A warm-up for solving noncompact sigma models: The Sinh-Gordon model
|
|
- Georgia Oliver
- 5 years ago
- Views:
Transcription
1 A warm-up for solving noncompact sigma models: The Sinh-Gordon model Jörg Teschner Based on A. Bytsko, J.T., hep-th/ , J.T., hep-th/
2 Goal: Understand string theory on AdS-spaces Building blocks: nonlin. sigma-models on super-groups. P SL(1, 1 2) AdS3 S3 P SL(2, 2 4) AdS5 S5 To write out worldsheet-actions, use Poincare coordinates: ds 2 = 1 z 2(dz2 + d x 2 ). Writing z = e ϕ world-sheet Hamiltonian of the form H =... + e 2ϕ (... ) + e 2ϕ (... ) +... Exponential interactions! On the other hand: Hope for integrability! (Bena, Polchinski, Roiban)
3 Nice recent story concerning limit (Hofman, Maldacena) J, E J = fixed, y R 1 AdS = fixed Gauge-fixing: J R = J/ y, R: world-sheet radius. Hg.f. = 2πR dσ (... + e 2ϕ (... ) + e 2ϕ (... ) +... ) 0 Conjecture: Hg.f. is integrable. Proposal for factorized scattering theory in infinite volume R ( J ) (Beisert, Staudacher, Hofman, Maldacena...) : Elementary excitations ( magnons ) 8B 8F. Proposal for 2 2 scattering matrix (for all y!!!) Proposal has passed highly nontrivial tests!
4 Important problem: Spektrum for finite R finite J? Asymptotic Bethe ansatz (Arutyunov, Frolov, Staudacher) Pretend magnons are free except for magnons crossing (δ-like interactions), Ansatz for coord. space wave-fct.: Plane-waves + crossings (S-matrix) Periodicity of wave-fct. e ip ar = b a S(pa, pb). Problem: Interactions are not δ-like (vacuum polarization) Expect corrections to As. Bethe ansatz!
5 Classes of integrable models: (A) Compact integrable models. Models associated to compact groups: WZNW, σ-models, XXX, XXZ, Ising, Interactions e ibφ : Sine-Gordon... (B) Noncompact integrable models - Real type Models associated to non-compact groups: WZNW, σ-models, XXX, XXZ... Interactions e bφ : Liouville theory, Sinh-Gordon, (Affine) Toda... Class (B) is important: String theory: Strings on noncompact target spaces (black holes, cosmology), Gauge theory - via AdS-CFT correspondence, Condensed matter: Integer quantum Hall, electron systems with disorder. But: Class (B) is very different from (A): Prediction: Bethe ansatz fails in class (B)!
6 Consider prototype: Sinh-Gordon model (on circle with circumference R) HShG = R dx { 4π Π π ( σϕ) 2 + 2µ cosh(bϕ) }. 0 Infinite volume R : Spectrum: One massive particle, E = m cosh ϑ, p = m sinh ϑ. Scattering: S-matrix factorizes into two-particle scattering S(ϑ1 ϑ2) S(ϑ) = sinh ϑ i sin ϑ 0. sinh ϑ + i sin ϑ0 Fields, correlation functions:...?! Finite volume R: Spectrum: Conserved quantities, classification of eigenvectors - today! First example for QFT from class (B) where spectrum was understood for finite volume!
7 First step: Construct integrable lattice regularization of the Sinh-Gordon model: Hilbert space H Hamiltonian H, set Q = {T0, T1,... } of commuting conserved charges 1. Definition of H: Discretize Sinh-Gordon variables as Πn Π(x), Φn Φ(x), x = n. Quantize: [ Πn, Φn ] = 1 i δ n,m. Hilbert space H (L 2 (R)) N.
8 2. Construction of Q. Let M(u) ( A(u) B(u) C(u) D(u) ) LN(u)... L2(u) L1(u), where L(u) ( L11(u) L12(u) L21(u) L22(u) ), ( ) L11(u) = e +β 8 (Π n+2s) 1 + e β(φ n+s) e +β 8 (Π n+2s) L12(u) = sinh ( πbu + β 2 Φ n ) L21(u) = sinh ( πbu β 2 Φ ) β = b 8π n ( ) L22(u) = e β 8 (Π n 2s) 1 + e +β(φ n s) e β 8 (Π n 2s) Consider the one parameter family of operators: T(u) = tr ( M(u) ) = A(u) + D(u). The operators Tm which are defined by T(u) = e πbnu N m=0 ( e 2πbu ) m Tm, are positive self adjoint and commuting, [Tm, Tn] = 0.
