Towards Canonical Quantization of Non-Linear Sigma-Models Vladimir Bazhanov The Australian National University joint work with Sergei Lukyanov (Rutgers), Gleb Kotousov (ANU/Rutgers) Integrability in Gauge and String Theories Paris, July 2017 V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 1 / 23
Outline Integrability structures in classical & quantum 2D field theory Classical and Quantum Inverse Scattering Method (CISM/QISM) Non-linear sigma-models (NLSM). Canonical quantization of (deformed) O(3) sigma-model (sausages and cigars) Connections to (lattice) parafermions ODE/IQFT correspondence — a powerful extension of QISM Non-Linear Integral Equations for vacuum eigenvalues Future developments V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 2 / 23
Integrability in 2D Classical Field Theory Zero-curvature representation (ZCR): A flat Lie algebra-valued world sheet connection, depending on an auxillary “spectral parameter”, such that ∂ µ A ν − ∂ ν A µ − [ A µ , A ν ] = 0 ⇒ Euler-Lagrange equations Classical Inverse Scattering Method (CISM): t Wilson loop generates infinite family of conserved quantities � ← � � A µ d x µ T = Tr P exp C It is unchanged under continous deformation of the contour. x ∼ x + R V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 3 / 23
Integrability in 2D Quantum Field Theory (QFT) Quantum Inverse Scattering Method (QISM) – (Faddeev-Sklyanin- Takhtajan,’79) Baxter’s commuting transfer matrices (1972) – quantum counterpart of the classical Wilson loop. ULTRALOCALITY Elementary transport matrices � x n +1 ← A µ d x µ M n = P exp x n commute for different segments of the discretized path! Yang-Baxter algebras & Quantum Groups and their representation theory. Architypal example: Sine-Gordon model. Discretization, canonical quantization, Bethe ansatz, filling of the vacuum state, continuous limit, . . . V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 4 / 23
Harmonic maps, Non-linear Sigma Models & Ricci flow The map z ′ = f ( z ) with analytic f ( z ) is a harmonic map C → C Define harmonic map from C to a Riemann manifold with coordinates X µ ∂ z ∂ z X µ + Γ µ νσ ∂ z X ν ∂ z X σ = 0 For symmetric connection Γ µ νσ = Γ µ σµ it is a stationary point of the action, defining non-linear sigma-models � d 2 x G µν ( X ) ∂ z X µ ∂ z X ν , A = When it is integrable? (in classical/quantum case) Can one define a consistent quantum theory for this equation? Renormalization group equation (Ricci flow) ∂ ∂ t G µν = −R µν V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 5 / 23
Non-Linear Sigma Models (NLSM) • The QISM fails when it is applied to NLSM. For instance, for the O (3) NLSM, � ( ∂ t n ) 2 − ( ∂ x n ) 2 � n ∈ S 2 L = 1 , 2 the flat connection is (Zakharov-Mikhailov’78) 3 � 2 ( A t ± A x ) = ∂ ± g g − 1 A ± = 1 , g = i n a σ a ∈ SU (2) , σ a = Pauli matrices 1 ± ζ a =1 • Non-ultralocal equal-time Poisson bracket { A µ ( x ) ⊗ , A ν ( y ) } = C (0) µν ( x ) δ ( x − y ) + C (1) µν ( x ) δ ′ ( x − y ) , hampers the application of QISM and the first-principle quantization of the model. • Plausibly one can use extended algebraic structures & “dynamical Yang-Baxter equation” (Maillet’86) V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 6 / 23
What do we know about O (3) NLSM and its extensions? ✓ Renormalizable: Asymptotic freedom (Polyakov’76), Renormalization group equations as the Ricci flow (Friedan’80) � � � � � O ( n ) � = 1 − 1 L [ n ] d 2 x D n O ( n ) exp f 2 Z 0 Dimensional transmutation: RG invariant scale � � − 4 π E ∗ = Λ exp f 2 0 (Λ) ✓ Instantons (Belavin-Polyakov’75) ✓ Generalizations of O(3) and O(4) NLSM: 1-param. deformation of O(3) NLSM (2D sausage), Fateev-Onofri-Zamolodchikov’93 2-param. deformation of O(4) NLSM (3D sausage), Fateev’96, 4-param. deformation of O(4) NLSM (torsion fields), Lukyanov’12 ✓ Classically integrable: Pohlmeyer’76, Zakharov & Mikhailov’78, Cherednik’81, Lukyanov’12, Climˇ cik’14 V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 7 / 23
