超対称ゲージ理論の 厳密計算に関する話題

1. 超対称ゲージ理論の 厳密計算に関する話題 Seiji Terashima (YITP) 21 February 2015 at Riken based on the paper:arXiv:1410.3630, to appear in JHEP

2. これからの弦理論 （例えば今後考えるべき問題や盛り上げていきたいトピックなど)

3. やっぱり 　わかりませんでした。 　申し訳ありません。

4. 様々な分野で、 大きな発展の芽が、 生まれているように、 思います。

5. 一人一人が、 　面白いと思うことを、 　やるのが良い（？）

6. （“超”主観的な） 　僕が面白いと思っている事：

7. １、場の理論の双対性、高次元場の理論 　　　Ｍ５ブレーンとＭ２ブレーン、 p-form ゲージ理論、３代数、、、 ２、量子重力に対応する幾何 　　　非可換幾何、反Ｄブレーン、行列模型、 　　　ＡｄＳ／ＣＦＴ対応、、、 ３、ブラックホール Fuzzball予想、Firewall予想、、、

8. 共通することの一つ： 非摂動な場の理論の信頼できる解析が重要 計算手法の理解そのものも深めたい！

9. 前置き、おわり。

10. Introduction

11. Analytic computations in QFT are hopeless, but, some exceptions: Important examples: SUSY field theories

12. How to get exact results? There are two major techniques: 1. Holomorphy 2. Localization (and index)

13. Holomorphy Superpotential of 4D N=1 SUSY gauge theory should be holomorphic “function” of chiral superfields Seiberg determined the moduli space of vacua. The gaugino condensation was also computed For 4D N=2 SUSY, prepotential should be holomorphic. Seiberg-Witten theory

14. Holomorphy Very powerful and simple, But, Indirect and superspace is needed

15. Localization (in SUSY field theories) Deformations of the action which do not change some correlation functions Weak coupling limit becomes exact, Thus, computable

16. Examples of Localization • Witten index • Topological field theory by Witten • Nekrasov partition function • SUSY theory on curved space by Pestun • Etc.

17. Localization Direct, systematic and superspace is not needed

18. An expectation: Localization technique is powerful enough to compute any exactly computable quantities in SUSY gauge theories For example, Seiberg-Witten prepotential can be reproduced from Nekrasov partition function

19. Important exception: Gaugino condensation Localization has not been applied to a theory in confining phase nor computation of local operators like gaugino bi-linear. Thus, it is important and interesting to compute gaugino condensation in 4d N = 1SUSY QCD by the localization There would be other interesting applications of the technique.

20. What I will show Computing the gaugino condensation directly by the localization technique with a choice of a SUSY generator.

21. Plan • Introduction • Localization • 4D N=1SUSY gauge theories • Computation of gaugino condensation • Non-perturvative proof of Dijkgraaf-Vafa conjecture

22. Localization

23. Let us consider a correlation function:

25. Original idea: BRST gauge fixing Kugo-Ojima Addition of the gauge fixing term do not change correlators of physical observables which satisfy

26. Path-integral is localized on the saddle points where Saddle point approximation is exact in the limit.

27. This is like WKB. We also need 1-loop factor for Thus, typically, ordinary integral, instead of path-integral !

28. SUSY field theories: We can always apply this localization technique to SUSY theories by choosing δ = a SUSY generator !

29. δ = a SUSY generator then, We define the regulator action as then, we see

30. In this case, they are NOT topological. Example : 4D N=2 gauge theory on four-sphere Many more examples Pestun

31. Localization of 4D N=1SUSY gauge theories ST

32. Let us first consider From this, by taking appropriate limit we can get

33. 4d N=1 SUSY (4 SUSYs) Vector multiplets Fields:

34. SUSY transf.

35. The SUSY invariant Lagrangian takes We will also use

36. Applying Localization technique, we need to choose a pair of spinors Then, a SUSY transformation is fixed. For the regulator Lagrangian, we will take the standard one:

37. First, we will take This was taken also for the SUSY theory on Then, we find where we have used Including the other contribution,

38. Saddle points are Combined Lagrangian includes Thus, the theory becomes weak coupling. (No gauge coupling dependence.) This gives Supercomformal index Imamura Dolan-Spiridonov-Vartanov Gadde-Yan

39. Let us consider a NEW CHOICE: SUSY transformation:

40. For this choice, we have This is ( Yang-Mills + pure imaginary θ term). Thus, weak coupling in the limit Saddle points are Instantons(=ASD connections)!

41. The original action at saddle points This is the usual instanton factor, thus gauge coupling dependent! (The holomorphy of superpotential can also be derived by this localization.)

42. Gaugino condensation

43. Let us consider the N=1 SUSY Yang-Mills theory with gauge group G on The gaugino condensation is known to be non-vanishing due to the strong coupling effects (confinement effects) It have bee shown by holomorphy and also R → 0 limit Novikov-Shifman-Vainshtein -Zakharov Afflek-Dine-Seiberg Seiberg Davies-Hollowood-Khoze-Mattis

44. Configurations are classified by • Wilson loop This breaks G to • Instanton charge • monopole charges for

45. Precisely two fermion zero modes are needed for non-vanishing gaugino bi-linear. (Instanton has more zero modes) We will call them “fundamental monopoles” which are ASD connections with two zero modes Davies-Hollowood-Khoze-Mattis

46. “fundamental monopoles” r BPS monopoles (more precisely T-dual of BPS monopoles =fractional instantons) and KK monopole (Instanton (=Caloron) is the bound state of these (r+1) monopoles ) Lee-Yi

47. Brane picture (SU(3) case)

48. BPS monopoles magnetic charge: Instanton charge: classical action:

49. KK monopoles magnetic charge: Instanton charge: classical action: is the lowest root which satisfies with is the Kac labels.

50. There are more than 2 non-compact dimensions, we need to find vacua of t → ∞ limit In this weak coupling limit, massless fields are vector multiplets with zero KK momentum reduction to 3d