9 3. Construction of H There exists an operator H which has the following properties. (a) The operator H is local: N H = n=1 Hn,n+1. (b) H commutes with the conserved charges Tk: [ H, Tk ] = 0, for k = 0,..., N. (c) Classical continuum limit (Πn Π(x), Φn Φ(x), x = n.) 0 N with { } R = N /2π fixed. m = 4 e πbs 1 H 0 HG,cl n n,n+1 2πR 0 dx ( 1 2 Π ( xφ) 2 + m2 β 2 cosh βφ ) + const
10 Good news: We do have integrable lattice regularization Strategy: Diagonalize T(u) Diagonalization of H Bad news: The Bethe ansatz fails: one would need pseudo-vacuum Ω annihilated by B(u) Since interactions involve real exponentials e bϕ : B(u)Ω = 0 leads to equations like e φ Ω = 0 no normalizable solution Ω H!!! Reasons: exponential interactions e φ, non-compactness of target space! However, there is something better than Bethe ansatz: Q-operators and separation of variables
11 Assume we have an operator Q(u) related to T(u) via the Baxter equation Q(u)T(u) = ( a(u) ) N Q(u ib) + ( d(u) ) NQ(u + ib), which furthermore satisfies (a) Q(u) is normal, Q(u)Q (v) = Q (v)q(u), (b) Q(u) Q(v) = Q(v) Q(u), (c) Q(u) T(v) = T(v) Q(u), Q(u) is repackaging of conserved quantities Diagonalization of Q(u) diagonalization of T(u). Eigenvalues q(u) of Q(u) must satisfy the Baxter equation. Explicit construction of Q(u) (Bytsko, J.T.) analytic and asymptotic properties of q(u): Quantization conditions!
12 Bytsko,J.T. Necessary conditions on the spectrum: A function t(u) can be an eigenvalue of the transfer matrix T(u) only if there exists a function qt(u) which satisfies (ii) qt(u) is meromorphic in C, with poles in ± Υ s, e +iπn σ u iπ 2 Nu2 for u, arg(u) < π 2 (iii) qt(u) e iπn σ u iπ 2 Nu2 for u, arg(u) > π 2. where d(u) = a( u) = 1 + ( m 4 )2 e πb(2u+ib), (i) t(u) qt(u) = ( a(u) ) N q t(u ib) + ( d(u) ) N q t(u + ib),
13 Separation of Variables Main idea (Sklyanin): Diagonalize B(u), parametrize eigenvalues b(u) as b(u) sinh 2πb(u yk). wave-functions Ψ(y1... yn). Key observations: (Sklyanin) T(u)Ψt = t(u)ψt(u) t(yk)ψ(... yk... ) = (a(yk)) N Ψ(... yk ib... ) + (d(yk)) N Ψ(... yk + ib... ) (Bytsko, J.T.) Properties (i)-(iii) Ansatz N Ψt = k=1 qt(yk) yields normalizable eigenstates of T(u).
14 Bytsko,J.T. Full characterization of the spectrum: A function t(u) is eigenvalue of the transfer matrix T(u) if and only if there exists a function qt(u) which satisfies (i) t(u) qt(u) = ( a(u) ) N q t(u ib) + ( d(u) ) N q t(u + ib), where d(u) = a( u) = 1 + ( m 4 )2 e πb(2u+ib), (ii) qt(u) is meromorphic in C, with poles in ± Υ s, e +iπn σ u iπ 2 Nu2 for u, arg(u) < π 2 (iii) qt(u), e iπn σ u iπ 2 Nu2 for u, arg(u) > π 2. Task: Classify set Q of solutions to the Baxter equation (i)
15 Let YM be the set of all functions Y (ϑ) which satisfy the integral equation (I)L log Y (ϑ) = dϑ C 4π σ(ϑ ϑ ) log(w (ϑ ) + Y (ϑ )) ( ) ( ) cosh(ϑ + iτ) cosh(ϑ iτ) N arctan N arctan sinh σ sinh σ M + log S(ϑ ϑa i π 2 ) + 1 M log S(ϑ ϑa i π 2 2 ). a=1 a=m +1 and which have the properties (Y1)L log Y (ϑ) iδn((ϑ σ i π 2 )2 τ 2 ) for u, arg(±u) < π 2, (Y2)L Y (ϑ) is meromorphic with poles of maximal order N in ± Υ s±iτ, (Y3)L There exist complex numbers ϑa S, a = 1,..., M, such that W (ϑ) + Y (ϑ) = 0 if ϑ = ϑa ± i π 2. ( σ(ϑ) 4 sin ϑ 0 cosh ϑ, ϑ0 π b cosh 2ϑ cos 2ϑ0 2 δ, σ π s 2 δ, τ π δ ) 2 δ.
16 Theorem 1. There is a one-to-one correspondence between the solutions Y (ϑ) YM of the integral equations (I)L and the elements Q QM. Given Q QM, get Y (ϑ): W (ϑ) + Y (ϑ) = Q(ϑ + i π 2 )Q(ϑ iπ 2 ). Z = {ϑ1,..., ϑm}: set of zeros of Q(ϑ) within S. Given Y (ϑ) YM (zeros: ϑa S \ S, ϑ b S) get Q Q M as (Q)L log Q(ϑ) = + C M a=1 dϑ 4π ϑ Ca log(w (ϑ ) + Y (ϑ )) cosh(ϑ ϑ ) dϑ 1 sinh(ϑ ϑa) ( ) cosh ϑ N arctan sinh σ M M b=1 ϑ Ca dϑ 1 sinh(ϑ ϑ b ).
