5d SCFTs and instanton partition functions

Transcription

1 5d SCFTs and instanton partition functions Hirotaka Hayashi (IFT UAM-CSIC) Hee-Cheol Kim and Takahiro Nishinaka [arxiv: ] Gianluca Zoccarato [arxiv: ] Yuji Tachikawa and Kazuya Yonekura [arxiv: ] 23 th of February, 2015 Munich

2 1. Introduction

3 Understanding strongly-coupled superconformal field theories (SCFTs) in various dimensions. 6d N = (2,0) or (1,0) SCFTs string theory construction : M5-branes, F-theory etc 6d N = (2,0) theories compactified on a Riemann surface with punctures 4d class S theories in the low energy limit. Gaiotto 09

4 The basic building block is a sphere with three full punctures. SU(N) SU(N) SU(N) T N theory : 4d SCFT with SU(N) 3 flavor symmetries. non-lagrangian The 5d version of the T N theory can be constructed by a web of external N D5-branes, N NS5-branes and N (1, 1) 5-branes in type IIB string theory. Benini, Benvenuti, Tachikawa 09

5 When N = 3, the SCFT has an E 6 flavor symmetry. It is supposed to be an ultraviolate completion of an SU(2) gauge theory with 5 flavors. Seiberg 96 Morrison, Seiberg 96 5-brane webs can also realize some low energy theory in a Higgs branch of the T N theory. In this case, some 5- branes jump over other 5-branes. This class includes SU(2) gauge theories with N f = 6, 7, whose ultraviolate completions are conjectured to exist as 5d SCFTs with E 7, 8 flavor symmetries.

6 Q. How can we compute the partition functions of the 5d T N theory and also Higgsed 5d T N theories? A. We can use the refined topological vertex by using the duality between a web and a toric CY 3.? However, the story is not that simple. If we simply apply the refined topological vertex to a web diagram, we will not see the expected flavor symmetry in the superconformal index. U(2) vs Sp(1) ( SU(2)). Kim, Kim, Lee 12

7 When some 5-branes jump over other 5-branes, the web corresponds to a non-toric CY 3. In this talk, we will solve the two problems. A general prescription to compute the partition function of a theory realized by any web diagram. As a by-product, we will see an interesting relation between the non-lagrangian T N theory and a Lagrangian linear quiver theory.

8 1. Introduction 2. Brane construction of 5d SCFTs 3. The solution to the 1 st problem 4. The solution to the 2 nd problem 5. The relation to a linear quiver theory 6. Summary

9 5-brane web in type IB string theory. Aharony, Hanany 97 Aharony, Hanany, Kol 97 DeWolfe, Iqbal, Hanany, Katz 99 Ex. SU(2) gauge theory D5-brane F1 string W-boson D1 string Instanton (1, 1) 5-brane NS5-brane x 6 x 5

10 5d T N theory Benini, Benvenuti, Tachikawa 09 N D5-branes N (1,1) 5-branes SU(N) SU(N) SU(N) N NS5-branes SU(N) 3 flavor symmetry is realized on 7-branes. The dimensions of the Coulomb and Higgs branch moduli spaces agree with the known results.

11 5d T 3 theory (SU(2) gauge theory with 5 flavors) SU(3) x SU(3) x SU(3) E 6 C 3 [1,1]X 3 [0,1]A 3 [1,0] (C [1,1] B [1,-1] ) 2 A 5 [1,0] 7-branes for affine E 6

12 Higgsed 5d T N theories. A Higgs branch of a theory appears by tuning parameters of the theory. In particular, we consider a tuning of putting several 5- branes on one 7-brane.

13 The supersymmetric condition implies that we cannot put more than one D5-brane between a D7-brane and an NS5-brane. Hanany, Witten 96 Benini, Benvenuti, Tachikawa 09 SUSY Non-SUSY Putting several 5-branes on the same 7-brane can be classified by Young diagrams.

14 The E 7 theory is realized by Higgsing the T 4 theory. SU(2) x SU(4) x SU(4) E 7 Benini, Benvenuti, Tachikawa 09

15 The E 8 theory is realized by Higgsing the T 6 theory. SU(2) x SU(3) x SU(6) E 8 Benini, Benvenuti, Tachikawa 09

16 Once we obtain a web diagram (without any jump), we can apply the refined topological vertex to the diagram as if it is a toric diagram. This is due to the following duality. A web in type IIB string theory Leung, Vafa 97 a toric CY 3 in M-theory compactification Then, Z Nekrasov = Z topological.?

17 3. The solution to the 1 st problem arxiv: with Hee-Cheol Kim and Takahiro Nishinaka

18 Ex. SU(2) gauge theory with 4 flavors. Z top = Z N [U(2) with 4 flavors] Z = Q 2 m Q 1 m q = Exp[-iε 2 ], t = Exp[iε 1 ] 2 differences : 1. U(2) partition function, not Sp(1) 2. extra singlet factor Z =

19 We can manually remove Z =. In the same way, we should remove the factor associated to the parallel vertical legs. H.H., Kim, Nishinaka 13 Bao, Mitev, Pomoni, Taki, Yagi 13 The 5d particles are not charged under the SU(2). Let the contributions be Z. Then, Z top /(Z = Z )= Z N [Sp(1) with 4 flavors]!

