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Old roots in new equations Christian Marboe Based on [1608.06504] - PowerPoint PPT Presentation

Feb 13, 2023 •107 likes •1.65k views

Old roots in new equations Christian Marboe Based on [1608.06504] + [1701.03704] + [17.] with Dmytro Volin [1706.02320] with Rouven Frassek, David Meidinger IGST 2017 1 / 32 The perturbative spectral problem in planar N = 4 SYM D O =

  1. Extension of Young diagrams u (3 | 3) 8 / 32

  2. Extension of Young diagrams u (4 | 1) 8 / 32

  3. Extension of Young diagrams u (0 | 3) 8 / 32

  4. Extension of Young diagrams u (3) C = 1 ← at each site L = 8 ✇ f † 1 | 0 � ⊗ f † 1 | 0 � ⊗ f † 1 | 0 � ⊗ f † 1 | 0 � ⊗ f † 2 | 0 � ⊗ f † 2 | 0 � ⊗ f † 2 | 0 � ⊗ f † 3 | 0 � 8 / 32

  5. Extension of Young diagrams u (3) C = 1 ← at each site L = 8 ⊗ 8 ✇ f † 1 | 0 � ⊗ f † 1 | 0 � ⊗ f † 1 | 0 � ⊗ f † 1 | 0 � ⊗ f † 2 | 0 � ⊗ f † 2 | 0 � ⊗ f † 2 | 0 � ⊗ f † 3 | 0 � 8 / 32

  6. Extension of Young diagrams | 0 � → f † 0 | ˜ u (4) 0 � C = 2 C → C + 1 L = 8 ✇ f † 1 f † 0 � ⊗ f † 1 f † 0 � ⊗ f † 1 f † 0 � ⊗ f † 1 f † 0 � ⊗ f † 2 f † 0 � ⊗ f † 2 f † 0 � ⊗ f † 2 f † 0 � ⊗ f † 3 f † 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 � 8 / 32

  7. Extension of Young diagrams | 0 � → f † 0 | ˜ u (4) 0 � C = 2 C → C + 1 L = 8 ⊗ 8 ✇ f † 1 f † 0 � ⊗ f † 1 f † 0 � ⊗ f † 1 f † 0 � ⊗ f † 1 f † 0 � ⊗ f † 2 f † 0 � ⊗ f † 2 f † 0 � ⊗ f † 2 f † 0 � ⊗ f † 3 f † 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 | ˜ 0 � 8 / 32

  8. Extension of Young diagrams -1 | ˜ | 0 � → f † 0 f † ˜ u (5) 0 � C = 3 C → C + 2 L = 8 ✇ 8 / 32

  9. pu (2 , 2 | 4) b 1 , b 2 a 1 , a 2 f 1 , f 2 , f 3 , f 4 9 / 32

  10. pu (2 , 2 | 4) b 1 , b 2 a 1 , a 2 f 1 , f 2 , f 3 , f 4 n f + n a − n b = 2 9 / 32

  11. pu (2 , 2 | 4) b 1 , b 2 a 1 , a 2 f 1 , f 2 , f 3 , f 4 n f + n a − n b = 2 α a † a † F αβ ≡ β | 0 � α f † a † Ψ α i ≡ i | 0 � f † i f † α b † α ≡ a † ≡ j | 0 � D α ˙ Φ ij ˙ α ǫ ijkl b † α f † j f † k f † ¯ Ψ ˙ ≡ l | 0 � α i ˙ b † α b † β f † 1 f † 2 f † 3 f † ¯ F ˙ ≡ 4 | 0 � α ˙ β ˙ ˙ 9 / 32

  12. Non-compact Young diagrams pu (2 , 2 | 4) [G¨ unaydin, Volin] ✻ ⑥ n a 1 n a 2 ✻ ② ❄ ❄ ✛ ✲ ⑥ n f 4 ✛ ✲ ⑥ n f 3 ✛ ✲ ⑥ n f 2 ✛ ✲ ⑥ n f 1 ✻ ✻ ⑥ n b 1 n b 2 ⑥ ❄ ❄ 10 / 32