17 Continuum limit: Easy! Continuum limit: N, s such that mr 2 sin π b 2δ 2N exp ( π ) 2δ s is kept constant. Main claim:
18 Main claim: The Hilbert space of the Sinh-Gordon model contains (is equal to!?) HTBA = M=0 H M. The spaces HM have ONB ek labelled by k = (k1,..., km) Z M,, k1 > > km, ek: eigenvect. to the conserved quantities of the model. The corresponding function Qk(ϑ) can be represented as (Q) log Qk(ϑ) = R dϑ 2π log(w (ϑ ) + YT k (ϑ )) mr cosh ϑ M + cosh(ϑ ϑ ) 2 sin ϑ0 a=1 ϑ Ca dϑ 1 sinh(ϑ ϑa). where Tk = [ϑ1,..., ϑm] is the unique solution of (B) 2πka+mR sinh ϑa+ M arg S(ϑa ϑb)+i b=1 b a R dϑ 2π σ(ϑ a ϑ+i π 2 ) log(1+y T k (ϑ)) = 0, with YT(u) being defined as the unique solution to dϑ (A) log YT(ϑ) R 2π σ(ϑ ϑ ) log(1+yt(ϑ ))+mr cosh ϑ+ M log S(ϑ ϑa i π 2 ) = 0. a=1
19 The energies are calculated from YT(ϑ) as follows: Ek = M m cosh ϑa m a=1 R dϑ 2π cosh ϑ log(1 + Y T(ϑ)) Excited state TBA! (Generalizes work of Al.B. Zamolodchikov, S. Lukyanov for the ground state)
20 The IR limit R Considering IR limit R, notice that YT(ϑ) = O(e mr ). Particle picture: M: number of particles, ϑa: Rapidity of particle a, Ea = m cosh ϑa, pa = m sinh ϑa, ϑa quantized by asymptotic Bethe ansatz equations. e irp a = M b=1 b a S(ϑa ϑb)
21 For finite R: where Φa R e ir(p a+φa) = M b=1 b a S(ϑa ϑb), dϑ 2πR σ(ϑ a ϑ + i π 2 ) log(1 + Y T(ϑ) = O(e mr ). Φa: Effects of vacuum polarization! The UV limit R 0: Connection with Liouville theory!
22 Compare with String Bethe Ansatz : Valid for J, J: angular momentum on S5 Radius of world-sheet cylinder in gauge-fixed action. For finite J: corrections O ( e 2πJ/ λ ) to String Bethe Ansatz (Schäfer-Nameki, Zamaklar, Zarembo) counterparts of e mr -corrections in Sinh-Gordon Gauge theory side: Wrapping interactions : (see Kotikov, Lipatov, Rej, Staudacher, Velizhanin) Prediction: Bethe ansatz will fail for AdS5 sigma model, finite J. But don t forget: There is sthg. better than Bethe ansatz!
AdS/CFT duality, spin chains and 2d effective actions
AdS/CFT duality, spin chains and 2d effective actions R. Roiban, A. Tirziu and A. A. Tseytlin Asymptotic Bethe ansatz S-matrix and Landau-Lifshitz type effective 2-d actions, hep-th/0604199 also talks
More informationS Leurent, V. Kazakov. 6 July 2010
NLIE for SU(N) SU(N) Principal Chiral Field via Hirota dynamics S Leurent, V. Kazakov 6 July 2010 1 Thermodynaamic Bethe Ansatz (TBA) and Y -system Ground state energy Excited states 2 3 Principal Chiral
More informationFinite temperature form factors in the free Majorana theory
Finite temperature form factors in the free Majorana theory Benjamin Doyon Rudolf Peierls Centre for Theoretical Physics, Oxford University, UK supported by EPSRC postdoctoral fellowship hep-th/0506105
More informationY-system for the Spectrum of Planar AdS/CFT: News and Checks
-,,,,, Y-system for the Spectrum of Planar AdS/CFT: News and Checks Vladimir Kazakov (ENS, Paris) Collaborations with N.Gromov, A.Kozak, S.Leurent, Z.Tsuboi and P.Vieira Outline Integrability for QFT in
More informationContinuum limit of fishnet graphs and AdS sigma model
Continuum limit of fishnet graphs and AdS sigma model Benjamin Basso LPTENS 15th Workshop on Non-Perturbative QCD, IAP, Paris, June 2018 based on work done in collaboration with De-liang Zhong Motivation
More informationString hypothesis and mirror TBA Excited states TBA: CDT Critical values of g Konishi at five loops Summary. The Mirror TBA
The Mirror TBA Through the Looking-Glass, and, What Alice Found There Gleb Arutyunov Institute for Theoretical Physics, Utrecht University Lorentz Center, Leiden, 12 April 2010 Summary of the mirror TBA
More informationIntegrability and Finite Size Effects of (AdS 5 /CFT 4 ) β
Integrability and Finite Size Effects of (AdS 5 /CFT 4 ) β Based on arxiv:1201.2635v1 and on-going work With C. Ahn, D. Bombardelli and B-H. Lee February 21, 2012 Table of contents 1 One parameter generalization
More informationNikolay Gromov. Based on
Nikolay Gromov Based on N. G., V.Kazakov, S.Leurent, D.Volin 1305.1939,????.???? N. G., F. Levkovich-Maslyuk, G. Sizov, S.Valatka????.???? A. Cavaglia, D. Fioravanti, N.G, R.Tateo????.???? December, 2013
More informationExact anomalous dimensions of 4-dimensional N=4 super-yang-mills theory in 'thooft limit
PACIFIC 2015 UCLA Symposium on Particle Astrophysics and Cosmology Including Fundamental Interactions September 12-19, 2015, Moorea, French Polynesia Exact anomalous dimensions of 4-dimensional N=4 super-yang-mills
More informationScaling dimensions at small spin
