We start with a simple remark about amplitudes The 1 1 amplitude in - PowerPoint PPT Presentation

We start with a simple remark about amplitudes

The 1 à 1 amplitude in String Theory

<latexit sha1_base64="jwbfl/j3VZcjoR673pAQOwX1R0=">ACBXicbVDLSsNAFJ3UV42vqMtuBlvRVUnqQjdC0Y3LCvYBTSw30k7dPJgZiKU0IUrP8WVoCBu/QhX/o3TNgtPXDhzDn3MvceP+FMKtv+Ngorq2vrG8VNc2t7Z3fP2j9oyTgVhDZJzGPR8UFSziLaVEx2kEhdDntO2Prqd+4EKyeLoTo0T6oUwiFjACgt9awSruDOvY0vsQs8GcIJHumXqyDFlZ5Vtqv2DHiZODkpoxyNnvXl9mOShjRShIOUXcdOlJeBUIxwOjHdVNIEyAgGtKtpBCGVXjY7YoKPtdLHQSx0RQrP1N8TGYRSjkNfd4aghnLRm4r/ed1UBRdexqIkVTQi84+ClGMV42kiuM8EJYqPNQEimN4VkyEIErnZpo6BWfx5mXSqlWds2rtlauX+V5FEJHaFT5KBzVEc3qIGaiKBH9Ixe0ZvxZLwY78bHvLVg5DOH6A+Mzx+cGJU+</latexit> <latexit sha1_base64="cGn1xMe4IUJXLiqNUil7Xig0evc=">ADL3icbVJNj9MwE3C1xI+tgtHLiMqlvSwVRIOcFlpgR4cCgs7Vaq28hx3a5VxwmOU6lE+UWc+CmIAwIkxJV/wbQNULqM5Oj5zcwbz2TiTIrc+P4X27l0+crVa3vX3Rs3b93ebxzc6edpoRnvsVSmehDTnEuheM8I/kg05wmseRn8fz5yn+24DoXqXpjlhkfJXSmxFQwapCKDuwOiflMqJK/VRruqygJIxKeFpFZQDEpBABXB4fAhEKAMlQGcAJEVRWKTS656+9MIWtDBm7Z+MQ3gH/T+XFQbg4xKO4LSCY3AJ0y7KubVcB7bkwIPXWFQkPIeF6AuFu97fgs/GD81y9ZfLaJSQ2e/9VBqU3MQ+bXg5k4MLVYteCGQTLTGZecowOQJl4Z6QBacwTwKULKG2A2RVM0k361M9Jp+iD1wNdkaWdRo+m1/bXARBDVoWrV1o8YnMklZkXBlmKR5Pgz8zIxKqo1gklcuKXKeUTanMz5EqCgOZFSuf3cFD5CZwDTVeLDBNbudUdIkz5dJjJEJNef5rm9F/s83LMz0yagUKisMV2xTaFpIwCVY7Q5MhObMyCUCyrTAtwI7p5oygxvmujiFYLfni6AftoNH7fBV2Dx5Vs9jz7pn3bc8K7AeWyfWC6tr9Sxmv7c/2l/tb84H57Pz3fmxCXsOueu9Y85P38B4X3y3w=</latexit> <latexit sha1_base64="X5bt+WEe59aq2GM21znbZWpMvnA=">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</latexit> The 1 à 1 amplitude in String Theory DX Z Z Z d 2 zV e � S = d 2 zV = A 1 ! 1 vol ( PSL (2)) DX Z vol ( R ⇥ U (1)) V (0) V ( 1 ) e � S = R dX 0 d ⌧ (2 ⇡ ) D � 1 � ( ~ k 1 � ~ k 2 ) h V (0) V ( 1 ) i 0 = R Energy is automatically conserved if we have on shell operators V. X 0 = α 0 k 0 τ A 1 → 1 = 2 k 0 (2 ⇡ ) D − 1 � D − 1 ( ~ k 1 − ~ k 2 ) Similar story for two point functions in AdS.