  13. Extension and counting s 11 / 32

  14. Extension and counting s 11 / 32

  15. Extension and counting s 11 / 32

  16. Extension and counting u (2 , 3 | 2) C = − 1 ✉ s ✉ 11 / 32

  17. Extension and counting ✉ u (1 , 4 | 12) C = 3 s ✉ 11 / 32

  18. Extension and counting u (9) ✉ C = 5 s ✉ 11 / 32

  19. Extension and counting u (2 , 2 | 4) C = 0 ✉ s ✉ Multiplet counting with so ( N ) characters (susy, non-compact) � Beisert, Bianchi, � Morales, Samtleben ’04 11 / 32

  20. Extension and counting u (9) ✉ C = 5 s Simply use u ( N ) characters [CM, Volin ’17] Multiplet counting ✉ with so ( N ) characters (susy, non-compact) � Beisert, Bianchi, � Morales, Samtleben ’04 11 / 32

  21. The spectrum ∆ 0 2 3 4 2 × 2 × 2 × . . . 5 27 multiplets 4 × 12 / 32

  22. The spectrum ∆ 0 2 3 4 2 × 2 × 2 × . . . 6 144 multiplets 16 × 12 / 32

  23. The spectrum ∆ 0 2 3 4 2 × 2 × 2 × . . . 7 918 multiplets 74 × 12 / 32

  24. The spectrum ∆ 0 2 3 4 2 × 2 × 2 × . . . 8 6918 multiplets 376 × 12 / 32

  25. Q-systems The 1-loop problem Getting the machine started 13 / 32

  26. 4 | 4 Q-system Q 1234 |∅ Q 1234 | i Q 1234 | ij Q 1234 | ijk Q 1234 | 1234 256 Q ’s Q abc |∅ Q abc | i Q abc | ij Q abc | ijk Q abc | 1234 Q = Q ( u ) Q ab |∅ Q ab | i Q ab | ij Q ab | ijk Q ab | 1234 Q a |∅ Q a | i Q a | ij Q a | ijk Q a | 1234 multiplet � solution Q ∅|∅ Q ∅| i Q ∅| ij Q ∅| ijk Q ∅| 1234 14 / 32

  27. 4 | 4 Q-system Q abc | i Q ab | i QQ = Q − Q + − Q + Q − Q ac | i Q a | i Q ± = Q ( u ± i 2 ) 14 / 32

  28. 4 | 4 Q-system QQ = Q − Q + − Q + Q − Q a | ij Q a | i Q a | ijk Q a | ik Q ± = Q ( u ± i 2 ) 14 / 32

  29. 4 | 4 Q-system QQ = Q − Q + − Q + Q − Q ab | i Q ab | ij Q a | i Q a | ij Q ± = Q ( u ± i 2 ) 14 / 32

  30. 4 | 4 Q-system Boundary conditions g → 0 weights � ( u − u k ) Q = up to factors of u ± L 14 / 32

  31. 4 | 4 Q-system Q 4 , 0 Q 4 , 1 Q 4 , 2 Q 4 , 3 Q 4 , 4 Q 3 , 0 Q 3 , 1 Q 3 , 2 Q 3 , 3 Q 3 , 4 Distinguished Q-functions Q 2 , 0 Q 2 , 1 Q 2 , 2 Q 2 , 3 Q 2 , 4 ≡ polynomials of lowest degree Q 1 , 0 Q 1 , 1 Q 1 , 2 Q 1 , 3 Q 1 , 4 Q 0 , 0 Q 0 , 1 Q 0 , 2 Q 0 , 3 Q 0 , 4 14 / 32

  32. How to solve the Q-system? 15 / 32

  33. How to solve the Q-system? Q 1234 | 123 Q 123 | 123 Bethe/Baxter Q 12 | 1 Q 12 | 12 Q 12 | 123 equations Q 1 | 1 Q ∅| 1 15 / 32