Princeton Center for Theoretical Science Princeton University March 2, 2012 Great Lakes String Conference, Purdue University arxiv:1109.3154 Spectral problem and AdS/CFT correspondence Spectral problem
More informationMagnon kinematics in planar N = 4 Yang-Mills
Magnon kinematics in planar N = 4 Yang-Mills Rafael Hernández, IFT-Madrid Collaboration with N. Beisert, C. Gómez and E. López Outline Introduction Integrability in the AdS/CFT correspondence The quantum
More informationIntegrability of the planar N = 4 gauge theory. and the Hubbard model. A. Rej, M. Staudacher AEI Golm D. Serban SPhT Saclay
Integrability of the planar N = 4 gauge theory and the Hubbard model A. Rej, M. Staudacher AEI Golm D. Serban SPhT Saclay Banff, February 16, 2006 Overview Integrability of the All loop integrability and
More informationSolving the AdS/CFT Y-system
Solving the AdS/CFT Y-system Dmytro Volin Nordita, Stockholm IHP, 2011 1110.0562 N.Gromov, V.Kazakov, S.Leurent. D.V. Setup: = IIB, AdS 5 xs 5 0 This is a talk about the spectral problem: Conformal dimension
More informationTHE SPECTRUM OF BOUNDARY SINE- GORDON THEORY
THE SPECTRUM OF BOUNDARY SINE- GORDON THEORY Z. Bajnok, L. Palla and G. Takács Institute for Theoretical Physics, Eötvös University, Budapest, Hungary Abstract We review our recent results on the on-shell
More informationSix-point gluon scattering amplitudes from -symmetric integrable model
YITP Workshop 2010 Six-point gluon scattering amplitudes from -symmetric integrable model Yasuyuki Hatsuda (YITP) Based on arxiv:1005.4487 [hep-th] in collaboration with K. Ito (TITECH), K. Sakai (Keio
More informationN=4 Super-Yang-Mills in t Hooft limit as Exactly Solvable 4D Conformal Field Theory
PACIFIC 2014 UCLA Symposium on Particle Astrophysics and Cosmology Including Fundamental Interactions September 15-20, 2014, Moorea, French Polynesia N=4 Super-Yang-Mills in t Hooft limit as Exactly Solvable
More informationKentaroh Yoshida (Kyoto Univ.)
2014/03/04 ``Progress in the synthesis of integrabilities arising from gauge string duality Recent progress on q deformations of the AdS 5 5 x S superstring Kentaroh Yoshida (Kyoto Univ.) In collaboration
More informationSystematic construction of (boundary) Lax pairs
Thessaloniki, October 2010 Motivation Integrable b.c. interesting for integrable systems per ce, new info on boundary phenomena + learn more on bulk behavior. Examples of integrable b.c. that modify the
More informationClassical AdS String Dynamics. In collaboration with Ines Aniceto, Kewang Jin
Classical AdS String Dynamics In collaboration with Ines Aniceto, Kewang Jin Outline The polygon problem Classical string solutions: spiky strings Spikes as sinh-gordon solitons AdS string ti as a σ-model
More informationGeometry and Physics. Amer Iqbal. March 4, 2010
March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries
More informationSine square deformation(ssd) and Mobius quantization of 2D CFTs
riken_ssd_2017 arxiv:1603.09543/ptep(2016) 063A02 Sine square deformation(ssd) and Mobius quantization of 2D CFTs Niigata University, Kouichi Okunishi Related works: Kastura, Ishibashi Tada, Wen Ryu Ludwig
More informationSpace from Superstring Bits 1. Charles Thorn
Space from Superstring Bits 1 Charles Thorn University of Florida Miami 2014 1 Much of this work in collaboration with Songge Sun A single superstring bit: quantum system with finite # of states Superstring
More informationAugust 2008 Solitons and AdS string solutions 2
* In collaboration with Kewang Jin, Chrysostomos Kalousios and AnastasiaVolovich. Outline Classical string solutions: spiky strings Spikes as sinh Gordon solitons AdS string as a σ model Inverse scattering
More informationMicrostates of AdS black holes and supersymmetric localization
Microstates of AdS black holes and supersymmetric localization Seyed Morteza Hosseini Università di Milano-Bicocca IPM, Tehran, May 8-11, 2017 Recent Trends in String Theory and Related Topics in collaboration
More informationTOPIC V BLACK HOLES IN STRING THEORY
TOPIC V BLACK HOLES IN STRING THEORY Lecture notes Making black holes How should we make a black hole in string theory? A black hole forms when a large amount of mass is collected together. In classical
More informationIntroduction to Integrability
Introduction to Integrability Problem Sets ETH Zurich, HS16 Prof. N. Beisert, A. Garus c 2016 Niklas Beisert, ETH Zurich This document as well as its parts is protected by copyright. This work is licensed
More informationOperator-algebraic construction of integrable QFT and CFT
Operator-algebraic construction of integrable QFT and CFT Yoh Tanimoto University of Rome Tor Vergata, supported by Programma Rita Levi Montalcini June 1st 2018, ICFT 22 Cardiff Y. Tanimoto (Tor Vergata
