Towards Canonical Quantization of Non-Linear Sigma-Models Vladimir - - PowerPoint PPT Presentation

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

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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 Aν − ∂ν Aµ − [Aµ, Aν] = 0 ⇒ Euler-Lagrange equations

x ∼ x + R t

Classical Inverse Scattering Method (CISM): Wilson loop generates infinite family of conserved quantities T = Tr ← P exp

• C

Aµ dxµ

• It is unchanged under continous deformation of

the contour.

• 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 Mn =

←

P exp xn+1

xn

Aµdxµ 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, . . .

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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 + Γµ

νσ ∂z X ν ∂zX σ = 0

For symmetric connection Γµ

νσ = Γµ σµ it is a stationary point of the

action, defining non-linear sigma-models A =

• d2x Gµν(X) ∂zX µ∂zX ν ,

When it is integrable? (in classical/quantum case) Can one define a consistent quantum theory for this equation? Renormalization group equation (Ricci flow) ∂ ∂t Gµν = −Rµν

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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, L = 1

2

• (∂tn)2 − (∂xn)2

, n ∈ S2 the flat connection is (Zakharov-Mikhailov’78) A± = 1

2(At ± Ax) = ∂±g g −1

1 ± ζ , g = i

3

• a=1

naσa ∈ SU(2) , σa = Pauli matrices

• Non-ultralocal equal-time Poisson bracket

{Aµ(x) ⊗ , Aν(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)

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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 Z

• Dn O(n) exp
• − 1

f 2

• L[n]d2x
• Dimensional transmutation: RG invariant scale

E∗ = Λ exp

• − 4π

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

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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) = −Sa1(θ) ⊗ Sa2(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)

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Quantization of NLSM Paris, July 2017 8 / 23

In the standard spherical coordinates θ and φ, dℓ2 = dθ2 + sin2 θ dφ2 A = 1 2 (∂tn)2 − (∂xn)2 dtdx = 1 2

• (∂+θ ∂−θ + sin2 θ ∂+φ ∂−φ)dx+dx−

πφ = δA δ∂tφ = sin2 θ ∂tφ, πθ = δA δ∂tθ = ∂tθ (Canonical momenta) Equal-time Poisson brackets {πφ(x), φ(y)} = δ(x − y), {πθ(x), θ(y)} = δ(x − y), Zakharov-Mikhailov flat connection A± = 1 1 ± ζ   − sin2 θ ∂±φ e−iφ ( 1

2 sin 2θ ∂±φ + i∂±θ)

eiφ ( 1

2 sin 2θ ∂±φ − i∂±θ)

sin2 θ ∂±φ   The Poisson brackets {∂±φ, ∂±φ} and {∂±θ, ∂±θ} contain δ′(x − y) 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± → A± = G A± G −1 + G ∂± G −1 Taking the following simple matrix G =

• ζ + cos θ

e−iφ sin θ eiφ sin θ ζ − cos θ

• .
• ne gets an ultralocal connection
• A± =
• sin θ ∂±φ ± i∂±θ
• 2(1 ± λ)
• − sin θ

e−iφ(cos θ ± 1) eiφ(cos θ ± 1) sin θ

• ,

λ = 1

2(ζ + ζ−1)

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)

L = 1 2 ∂µn ∂µn (κ−1 − κ n2

3) ,

0 < κ < 1

• 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)

∼ log 1+κ

1−κ

• In elliptic coordinates

n1 = sd(θ, κ) cos φ, n2 = sd(θ, κ) sin φ, n1 = cd(θ, κ) the sausage metric is κ

• dθ2 + sn2(θ, κ)dφ2

,

• 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, r(µ) = 1 µ − µ−1

• 2t1 ⊗ t1 + 2t2 ⊗ t2 +
• µ + µ−1

t3 ⊗ t3

• ,

[ta, tb] = iǫabctc Conserved charges (Wilson loops) Tj(µ) = Tr

• e−iπkh Mj(µ)
• ,

Mj = πj ← P exp

• C

dx Ax

• .

Poisson commute

• Tj(µ), Tj′(µ′)
• = 0 .

