A Localization Computation in Confining Phase

Transcription

1 A Localization Computation in Confining Phase Seiji Terashima (YITP) 20 January 2015 at Osaka based on the paper: arxiv:

2 Introduction 2

3 Analytic computations in QFT are hopeless, but, some exceptions: Important examples: SUSY field theories 3

4 How to get exact results? There are two major techniques: 1. Holomorphy 2. Localization (and index) 4

5 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 5

6 Holomorphy Very powerful and simple, But, Indirect and superspace is needed 6

7 Localization (in SUSY field theories) Deformations of the action which do not change some correlation functions Weak coupling limit becomes exact 7

8 Examples of Localization Witten index Topological field theory by Witten Nekrasov partition function SUSY theory on curved space by Pestun Etc. 8

9 Localization Direct, systematic and superspace is not needed 9

10 An expectation: Localization technique is powerful enough to compute any exactly computable quantities in SUSY gauge theories Indeed, Seiberg-Witten prepotential can be reproduced from Nekrasov partition function 10

11 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. 11

12 What I will show Computing the gaugino condensation directly by the localization technique with a choice of a SUSY generator. 12

13 Plan Introduction Localization 4D N=1SUSY gauge theories Computation gaugino condensation 13

14 Localization 14

15 Let us consider a correlation function: 15

16 16

17 Path-integral is localized on the saddle points where Saddle point approximation is exact in the limit. We also need 1-loop factor for Thus, 17

18 Example 1: BRST gauge fixing term In Rξ gauge, 1/ξ is unitary gauge, i.e. localized to physical modes 18

19 Example 2: Witten s topological field theory 4d N=2 SUSY gauge theory action is invariant under a SUSY transformation which becomes scalar by the twist. In this case, and the action and energy momentum tensor are δ-exact. Topological 19

20 Example 3 : curved manifold 4D N=2 gauge theory on four-sphere 3D N=2 gauge theory on three-sphere 4D N=1 gauge theory on Many more examples Pestun Hama-Hosomichi Nosaka-Terashima Kapustin-Willett-Yaakov Jefferis Hama-Hosomichi-Lee Imamura Dolan-Spiridonov-Vartanov Gadde-Yan SYM action and SUSY transformation are modified from flat space case. 20

21 In this case, they are NOT topological. By choosing a SUSY generator δ, we have Then, define we have 21

22 Localization of 4D N=1SUSY gauge theories ST 22

23 Let us first consider From this, by taking appropriate limit we can get and 23

24 4d N=1 SUSY (4 SUSYs) Vector multiplets Fields: 24

25 SUSY transf. 25

26 SUSY transf. is same form: 26

27 However, derivative is changed: Killing spinors should satisfy: 27

28 The SUSY invariant Lagrangian takes same form: We will also use 28

29 Applying Localization technique, we need to choose a pair of Killing spinors Then, a SUSY transformation is fixed. For the regulator Lagrangian, we will take the standard one: 29

30 First, we will take This was taken for the SUSY theory on Then, we find where we have used Including the other contribution, 30

31 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 31

32 Let us consider a NEW CHOICE: SUSY transformation: 32

33 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)! 33

34 The original action at saddle points This is the usual instanton factor, thus gauge coupling dependent! 34

35 We will consider SUSY theory on by taking the flat limit. This theory is easier to compute 35

36 Gaugino condensation 36

37 Let us consider the N=1 SUSY Yang-Mills theory with gauge group G on The gaugino condensation known to be non-vanishing due to the strong coupling effects (confinement effects) It have bee shown by holomorphy and also R 0 limit is Novikov-Shifman-Vainshtein -Zakharov Afflek-Dine-Seiberg Seiberg Davies-Hollowood-Khoze-Mattis 37

38 Configurations are classified by Wilson loop This breaks G to Instanton charge monopole charges for 38

39 Precisely two fermion zero modes are needed for non-vanishing gaugino bi-linear. (Instanton has more zero modes) We will call fundamental monopoles as ASD connections with two zero modes Davies-Hollowood-Khoze-Mattis 39

40 fundamental monopoles r BPS monopoles (more precisely T-dual of BPS monopoles =fractional instantons) and KK monopole Lee-Yi (Instanton (=Caloron) is the bound state of these (r+1) monopoles ) 40

41 Brane picture (SU(3) case) 41

42 BPS monopoles magnetic charge: Instanton charge: classical action: 42

43 KK monopoles magnetic charge: Instanton charge: classical action: is the lowest root which satisfies with is the Kac labels. 43

44 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 44

45 Abelian vector multiplet in 3d contains Wilson loop scalar and dual photon scalar : combined into complex scalar (classical moduli): N=1 chiral superfield in 3d with the action: 45

46 To find vacua, scalar potential should be computed superpotential fermion bi-linear term two fermion zero modes are needed only fundamental monopoles contribute 46

47 Fundamental monopole classical action: path-integral measure of zero modes: a is the position on R³ Ω is U(1) phase ξ are fermion zero modes 47

48 zero modes at spacial infinity We find the superpotential: 48

49 The vacua defined by is solved as: 49

50 Gaugino condensation and superpotential Finally, we find where Л is the dynamical scale in Pauli-Villars renormalization scheme 50

51 This is correct exact results! 51

52 Now we will consider theory at t=0 We will denote as the correlator of the original theory, but choosing the vacua of the theory with the regulator action with t. Then, we find But, the vauca are descrete, then, vacua is unchanged by changing t 52

53 The topics not presented today: chiral multiplets proof of Dijkgraaf-Vafa conjecture holomorphy of superpotential in view of localization SUSY gauge theory on Localization for theory on 53

54 Conclusion New choice of SUSY generator for 4d N=1 SUSY localization localization holomorphy Computing the gaugino condensation directly by the localization technique 54

55 Fin. 55

56 Backups 56