1 2015/12/21 Lauderdale, Florida Towards the gravity/CYBE correspondence Kentaroh Yoshida ( Dept. of Phys., Kyoto U.) Based on collaboration with Takuya Matsumoto (Nagoya U.) Io Kawaguchi, Takashi Kameyama, Marcos Crichigno, Domenico Orlando, Susanne Reffert, Jun-ichi Sakamoto, Hideki Kyono, Andrzej Borowiec, Jerzy Lukierski 2 type IIB string on AdS 5 x S 5 4D SU(N) SYM The AdS/CFT correspondence Recent progress: the discovery of integrability The integrable structure provides powerful tools. It enables us to check the conjectured relations in the AdS/CFT without SUSY, even at finite coupling. the classical integrability on the string-theory side. EX anomalous dims, amplitudes etc. A huge amount of works have been done, so far. The existence of Lax pair (kinematical integrability) [For a big review, Beisert et al., ] Motivated by this success, our interest here is 3 The coset structure of AdS 5 x S 5 is closely related to the integrability. Z 2 -grading classical integrability : symmetric coset The classical integrability of the AdS 5 x S 5 superstring Including fermions : super coset Z 4 -grading classical integrability elucidated by [Bena-Polchinski-Roiban, 2003] This fact is the starting point of our later argument. [Metsaev-Tseytlin, 1998] 4 Integrable deformations of the AdS 5 x S 5 superstring The next issue Obs. Various kinds of integrable deformations may lead to the associated sols. of type IIB SUGRA. Integrability techniques solution generation techniques EX Integrable twists EX TsT transformations Integrable deformations deformed AdS 5 x S 5 geometries Solutions of type IIB SUGRA (as exactly solvable models) A classification of SUGRA sols. based on integrable deformations Motive 5 A systematic deformation scheme An integrable deformation is specified by a classical r-matrix. satisfying modified classical Yang-Baxter eq. (mCYBE) R : a linear op. a classical r-matrix Yang-Baxter (YB) deformation [Klimcik, 2002, 2008] Integrable deformation! : const Given a classical r-matrix, a Lax pair follows automatically. The insertion of is a characteristic of YB deformation. NOTE 6 R-operators and classical r-matrices A linear R-operator A skew-symmetric classical r-matrix Two sources of classical r-matrix 1) modified classical Yang-Baxter eq. (mCYBE) the original work by Klimcik 2) classical Yang-Baxter eq. (CYBE) a possible generalization ( ) 7 The list of generalizations of YB deformations (i) The standard q-deformation mCYBE (trigonometric) (ii) Jordanian deformations CYBE (rational) 1) Principal chiral model 2) Symmetric coset sigma model 3) Type IIB superstring on AdS 5 xS 5 1) Principal chiral model 2) Symmetric coset sigma model 3) Type IIB superstring on AdS 5 xS 5 [Delduc-Magro-Vicedo, ] [Kawaguchi-Matsumoto-KY, ] [Klimcik, hep-th/ , ] [Matsumoto-KY, ] [Delduc-Magro-Vicedo, ] (2 classes) Further direction: [Mikhailov, Popowicz-Chau Wang, Abdalla-Forger, Curtright-Zachos, Uematsu etc.] include world-sheet SUSY 8 solutions of type IIB SUGRA classical r-matrices Gravity/CYBE correspondence [Delduc-Magro-Vicedo, ] [Kawaguchi-Matsumoto-KY, ] Integrable deformations of the AdS 5 x S 5 superstring The moduli space of a certain class of solutions of type IIB SUGRA is described by the classical Yang-Baxter equation. Claim c.f., a droplet picture of free fermion in the bubbling geometry [Lin-Lunin-Maldacena, 2005] One may construct a new bubbling-like scenario. Droplet