Now to the topic of the talk

Symmetries Near the Horizon Juan Maldacena Institute for Advanced Study

Based on work with: Henry Lin Ying Zhao

Preliminaries

Black holes and quantum systems

Basic Assumption (central dogma) • A black hole seen from the outside can be described as a quantum system with order S degrees of freedom (qubits) (coupled to the rest of spacetime) = S = Area T Hawking 4 G N

Geometry of a Black Hole made from collapse Singularity Oppenheimer Snyder 1939 interior horizon star One exterior, one interior.

Full Schwarzschild solution singularity Eddington, Lemaitre, Einstein, Rosen, Finkelstein, Kruskal ER Right Left exterior exterior Vacuum solution. No exotic matter. Two exteriors, sharing the interior.

If one black hole = quantum system, What do these two connected black holes correspond to ?

Wormhole and entangled states Connected through the interior = W. Israel J.M. Entangled In a particular entangled state

<latexit sha1_base64="hAi71xbXN/JyWxrl50j5pfN4yGE=">ACD3icbVDLSgMxFM3UVx1foy7dXGwLdWGZGRe6LrpsoJ9QKeUTJpQzMPkoxQhn6BKz/FlaAgbl278m9M21lo9UDC4ZxzSe7xE86ksu0vo7C2vrG5Vdw2d3b39g+sw6O2jFNBaIvEPBZdH0vKWURbilOu4mgOPQ57fiTm7nfuadCsji6U9OE9kM8iljACFZaGliVcgaeTxUGL9Y5cMFLGMygCo2BgHN98zMoD6ySXbMXgL/EyUkJ5WgOrE9vGJM0pJEiHEvZc+xE9TMsFCOczkwvlTBZIJHtKdphEMq+9linRlUtDKEIBb6RAoW6s+JDIdSTkNfJ0OsxnLVm4v/eb1UBVf9jEVJqmhElg8FKQcVw7wbGDJBieJTARTP8VyBgLTJRu0DR1C87qzn9J2605FzX31i3Vr/M+iugEnaIqctAlqMGaqIWIugBPaEX9Go8Gs/Gm/G+jBaMfOY/YLx8Q1Z15jQ</latexit> The near horizon region Left exterior Right exterior X e − β E n / 2 | ¯ | TFD i = E n i L | E n i R n Boost = exact symmetry = β 2 π ( H r − H l ) (No simple bulk lattice discretizations in gravity)

<latexit sha1_base64="VRNju2j0WUridAdq3knxDqE6MGk=">ACWXicbVFNaxsxENVu8+FuvpzmIuICRQKZtc9pEeTXHJMIY4NXsfMymNHWCtpdmCWew/WXoI/Sm9VHb2kMYZEPN4740+nrJCSUdx/ByEH3Z29/YbH6ODw6Pjk+bpwdnSiuwJ4wydpCBQyU19kiSwkFhEfJMYT+b36z1/k+0Thp9T4sCRznMtJxKAeSpcdPwNMOZ1BX+0GAtLJZ8BjzlMymf+FVanMujHYEmpacp8Ly1SryLRo8JrUx2Tam2hDMohT15NXe0bjZitvxpvg2SGrQYnXdjZu/0okRZY6ahALnhklc0KgCS1IoXEZp6bAMYcZDj3UkKMbVZtglvzSMxM+NdYvTXzDvp6oIHdukWfemQM9ubfamnxPG5Y0/TaqpC5KQi1eDpqWivs01inzibQoSC08AGlvysXT2BkP+LaJ1C8vbN2+Ch06+tjvfO63udZ1Hg52zC/aZJeyKdktu2M9Jthv9jfYCXaDP2EQNsLoxRoG9cwZ+6/Cs38fvrG</latexit> Two other approximate symmetries X 0 X 1 Left exterior Right exterior X 0 → X 0 + constant E = global time translation symmetry P = Spatial translation X 1 → X 1 + constant

This talk will be about these two other (approximate) symmetries X 0 They move us behind the horizon They move us from one side to the other X 1 Left exterior Right exterior

• We will discuss this for near extremal black holes. • These are described by Nearly-AdS 2 gravity. • A similar structure appears in the SYK model. • We will now review both cases.

