Tensor Models for Black Hole Probe

Transcription

1 Tensor Models for Black Hole Probe IAP, Paris, 15th Workshop on Non-Perturbative Quantum Chromodynamics June 12, 2018 Swapnamay Mondal LPTHE, UPMC(Sorbonne Université) (based on arxiv: with Nick Halmagyi)

2 Summary Recently there have been models of near extremal black holes (and possibly examples of NAdS_2/NCFT_1). Such black holes Hawking radiate. Model for a probe interacting with such black holes? We propose a class of tensor models as models of black hole interacting with a probe. Non- trivial features: symmetries of near horizon geometry, black hole (partially) inflicts maximal chaos into the probe.

3 Outline Introduction Our Model Future directions

4 Introduction

5 Near-extremal black holes A. Pattern of symmetry breaking: Near horizon geometry of extremal black holes contains a AdS_2 factor. Near horizon geometry of near-extremal black holes contains a near-ads_2 AdS_2 => Asymptotic symmetries are all reparameterizations of asymptotic boundary circle, but AdS_2 metric preserves only a SL(2, R) subgroup of this. => Spontaneous breaking of reparameterization to SL(2,R). Near AdS_2 => spontaneous and explicit breaking of reparameterization symmetry. B. Maximal Chaos: (Non-extremal) black holes are maximally chaotic systems. explicit breaking of reparameterization symmetry should be related to maximal chaos.

6 <latexit sha1_base64="jrwog46q3nkbz2m79o6iz0w3c9i=">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</latexit> <latexit sha1_base64="jrwog46q3nkbz2m79o6iz0w3c9i=">aaacohicbvdlsgmxfm3uv62vqks3wsj1vwdqqtdcszuiobx6gryomttthmyejbmhhpksn36go3hjqhg3fohpa7xva4fzz7mxm3uckfehtfpjscwsli2vjfdta+sbm1vp7z26cckosq0hlobnbwncqe9qkkpgmienyhmyatid0shv3beuaobx5takhq/1fopsjksw7pr12val6mu1hmew7xke1uwscle36vgoh8ewhqpqk+tgh5rf2e8i68wuwrzb6yyzm8eaf4k1jrkwrcvop7a7ay484kvmkbatywxlryeukwyktrujqukeb6hhwpr6ycoio8ahx/bak13oblw/x8kx+ntciu+ioefotg/jvpj3ruj/xius7mlhut+mjphxzjebmsgdoeordiknwlkhjghzqv8kcr/p+ktooqvdsozp/kvq+zxl5qybqqz4po0jcfbapjgefjgbrvagfvadgnydz/ak3owh48v4nz4mrqljormlzmb8fgguc6uv</latexit> <latexit sha1_base64="jrwog46q3nkbz2m79o6iz0w3c9i=">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</latexit> <latexit sha1_base64="jrwog46q3nkbz2m79o6iz0w3c9i=">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</latexit> Tensor models for near-extremal black holes Certain tensor models exhibit the same pattern of symmetry breaking and maximal chaos, in large N limit and deep IR. Simplest such tensor model is Carozza-Tanasa-Klebanov- Tarnopolsky (CTKT) Model H CTKT = abc J 4N 3/2 is a real fermion. a,b,c = 1,,N abc ab 0 c 0 a 0 bc 0 a 0 b 0 c

7 Two point function in CTKT Model Simplest leading correction to free propagator Replace any propagator by the above diagram

8 Two point functions in CTKT model (contd.) Schwinger Dyson eqn: large N ) (t 1,t 2 )=J 2 G(t 1,t 2 ) 3 1 deep IR ) G(!) = i! (!) Z large N and deep IR ) J 2 dt G(t 1,t)G(t, t 2 ) 3 = (t 1 t 2 ) Reparameterization symmetry: G(t 1,t 2 )! df (t 1) dt 1 df (t 2 ) dt 2 1/4 G(f(t1 ),f(t 2 )) spontaneously broken down to SL(2,R) by solution G(t) = 1 sgn(t) (4 J 2 ) 1/4 t 1/2

9 Four point functions in CTKT model connected four point function 1 N 6 h abc(t 1 ) abc (t 2 ) a0 b 0 c 0 (t 3 ) a0 b 0 c 0 (t 4 )i = G(t 1,t 2 )G(t 3,t 4 )+ 1 N 3 F(t 1,t 2 ; t 3,t 4 ) Z F n+1 K F n where K = commutes with SL(2,R) gen. F =(1+K + K )F 0 = 1 1 K F 0 K has an eigenvalue 1, when evaluated in conformal limit. For this eigenspace, K must be evaluated away from conformal limit, thus reparameterization symmetry is explicitly broken.