More informationNumerics and strong coupling results in the planar AdS/CFT correspondence. Based on: Á. Hegedűs, J. Konczer: arxiv:
Numerics and strong coupling results in the planar AdS/CFT correspondence Based on: Á. Hegedűs, J. Konczer: arxiv:1604.02346 Outline Introduction : planar AdS/CFT spectral problem From integrability to
More informationRECENT DEVELOPMENTS IN STRING THEORY International Conference Ascona, July Quantum spectral curve of N=4 SYM and its BFKL limit
RECENT DEVELOPMENTS IN STRING THEORY International Conference Ascona, 21 25 July 2014 Quantum spectral curve of N=4 SYM and its BFKL limit Vladimir Kazakov (ENS, Paris) Collaborations with: Nikolay Gromov
More informationD = 4, N = 4, SU(N) Superconformal Yang-Mills Theory, P SU(2, 2 4) Integrable Spin Chain INTEGRABILITY IN YANG-MILLS THEORY
INTEGRABILITY IN YANG-MILLS THEORY D = 4, N = 4, SU(N) Superconformal Yang-Mills Theory, in the Planar Limit N, fixed g 2 N P SU(2, 2 4) Integrable Spin Chain Yangian Symmetry Algebra of P SU(2, 2 4) Local
More informationBogolyubov Limit of Bethe Ansatz. and Dark Solitons
Bogolyubov Limit of Bethe Ansatz and Dark Solitons Konstantin Zarembo (LPT) arxiv:0702:???? Journées de la FRIF, 1.02.2008 Large-N expansion of YM theory String theory AdS/CFT correspondence Maldacena
More informationQFT. Unit 1: Relativistic Quantum Mechanics
QFT Unit 1: Relativistic Quantum Mechanics What s QFT? Relativity deals with things that are fast Quantum mechanics deals with things that are small QFT deals with things that are both small and fast What
More informationComments on the Mirror TBA
Comments on the Mirror TBA Sergey Frolov Hamilton Mathematics Institute and School of Mathematics Trinity College Dublin Integrability and Gauge/String Duality, IAS at Hebrew University, April 3, 2012
More informationANALYTIC SOLUTION OF BREMSSTRAHLUNG TBA BASED ON WITH A.SEVER. IGST2012 Zurich
ANALYTIC SOLUTION OF BREMSSTRAHLUNG TBA BASED ON 1207.5489 WITH A.SEVER IGST2012 Zurich Introduction Big progress in understanding of spectrum of N=4 SYM [Lipatov] [Faddeev,Korchemsky] [Minahan,Zarembo]
More informationTopological insulator part II: Berry Phase and Topological index
Phys60.nb 11 3 Topological insulator part II: Berry Phase and Topological index 3.1. Last chapter Topological insulator: an insulator in the bulk and a metal near the boundary (surface or edge) Quantum
More informationAs we have seen the last time, it is useful to consider the matrix elements involving the one-particle states,
L7 As we have seen the last time, it is useful to consider the matrix elements involving the one-particle states, F (x, θ) = 0 σ(x/2)µ( x/2) A(θ). (7.1) From this, and similar matrix element F (x, θ),
More informationNikolay Gromov. Based on works with V.Kazakov, S.Leurent, D.Volin F. Levkovich-Maslyuk, G. Sizov
Nikolay Gromov Based on works with V.Kazakov, S.Leurent, D.Volin 1305.1939 F. Levkovich-Maslyuk, G. Sizov 1305.1944 Gauge/Gravity duality July 29- August 2, 2013 Munich, Germany Motion of the string: The
More informationGluon Scattering Amplitudes from Gauge/String Duality and Integrability
Gluon Scattering Amplitudes from Gauge/String Duality and Integrability University of Tsukuba Yuji Satoh based on JHEP 0910:001; JHEP 1001:077 w/ K. Sakai (Keio Univ.) JHEP 1004:108; JHEP 1009:064, arxiv:1101:xxxx
More informationBubbling Geometries for Half BPS Wilson Lines. Satoshi Yamaguchi (IHES) S. Yamaguchi, hep-th/ S. Yamaguchi, to appear
Bubbling Geometries for Half BPS Wilson Lines Satoshi Yamaguchi (IHES) S. Yamaguchi, hep-th/0601089 S. Yamaguchi, to appear 1. Overview AdS5 CFT4 AdS5 x S5 Goal deform Supergravity Solutions 4dim N=4 Super
More informationTESTING ADS/CFT. John H. Schwarz STRINGS 2003
TESTING ADS/CFT John H. Schwarz STRINGS 2003 July 6, 2003 1 INTRODUCTION During the past few years 1 Blau et al. constructed a maximally supersymmetric plane-wave background of type IIB string theory as
More informationAn Algebraic Approach to Reflectionless Potentials in One Dimension. Abstract
An Algebraic Approach to Reflectionless Potentials in One Dimension R.L. Jaffe Center for Theoretical Physics, 77 Massachusetts Ave., Cambridge, MA 02139-4307 (Dated: January 31, 2009) Abstract We develop
More informationExact results for Wilson loops in N = 4 SYM
Exact results for Wilson loops in N = 4 SYM Diego H. Correa Universidad Nacional de La Plata - CONICET - Argentina Based on arxives: 1202.4455, 1203.1019 and 1203.1913 In collaboration with: J. Henn, J.