  34. How to solve the Q-system? Q 1234 | 123 Q 123 | 123 Bethe/Baxter Q 12 | 1 Q 12 | 12 Q 12 | 123 equations Q 1 | 1 ( u 1 , k + i 2 ) = Q 1 | 1 ( u 1 , k − i Q 1 | 1 2 ) Q ∅| 1 ( u 2 , j + i 2 ) Q 12 ( u 2 , j + i 2 ) = − Q 1 ( u 2 , j − i ) 2 ) Q ∅ ( u 2 , j − i 2 ) Q 12 ( u 2 , j − i Q 1 ( u 2 , j + i ) . . Q ∅| 1 . 15 / 32

  35. How to solve the Q-system? Q 1234 | 123 Q 123 | 123 Bethe/Baxter Q 12 | 1 Q 12 | 12 Q 12 | 123 equations Q 1 | 1 ( u 1 , k + i 2 ) = Q 1 | 1 ( u 1 , k − i Q 1 | 1 2 ) SLOW Q ∅| 1 ( u 2 , j + i 2 ) Q 12 ( u 2 , j + i 2 ) = − Q 1 ( u 2 , j − i ) 2 ) Q ∅ ( u 2 , j − i 2 ) Q 12 ( u 2 , j − i Q 1 ( u 2 , j + i ) . . Q ∅| 1 . 15 / 32

  36. How to solve the Q-system? Q 1234 | 123 Q 123 | 123 Bethe/Baxter Q 12 | 1 Q 12 | 12 Q 12 | 123 equations Q 1 | 1 ( u 1 , k + i 2 ) = Q 1 | 1 ( u 1 , k − i Q 1 | 1 2 ) SLOW Q ∅| 1 ( u 2 , j + i 2 ) Q 12 ( u 2 , j + i 2 ) = − Q 1 ( u 2 , j − i ) 2 ) Q ∅ ( u 2 , j − i 2 ) Q 12 ( u 2 , j − i Q 1 ( u 2 , j + i ) . . Q ∅| 1 . TOO MANY SOLUTIONS 15 / 32

  37. What’s the problem? u (1 | 2) again Q 12 |∅ Q 12 | 1 s s Q 2 |∅ Q 2 | 1 s s Q 1 |∅ Q 1 | 1 s s Q ∅|∅ Q ∅| 1 16 / 32

  38. What’s the problem? Q 12 |∅ Q 12 | 1 s s Bethe/Baxter equations only guarantee polynomiality on path Q 2 |∅ Q 2 | 1 s s Q 1 |∅ Q 1 | 1 s s Q ∅|∅ Q ∅| 1 16 / 32

  39. What’s the problem? Q 12 |∅ Q 12 | 1 s s Bethe/Baxter equations only guarantee polynomiality on path Q 2 |∅ Q 2 | 1 s s Q 1 |∅ Q 1 | 1 How to impose polynomiality s s Q ∅|∅ Q ∅| 1 of full Q-system efficiently? 16 / 32

  40. Q-systems on Young diagrams u (1 | 2) Q 12 |∅ Q 12 | 1 s s Q 2 |∅ Q 2 | 1 s s Q 1 |∅ Q 1 | 1 s s Q ∅|∅ Q ∅| 1 17 / 32

  41. Q-systems on Young diagrams Q 12 |∅ Q 12 | 1 s s Q 2 |∅ Q 2 | 1 s s Q 1 |∅ Q 1 | 1 s s Q ∅|∅ Q ∅| 1 17 / 32

  42. Q-systems on Young diagrams Q 123 |∅ Q 123 | 1 s s Q ab |∅ Q ab | i Q 12 | 12 s s s Q a |∅ Q a | i Q a | ij Q 1 | ijk Q 1 | 1234 s s s s s Q ∅|∅ Q ∅| i Q ∅| ij Q ∅| ijk Q ∅| 1234 s s s s s 17 / 32

