So Matsuura (Department of Physics Hiyoshi, - PowerPoint PPT Presentation

球面上の超対称ゲージ理論の数値実験 So Matsuura (Department of Physics Hiyoshi, Keio University) 1 Based on work with K. Ohta, T. Misumi and S. Kamata 島根大学松江キャンパス Numerical Experiment of Supersymmetric Gauge Theory on 2-Sphere 2019/9/10

Continuum theory 島根大学松江キャンパス 2 Starting point bosons: fermions: global symmetries SUSY transformation 2D 𝑂 = (2,2) SYM theory 𝐵 ( , 𝜚, * 𝜚 𝑉 1 - transformation 𝑉 1 . transformation 2019/9/10

Seiberg 2011, 2012 : scalar 3 Derivatives on a curved background spin connection: Special background or Topological twisting 島根大学松江キャンパス : vector 𝜖 ( → ∇ ( 𝜕 34 ≡ 𝜕 Derivatives on a curved background + background 𝑉 1 . field 𝐶 ( 𝜖 ( + 𝑗 2 𝐶 ( − 𝜕 ( 𝜏 4 𝜔 → 𝜖 ( 𝜔 𝐶 ( = 𝜕 ( 𝜖 ( + 𝑗 𝜔 → 𝜖 ( + 𝑗𝜕 ( 𝜏 4 * * 2 𝐶 ( + 𝜕 ( 𝜏 4 𝜔 2019/9/10

島根大学松江キャンパス 4 Continuum action Natural renaming in this background : scalar : vector 𝑉 1 - symmetry 𝑉 1 . symmetry 2019/9/10

̅ 5 SUSY transformation of the action (1) scalar SUSY transformations: ̅ 島根大学松江キャンパス (2) vector SUSY transformations: ̅ preserve iff 𝜗 ( = 0 𝜗 ? = 𝜗 ? (𝑦) preserve for 𝜗 ? = const. 𝜗 ( = ̅ 𝜗 ( (𝑦) 𝜗 ? = 0 𝜗 ( is covariantly const. 2019/9/10

(2) keep Q-symmetry (1) assign bosons on the lattice corresponding 6 Continuum action in Q-exact form Observation toward discretized theory Bosonic fields on lattice 島根大学松江キャンパス to their vector structure site variable link variable face variable requirement scalar 𝜚(𝑦) vector A ( (𝑦) field tensor 𝐺 (C (𝑦) 2019/9/10

7 7 島根大学松江キャンパス Fields on lattice Discretized action cf) Sugino 2003 u t s 2019/9/10

島根大学松江キャンパス 8 Topological information is preserved on lattice (1) The same localization technique with the continuum theory works also on lattice non-trivial fixed point equations Euler characteristic ! 1-loop contribution is exact 𝑎 ∼ (2) The 𝑉 1 - anomaly appears in the measure N S N L N F D � D Φ s D ¯ � � � : U(1) R neutral B = ( Φ s )( D U l )( D Y f ) 𝑉 1 - symmetry s =1 l =1 f =1 N S N L N F D � � � � F = ( D � s )( D � l )( D � f ) s =1 l =1 f =1 � � e i ( N S − N L + N F )( N 2 − 1) α D � F → 2019/9/10

(1) Tree level continuum limit reproduces the continuum action: from power counting point of view 1-loop tree <latexit sha1_base64="4mzDHAsGfoDq16kE0cVk0mAqH6U=">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</latexit> 島根大学松江キャンパス <latexit sha1_base64="gfav4GOh/ZEyFXU31xV18XOJus0=">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</latexit> (It gives a “definition” of the SUSY theory on curved background!) We have to check it non-perturbatively. YES! Natural question : Can we take the continuum limit ? 9 2-loop (2) There is no Q, 𝑉 1 - and gauge-invariant radiative correction which spoils the geometry. ✓ a p − 4 ◆ Z + c 1 p p − 2 + c 2 a p g 2 + · · · O ∼ B or B 2 d 2 x √ g O p ( x ) g 2 It is expected that the continuum theory will be obtained by simply taking 𝑏 → 0 . 2019/9/10

島根大学松江キャンパス 10 What should we check? They must be restored in the continuum limit. fact These symmetries are related with each other in the continuum limit. <latexit sha1_base64="f5hL2nPISl2kQRmrFe0KD6qy+I=">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</latexit> <latexit sha1_base64="B4EZVkTOjSOF3OBejB6bPIGM/Ic=">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</latexit> The 𝑉 1 . symmetry and G 𝑅 -symmetry are broken by discretization. dJ Q = 0 dJ ˜ Q = 0 It is sufficient to check the 𝑉 1 . symmetry 2019/9/10

Model 3 parameter tuning M(N-1)+2 ・・・ M-1 M 2 1 11 島根大学松江キャンパス (M,N)-polygon decomposition of 𝑇 4 𝑒𝑡 4 = 𝑆 4 (𝑒𝜄 4 + cos 4 𝜄 𝑒𝜒 4 ) (− ⁄ ⁄ 𝜌 2 < 𝜄 < 𝜌 2) MN/2 M(N-1)+1 2019/9/10

島根大学松江キャンパス coming from the discretization Ignore only the artificial phase 12 Anomaly-phase-quench method vev in the continuum theory U(1) charge: ( N 2 − 1) χ h phase quench method in usually used in Monte Carlo method NOT A GOOD U(1) charge: ZERO APPROXIMATION Observation Pf( D ) = | Pf( D ) | e i θ A + i θ philosophy of the phase quench 1.U(1) R phase θ A 2.lattice artifact θ We should ignore only θ 2019/9/10

島根大学松江キャンパス 13 Compensator Kamata-Misumi-Ohta-S.M. 2016 A : an operator with anomaly-phase-quench method Q A = 0 • = 1 � D � BO| Pf ( D ) | e i θ A [ A ] = − ( N 2 − 1) χ h • Z q A ≡ |A| e − i θ A • determinant type trace type Izykson-Zuber type 2019/9/10

島根大学松江キャンパス 14 without compensator with trace compensator Trivial WT identity for 𝑅 -symmetry basic relation (for 𝜈 = 0 ) compensator ダメダメ 超大事 2019/9/10

島根大学松江キャンパス 15 Site action Face action <latexit sha1_base64="SCd4B/A6/RT9tPx0El1QhOu42oY=">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</latexit> Identity 1 from 𝑉 1 . symmetry 𝑅 -exact 𝑉 1 . doublet → 1 2 N F ( N 2 c − 1) ( a → 0) 2019/9/10

島根大学松江キャンパス 16 Identity 2 from 𝑉 1 . symmetry 𝑅 -exact 𝑉 1 . doublet + 𝑏 → 0 → 2019/9/10

島根大学松江キャンパス 17 In the continuum theory = 0 problem 1 There are infinitely many ways to construct composite operators on the lattice. How to define Rotation and Divergence on lattice? problem 2 We should check 𝑉 1 . WT identity For a 𝑉 1 . invariant operator 𝒫(𝑦) , 2019/9/10