classical r-matrix free fermion CYBE (Just an analogy ) The current status of two kinds of YB deformations 1) Standard q-deformation mCYBE 9 [Delduc-Magro-Vicedo, ] A deformed classical action ( -model ) The metric and B-field [Arutyunov-Borsaro-Frolov, ] [Kameyama-KY, , 1512.NNNNN] minimal surfaces, quark-antiquark potential [Arutyunov-Borsaro-Frolov, ] [Arutyunov-Frolov-Hoare -Roiban-Tseytlin, ] Supercoset construction Scale invariance on the world-sheet (shown) Not Weyl invariant (probably) NOT satisfy the e.o.m. of type IIB SUGRA Modifications of SUGRA (conjecture) c.f., [Hoare-Tseytlin, ] + many other works with classical r-matrix of Drinfeld-Jimbo type 10 [Kawaguchi-Matsumoto-KY, ] 2) Jordanian deformations CYBE i) Partial deformations are possible (i.e., only AdS 5 or only S 5 ) ii) Many r-matrices have been identified with solutions of type IIB SUGRA [Kawaguchi-Matsumoto-KY, ] A lot of classical r-matrices [Matsumoto-KY, , , , ] In the following, we will concentrate on Jordanian deformations. The current status of two kinds of YB deformations This is an advantage of Jordanian deformations (a) The deformed classical action (b) Some examples of classical r-matrices 11 Jordanian deformation of the AdS 5 x S 5 superstring [Kawaguchi-Matsumoto-KY, ] Lax pair is constructed : classical integrability Kappa invariance : a consistency as string theory R Jor satisfies CYBE. The undeformed limit: the Metsaev-Tseytlin action [Metsaev-Tseytlin, hep-th/ ] The replacement: mCYBE CYBE modifies Lax pair and kappa-transformation. This is a generalization of the work for mCYBE by Delduc-Magro-Vicedo. NOTE 12 3 Examples: i) gamma-deformations of S 5 Metric: Abelian r-matrix: Identification of parameters: B-field: [Matsumoto-KY, ] [Lunin-Maldacena, Frolov, 2005] 13 ii) Gravity duals for SYM on non-commutative space [Hashimoto-Itzhaki, Maldacena-Russo, 1999] Abelian Jordanian r-matrix: where,, Identification of parameters: Note: [Matsumoto-KY, ] Metric: B-field: This background can be reproduced as a special limit of -deformed AdS 5 [Arutyunov-Borsaro-Frolov, ] [Kameyama-Kyono-Sakamoto-KY, ] 14 iii) Schrdinger spacetimes [Matsumoto-KY, ] [Bobev-Kundu, ] Metric: B-field: Mixed r-matrix: [Herzog-Rangamani-Ross, ] [Maldacena-Martelli-Tachikawa, ] [Adams-Balasubramanian-McGreevy, ] S 5 -coordinates: 1) 3-parameter cases 2) general Melvin twists [Bobev-Kundu-Pilch, ] NOTEWe have also found classical r-matrices for 15 AdS pp-wave background Other examples: NC deformations of global AdS 5 [Hubeby-Rangamani-Ross, hep-th/ ] [Dhokarh-Haque-Hashimoto, ] Dipole backgrounds [Kawaguchi-Matsumoto-KY, ] [Bergman-Dasgupta-Ganor-Karzmarek-Rajesh, hep-th/ ] [Matsumoto-KY, ] [McLaughlin-Swanson, hep-th/ ] Schrdinger AdS pp-wave 2) RR sector and dilaton have not been confirmed yet. Schrdinger spacetimes are good candidates. The dilaton is const. and the RR sector is the same as the usual AdS 5 x S 5. Remarks 1) Solutions can also be derived as TsT transformations of AdS 5 x S 5. TsT trans. are well described by Yang-Baxter deformations. In fact, 3 examples presented here can be interpreted as the undeformed theory with twisted b.c. c.f., Frolov, Alday-Arutyunov-Frolov, [Kameyama-Kyono-Sakamoto-KY, ] [S.J. van Tongeren, ] 16 Summary Towards the gravity/CYBE correspondence EX Lunin-Maldacena-Frolov, Maldacena-Russo, Schrdinger spacetimes 1) YB deformations of non-integrable backgrounds: EX AdS 5 x T 1,1 : non-integrability & chaos [Crichigno-Matsumoto-KY, ] YB deformations can capture TsT trans. of T 1,1 as well. [Basu-Pando Zayas, ] 2) YB deformations of 4D Minkowski space: [Matsumoto-Orlando-Reffert-Sakamoto-KY, ] [Asano-Kawai-Kyono-KY, ] Further generalizations Need to make efforts for the RR-sector and dilaton [Borowiec-Kyono-Lukierski-Sakamoto-KY, ] Melvin, pp-wave, T-duals of (A)dS, Hashimoto-Sethi, Spradlin-Takayanagi-Volovich, etc. 17 Thank you! The limitation of the gravity/CYBE correspondence? To what extent the correspondence can work? What is the mathematical origin of Yang-Baxter deformations? What are the associated deformations of N=4 SYM? It is of great significance to reveal the fundamental aspects of Yang-Baxter deformations There are many open problems. 18 Back up 19 A list of classical r-matrices and backgrounds As a result, this part has been reproduced by kappa-Poincare classical r-matrices! Class (b) 2. (class (a) 2.) What is the reasoning behind this coincidence? 20 How prevalent is integrability in various kinds AdS/CFTs? NOTE: Integrable subsectors are ubiquitous. EX integrable subsectors in large N QCD in 4 D Question There are various kinds of AdS/CFT Real (or complex) beta-deformations [Lunin-Maldacena, Frolov] Gravity duals for NC gauge theories [Hashimoto-Itzhaki, Maldacena-Russo] AdS 5 x T 1,1 [Klebanov-Witten] AdS BH [Horowitz-Strominger] AdS solitons [Witten, Horowitz-Myers] AdS/NRCFT [Son,Balasubramanian-McGreevy, Kachru-Liu-Mulligan] q-deformation of AdS 5 xS 5 [Delduc-Magro-Vicedo, Arutyunov-Borsato-Frolov] Klebanov-Strassler, Maldacena-Nunez AdS 5 x Y p,q [Gauntlett-Martelli-Sparks-Waldram] So we will concentrate on the full integrability below. 21 A classification list Integrable backgrounds Non-integrable backgrounds [Bai-Chen-Lee-Moon, ] Real beta-deformations Complex beta-deformations [Giataganas-Pando Zayas-Zoubos, ] Gravity duals for NC gauge theories AdS 5 x T 1,1 [Basu-Pando Zayas, ] AdS solitons [Basu-Das-Ghosh, ] AdS BH [Pando Zayas-Terrero Escalante, ] AdS 5 x Y p,q [Basu-Pando Zayas, ] [Matsumoto-KY, ] [Frolov, hep-th/ ] (would not be complete) Lifshitz space (with hyper-scaling violation) [Giataganas-Sfetsos, ] p-brane backgrounds [Stepanchuk-Tseytlin, ] [Chervonyi-Lunin, ] q-deformation of AdS 5 x S 5 [Delduc-Magro-Vicedo, , Arutyunov-Borsato-Frolov, ] Schrdinger spacetime with [Giataganas-Sfetsos, ] TsT transformatios of AdS 5 [Kawaguchi-Matsumoto-KY, ] [Hubeny-Rangamani-Ross, hep-th/ ] [Dhokarh-Haque-Hashimoto, ] Klebanov-Strassler, Maldacena-Nunez [Basu-Das-Ghosh-Pando Zayas, ] 22 Deformed AdS 5 xS 5 superstring action substitute from CYBE A schematic picture Note: The metric & NS-NS 2-form have successfully been obtained so far. However, further studies are necessary for the RR sector and dilaton, though the kappa-symmetry may ensure that they can be reproduced as well. The metric (string frame) & NS-NS 2-form rewrite the action & read off A gravity solution of type IIB SUGRA compare! correspondence (to see the gravity/CYBE correspondence) A classical r-matrix 23 Jordanian deformation of the AdS 5 x S 5 superstring, Maurer-Cartan 1-formProjection on the group manifold Projection on the world-sheet R Jor satisfies CYBE. 24 Exactly solvable Yang-Baxter deformations of 4D Minkowski spacetime Integrable Integrable? The results obtained so far support the integrability of YB-deformed action.