1bAAl9C0x8eZn7zApQWUXiKoxg6ATsPRV9whkbqFjau6D71eKRLHg4A2Tb14DIuCS8eiO1uoWiX93arbqVK7bJt1xzXyYhbq+xUqGOUDEUyxnG38Or1Ip4EECKXTOu2Y8fYSZlCwSVc571EQ8z4kJ1D29CQBaA76ccX13TLKD3aj5SpEOmH+n0iZYHWo8A3nQHDgf7tZeJfXjvB/m4nFWGcIIT8c1E/kRQjmkVCe0IBRzkyhHElzK2UD5hiHE1weRPC16f0f3Lmlh3DTyrF+sE4jhxZJ5ukRBxSI3VyRI5Jg3ByQ+7IA3m0bq1768l6/mydsMYza+QHrJd3zQCXgA==

AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg== =1 Sits at origin of global AdS coordinates.

⇢AAAB7XicdVDLSgMxFM3UV62vqks3wSK4Kpk6tHUhFN24rGAf0A4lk2ba2EwyJBmhDP0HNy4Ucev/uPNvzLQVVPTAhcM593LvPUHMmTYIfTi5ldW19Y38ZmFre2d3r7h/0NYyUYS2iORSdQOsKWeCtgwznHZjRXEUcNoJJleZ37mnSjMpbs00pn6ER4KFjGBjpXZfjeUFGhRLqHxer1a8KkRlhGpuxc1IpeadedC1SoYSWKI5KL73h5IkERWGcKx1z0Wx8VOsDCOczgr9RNMYkwke0Z6lAkdU++n82hk8scoQhlLZEgbO1e8TKY60nkaB7YywGevfXib+5fUSE9b9lIk4MVSQxaIw4dBImL0Oh0xRYvjUEkwUs7dCMsYKE2MDKtgQvj6F/5N2pexafuOVGpfLOPLgCByDU+CCGmiAa9AELUDAHXgAT+DZkc6j8+K8LlpzznLmEPyA8/YJcl2PCg== =0 Dual giant: analytic continuation

⇠⇠¯ y˜2 = 1 i

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 i ⇠ = coshX⇢ exp(it)

AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv2h9cTOI0J15ghpTquk2h/jKSmmJFJyUsVSRAeoD7pGMtRTJQ/np4wgYeG9GAopHlcwyn9PjFGsVKjODCdMdKR+l3L4F+1TqrDc39MeZJqwvFsUZgyqAXM8oA9KgnWbGQMwpKav0IcIYmwNqmVTAjzS+H/pnlcdY2/OSnXLvM4imAfHIAKcMEZqIFrUAcNgME9eATP4MV6sJ6sV+tt1lqw8pld8EPW+xd6H5Yusha1_base64="ck8pdC+ekZH4nUmSP+ZG7r8lEyk=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odOn4MoA7ncAFXEMIN3MEDdKALAhJ4hXdv4r15H6uuat66tDP4I+/zBzjGijg=sha1_base64="32Jo/CzqMvY+HKRgTOn0PWRR5wo=">AAAB93icbZBLSwMxFIXv+Ky16uhO3ASLUDdlxo1uBMGNywr2AZ2hZNJMJzSTDElGWobixr/ixoUi/hN3/hvTx0JbDwQ+zkm4uSfKONPG876dtfWNza3t0k55t7K3f+AeVlpa5orQJpFcqk6ENeVM0KZhhtNOpihOI07b0fB2mrcfqdJMigczzmiY4oFgMSPYWKvnHgcjdo0CInWCApVIFNBRVmPInPfcqlf3ZkKr4C+gCgs1eu5X0JckT6kwhGOtu76XmbDAyjDC6aQc5JpmmAzxgHYtCpxSHRazFSbozDp9FEtljzBo5v5+UeBU63Ea2ZspNolezqbmf1k3N/FVWDCR5YYKMh8U5xwZiaZ9oD5TlBg+toCJYvaviCRYYWJsa2Vbgr+88iq0Luq+5XsPSnACp1ADHy7hBu6gAU0g8AQv8AbvzrPz6nzM61pzFr0dwR85nz/2zJS3sha1_base64="ZdZpD4kWHpDg34n7mB1VVQwx4kU=">AAAB93icdZBPS8MwGMbfzn9zTq3exEtwCPMyOkH0IghePE5wf2AtI83SNSxtSpLKRhle/CpePCjiN/HmtzHdOlDRBwIPvyfhzfv4CWdKO86nVVpZXVvfKG9WtqrbO7v2XrWjRCoJbRPBhez5WFHOYtrWTHPaSyTFkc9p1x9f53n3nkrFRHynpwn1IjyKWcAI1gYN7AN3wi6RS4QKkStDgVw6SeoM6ZOBXXMaZ04u5DScpSlIsyA1KNQa2B/uUJA0orEmHCvVbzqJ9jIsNSOczipuqmiCyRiPaN/YGEdUedl8hRk6NmSIAiHNiTWa0+8vMhwpNY18czPCOlS/sxz+lfVTHVx4GYuTVNOYLAYFKUdaoLwPNGSSEs2nxmAimfkrIiGWmGjTWsWUsNwU/W86p42m8bcOlOEQjqAOTTiHK7iBFrSBwAM8wQu8Wo/Ws/W2qKtkFb3tww9Z718E/5TBsha1_base64="CTa2+xbyceepiJsiPBofYdZTmaM=">AAACAnicdZDLSsNAFIYn9VbrLepK3AwWoW7KRBDdCEU3LivYCzShTKaTZugkE2Ym0hKKG1/FjQtF3PoU7nwbJ20EFf1h4Oc753Dm/H7CmdIIfVilhcWl5ZXyamVtfWNzy97eaSuRSkJbRHAhuz5WlLOYtjTTnHYTSXHkc9rxR5d5vXNLpWIivtGThHoRHsYsYARrg/r2njtm59AlQoXQlaGALh0nNQb1Ud+uovoJygVRHX2ZgjgFqYJCzb797g4ESSMaa8KxUj0HJdrLsNSMcDqtuKmiCSYjPKQ9Y2McUeVlsxOm8NCQAQyENC/WcEa/T2Q4UmoS+aYzwjpUv2s5/KvWS3Vw5mUsTlJNYzJfFKQcagHzPOCASUo0nxiDiWTmr5CEWGKiTWoVE8LXpfB/0z6uO8Zfo2rjooijDPbBAagBB5yCBrgCTdACBNyBB/AEnq1769F6sV7nrSWrmNkFP2S9fQJ435Yq

Sits at “origin” in z

✓AAAB73icbZA9SwNBEIbn4leMX1FLm8UgWIU7EbQRgjaWEUwMJEfY28wlS/Y+3J0TQsifsLFQxNa/Y+e/cZNcoYkvLDy8M8POvEGqpCHX/XYKK6tr6xvFzdLW9s7uXnn/oGmSTAtsiEQluhVwg0rG2CBJClupRh4FCh+C4c20/vCE2sgkvqdRin7E+7EMpeBkrVaHBkj8yu2WK27VnYktg5dDBXLVu+WvTi8RWYQxCcWNaXtuSv6Ya5JC4aTUyQymXAx5H9sWYx6h8cezfSfsxDo9FibavpjYzP09MeaRMaMosJ0Rp4FZrE3N/2rtjMJLfyzjNCOMxfyjMFOMEjY9nvWkRkFqZIELLe2uTAy45oJsRCUbgrd48jI0z6qe5bvzSu06j6MIR3AMp+DBBdTgFurQAAEKnuEV3pxH58V5dz7mrQUnnzmEP3I+fwCYO4+p =0

For giants:

