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 <latexit sha1_base64="dwjOTb3RbH0JYMRdX3tlywlkgXQ=">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</latexit> 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,¯ φi=1,...4,Aµ J=1,...4 ¯↵˙ , <latexit sha1_base64="D02ycfgCBAT+iB3lYvs/3MPN0yM=">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</latexit> ↵ 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) <latexit sha1_base64="bSv7JiK7qZzAvesk8k0YvWdQM9E=">AAACA3icdVDLSgMxFM3UV62vqjvdBItQQcrMWNq6K7pxWaEvbMchk6ZtaCYzJhmhjAU3/oobF4q49Sfc+TdmbAUVPRA4nHMPN/d4IaNSmea7kZqbX1hcSi9nVlbX1jeym1tNGUQCkwYOWCDaHpKEUU4aiipG2qEgyPcYaXmj08RvXRMhacDrahwSx0cDTvsUI6UlN7tTF3l4cckPbtyYdwfkCtqHsNsLlJy42ZxZOK6U7GIJmgXTLFu2lRC7XDwqQksrCXJghpqbfdNBHPmEK8yQlB3LDJUTI6EoZmSS6UaShAiP0IB0NOXIJ9KJP2+YwH2t9GA/EPpxBT/V74kY+VKOfU9P+kgN5W8vEf/yOpHqV5yY8jBShOPpon7EoApgUgjsUUGwYmNNEBZU/xXiIRIIK11bRpfwdSn8nzTtgqX5eTFXPZnVkQa7YA/kgQXKoArOQA00AAa34B48gifjzngwno2X6WjKmGW2wQ8Yrx+Wmpbc</latexit> 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 φiZ − φi0 ) X<latexit sha1_base64="1IEKM8ggKUesZ+JeCujQkJhJBHQ=">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</latexit> 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) <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/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 <latexit sha1_base64="2ZQovfQXXK9s8Cn/AFbm3OG8DDI=">AAACCHicdZDLSsNAFIYnXmu9RV26cLAIglCSUtGNUHTjsoK9QJuGyXTSDp1MwsxESEOXbnwVNy4UcesjuPNtnLQpqOgPAz/fOYcz5/ciRqWyrE9jYXFpeWW1sFZc39jc2jZ3dpsyjAUmDRyyULQ9JAmjnDQUVYy0I0FQ4DHS8kZXWb11R4SkIb9VSUScAA049SlGSiPXPBjDrocEHMMT2JVx4Kb0wp70qjDpVVxtXbNklU+tTNAqW3OTEzsnJZCr7pof3X6I44BwhRmSsmNbkXJSJBTFjEyK3ViSCOERGpCOthwFRDrp9JAJPNKkD/1Q6McVnNLvEykKpEwCT3cGSA3l71oG/6p1YuWfOynlUawIx7NFfsygCmGWCuxTQbBiiTYIC6r/CvEQCYSVzq6oQ5hfCv83zUrZ1v6mWqpd5nEUwD44BMfABmegBq5BHTQABvfgETyDF+PBeDJejbdZ64KRz+yBHzLevwDQTZfr</latexit> i=1 X Consider a 3-sphere at fixed z and move it as follows in time z(t)=z exp(it) <latexit sha1_base64="A/UxrIlG2UqeY/zKDZBlrRddy0o=">AAAB/XicdVBNSwJRFH1jX2Zf9rFr80gC3cjMJGqLQGrT0iA10GF483zqwzcfvHcnUpH+SpsWRbTtf7Tr3/RGDSrqwIXDOfdy7z1eJLgC0/wwUkvLK6tr6fXMxubW9k52d6+pwlhS1qChCOWNRxQTPGAN4CDYTSQZ8T3BWt7wIvFbt0wqHgbXMIqY45N+wHucEtCSmz0Y56GAz/DYNXGH3UV5jqHgZnNm8bRatktlbBZNs2LZVkLsSumkhC2tJMihBepu9r3TDWnsswCoIEq1LTMCZ0IkcCrYNNOJFYsIHZI+a2saEJ8pZzK7foqPtdLFvVDqCgDP1O8TE+IrNfI93ekTGKjfXiL+5bVj6FWdCQ+iGFhA54t6scAQ4iQK3OWSURAjTQiVXN+K6YBIQkEHltEhfH2K/ydNu2hpflXK1c4XcaTRITpCeWShCqqhS1RHDUTRGD2gJ/Rs3BuPxovxOm9NGYuZffQDxtsn4BeTkQ==</latexit> 