More informationPerturbative Integrability of large Matrix Theories
35 th summer institute @ ENS Perturbative Integrability of large Matrix Theories August 9 th, 2005 Thomas Klose Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute), Potsdam, Germany
More informationSine square deformation(ssd)
YITP arxiv:1603.09543 Sine square deformation(ssd) and Mobius quantization of twodimensional conformal field theory Niigata University, Kouichi Okunishi thanks Hosho Katsura(Univ. Tokyo) Tsukasa Tada(RIKEN)
More informationBrane decay in curved space-time
Brane decay in curved space-time Dan Israël, IAP, Univ. Pierre et Marie Curie From D. I. & E. Rabinovici, JHEP 0701 (2007) D. I., JHEP 0704 (2007) D. Israël, Brane decay in curved space-time 1 Outline
More informationAdS/CFT Correspondence and Entanglement Entropy
AdS/CFT Correspondence and Entanglement Entropy Tadashi Takayanagi (Kyoto U.) Based on hep-th/0603001 [Phys.Rev.Lett.96(2006)181602] hep-th/0605073 [JHEP 0608(2006)045] with Shinsei Ryu (KITP) hep-th/0608213
More informationTime-dependent backgrounds of compactified 2D string theory:
Time-dependent backgrounds of compactified 2D string theory: complex curve and instantons Ivan Kostov kostov@spht.saclay.cea.fr SPhT, CEA-Saclay I.K., hep-th/007247, S. Alexandrov V. Kazakov & I.K., hep-th/0205079
More informationEntanglement in Quantum Field Theory
Entanglement in Quantum Field Theory John Cardy University of Oxford Landau Institute, June 2008 in collaboration with P. Calabrese; O. Castro-Alvaredo and B. Doyon Outline entanglement entropy as a measure
More informationNikolay Gromov. Based on
Nikolay Gromov Based on N. G., V. Kazakov, S. Leurent, D. Volin 1305.1939, 1405.4857 N. G., F. Levkovich-Maslyuk, G. Sizov, S. Valatka 1402.0871 A. Cavaglia, D. Fioravanti, N. G., R. Tateo 1403.1859 N.
More informationHigher-Spin Black Holes and Generalised FZZ Duality
Higher-Spin Black Holes and Generalised FZZ Duality Batsheva de Rothschild Seminar on Innovative Aspects of String Theory, Ein Bokek, Israel, 28 February 2006 Based on: Anindya Mukherjee, SM and Ari Pakman,
More informationarxiv: v3 [hep-th] 13 Jan 2012
Prepared for submission to JHEP arxiv:1109.6305v3 [hep-th] 13 Jan 2012 Deeper Look into Short Strings Nikolay Gromov a,b Saulius Valatka a a Mathematics Department, King s College London, The Strand, London
More informationThe TT Deformation of Quantum Field Theory
The TT Deformation of Quantum Field Theory John Cardy University of California, Berkeley All Souls College, Oxford ICMP, Montreal, July 2018 Introduction all QFTs that we use in physics are in some sense
More informationIntegrability of spectrum of N=4 SYM theory
Todai/Riken joint workshop on Super Yang-Mills, solvable systems and related subjects University of Tokyo, October 23, 2013 Integrability of spectrum of N=4 SYM theory Vladimir Kazakov (ENS, Paris) Collaborations
More informationSpiky strings, light-like Wilson loops and a pp-wave anomaly
Spiky strings, light-like Wilson loops and a pp-wave anomaly M. Kruczenski Purdue University Based on: arxiv:080.039 A. Tseytlin, M.K. arxiv:0804.3438 R. Ishizeki, A. Tirziu, M.K. Summary Introduction
More information1 Equal-time and Time-ordered Green Functions
1 Equal-time and Time-ordered Green Functions Predictions for observables in quantum field theories are made by computing expectation values of products of field operators, which are called Green functions
More informationNon-relativistic Quantum Electrodynamics
Rigorous Aspects of Relaxation to the Ground State Institut für Analysis, Dynamik und Modellierung October 25, 2010 Overview 1 Definition of the model Second quantization Non-relativistic QED 2 Existence
More informationAPPLICATION OF MASSIVE INTEGRABLE QUANTUM FIELD THEORIES TO PROBLEMS IN CONDENSED MATTER PHYSICS
APPLICATION OF MASSIVE INTEGRABLE QUANTUM FIELD THEORIES TO PROBLEMS IN CONDENSED MATTER PHYSICS FABIAN H.L. ESSLER The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road,
More informationHolographic Entanglement and Interaction
Holographic Entanglement and Interaction Shigenori Seki RINS, Hanyang University and Institut des Hautes Études Scientifiques Intrication holographique et interaction à l IHES le 30 janvier 2014 1 Contents
More informationString/gauge theory duality and QCD
String/gauge theory duality and QCD M. Kruczenski Purdue University ASU 009 Summary Introduction String theory Gauge/string theory duality. AdS/CFT correspondence. Mesons in AdS/CFT Chiral symmetry breaking
More informationAccepted Manuscript. Temperature quantization from the TBA equations. Sergey Frolov, Ryo Suzuki
Accepted Manuscript Temperature quantization from the TBA equations Sergey Frolov, Ryo Suzuki PII: S0370-2693(09)00783-7 DOI: 10.1016/j.physletb.2009.06.069 Reference: PLB 25985 To appear in: Physics Letters
More informationNikolay Gromov. Based on
Nikolay Gromov Based on N. G., V. Kazakov, S. Leurent, D. Volin 1305.1939, 1405.4857 N. G., F. Levkovich-Maslyuk, G. Sizov, S. Valatka 1402.0871 A. Cavaglia, D. Fioravanti, N. G., R. Tateo 1403.1859 N.