Here πj (j = 0, 1

2, 1, . . .) is the spin-j representaion of sl(2) and k is the twist

parameter n3(x + R) = n3(x), n±(x + R) = e±2πikn±(x), n± = n1 ± in2

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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 = 2π (1 − κ2) + O(2) , (1) where is the (dimensionless) Planck constant. Integrating this equation leads to 1 − κ 1 + κ = (E∗/Λ)ν , ν = π (2) Here E∗ is an RG invariant energy scale. So κ → 1− as Λ → ∞. Stereographic projection. Introduce real fields φ and α e2φ+2iα = 1 − n3 1 + n3 n1 + in2 n1 − in2 ,

• V. Bazhanov (ANU)

Quantization of NLSM Paris, July 2017 13 / 23

UV-limit: long “sausages” and “cigars”

∼ log 1+κ

1−κ

• For κ → 1 the sausage turns into two cigars.

P(in)

1

P(out)

1

with the metric (Hamilton’88) L = 1

2

• (∂µφ)2 + tanh2(φ) (∂µα)2

, φ → φ + 1

2 log(1 − κ

1 + κ)

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Quantization of NLSM Paris, July 2017 14 / 23

Chiral transfer-matrices for the cigar

R t0 t x

Infinite momentum frame: Move contour to the light cone.

←

P exp dx+ A+ ← P exp dx− A−

• = L+(λ+) L−(λ−),

λ+λ− = 1 − κ 1 + κ For κ → 1 one factor becomes trivial. Lj(λ+) = πj ← P exp

• iλ+

t0+R

t0

dx+

• V + e+ + V − e−
• e−πP1h
• V ± = (∂+φ+ − i ∂+α+) e±2φ+ ,

φ+(t0 + R) = 2πP1 + φ+(t0) , α+(t0 + R) = 2πP2 + α+(t0) Here {h, e, f} is the Cartan-Weyl basis of sl(2).

• V. Bazhanov (ANU)

Quantization of NLSM Paris, July 2017 15 / 23

Quantization of the chiral transfer-matrices

Quantize the Lie algebra: sl(2) → Uq

• sl(2)
• [h, e±] = ±2 e± ,

[e+, e−] = qh − q−h q − q−1 , ≡ 2π

n ,

q = e

iπ n .

Replace α+(x) and θ+(x) with quantum chiral bose fields Then the quantum L-operator is (the same form as for quantum KdV (BLZ)) Lj(λ+) = πj ← P exp

• iλ+

t0+R

t0

dx

• V + q

h 2 e+ + V − q− h 2 e−

• e−πP1h
• .

Quantum Yang-Baxter algebra R(λ′

+/λ+)

• L(λ+) ⊗ 1
• (1 ⊗ L(λ′

+)

• =
• 1 ⊗ L(λ′

+)

L(λ+) ⊗ 1

• R(λ′

+/λ+) ,

where R(λ) is the trigonometric R-matrix for Uq

• sl(2)
• .

Chiral transfer-matrices form commuting family τj(λ+) = Tr

• Lj(λ+) e−πP1h

, [ τj(λ+), τj′(λ′

+) ] = 0 ,

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Quantization of NLSM Paris, July 2017 16 / 23

Quantization of the O(3)/sausage model

✓ Conserved quantities and ultralocal canonical Poisson structure ✓ Construction of commuting transfer-matrices in QFT. (start with CFT = RG fixed point at short distances (UV)). Quantum cigar. ✓ Discretization of quantum chiral transfer matrices, functional

• relations. Lattice parafermions, Fateev-Zamolodchikov model & chiral

Potts model. ✓ Identification of functional relations with those from the “paperclip model” (Lukyanov-Vitchev-Tsvelick-Zamolodchikov) ✓ ODE/IQFT correspondence. Eigenvalues of chiral transter-matrices = connection coefficients for an ODE. (Voros’82, Dorey-Tateo’98, BLZ’98 , Suzuki’99, Feigin-Frenkel’07, Fioravanti’05, Dunning’03, Masoero’06, Raimondo, Valeri, Faldella, Negro, Ito, Locke. . . ) ✓ ODE/IQFT correspondence for (massive) models. Modified sinh-Gordon equation (Lukyanov-Zamolodchikov’10)