What do we know about O (3) NLSM and its extensions? ✓ Factorized scattering: exact 2-particle S-matrix (Zamolodchikov(2)’78, Fateev-Onofri-Zamolodchikov’93, O(4)-model: Polyakov & Wiegmann’85, Faddeev & Reshetikhin ’85, Fateev’96 ) S ( θ ) = − S a 1 ( θ ) ⊗ S a 2 ( θ ) ✓ Thermodynamic Bethe Ansatz (TBA) Wiegmann’84, Zamolodchikov(2)’92, Fateev-Onofri-Zamolodchikov’93, Fateev’96, Balog-Hegedus’04, Gromov-Kazakov-Vieira’09, Ahn-Balog-Ravanini’07 ✗ QISM: Yang-Baxter structure, discretization, commuting transfer matrices, Bethe Ansatz — this talk and arXiv:1706.09941 (VB.-Kotousov-Lukyanov) V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 8 / 23
In the standard spherical coordinates θ and φ , d ℓ 2 = d θ 2 + sin 2 θ d φ 2 � � � ( ∂ t n ) 2 − ( ∂ x n ) 2 � A = 1 dtdx = 1 ( ∂ + θ ∂ − θ + sin 2 θ ∂ + φ ∂ − φ ) dx + dx − 2 2 π φ = δ A π θ = δ A δ∂ t φ = sin 2 θ ∂ t φ, δ∂ t θ = ∂ t θ (Canonical momenta) Equal-time Poisson brackets { π φ ( x ) , φ ( y ) } = δ ( x − y ) , { π θ ( x ) , θ ( y ) } = δ ( x − y ) , Zakharov-Mikhailov flat connection − sin 2 θ ∂ ± φ e − i φ ( 1 2 sin 2 θ ∂ ± φ + i ∂ ± θ ) 1 A ± = 1 ± ζ e i φ ( 1 sin 2 θ ∂ ± φ 2 sin 2 θ ∂ ± φ − i ∂ ± θ ) The Poisson brackets δ ′ ( x − y ) { ∂ ± φ, ∂ ± φ } and { ∂ ± θ, ∂ ± θ } contain Reason for non-ultralocality: A ± – components of Lorentz vector, they must contain field derivatives, since the spectral parameter is a scalar. V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 9 / 23
Zero curvature representation: ultralocal gauge Zero curvature representation is not affected by gauge transformations. A ± = G A ± G − 1 + G ∂ ± G − 1 A ± → � Taking the following simple matrix � � e − i φ sin θ ζ + cos θ G = . e i φ sin θ ζ − cos θ one gets an ultralocal connection � � � � e − i φ (cos θ ± 1) − sin θ sin θ ∂ ± φ ± i ∂ ± θ � λ = 1 2 ( ζ + ζ − 1 ) A ± = , 2(1 ± λ ) e i φ (cos θ ± 1) sin θ Still do not know how to quantize with QISM, because of complicated mixture of the quasi-classical limit ( � → 0) and the UV fixed point ( f 2 0 (Λ) → 0, asymptotic freedom). V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 10 / 23
Deformed O (3) NLSM (2D Sausage model) ∂ µ n ∂ µ n L = 1 3 ) , 0 < κ < 1 ( κ − 1 − κ n 2 2 • For κ → 0 reduces to the O(3) NLSM. Target manifold is a round sphere. • For κ → 1 − the target manifold is a long sausage with the length ∝ log( 1+ κ 1 − κ ) � 1+ κ � ∼ log 1 − κ In elliptic coordinates n 1 = sd ( θ, κ ) cos φ, n 2 = sd ( θ, κ ) sin φ, n 1 = cd ( θ, κ ) the sausage metric is � d θ 2 + sn 2 ( θ, κ )d φ 2 � κ , V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 11 / 23
ZCR for the sausage model (Lukyanov’12) The “ultralocal gauge” also exists! (take the undeformed case and replace trig function by the elliptic ones). The r -matrix Poisson brackets � � � � A ± ( x | µ ) ⊗ , A ± ( x ′ | µ ′ ) A ± ( x | µ ) ⊗ 1 + 1 ⊗ A ± ( x ′ | µ ′ ) , r ( µ/µ ′ ) δ ( x − x ′ ) = ± with the classical r -matrix of the 6-vertex model, � � � µ + µ − 1 � 1 r ( µ ) = 2 t 1 ⊗ t 1 + 2 t 2 ⊗ t 2 + t 3 ⊗ t 3 , [ t a , t b ] = i ǫ abc t c µ − µ − 1 Conserved charges (Wilson loops) � ← � � � � e − i π k h M j ( µ ) T j ( µ ) = Tr , M j = π j P exp d x A x . C Poisson commute � � T j ( µ ) , T j ′ ( µ ′ ) = 0 . Here π j ( j = 0 , 1 2 , 1 , . . . ) is the spin- j representaion of sl (2) and k is the twist parameter n ± ( x + R ) = e ± 2 π i k n ± ( x ) , n 3 ( x + R ) = n 3 ( x ) , n ± = n 1 ± i n 2 V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 12 / 23
Continuous version of QISM. Continuous version of QISM. VB-Lukyanov-Zamolodchikov (BLZ) BLZ approach starts with the analysis of the UV fixed point of RG equations Λ ∂κ ∂ Λ = � 2 π (1 − κ 2 ) + O ( � 2 ) , (1) where � is the (dimensionless) Planck constant. Integrating this equation leads to 1 − κ ν = � 1 + κ = ( E ∗ / Λ) ν , (2) π Here E ∗ is an RG invariant energy scale. So κ → 1 − as Λ → ∞ . Stereographic projection. Introduce real fields φ and α e 2 φ +2 i α = 1 − n 3 n 1 + i n 2 , 1 + n 3 n 1 − i n 2 V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 13 / 23
UV-limit: long “sausages” and “cigars” � 1+ κ � ∼ log 1 − κ For κ → 1 the sausage turns into two cigars. P ( out ) 1 P ( in ) 1 with the metric (Hamilton’88) � ( ∂ µ φ ) 2 + tanh 2 ( φ ) ( ∂ µ α ) 2 � 2 log(1 − κ L = 1 φ → φ + 1 , 1 + κ ) 2 V. Bazhanov (ANU) Quantization of NLSM Paris, July 2017 14 / 23
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