10 Spectrum of primaries K commutes with SL(2,R) generators. Utilise conformal symmetry: F(t 1,t 2 ; t 3,t 4 )=G (t 12 )G (t 34 )F( ), where = t 12t 34 t 13 t 24 In the limit! 0, F( ) primary of dimension h n ) bulk scalar of mass m 2 n = h n (h n 1) NOT a sparse spectrum. No truncation to SUGRA expected. (see , for discussion on holographic dual) 1X m=1 c 2 m h m

11 A Tensor Model for Black Hole Probe

12 Guessing a probe model Had these tensors been matrices, (which they are not), one would think of the black hole as a bound state of D0 branes and them matrices describing strings stretched between D0 branes. An external D0 brane would be a probe. probes would be described by vectors. Naive tensor analog: probes could be tensors of lower rank.

13 Desirable properties for the Black Hole Probe Probe does not affect the black hole dynamics to leading order. Emergent reparameterization symmetry and the pattern of its breaking should be intact. non trivial because this symmetry is not visible in the Lagrangian. Infliction of maximal chaos.

14 <latexit sha1_base64="pua2idvegsfwpvnx+t6dk5iis7y=">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</latexit> <latexit sha1_base64="pua2idvegsfwpvnx+t6dk5iis7y=">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</latexit> <latexit sha1_base64="pua2idvegsfwpvnx+t6dk5iis7y=">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</latexit> <latexit sha1_base64="pua2idvegsfwpvnx+t6dk5iis7y=">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</latexit> Our Model removing an open line do not change the N dependence. One can go on inserting this diagram in internal propagators to get new leading diagrams. These diagrams can come from a vertex like Above method generates all leading diagrams Suggests the Hamiltonian H = H CTKT + g 1 N 3/2 abc a 0 b 0 capple ab 0apple a 0 b

15 physics intact in large N limit <latexit sha1_base64="17uenevnbgzaf0rc1eqqvr4zn7q=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3njfmyu3fs7gnhyf+wsvde1j9k579xl7lcex8mpn6bywzelahuro9/e6w19y3nrfj2zwd3b/+genjunngqgbzylgldiahbwrw2llcco4lgkiobj9hknvcfn1abhqsho00wlhsk+jazanoplxjer9b8uj8hwsvbqwpqonmvfvugmuslkssenayb+iknm6otzwjnlv5qmkfsqkfydvrrisbm5rfoyjltbmqya1fkkrn6eykj0pipjfynphzslr1c/m/rpnz4hwzcjalfxralhqkgnib542tantirpo5qprm7lbax1zrzf0/fhrasv7xk2hf1wk8h95e1xk0rrxlo4btoiyaramadnkefdmbwdk/w5knvxxv3phatja+yoyy/8d5/acq/jko=</latexit> Leading contributions to two and four point functions can be obtained by cutting melons. melons are subleading compared to <latexit sha1_base64="qasuf/1ialaqb3i2yf1c2veixku=">aaab7xicbvdlsgnbeoynrxhfuy9ebopgkeykomegf48rzaosjfrojsmy2dlhzlyis/7biwdfvpo/3vwbj8kenlggoajqprsruoib6/vfxmftfwnzq7hd2tnd2z8ohx41tzjqyho0eylur2iy4ji1llectzvmgeectalx7cxvptftecif7esxmmah5ano0tqp2r2jutgrv/yqpwdzjufokpcj3it/dfsjtwmmlrvotcfwlq0z1jztwaalbmqyqjrgies4kjfmjszm107jmvp6zjbov9ksufp7ispymekcuc4y7cgsezpxp6+t2sf1mhgpusskxswapilyhmxej32ugbvi4ghszd2thi5qi7uuojilivh+ezu0l6qbxw3ulyu1mzyoipzakzxdafdqgzuoqwmopmizvmkbl3gv3rv3swgtepnmmfyb9/kdliyphw==</latexit> <latexit <latexit sha1_base64="17uenevnbgzaf0rc1eqqvr4zn7q=">aaab63icbva9swnbej2lxzf+rs1tfongfe5e0djoyxnbxebyhl3njfmyu3fs7gnhyf+wsvde1j9k579xl7lcex8mpn6bywzelahuro9/e6w19y3nrfj2zwd3b/+genjunngqgbzylgldiahbwrw2llcco4lgkiobj9hknvcfn1abhqsho00wlhsk+jazanoplxjer9b8uj8hwsvbqwpqonmvfvugmuslkssenayb+iknm6otzwjnlv5qmkfsqkfydvrrisbm5rfoyjltbmqya1fkkrn6eykj0pipjfynphzslr1c/m/rpnz4hwzcjalfxralhqkgnib542tantirpo5qprm7lbax1zrzf0/fhrasv7xk2hf1wk8h95e1xk0rrxlo4btoiyaramadnkefdmbwdk/w5knvxxv3phatja+yoyy/8d5/acq/jko=</latexit> melons.