• V. Bazhanov (ANU)

Quantization of NLSM Paris, July 2017 17 / 23

New connection between quantum and classical systems

Quasiclassical approximation, when Planck constant → 0 Quantum theory ⇒ Classical theory A new type connection for finite values of Integrable Quantum Field Theory ⇔ Integrable Classical Field Theory Stationary quantum QFT states are described by flat connections on the punctured Riemann sphere (involving special solutions of non-linear PDE). The energy of the state coincides with the (regularized) area of a CMC surface embedded into AdS3, described by sinh-Gordon equation (Lukyanov, VB, Kotousov, Runov, 2013-2017)

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Quantization of NLSM Paris, July 2017 18 / 23

Quantum/classical correspondence

The modified Sinh-Gordon (MShG) equation ∂z∂¯

zη − e2η + ρ4P(z) ¯

P(¯ z) e−2η = 0, P(z) = (z − z2)n (z − z1)2+n(z − z3)2 e−η ∼ |P(z)|−1/2 z → z1,3, e−η ∼ |z − z2|1−(n+2)k z → z2 This equation is the compatibility condition of the linear problem (∂z − Az)Ψ = 0 , (∂¯

z − A¯ z)Ψ = 0 ,

λ = ρeθ, ¯ λ = ρe−θ Az = − 1

2 ∂zη σ3 +

• σ+ eη + σ− λ2P(z) e−η

A¯

z

=

1 2 ∂¯ zη σ3 +

• σ− e−η + σ+ ¯

λ2 ¯ P(¯ z) eη . (3) Connection coefficients for these ordinary differential operators coincide with vacuum eigenvalues various transfer-matrices in sausage model

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Quantization of NLSM Paris, July 2017 19 / 23

Non-Linear Integral Equations for sausage model

ε(θ − iγ) = r sinh(θ − iγ) − 2πk + ∞

−∞

dθ′ 2π G1(θ − θ′ − iγ) log

• 1 + e−ω(θ′)

+ ∞

−∞

dθ′ 2πi

• G(θ − θ′ − 2iγ)
• L(θ′ − iγ)

∗ − G(θ − θ′) L(θ′ − iγ)

• ω(θ)

= r cosh(θ) + Im ∞

−∞

dθ′ π G1(θ − θ′ + iγ) L(θ′ − iγ)

• −

∞

−∞

dθ′ π G2(θ − θ′) log

• 1 + e−ω(θ′)

L(θ) = log

• 1 + e−iε(θ)

. Effective central charge, F = −ceff (r)/6, r = MR F(r, k) = r 2π2 ∞

−∞

dθ

• ± 2 Imm
• e±(θ−iγ) L(θ − iγ)
• − e±θ log
• 1 + e−ω(θ)
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Quantization of NLSM Paris, July 2017 20 / 23

F g k = 0 k = 0.2 k = 0.4

• ×

× × × × × × × × × × × × × × × × × × × × × × ×

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

0.1 0.2 0.3 0.4

• 0.3
• 0.2
• 0.1

0.1

Figure: A plot of F(r, k) with k = 0, 0.2 and 0.4 as a function of the running coupling constant g(r) for the O(3) sigma model g −1 e− 1

g =

1 32π eγE +1 r

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Quantization of NLSM Paris, July 2017 21 / 23

Future tasks

Reproduce the same results in a generic non-ultralocal gauge for the flat connection. Construct infinite-dimensional representations for vertex operators of the cigar/sausage models. Study fusion algebra of quantum transfer-matrices in the most general case (not just Zn parafermions). Fascinating relations to affine superalgebras are expected. Apply the approach to more interesting sigma-models. For instance, the sausage model with θ-term, deformed O(4) NLSM, AdS/CFT, etc.

• V. Bazhanov (ANU)

Quantization of NLSM Paris, July 2017 22 / 23

Thank you!

• V. Bazhanov (ANU)

Quantization of NLSM Paris, July 2017 23 / 23