Aspects of integrability in classical and quantum field theories

Transcription

1 Aspects of integrability in classical and quantum field theories Second year seminar Riccardo Conti Università degli Studi di Torino and INFN September 26, 2018

2 Plan of the talk PART 1 Supersymmetric 3D and 4D QFTs Integrability AdS/CFT Vocabulary : QFT = Quantum Field Theory CFT = Conformal Field Theory AdS = anti de Sitter Integrable deformations of 2D QFTs PART 2

3 Integrability in a nutshell INTEGRABILITY EXACT SOLVABILITY Physical observables are found analytically or by quadrature for any value of the parameters. completely non perturbative approach (no Feynman diagrams!) Necessary conditions : infinite number of conservation s laws (in field theories). Drawback : few systems are integrable! Usually low dimensional systems D 2 Examples : harmonic oscillator, Korteweg devries (KdV), sine Gordon, Heisenberg spin chain, (hydrodynamics) (Kondo effect) (ferromagnets) However several applications to low dimensional physics; studying simpler (integrable) models allows to better understand more sophisticated ones.

4 PART 1 Supersymmetric 3D and 4D QFTs

5 Integrability in 3D and 4D QFTs N 4super Yang Mills (SYM) in D 4 Quantum Chromodinamic (QCD). N 6super Chern Simons (ABJM) in D 3 Fractional Quantum Hall effect (FQHE) (supersymmetry conformal symmetry) 1 to get insights about very complicated physical theories; Why studying them? 2 to study the gauge gravity correspondence AdS/CFT; 1 conformal dimensions h What are the observables? 2 structure constants c h How does integrability emerge? connection with 2D integrable systems (spin chains) h h E h E E is the energy of a spin chain with interaction between n neighbouring sites.

6 Conformal dimensions in ABJM Quantum Spectral Curve (QSC) Bombardelli, Cavaglià, Conti, Tateo, /JHEP04(2018)117 Finite set of functional equations involving 6 6multi valued functions, P u,, and Q u,, INPUT Riemann Hilbert problem OUTPUT u plane Large u behavior of P u and Q u QSC P u and Q u in the whole complex u plane The conformal dimension h is hidden in the asymptotics! Idea : fix h imposing the QSC equations

7 Numerical results Bombardelli, Cavaglià, Conti, Tateo, /JHEP04(2018)117 Method to compute conformal dimensions of any operator at any value of the coupling constant h with high precision; Confirmation of the validity of the AdS/CFT correspondence. h~ CFT predictions h h h~2g / AdS predictions g / g 2πh log2 g / Numerics

8 Analytic structure of Bombardelli, Cavaglià, Conti, Tateo (to appear) 3i/4 i/2 Re h i/4 Motivation : phase transition AdS ds when h i h(polyakov conjecture). study the critical exponent of h around branch point singularities in the complex h plane. Hinted presence of a logarithmic term Im h h ~α α β log 16h 1 16h 1 /

9 PART 2 Integrable deformations of 2D QFTs

10 Integrable deformations of 2D QFTs Generic 2D QFT with finite size (R) spectrum E R, 0 CDD factor evolution equation for the finite size spectrum E R, τ S matrix : S S S e i Burgers equation E R, τ E R, τ P R Solution E R, τ P R E R,0 P R, where R R τ E R,τ τ P R E R, τ is uniquely determined from the initial condition E R, 0 (integrable deformation) Example : starting from the spectrum of a CFT, E R, 0 E R, τ R 2τ 4τ 1 1 4τ A R R P R Nambu Goto energy levels (quantum gravity theory)

11 deformation What is the generator of this deformation? TT operator : TT TT Θ Factorization property : ntt n ntn nt n nθn nθn Burgers equation implies the flow equation : τ L TT L τ 0 TT τ L τ dτ Example : starting from N free bosons L g φ φ, μ, ν 1,2, i 1,, N L τ det g det g τ h, h φ φ Nambu Goto Lagrangian in static gauge

12 Geometric interpretation Conti, Iannella, Negro, Tateo, arxiv: [hep th] Conti, Negro, Tateo, arxiv: [hep th] Extension to a generic interacting bosonic theory : L g φ φ V φ, φ φ,,φ L τ det g det g τ h, h φ φ, τ τ 1 τ V the equations of motion (EoMs) associated to L τ are really complicated! There exists a field dependent change of variables x,x y x,x, y x,x such that EoMs of L τ EoMs of L φ x,x τ φ y x,x, y x,x τ 0 TT deformed solutions on flat spacetime g η undeformed solutions on curved spacetime g η τ ε ε 2T τ T Jackiw Teitelboim (JT) topological gravity in 2D

13 Study case : sine Gordon model sine Gordon (sg) model is a 2D integrable theory described by the Lagrangian L φ φ V φ, V φ 4 sin solutions of sg are well known! solitons stationary 1 kink solution stationary breather solution

14 Deformed 1 kink solution shock wave singularity at τ explained as singularity of the change of variables typical phenomenon of Burgers equation

15 Deformed breather solution (dilation)

16 Deformed breather solution (contraction)

17 Future directions Conti, Iannella, Negro, Tateo, arxiv: [hep th] Extension to higher dimensions : 4D Maxwell Born Infeld theory (non linear electrodynamics) asa TT deformation of Maxwell electrodynamics τ L det T Tr T Tr T L τ det g det g 2τ F, F A A Goal : find a renormalization scheme to quantize Maxwell Born Infeld. Problem : no factorization property of TT in dimension higher than 2!

18 List of publications, talks and posters Publications : 1) D. Bombardelli, A. Cavaglià, R. Conti and R. Tateo, Exploring the spectrum of planar AdS /CFT at finite coupling, JHEP 04 (2018) 117 arxiv: [hep th]. 2) R. Conti, L. Iannella, S. Negro and R. Tateo, Generalised Born Infeld models, Lax operators and the TT perturbation, arxiv: [hep th]. 3) R. Conti, S. Negro, R. Tateo, The TT perturbation and its geometric interpretation, arxiv: [hep th]. 4) D. Bombardelli, A. Cavaglià, R. Conti, R. Tateo, On the analytic structure of anomalous dimensions in planar AdS /CFT, (to appear). Talks and posters : 1) Talk at Young Researcher Integrability School and Workshop 2017 with the title Numerical results in ABJM theory. 2) Poster at Young Researcher Integrability School and Workshop 2018 with the title Exploring the spectrum of AdS /CFT at finite coupling. 3) Poster at Integrability in Gauge and String Theory 2018 with the title Generalised Born Infeld models, Lax operators and the TT perturbation.