E = N(1 z 2)=N(1 cos2(✓)) AAACDHicdVDLSgMxFM34rPVVdekmWIR2YZkpim6EogiupIJ9QKctmTRtQzMPkjtCnfYD3Pgrblwo4tYPcOffmGmnoKIHAifnnEtyjxMIrsA0P425+YXFpeXUSnp1bX1jM7O1XVV+KCmrUF/4su4QxQT3WAU4CFYPJCOuI1jNGZzHfu2WScV97waGAWu6pOfxLqcEtNTOZC9O8VXOOhjdjVrFPJ5ebOqrVjFnQ58Byed1yiwcmTGwWTBnJFGsRMmiBOV25sPu+DR0mQdUEKUalhlAMyISOBVsnLZDxQJCB6THGpp6xGWqGU2WGeN9rXRw15f6eIAn6veJiLhKDV1HJ10CffXbi8W/vEYI3ZNmxL0gBObR6UPdUGDwcdwM7nDJKIihJoRKrv+KaZ9IQkH3l9YlzDbF/5NqsWBpfn2YLZ0ldaTQLtpDOWShY1RCl6iMKoiie/SIntGL8WA8Ga/G2zQ6ZyQzO+gHjPcvhVWYIg== | |

For dual giants:

E = N( ⇠ 2 1) = N(cosh2(⇢) 1) AAACDXicdZDLSgMxFIYz9VbrbdSlm2AV2oVlpii6EYoiuJIK9gKdtmTStBOamQxJRixtX8CNr+LGhSJu3bvzbcy0U1DRHwJ/vnMOyfndkFGpLOvTSM3NLywupZczK6tr6xvm5lZV8khgUsGccVF3kSSMBqSiqGKkHgqCfJeRmts/j+u1WyIk5cGNGoSk6aNeQLsUI6VR29y7OIVXuZFzR0et4oGdh/HVwVx6rWLOER7Pa9g2s1bhyIoFrYI1MwmxE5IFicpt88PpcBz5JFCYISkbthWq5hAJRTEj44wTSRIi3Ec90tA2QD6RzeFkmzHc16QDu1zoEyg4od8nhsiXcuC7utNHypO/azH8q9aIVPekOaRBGCkS4OlD3YhBxWEcDexQQbBiA20QFlT/FWIPCYSVDjCjQ5htCv831WLB1v76MFs6S+JIgx2wC3LABsegBC5BGVQABvfgETyDF+PBeDJejbdpa8pIZrbBDxnvX07jmIo= | | Energies up to

O(N)

AAAB7HicdVBdSwJBFL1rX2ZfVo+9DElgL7Jri9qb1EtPZdCqoCKz46wOzs4uM7OBiL+hlx6K6LUf1Fv/plk1qKgDFw7n3Mu99/gxZ0rb9oeVWVldW9/Ibua2tnd29/L7B00VJZJQj0Q8km0fK8qZoJ5mmtN2LCkOfU5b/vgy9Vv3VCoWiTs9iWkvxEPBAkawNpJ3U0TXp/18wS6d1yplt4Lskm1XnbKTknLVPXORY5QUBVii0c+/dwcRSUIqNOFYqY5jx7o3xVIzwuks100UjTEZ4yHtGCpwSFVvOj92hk6MMkBBJE0Jjebq94kpDpWahL7pDLEeqd9eKv7ldRId1HpTJuJEU0EWi4KEIx2h9HM0YJISzSeGYCKZuRWREZaYaJNPzoTw9Sn6nzTLJcfwW7dQv1jGkYUjOIYiOFCFOlxBAzwgwOABnuDZEtaj9WK9Lloz1nLmEH7AevsEv56N/A== 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

Zi1 ...Zik i0 i0 AAACDHicdZC9TsMwEMcvfJbyFWBksagQTFWCQDBWsDAWiX6INkSO67RWHSeyHaQq6gOw8CosDCDEygOw8TY4bSoBgpOs++t3d7bvHyScKe04n9bc/MLi0nJppby6tr6xaW9tN1WcSkIbJOaxbAdYUc4EbWimOW0nkuIo4LQVDC/yeuuOSsVica1HCfUi3BcsZARrg3y7cnObMd8d+xk7MAl1e7FWaAKHU2iSXXGqJ04eyKk6M1EQtyAVKKLu2x/mGpJGVGjCsVId10m0l2GpGeF0XO6miiaYDHGfdowUOKLKyybLjNG+IT0UxtIcodGEfp/IcKTUKApMZ4T1QP2u5fCvWifV4ZmXMZGkmgoyfShMOdIxyp1BPSYp0XxkBCaSmb8iMsASE238KxsTZpui/0XzqOoafXVcqZ0XdpRgF/bgEFw4hRpcQh0aQOAeHuEZXqwH68l6td6mrXNWMbMDP8J6/wJCKJsd 1 k