16 Propagators New diagrams give subleading contribution to propagators. G (t) = 1 sgn(t) (4 g0 2)1/4 t 1/2 G apple satisfy same (up to constants) Schwinger Dyson equation as G G apple (t) = g 0 g 1 G (t)

17 h i 1 1 h ab (t1 ) ab (t2 ) ab (t3 ) ab (t4 )i = G (t12 )G (t34 ) + 3 F (t1, t2 ; t3, t4 ) 6 N N Diagrams contributing to F are 2 g 1 0 F = 2 F0 1 g1 1 3 K <latexit sha1_base64="2c5xwmqtqi3xnrql43r2nup55gm=">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</latexit> No breaking of conformal symmetry. F ( 1,..., 4 ) = G ( 12 )G ( 34 )F ( ) 1 X 2 e lim F ( ) = e c m hm <latexit sha1_base64="x/7ysxmmoadirygusj8bou+ztfw=">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</latexit>!0 m=1 Primaries of different dimensions, but in natural correspondence with those of the black hole.

18 1 X h 5 N h i 1 F abc (t1 ) abc (t2 ) a0 b0 (t3 ) a0 b0 (t4 )i = G (t1, t2 )G (t3, t4 ) + 3 N F <latexit sha1_base64="kdecz23w6cy7vgs5lfpl1xvzoj0=">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</latexit> 2g0 = 3g1 (1 K K)(1 F0 1 3 K) F ( 2, 2, 3, 4 ) = F ( ) + F conf non conf ( 1, 2, 3, 4 ) G ( 12 )G ( 34 ) 1 h i X e lim Fconf ( )= c2m hm e c2m hm <latexit sha1_base64="kt8spu49ixjc3vg1u1yr/jqlps0=">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</latexit>!0 m=1 primaries of same dimensions appear.

19 e t Chaos Replace position and momenta by general operators and separate two commutators along thermal circle. )h[p(0),q(t)] 2 i e t tr [V (t +3i /4) W (i /2) V (t + i /4) W (0)] This is an Out of Time Order Correlator (OTOC). For us case 1: V =,W = case 2: V = apple,w = apple case 3: V =,W = apple

20 Chaos continues For four point functions are of the form f (K) F= F0 l (1 K) <latexit sha1_base64="g+/objjvd1nsi4u1knnujzbk8i8=">aaacgnicbvdlssnafj34rpuvdelmsajtwpkiobuhkijqtqx7gcawyxtsdp1mwsxekchf4czfcencexfixr9x0mzrww8mhm45l7n3ebgjulnwj7g0vlk6tl7ykg5ube/smnv7lrngapmmdlkooh6shffomooqrjqricjwggl7o+vmbz8siwni79u4im6abpz6fcolpz5powfsq4xycppcs+j4auhel9cravk2t+qvb5bcmujpgj2zzfwtceaisxnsajkapfpl6yc4dghxmcepu7yvktdbqlhmsfp0ykkihedoqlqachqq6sat01j4rju+9eohh1dwos5ojciqchx4opltkee9tpzp68bkv3atyqnyey6nh/kxgyqewu+wtwxbio01qvhqvsveq6tbubrnoi7bnj95kbroq7zvte/osrwrvi4coarhoaxsca5q4by0qbng8arewbt4n56nv+pd+jxgl4x85gd8gfh9c9whn3c=</latexit> corresponding OTOC is maximally chaotic. F, F are maximally chaotic. F turns out to be non-chaotic. <latexit sha1_base64="galvdm9dde3qxyls1sir0/sdkq4=">aaacinicbvdlsgmxfm3uv62vqks3wsk4kdijgrorcukygn1azyx30kwbmpkjsuyow3ylg3/fjqtfxql+jjm2i9p6iha4597k5picm6vt+9sqlc2vrk4v10sbm1vbo+xdvaake0log8q8lm0ffousog3nnkdtismepqctf3id+61hkhwlo3s9etqlor+xgbhqruqwl90q9iaat2+yb1codpk641ttnyc0w7n2mvvyhyiqkgxdcswu2mpgrejmsqvnue+wp91etjkqrppwukrj2ej7kujnckdzyu0ufucg0kcdqymiqflsczgmhxmlh4nymhnppfznn1iilrqfvpnme6t5lxf/8zqjdi68leui0tqik4echgmd47wv3goses1hhgcrzgtfzaasidatlkwjzvyxf0nztoryvefurfk7mtzrrafoeb0jb52jgrpfddrabd2hf/sg3q1n69x6sl4mowvrurop/sd6+qxkcqw5</latexit> <latexit sha1_base64="ia+rc35bi+ev/o2m5yprao92otu=">aaab+3icbvdlssnafj34rpuv69jnsaiusikcloucukxgh9decjodtemnk2fmipaqx3hjqhg3/og7/8zjm4w2hhg4nhmv98wjbanku+63tbk6tr6xwdmqbu/s7u3bb7woslkjsrsnljg9ebrhljo2ppqrnpae4pcrbji5lvzui5gkjvxetwujyhhxglem2kgdu+bhomcywhatp/gteaigdt1tudm4y8qrsr2vaa3sl3+y4dqmxgmgsvu9v+gga6kpzisv+qkiavaerqrvkieyqccbzc+de6mmnsir5nhtzntfgxnesk3j0ewwsdwiv4j/ef1ur5dbrrlinef4fihkmamtpyjcgvjjsgztqwblari6eawssdz1vu0j3ukxl0nnrog5de/uvn68kuuoocn0je6rhy5qe92ifmojjj7qm3pfb1zuvvjv1sd8dmuqdw7rh1ifp1asljo=</latexit>

