A lattice study of N =2 Landau-Ginzburg model using a Nicolai map - PowerPoint PPT Presentation

a A lattice study of N =2 Landau-Ginzburg model using a Nicolai map based on arXiv:1005.4671 Hiroki Kawai (in collaboration with Y.Kikukawa) Institute of Physics, The University of Tokyo

1 Purposeg 2d N =2 Landau-Ginzburg model (LG model) ( ∫ ) ∫ d 2 xd 4 θ K (Φ , ¯ d 2 xd 2 θ W (Φ) + c.c. S = Φ) + Φ … chiral superfield At the IR fixed point, W (Φ) = λ Φ k is believed to describe... ✕ ✁ { N = 2 minimal model ← check for K (Φ , ¯ Φ) = ¯ ΦΦ (WZ model) ✁ → Gepner model (compactified string), ... ֒ λ eff → ∞ , lattice ! Why it is believed that LG models describe CFTs ? ● 2d bosonic case ’86 A.B.Zamolodchikov … φ 2 p − 3 6 (2 , 2) ∝ ∂ 2 φ (2 , 2) In the c = 1 − p ( p +1) minimal model, the fusion rule implies … φ 2 p − 3 ∝ ∂ 2 φ In the 2d bosonic LG model L = 1 2 ∂ µ φ∂ µ φ + gφ 2 p − 2 , EOM is conjecture ⇒ φ = φ (2 , 2) at the IR fixed point. ⇒ Extending this idea, ...

How to check the conjecture ● early studies → ’89 Kastor, Martinec and Shenker RG flow of c -functions → ’89 Vafa and Warner → For W (Φ) = λ Φ k , catastrophe theory  → ’89 Howe and West c = 3(1 − 2 ϵ -expansion k )    → ’93 Witten Φ : ( h, ¯  h ) = ( 1 2 k , 1 elliptic genus, SCA 2 k )    Φ 2 : ( h, ¯ h ) = ( 2 2 k , 2 2 k ) ...  ：     Φ k − 2 : ( h, ¯  h ) = ( k − 2 2 k , k − 2 2 k )  ● We computed correlation functions non-perturbatively for W (Φ) ∝ Φ 3 . susceptibility of CFT: ∫ ∫ 1 finite volume h ∝ V 1 − h − ¯ d 2 x 〈 φ ( x ) φ ∗ (0) 〉 d 2 x h χ ≡ − → | x | 2 h +2¯ V ⇒ log χ = (1 − h − ¯ h ) log V + const. ✁ ✕ ✁ ✁ 6 = 0.666... For the present W (Φ) ∝ Φ 3 , 1 − h − ¯ h = 1 − 1 6 − 1

2 Lattice Formulation of WZ modelg lattice action: ’83 Sakai and Sakamoto ’02 Catterall and Karamov ’02 Kikukawa and Nakayama φ ∗ Tφ + W ∗ (1 − a 2 ’09,’10 Kadoh and Suzuki,.. ∑ { ( ) W ′ ( − S 1 + iS 2 ) φ + c.c. S = 4 T ) W + ( + 1 − γ 3 W ′′∗ 1 − ˆ ) } D + 1 + γ 3 W ′′ 1 + ˆ γ 3 γ 3 + ¯ ψ ψ 2 2 2 2 D = 1 [ ] X W = λ 3 Φ 3 √ where 1 + = T + γ 1 S 1 + γ 2 S 2 , E 1 2 X † X ✻ a enough λ { modes ! continuum limit : aλ → 0 ✏ ✶ ✏ ● λ is the unique mass parameter (besides a ) ⇒ To see CFT, L ≫ ( aλ ) − 1 is needed. 0 { ◎ one SUSY Q ● no extra fine-tunings ⇐ ◎ Z 3 R-symmetry ← overlap fermion ● This lattice model faces the sign problem | D + F | is real, but can be negative. ⇐ γ 1 ( D + F ) γ 1 = ( D + F ) ∗

3 Simulationg We utilized the Nicolai map : η = W ′ + ( φ − a 2 W ′ ) T + ( φ ∗ − a g 2 W ∗′ )( S 1 + iS 2 ) . 〈O〉 = 〈 ∑ N ( η ) i =1 O ( φ i ) sgn | D + F ( φ i ) |〉 η 〈 ∑ N ( η ) a → 0 i =1 sgn | D + F ( φ i ) |〉 η → Witten index ∆ = 2 (cubic potential) x | η | 2  η X e − P R D η D ¯ 〈 X 〉 η ≡  x | η | 2 η e − P R  D η D ¯ where  N ( η ) counts the solutions of the Nicolai map φ 1 , .., φ N ( η )  1. Assigning { η , η ∗ } as the standard normal distribution, 2. Solving the Nicolai map by the Newton-Raphson algorithm, 3. Sample the configurations of { φ , φ ∗ } . … no autocorrelation ● advantage … N ( η ) ● difficulty

Susceptibility: χ φ ≡ ∑ x ≥ 3 〈 φ ( x ) φ (0) 〉 W (Φ) = λ 3 Φ 3 , aλ = 0 . 3 , L = 18 , 20 , .., 32 (Newton iter. from 100 initial config. for each noise) × 320 noises 4.8 linear fit by least-square-method 4.5 ❅ ❅ ❅ ❘ ln χ φ 4.2 3.9 6 6.5 7 ln L 2 ⇒ χ φ ∝ V 0.660 ± 0.011 ⇒ consistent with the conjecture χ φ ∝ V 0.666... ◎

4 Summary and future plan Summary ∫ V d x 2 〈 φ ( x ) φ ∗ (0) 〉 in the cubic potential case, and got the consistent result • We observed χ = with the conjecture χ ∼ V 0 . 660 ± 0 . 011 . • We also extracted the effective coupling constant K of the Gaussian model, and obtained 3 K = 0 . 242 ± 0 . 010 which is consistent with the N = 2 SUSY point K = 4 π = 0 . 238 .. . This implies the restoration of all supersymmetries in the IR. (see more detail in arXiv:1005.4671) Future Plan • further check of the A-D-E classification: → A 3 model ？ Φ 4 W = Φ 3 + Φ ′ 4 → E 6 = A 2 ⊗ A 3 model ？ Φ 3 + ΦΦ ′ 2 → D 4 model ？ • c-function → central charge, c-theorem • 2d N = 1 LG model with W ∝ Φ 3 ( infrared → tricritical ising model) ⇒ dynamical SUSY breaking