Free theory has a symmetry of upper and lower indices

SU(N) SU(N) AAAB/nicdVDLSsNAFL3xWesrKq7cDBahbkoiii6LunAlFU1baEOYTCft0MmDmYlQQsFfceNCEbd+hzv/xkmbgooeGDj3nHu5d46fcCaVZX0ac/MLi0vLpZXy6tr6xqa5td2UcSoIdUjMY9H2saScRdRRTHHaTgTFoc9pyx9e5H7rngrJ4uhOjRLqhrgfsYARrLTkmbu3TvX60HO6ioVUoml16ZkVq3Zi5UBWzZqRQrELpQIFGp750e3FJA1ppAjHUnZsK1FuhoVihNNxuZtKmmAyxH3a0TTCepmbTc4fowOt9FAQC/0ihSbq94kMh1KOQl93hlgN5G8vF//yOqkKztyMRUmqaESmi4KUIxWjPAvUY4ISxUeaYCKYvhWRARaYKJ1YWYcw+yn6nzSParbmN8eV+nkRRwn2YB+qYMMp1OEKGuAAgQwe4RlejAfjyXg13qatc0YxswM/YLx/AbxDlAo= U ⇥ D

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).

Idea: intertwine indices of extra letters with the Z in a column or row to produce open strings spin chains with interesting boundary conditions.

Berenstein, Balasubramanian, Feng, Huang, hep-th/0411205 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, one can associate a central charge to the spin chain (Beisert).

Which shows effectively as follows

Y [Y,Z] ! Evaluating one gets

Q, Q exp(ip) { } /

This is evaluated on a single magnon. Produces a non-trivial dispersion relation for impurity

J 1+h()sin2 p/2 ' q

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

(Hofmann -Maldacena)

S3 S5 ! Same coordinates as before # D2

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

Solutions of type IIB supergravity, that preserve half the SUSY

1 2 y 4 z ds2 = (dt + V dxi)2 + (dy2 + dxidxi) i q 1 z2 y 4 q 1 z 1 + z 2 2 2 ˜ 2 +y d⌦3 + y d⌦3 s 1 + z s 1 z 2 ! 2 ! Lin, Lunin, Maldacena, JHEP 0410 (2004) 025

+ attendant ﬂuxes depend on a single function z(~x, y)

Satisﬁes an ordinary linear PDE @ z @ @ z + y@ y =0 i i y y ✓ ◆ Reality of the metric requires that

1 z | |  2 Degeneration at y=0

Absence of conical singularity forces

1 z(0, ~x)= ±2

This provides a two coloring of the plane.

These are all horizon free smooth 10 D geometries. 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. They have a conserved “ﬂux”

N = ⇢(~x)d2x Z Area of the droplet

An energy function

E = ⇢(~x)~x2d2x E Z Complete classiﬁcation Has energies of order

O(N 2)

AAAB7nicdVDLSgNBEOz1GeMr6tHLYBDiJeyuSxJvQS+eNIJ5QLKG2clsMmT2wcysEJZ8hBcPinj1e7z5N84mEVS0oKGo6qa7y4s5k8o0P4yl5ZXVtfXcRn5za3tnt7C335JRIghtkohHouNhSTkLaVMxxWknFhQHHqdtb3yR+e17KiSLwls1iakb4GHIfEaw0lL7uoSu7uyTfqFols9qFdupILNsmlXLtjJiV51TB1layVCEBRr9wntvEJEkoKEiHEvZtcxYuSkWihFOp/leImmMyRgPaVfTEAdUuuns3Ck61soA+ZHQFSo0U79PpDiQchJ4ujPAaiR/e5n4l9dNlF9zUxbGiaIhmS/yE45UhLLf0YAJShSfaIKJYPpWREZYYKJ0Qnkdwten6H/SssuW5jdOsX6+iCMHh3AEJbCgCnW4hAY0gcAYHuAJno3YeDRejNd565KxmDmAHzDePgHmf46g 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.