Review of nearly AdS 2 gravity

Near extremal black holes M ≥ Q M ∼ Q N-AdS 2 x S 2 horizon

<latexit sha1_base64="TM2gil7N0TNGD4nmOYLp06B0vqU=">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</latexit> Nearly-AdS 2 gravity Jackiw-Teitelboim Almheiri-Polchinski Z � Z Z Z S = φ 0 R + 2 + φ ( R + 2) + 2 φ b K + S m [ g µ ν , χ ] K Fixes Only extermal entropy S 0 Metric to AdS 2 Matter moves in a rigid AdS 2 spacetime Boundary becomes dynamical = boundary graviton = only physical degree of freedom for gravity

The surprisingly simple gravitational dynamics of N-AdS 2 NAdS 2 = AdS 2 + location of boundary AdS 2 ds 2 = − d τ 2 + d σ 2 matter sin 2 σ Dynamics of the boundary is SL(2) invariant. Proper time along the boundary = time of the asymptotically flat region = time of the quantum system

<latexit sha1_base64="9KaCP4Ku/Gh4UEqlb86oj5/isRk=">ACLHicbZC7TsMwFIadcivhFmBksWiQ2qUkYCxKksHhiLoRWqryHd1qpzke0gVHeh4lHYQEJEGLlOXDaDNDyS5Z+fecHZ/fixgV0rI+tMLa+sbmVnFb39nd2z8wDo/aIow5Ji0cspB3PSQIowFpSoZ6UacIN9jpONr7N654FwQcPgXs4iMvDROKAjipFUyDXq0DRhOYF9jBhspC6DfUl9ImCyIDB1/SWUuhxWzu9uyk7FHZum7holq2rNBVeNnZsSyNV0jZf+MSxTwKJGRKiZ1uRHCSIS4oZSfV+LEiE8BSNSU/ZAKnlg2R+awrPFBnCUcjVCySc098TCfKFmPme6vSRnIjlWgb/q/ViOboaJDSIYkCvFg0ihmUIcyCg0PKCZspgzCnKq/QjxBHGp4tWzFOzlm1dN26naF1Xn1inV6nkeRXACTkEZ2OAS1EADNELYPAInsEbeNetFftU/tatBa0fOY/JH2/QMW86Ob</latexit> ds 2 = − d τ 2 + d σ 2 sin 2 σ AdS 2 Simplest case = Boundaries are infinitely far away. matter Matter moves effectively in all of AdS 2 and is not affected by the motion of the boundary. The boundary is still dynamical and tells us how to translate to the physical boundary time = time of the dual quantum system. Then the Hilbert space splits as: ( H l × H m × H r ) /SL (2) g Common motion of the three of them is not physical

<latexit sha1_base64="izWPX92KH3Q13YxVrxFXVEk7yI=">ACBnicbVC7SgNBFJ2Nr7i+Vi21GEwCNobdWGgjBG0sI5iHJGu4O5lNhsw+mJkVwpLGyk+xEhTE1n+w8m+cTVJo4oELh3Pu5d57vJgzqWz728gtLa+sruXzY3Nre0da3evIaNEFonEY9EywNJOQtpXTHFaSsWFAKP06Y3vMr85gMVkXhrRrF1A2gHzKfEVBa6lqHuFjEHaqgm4I3xnf3oMvDFydOsWh2rYJdtifAi8SZkQKaoda1vjq9iCQBDRXhIGXbsWPlpiAUI5yOzU4iaQxkCH3a1jSEgEo3nXwxiWt9LAfCV2hwhP190QKgZSjQF9ZCkAN5LyXif957UT527KwjhRNCTRX7CsYpwFgnuMUGJ4iNgAimb8VkAKI0sGZWQrO/M+LpFEpO6flyk2lUL2c5ZFHB+gIHSMHnaEqukY1VEcEPaJn9IrejCfjxXg3PqatOWM2s4/+wPj8ATvylYs=</latexit> <latexit sha1_base64="iVP3wu5qDp/mt4VolGE+1EXF098=">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</latexit> • Each of the three systems is described by an SL(2) g invariant action. • The matter moves in a rigid AdS 2 • Each boundary is like a massive particle with spin (or in an electric field in AdS 2 ) η ab Y a Y b = − 1 Embedding coordinates Rescaled embedding coordinates X · ˙ ˙ X a ( u ) , X · X = 0 X = − 1 Proper time along the boundary = time of the dual quantum system.