21 Some Cousin Models Restoring permutation symmetry Adding vector fields Coloured probe model

22 Disordered probe model (arxiv: 1806.???? )

23 Probe fields are n in number. 1<<n<<N. 0 H HSY K + j q p p <latexit sha1_base64="dduxeavdupcxq84efhtyq4m9zxm=">aaacehicbvdlssnafj3uv62vqes3g0uqicurqzdfnwu3fe1dmjrmptn27cqzzyzccfken/6kgxekuhxpzr9x2mah1qmxdufcy733+jxrqszry8jnzs8sluwxcyura+sb5uzwq0axwksoixajlo8kytqkduuviy0ucap8rpr+8hzsn++jkdqkr9wiezda/zd2kezks55zqkjh0gbwvetq5ikfb/c25hbjo8ndiu+hm0scow6hnlm0ytye8c+xm1iegwqe+el0ixwhjfsyisnbtswvmychkgykltixjbzhieqttqyhcoh0k8ldkdztshf2iqervhci/pxiucdlkpb1z4duqm56y/e/rx2r3qmb0jdhior4uqgxm6gioe4hdqkgwlgrjgglqm+feiaewkpnwnah2lmv/ywno7jtle3l42lllisjd3baltghnjgbfvafnvahgdyaj/acxo1h49l4m96nrtkjm9kgv2b8fam3b5tz</latexit> F and F are maximally chaotic. has Lyapunov coefficient. smaller than, but F order of the maximal value, 2 T. <latexit sha1_base64="2ejtlzqzhonrazotelyguiabrre=">aaacanicbzdlssnafiynxmu8rv2jm8fscfusexrzfirlcvyctvom00k7ddizzizcccwnr+lghsjufqp3vo3tnatt/whg4z/ncob8owbuadf9tlzw19y3nktb9vbo7t6+c3dyukkqmwnihcwyeyjfgowkqalmpcmkqxhisdsc38zq7qcifu34vz4iesroyglemdlg6jvh9z4vfiv2bdz7wy7+gambpn2n7fbdxhazvalkofcj73z5gwsnmeeam6ru13ofdjikncwmtg0/vuqgpezd0jxiuuxukountghfoamyjdi8rmhu/p7iukzuja5nz4z0sc3wzuz/tw6qo6sgo1ykmna8xxsldooezvkaayoj1mxiagfjzv8hhigjsdap2syeb/hkzwidvz236t1dlgvxrrwlcajowrnwwcwogvvqae2awsn4bq/gzxqyxqx362peumivm0fgj6zph28fli0=</latexit> <latexit sha1_base64="0gyjjdao1cmfyzraydghtbckq9e=">aaacanicbzdlssnafiynxmu8rv2jm8fscfusexrzfirlcvyctvom00k7ddizzizcccwnr+lghsjufqp3vo3tnatt/whg4z/ncob8owbuadf9tlzw19y3nktb9vbo7t6+c3dyukkqmwnihcwyeyjfgowkqalmpcmkqxhisdsc38zq7qcifu34vz4iesroyglemdlg6jvhlxrpf4pcg9z7wu7+gambpn2n7fbdxhazvalkofcj73z5gwsnmeeam6ru13ofdjikncwmtg0/vuqgpezd0jxiuuxukountghfoamyjdi8rmhu/p7iukzuja5nz4z0sc3wzuz/tw6qo6sgo1ykmna8xxsldooezvkaayoj1mxiagfjzv8hhigjsdap2syeb/hkzwidvz236t1dlgvxrrwlcajowrnwwcwogvvqae2awsn4bq/gzxqyxqx362peumivm0fgj6zph23rli0=</latexit> <latexit