More information(m, n) ZZ branes and the c = 1 matrix model
Physics Letters B 604 (2004) 115 122 www.elsevier.com/locate/physletb (m, n) ZZ branes and the c = 1 matrix model Sergei Alexandrov Institute for Theoretical Physics & Spinoza Institute, Utrecht University,
More informationMarta Orselli. Based on: hep-th/ , hep-th/ hep-th/ , arxiv: arxiv: , arxiv: work in progress
Marta Orselli Based on: hep-th/0605234, hep-th/0608115 hep-th/0611242, arxiv:0707.1621 arxiv:0806.3370, arxiv:0912.5522 + work in progress IDEA: obtain an understanding of the quantum mechanical nature
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 12, 2011 1:00PM to 3:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of this
More information8.821 F2008 Lecture 12: Boundary of AdS; Poincaré patch; wave equation in AdS
8.821 F2008 Lecture 12: Boundary of AdS; Poincaré patch; wave equation in AdS Lecturer: McGreevy Scribe: Francesco D Eramo October 16, 2008 Today: 1. the boundary of AdS 2. Poincaré patch 3. motivate boundary
More informationY-system for the exact spectrum of N = 4 SYM
Universitȁt von Bonn, 6 Juni 2011 Y-system for the exact spectrum of N = 4 SYM Vladimir Kazakov (ENS,Paris) Theoretical Challenges in Quantum Gauge Field Theory QFT s in dimension>2, in particular 4D Yang-Mills
More informationHolographic Geometries from Tensor Network States
Holographic Geometries from Tensor Network States J. Molina-Vilaplana 1 1 Universidad Politécnica de Cartagena Perspectives on Quantum Many-Body Entanglement, Mainz, Sep 2013 1 Introduction & Motivation
More informationThe continuum limit of the integrable open XYZ spin-1/2 chain
arxiv:hep-th/9809028v2 8 Sep 1998 The continuum limit of the integrable open XYZ spin-1/2 chain Hiroshi Tsukahara and Takeo Inami Department of Physics, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551
More information8.821 String Theory Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.81 String Theory Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.81 F008 Lecture 1: Boundary of AdS;
More informationIntroduction to Integrability in AdS/CFT: Lecture 4
Introduction to Integrability in AdS/CFT: Lecture 4 Rafael Nepomechie University of Miami Introduction Recall: 1-loop dilatation operator for all single-trace operators in N=4 SYM is integrable 1-loop
More informationIntegrability in Super Yang-Mills and Strings
13/9/2006 QFT06, Kyoto Integrability in Super Yang-Mills and Strings Kazuhiro Sakai (Keio University) Thanks for collaborations to: N.Beisert, N.Gromov, V.Kazakov, Y.Satoh, P.Vieira, K.Zarembo 1 Contents
More informationHexagons and 3pt functions
Hexagons and 3pt functions Benjamin Basso ENS Paris Current Themes in Holography NBI Copenhagen 2016 based on work with Vasco Goncalves, Shota Komatsu and Pedro Vieira The spectral problem is solved O(x)
More informationA Solvable Irrelevant
A Solvable Irrelevant Deformation of AdS $ / CFT * A. Giveon, N. Itzhaki, DK arxiv: 1701.05576 + to appear Strings 2017, Tel Aviv Introduction QFT is usually thought of as an RG flow connecting a UV fixed
More informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More informationEntanglement entropy and the F theorem
Entanglement entropy and the F theorem Mathematical Sciences and research centre, Southampton June 9, 2016 H RESEARH ENT Introduction This talk will be about: 1. Entanglement entropy 2. The F theorem for
More informationSeminar in Wigner Research Centre for Physics. Minkyoo Kim (Sogang & Ewha University) 10th, May, 2013
Seminar in Wigner Research Centre for Physics Minkyoo Kim (Sogang & Ewha University) 10th, May, 2013 Introduction - Old aspects of String theory - AdS/CFT and its Integrability String non-linear sigma
More informationarxiv:hep-th/ v1 14 Jul 2005
Nonlinear integral equations for the finite size effects of RSOS and vertex-models and related quantum field theories arxiv:hep-th/0507132v1 14 Jul 2005 Árpád Hegedűs Istituto Nazionale di Fisica Nucleare,
More informationOn the algebraic Bethe ansatz approach to correlation functions: the Heisenberg spin chain
On the algebraic Bethe ansatz approach to correlation functions: the Heisenberg spin chain V. Terras CNRS & ENS Lyon, France People involved: N. Kitanine, J.M. Maillet, N. Slavnov and more recently: J.