20 Unlike removing Z =, removing Z from the U(2) Nekrasov partition function is non-trivial since it depends on the instanton fugacity. This gives a new expression for the Sp(1) Nekrasov partition function without contour integrals. Our general claim for a web diagram dual to a toric CY 3 : Z Nekrasov = Z topological /Z parallel

21 T 3 theory Z N [T 3 ] = Z top /(Z = Z Z // ) = Z N [SU(2) with N f = 5]! We can generalize the computation to any web diagram like the one for a linear quiver with SU(N) gauge groups. Removing extra factors gives us new partition functions. H.H., Kim, Nishinaka 13 Bergman, Rodríguez-Gómez, Zafrir13

22 4. The solution to the 2 nd problem arxiv: with Hee-Cheol Kim and Takahiro Nishinaka arxiv: with Gianluca Zoccarato

23 How can we apply the technique to Higgsed theories? Ex. SU(2) gauge theory with 6 flavors The partition function of the T 4 theory can be computed as before.

24 In order to open up the Higgs branch, we need to tune parameters of the theory or Kähler parameters of the toric CY 3. Q1 Q2 Q 1 =?, Q 2 =?

25 A prescription to obtain an index of an IR theory in a Higgs branch of a UV theory. Take a residue of the UV index Rough idea : consider a UV index Gaiotto, Razamat 12 Gaiotto, Rastelli, Razamat 12 The residue becomes The shift of the charge implies the operator O gets VEV.

26 In this case, the Higgs vacuum is obtained by a vev of a mesonic operator. The simple pole can be found from the index of the perturbative part. Q 1 Q 2 = (q/t) Furthermore, the refined version of the geometric transition implies Q1 = (q/t) ½ Dimofte, Gukov, Hollands 10 Taki 10 Aganagic, Shakirov 11, 12 Combining the two results gives Q1 = (q/t) ½, Q2 = (q/t) ½ H.H., Kim, Nishinaka 13

27 For the E 7 theory, we set all the tuned Kähler parameters to be (q/t) ½. Then, we obtain the Nekrasov partition function of the SU(2) gauge theory with 6 flavors. Check: E 7 enhancement H.H., Kim, Nishinaka 13

28 Application to the E 8 theory In this case, we need to find a simple pole associated to an operator which carries an instanton charge. It is difficult to see it from the instanton partition function.

29 We can use the symmetric property of the web diagram. Q1 Q2 Therefore, we can use the same tuning for putting NS5- branes together. Q1 = (q/t) ½, Q2 = (q/t) ½

30 For the E 8 theory also, we set all the tuned Kähler parameters to be (q/t) ½. It gives the partition function of the SU(2) gauge theory with 7 flavors. Check: E 8 enhancement H.H., Zoccarato 14 The partition function of Higgsed 5d T N theories: 1. Obtain the partition function of the UV theory. 2. Set a pair of tuned parameters to be (q/t) ½ (or (t/q) ½ ). 3. Remove singlet factors.

31 5. The relation to a linear quiver arxiv: with Yuji Tachikawa and Kazuya Yonekura

32 The partition function of the T N theory. In fact, it is exactly the same as the partition function of the following linear quiver theory. SU(N) flavor SU(N-1) SU(N-2) - - SU(2) T 2 gauge groups There are also bi-fundamental hypermultiplets between the gauge groups. The flavor symmetry is SU(N) x U(1) 2N-2.

33 Our claim is Mass deformation SU(N) B x SU(N) C SU(N-1) B x SU(N-1) C x U(1) 2 RG flow SU(N) A SU(N-1) T N-1 Further mass deformation gives the linear quiver theory. Aganagic, Haouzi, Shakirov 13 Bergman, Zafrir 14 H.H., Tachikawa, Yonekura 14

34 Evidences H.H., Tachikawa, Yonekura 14 The Higgs branch moduli space of the UV theory agrees with that of the IR theory. The (4d) Seiberg-Witten curves agree with each other in a certain limit by exchanging the role of the two coordinates. etc When SU(N) A is Higgsed, the generic mass deformation again yields a linear quiver theory in IR. Bergman, Zafrir 14 H.H., Tachikawa, Yonekura 14

35 6. Summary

36 We have obtained a systematic method to compute the partition function of a theory realized by a web diagram (ex. T N theory, Higgsed T N theories.) Some computation predicts new partition functions or gives a new way to express the known Nekrasov partition functions. The relation between the non-lagrangian T N theory and the Lagrangian linear quiver theory is uncovered and we checked the non-trivial evidences.

37 4d limit? Ex. A proposal of the partition function of the 4d T N theory. Mitev, Pomoni 14 Other simple poles and their relation to the presence of a surface defect. The simplest case was analyzed. Gaiotto, Kim 14 Comparison with the B-model computation Application to 6d SCFTs? Huang, Klemm, Poretschkin 13