sha1_base64="isivi+nr6bxhkxk5imzgucgsxri=">aaab/xicbvdlssnafj34rpevhzs3g6xgqiqi6liofjcv7apatnxmj+3qstlmtiqair/ixouibv0pd/6n0zylbt1w4xdovdx7tya4u9p1v62v1bx1jc3clr29s7u37xwcnlsssklrjogjbawgkgcxrwumow0jssekog0go5up33ygurekvtdjqf0ibjelgqftpj5zxkp2o0ixbonqn+umqaiy9jyiw3znwmvey0kr5aj1nk9opyfprgnnocjv9lyh/qykzottid1jfrvarjcgbunjikjys9n1e1wysh+hitqvazxtf09keck1jgltgyeeqkvvkv7ntvmdxvkzi0wqauzmi8kuy53gars4zyqlmo8naskzurwtiugg2grmmxc8xzexseo87lll7+6iwlno4yige3skzpchllef3aiaqiochtezekvv1pp1yr1bh/pwfsufouj/yh3+anzlldm=</latexit> <latexit sha1_base64="9thokael5wd3phcpotqrybpvl50=">aaab7nicbvdlsgnbeoz1gemr6thlyba8hd0g6dhoxwoevcbzwuxkkgyznr1meoww5co8efdeq9/jzb9xkuxbewsaiqpuursilyvf3//2nja3tnd2c3vf/ypdo+psywnljqlhvmksmzhorc2xqvemcps8ow2ncsr5o5rcz/32ezdwjkqbu83dmi6ugapg0untkulpqrr9utmv+auqdrlkpaw56v3sv2+qsdtmcpmk1nydx2oyuyocst4r9lllnwutoujdrxwnuq2zxbkzcumuarkmxpvcslb/t2q0tnyar64zpji2q95c/m/rpji8dtohdipcsewiysojjmt+oxkiwxnkqsougefujwxmdwxoeiq6eilvl9djq1oj/erwef2u3evxfoacluakaribgjxahzrayalp8apvnvzevhfvy9m64euzz/ah3ucpe/wouw==</latexit> p In the limit q! 1,! 1 q ALL four point functions are maximally chaotic. <latexit sha1_base64="8eeambhz+kv7tpgc7t2u0ems6cs=">aaace3icbvbns8naej34wetx1koxxskisele0gpri8ck9goaudbbtbt0s0l3n0oj/q9e/ctepcji1ys3/43bngfb+mdg8d4mm/ochdolhefhwlpewv1bl2wun7e2d3btvf26ilnjai3epjbnacvkmaa1ztsnzurshawcnol+zdhvpfcpwczu9tchfos7gowmyg2ktn06qj5k3z7gusapyd1dxigxyzjrnhjnwm275jsdcdaicxnsghzvtv3tdwksrlrowrfslddjtj9hqrnhdft0ukuttpq4s1ugchxr5wetn0bo2cgdfmbslnboov6dyhck1daktgeedu/ne2pxp6+v6vdkz5hiuk0fms4ku450jmyboq6tlgg+naqtycytipswyusbgismbhf+5uvspy+7ttm9uyhvrvm4cnair3acllxcbw6hcjug8aqv8abv1rp1an1yn9pwjsufoyazwf+/pdgeaq==</latexit> Probe primaries make a copy of the BH primaries.

24 Scorecard Our models satisfy the non trivial requirement of symmetry breaking and partially fulfil that of maximal chaos. Probe spectrum is in natural correspondence with black hole spectrum. Significant improvement for disordered probe model.

25 Future Directions

26 Study thermalisation, i.e. test ETH. Higher point functions.

27 Merci Beaucoup!