More informationLanglands duality from modular duality
Langlands duality from modular duality Jörg Teschner DESY Hamburg Motivation There is an interesting class of N = 2, SU(2) gauge theories G C associated to a Riemann surface C (Gaiotto), in particular
More informationAn alternative to perturbative QFT: lattice discretization of the Sine-Gordon model
An alternative to perturbative QFT: lattice discretization of the Sine-Gordon model Rob Klabbers II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany,
More informationExact spectral equations in planar N=4 SYM theory
Euler Symposium on Theoretical and Mathematical Physics Euler International Mathematical Institute, St. Petersburg, July 12-17, 2013 Exact spectral equations in planar N=4 SYM theory Vladimir Kazakov (ENS,
More informationBlack holes with AdS asymptotics and holographic RG flows
Black holes with AdS asymptotics and holographic RG flows Anastasia Golubtsova 1 based on work with Irina Aref eva (MI RAS, Moscow) and Giuseppe Policastro (ENS, Paris) arxiv:1803.06764 (1) BLTP JINR,
More informationGlobal AdS Picture of 1/2 BPS Wilson Loops
Global AdS Picture of 1/2 BPS Wilson Loops arxiv:0912.1844v3 [hep-th] 4 Jan 2010 Kazumi Okuyama Department of Physics, Shinshu University Matsumoto 390-8621, Japan kazumi@azusa.shinshu-u.ac.jp We study
More informationIntroduction to AdS/CFT
Introduction to AdS/CFT D-branes Type IIA string theory: Dp-branes p even (0,2,4,6,8) Type IIB string theory: Dp-branes p odd (1,3,5,7,9) 10D Type IIB two parallel D3-branes low-energy effective description:
More informationQuantum Physics III (8.06) Spring 2008 Final Exam Solutions
Quantum Physics III (8.6) Spring 8 Final Exam Solutions May 19, 8 1. Short answer questions (35 points) (a) ( points) α 4 mc (b) ( points) µ B B, where µ B = e m (c) (3 points) In the variational ansatz,
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationShort spinning strings and AdS 5 S 5 spectrum
Short spinning strings and AdS 5 S 5 spectrum Arkady Tseytlin R. Roiban, AT, arxiv:0906.4294, arxiv:02.209 M. Beccaria, S. Giombi, G. Macorini, R. Roiban, AT, arxiv:203.570 70-s: origin of string theory
More informationEmergent Gauge Theory
Emergent Gauge Theory (Based on work with JiaHui Huang, Minkyoo Kim, Laila Tribelhorn and Jaco Van Zyl) Robert de Mello Koch South China Normal University and Mandelstam Institute for Theoretical Physics
More information10 Interlude: Preview of the AdS/CFT correspondence
10 Interlude: Preview of the AdS/CFT correspondence The rest of this course is, roughly speaking, on the AdS/CFT correspondence, also known as holography or gauge/gravity duality or various permutations
More informationGround state and low excitations of an integrable. Institut fur Physik, Humboldt-Universitat, Theorie der Elementarteilchen
Ground state and low excitations of an integrable chain with alternating spins St Meinerz and B - D Dorfelx Institut fur Physik, Humboldt-Universitat, Theorie der Elementarteilchen Invalidenstrae 110,
More informationNon-abelian statistics
Non-abelian statistics Paul Fendley Non-abelian statistics are just plain interesting. They probably occur in the ν = 5/2 FQHE, and people are constructing time-reversal-invariant models which realize
More informationOn sl3 KZ equations and W3 null-vector equations
On sl3 KZ equations and W3 null-vector equations Sylvain Ribault To cite this version: Sylvain Ribault. On sl3 KZ equations and W3 null-vector equations. Conformal Field Theory, Integrable Models, and
More informationComputation of the scattering amplitude in the spheroidal coordinates
Computation of the scattering amplitude in the spheroidal coordinates Takuya MINE Kyoto Institute of Technology 12 October 2015 Lab Seminar at Kochi University of Technology Takuya MINE (KIT) Spheroidal
More informationarxiv: v3 [hep-th] 17 Dec 2015
DMUS-MP-15/07 Integrable open spin-chains in AdS 3 /CFT 2 correspondences Andrea Prinsloo, Vidas Regelskis and Alessandro Torrielli Department of Mathematics, University of Surrey, Guildford, GU2 7XH,
More information129 Lecture Notes Relativistic Quantum Mechanics
19 Lecture Notes Relativistic Quantum Mechanics 1 Need for Relativistic Quantum Mechanics The interaction of matter and radiation field based on the Hamitonian H = p e c A m Ze r + d x 1 8π E + B. 1 Coulomb
More informationarxiv:hep-th/ v1 15 May 1996
DAMTP 96 4 KCL TH 96 7 May 13 1996 arxiv:hep-th/96514v1 15 May 1996 On the relation between Φ (1) and Φ (15) perturbed minimal models Horst Kausch 1 Gábor Takács Department of Applied Mathematics and Theoretical
More informationFour-point functions in the AdS/CFT correspondence: Old and new facts. Emery Sokatchev
Four-point functions in the AdS/CFT correspondence: Old and new facts Emery Sokatchev Laboratoire d Annecy-le-Vieux de Physique Théorique LAPTH Annecy, France 1 I. The early AdS/CFT correspondence The
More informationNew Phenomena in 2d String Theory
New Phenomena in 2d String Theory Nathan Seiberg Rutgers 2005 N.S. hep-th/0502156 J.L. Davis, F. Larsen, N.S. hep-th/0505081, and to appear J. Maldacena, N.S. hep-th/0506141 1 Low Dimensional String Theories
More informationTime dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012
Time dependent perturbation theory D. E. Soper University of Oregon May 0 offer here some background for Chapter 5 of J. J. Sakurai, Modern Quantum Mechanics. The problem Let the hamiltonian for a system
More informationQuantization of the open string on exact plane waves and non-commutative wave fronts
Quantization of the open string on exact plane waves and non-commutative wave fronts F. Ruiz Ruiz (UCM Madrid) Miami 2007, December 13-18 arxiv:0711.2991 [hep-th], with G. Horcajada Motivation On-going
More information