  43. Q-systems on Young diagrams All distinguished Q ’s polynomial ⇓ Q 3 , 0 Q 3 , 1 All Q ’s polynomial s s [CM, Volin ’16] Q 2 , 0 Q 2 , 1 Q 2 , 2 s s s Q 1 , 0 Q 1 , 1 Q 1 , 2 Q 1 , 3 Q 1 , 4 s s s s s Q 0 , 0 Q 0 , 1 Q 0 , 2 Q 0 , 3 Q 0 , 4 s s s s s 17 / 32

  44. Q-systems on Young diagrams #roots = #boxes right/above ② ② 0 0 ② ② ② 1 0 0 ② ② ② ② ② 3 1 0 0 0 ② ② ② ② ② 7 4 2 1 0 17 / 32

  45. Q-systems on Young diagrams RECIPE (1) Make poly. ansatz on path u 3 + d 2 u 2 + d 1 u + d 0 u + c ② ② ✏ 0 0 ✟ ✏ ✟ ✏ ✏ ✟ ✏ ✟ ✏ ✏ ✟ ✏ ✟ ✏ ✏ ✟ ✏ ✟ ✏ ② ② ② ✏ 1 0 ✟ 0 ✏ ✟ ✏ ✏ ✟ ✏ ✟ ✏ ✏ ✟ ✏ ✟ ✏ ✏ ✮ ✟ ✙ ② ② ② ② ② 3 1 0 0 0 ② ② ② ② ② 7 4 2 1 0 ■ ❅ u 7 17 / 32

  46. Q-systems on Young diagrams RECIPE (1) Make poly. ansatz on path ② ② u 3 + d 2 u 2 + d 1 u + d 0 u + c 0 0 ❅ ■ (2) Generate rest by polynomial division ② ❅ ② ② � � Q + Q − − Q − Q + 1 0 0 Q ∝ Quotient Q ❅ ■ ❅ ② ❅ ② ❅ ❘ ② ② ② 3 1 0 0 0 ❅ ❅ ❅ ❅ ② ❘ ❅ ② ❘ ❅ ② ❘ ❅ ② ❅ ❘ ② 7 4 2 1 0 17 / 32

  47. Q-systems on Young diagrams RECIPE (1) Make poly. ansatz on path ② ② u 3 + d 2 u 2 + d 1 u + d 0 u + c 0 0 ■ ❅ (2) Generate rest by polynomial division ② ❅ ② ② � � Q + Q − − Q − Q + 1 0 0 Q ∝ Quotient Q ■ ❅ ❅ ② ❅ ② ❘ ❅ ② ② ② (3) Impose vanishing remainders 3 1 0 0 0 � � Q + Q − − Q − Q + = 0 Remainder ❅ ❅ ❅ ❅ Q ② ❘ ❅ ② ❘ ❅ ② ❘ ❅ ② ❘ ❅ ② 7 4 2 1 0 17 / 32

  48. Non-compact case ✇ ✇ ✇ ② ✇ 0 2 4 11 0 same idea ✇ ✇ ✇ ✇ ✇ 0 1 2 8 0 ∼ factors of u ± L ✇ ✇ ✇ ✇ ✇ ✇ ✇ 0 0 0 5 0 0 0 ✇ ✇ ✇ ✇ ✇ 0 4 2 1 0 ✇ ✇ ✇ ✇ ✇ 0 3 4 2 0 ✇ ✇ ✇ ✇ ✇ 0 2 6 3 0 ✇ ✇ ✇ ✇ ✇ 0 1 4 4 0 ✇ ✇ ✇ ✇ ✇ ✇ 0 0 2 5 0 0 ✇ ✇ ✇ ✇ ✇ 0 1 7 1 0 ✇ ✇ ✇ ✇ ✇ ✇ 0 0 9 2 0 0 ✇ ② ✇ ✇ ✇ 0 12 4 1 0 ✇ ② ✇ ✇ ✇ 0 15 6 2 0 18 / 32