0 z = cos(✓)exp(iφ) <latexit sha1_base64="ErrbacEToOr+YriCTWQOgIoLKoI=">AAACBXicdVA9SwNBEN3zM8avqKUWi0GITbg7Q6KFELSxVDAmkAthbzMxS/Y+2J0T42Fj41+xsVDE1v9g579xTyOo6IOBx3szzMzzYyk02vabNTE5NT0zm5vLzy8sLi0XVlbPdJQoDg0eyUi1fKZBihAaKFBCK1bAAl9C0x8eZn7zApQWUXiKoxg6ATsPRV9whkbqFjau6D71eKRLHg4A2Tb14DIuCS8eiO1uoWiX93arbqVK7bJt1xzXyYhbq+xUqGOUDEUyxnG38Or1Ip4EECKXTOu2Y8fYSZlCwSVc571EQ8z4kJ1D29CQBaA76ccX13TLKD3aj5SpEOmH+n0iZYHWo8A3nQHDgf7tZeJfXjvB/m4nFWGcIIT8c1E/kRQjmkVCe0IBRzkyhHElzK2UD5hiHE1weRPC16f0f3Lmlh3DTyrF+sE4jhxZJ5ukRBxSI3VyRI5Jg3ByQ+7IA3m0bq1768l6/mydsMYza+QHrJd3zQCXgA==</latexit> ˙ φ<latexit sha1_base64="Qw7ebSUcFqFlDjRe39TChl9xSuw=">AAAB83icdVDLSsNAFJ34rPVVdelmsAiuSlJDWxdC0Y3LCvYBTSiT6aQdOpmEmRuhhP6GGxeKuPVn3Pk3TtoKKnpg4HDOPdw7J0gE12DbH9bK6tr6xmZhq7i9s7u3Xzo47Og4VZS1aSxi1QuIZoJL1gYOgvUSxUgUCNYNJte5371nSvNY3sE0YX5ERpKHnBIwkucNY8BeMub40hmUynblolGrujVsV2y77lSdnFTr7rmLHaPkKKMlWoPSu4nTNGISqCBa9x07AT8jCjgVbFb0Us0SQidkxPqGShIx7Wfzm2f41ChDHMbKPAl4rn5PZCTSehoFZjIiMNa/vVz8y+unEDb8jMskBSbpYlGYCgwxzgvAQ64YBTE1hFDFza2YjokiFExNRVPC10/x/6RTrTiG37rl5tWyjgI6RifoDDmojproBrVQG1GUoAf0hJ6t1Hq0XqzXxeiKtcwcoR+w3j4BM8eRIg==</latexit> =1 Sits at origin of global AdS coordinates. ⇢<latexit sha1_base64="3UAsdzuJ9FQ9O/8QIOSp6bI+u4E=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4Kpk6tHUhFN24rGAf0A4lk2ba2EwyJBmhDP0HNy4Ucev/uPNvzLQVVPTAhcM593LvPUHMmTYIfTi5ldW19Y38ZmFre2d3r7h/0NYyUYS2iORSdQOsKWeCtgwznHZjRXEUcNoJJleZ37mnSjMpbs00pn6ER4KFjGBjpXZfjeUFGhRLqHxer1a8KkRlhGpuxc1IpeadedC1SoYSWKI5KL73h5IkERWGcKx1z0Wx8VOsDCOczgr9RNMYkwke0Z6lAkdU++n82hk8scoQhlLZEgbO1e8TKY60nkaB7YywGevfXib+5fUSE9b9lIk4MVSQxaIw4dBImL0Oh0xRYvjUEkwUs7dCMsYKE2MDKtgQvj6F/5N2pexafuOVGpfLOPLgCByDU+CCGmiAa9AELUDAHXgAT+DZkc6j8+K8LlpzznLmEPyA8/YJcl2PCg==</latexit> =0 Dual giant: analytic continuation ⇠⇠¯ y˜2 = 1 − i − <latexit sha1_base64="T75+gMRpB8rOBI5nXmmMpD5gIus=">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</latexit> i ⇠ = coshX⇢ exp(it) <latexit sha1_base64="cF/kEWmm9Iau4L5am8nxnsb+4yo=">AAACAnicdZDLSgMxFIYz9VbrbdSVuAkWoW7KjCi6EYpuXFawF+gMJZNmOqGZZEgy0lKKG1/FjQtF3PoU7nwbM+0UVPSHwM93zuHk/EHCqNKO82kVFhaXlleKq6W19Y3NLXt7p6lEKjFpYMGEbAdIEUY5aWiqGWknkqA4YKQVDK6yeuuOSEUFv9WjhPgx6nMaUoy0QV17zxvSC+hhoSLoyUhAjwyTCoX6qGuXneqpkwk6VWducuLmpAxy1bv