  49. Non-compact case ✇ ✇ ✇ ② ✇ 0 2 4 11 0 same idea ✇ ✇ ✇ ✇ ✇ 0 1 2 8 0 ∼ factors of u ± L ✇ ✇ ✇ ✇ ✇ ✇ ✇ 0 0 0 5 0 0 0 ✇ ✇ ✇ ✇ ✇ 0 4 2 1 0 ✇ ✇ ✇ ✇ ✇ 0 3 4 2 0 ✇ ✇ ✇ ✇ ✇ 0 2 6 3 0 ✇ ✇ ✇ ✇ ✇ 0 1 4 4 0 ✇ ✇ ✇ ✇ ✇ ✇ 0 0 2 5 0 0 ✇ ✇ ✇ ✇ ✇ 0 1 7 1 0 ✇ ✇ ✇ ✇ ✇ ✇ 0 0 9 2 0 0 ✇ ② ✇ ✇ ✇ 0 12 4 1 0 ✇ ② ✇ ✇ ✇ 0 15 6 2 0 18 / 32

  50. Performance Give generic code 15 minutes to solve per diagram (on a worn-out MacBook Air) ∆ 0 # diagrams solved total # solutions found 2 1 / 1 1 / 1 3 1 / 1 1 / 1 4 7 / 7 10 / 10 5 13 / 13 27 / 27 5.5 12 / 12 36 / 36 6 39 / 39 144 / 144 6.5 36 / 36 276 / 276 7 68 / 77 600 / 918 7.5 54 / 84 694 / 2204 8 107 / 180 1395 / 6918 19 / 32

  51. Perturbative corrections Perturbative corrections Engine architecture 20 / 32

  52. [Gromov, Kazakov, Leurent, Volin ’13,’14] Quantum Spectral Curve Q 1234 |∅ Q 1234 | i Q 1234 | ij Q 1234 | ijk Q 1234 | 1234 Q abc |∅ Q abc | i Q abc | ij Q abc | ijk Q abc | 1234 256 Q ’s Q ab |∅ Q ab | i Q ab | ij Q ab | ijk Q ab | 1234 Q a |∅ Q a | i Q a | ij Q a | ijk Q a | 1234 Q ∅|∅ Q ∅| i Q ∅| ij Q ∅| ijk Q ∅| 1234 21 / 32

  53. Quantum Spectral Curve Cut structure Q abc | 1234 r r ✁ ✁ u − 2 g 2 g ✁ ✁ ✁ ✁ ✁ ✁ r r r r ✁ r r ✁ u r r ✁ ✁ r r ✁ ✁ ✁ ✁ Q a |∅ 21 / 32

  54. Quantum Spectral Curve Q 1234 | i Q 1234 | ij Q 1234 | ijk Cut structure Q abc | i Q abc | ij Q abc | ijk r r ✁ ✁ u r r ✁ ✁ r r ✁ ✁ ✁ ✁ r r Q ab | i Q ab | ij Q ab | ijk r r ✁ r r ✁ u ✁ ✁ ✁ ✁ ✁ ✁ Q a | i Q a | ij Q a | ijk Q ∅| i Q ∅| ij Q ∅| ijk 21 / 32

  55. Quantum Spectral Curve Analytic continuation Q abc | 1234 u s s Q a |∅ 21 / 32

  56. Quantum Spectral Curve Analytic continuation P a u s s P a 21 / 32

  57. Quantum Spectral Curve Analytic continuation s s P a u s s s s s s s s P a ˜ P a = µ ab P b 21 / 32

  58. Quantum Spectral Curve Analytic continuation s s P a u s s s s µ ab = ω ij Q − ab | ij s s s s P a ˜ P a = µ ab P b 21 / 32

  59. Perturbative strategies 22 / 32

  60. Perturbative strategies P µ , sl (2) [CM,Volin ’14] 22 / 32

  61. Perturbative strategies P µ , sl (2) [CM,Volin ’14] Q-system, general [ Gromov, Sizov ] Levkovich-Maslyuk ’15 22 / 32

  62. Perturbative strategies P µ , sl (2) [CM,Volin ’14] Q-system, general [ Gromov, Sizov ] Levkovich-Maslyuk ’15 P µ , general [CM,Volin ’17] 22 / 32

  63. Key property: structure of P u � � P a r r � � � � r r r r � u � r r ˜ r r P a r r � � � � 23 / 32

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