Small instanton transitions for M5 fractions Hiroyuki Shimizu - PowerPoint PPT Presentation

Small instanton transitions for M5 fractions Hiroyuki Shimizu (Kavli IPMU) Based on arXiv:1707.05785 with N.Mekareeya K.Ohmori, and A.Tomasiello @YITP workshop Field Theory and String 2017

Introduction 6d conformal matters [delZotto, Heckman, Tomasiello, Vafa ’14] ・An important example of 6d N=(1,0) theories: N M5 branes on ALE singularity . ・We have many example of 6d N=(1,0) theories. [Heckman, Morrison, (Rudelius,) Vafa ’13, ‘15] C 2 / Γ G

Introduction Present some new results about frozen conformal matter theories. ・Related to many other 6d theories via RG flow. Higgs deformation: T-brane theories Tensor deformation: frozen conformal matters [Heckman, Rudelius, Tomasiello ’16] [Mekareeya, Rudelius, Tomasiello ’17] Aim of this talk

Plan (review) (new) 1. Introduction 2. 6d conformal matters and their frozen variants 3. Properties of frozen conformal matter theories 4. Conclusions

6d conformal matters and their frozen variants

6d (G,G) conformal matters N M5 branes x 6 x 7 ~x 10 7d SYM 7d SYM [del Zotto, Heckman, Tomasiello, Vafa ’14] ・Worldvolume theory of N M5-banes on top of the singularity. ・ flavor symmetry. C 2 / Γ G C 2 / Γ G G × G G = SU ( k ) , SO (2 k ) , E k

F-theory dual ・F-theory is useful to examine the tensor branch. ・Dual to F-theory on non-compact elliptic CY3: Z_N singularity on B2. G-type 7-brane wrapping these two non-compact curves B 2 T 2 CY 3 B 2

・We’d like to know the tensor structure of theory. → Repeated blowup on the base. Base blowup minus of self-intersection # of CP^1 1 3 1 SU(3) φ φ [E6] [E6] [E6] [E6] ・Quiver gauge theory is hidden at intersection. CP 1 CP 1 CP 1 ・Blowup rule: m, n → ( m + 1) , 1 , ( n + 1)

・We have the following sequence of CP^1s. (Quiver gauge theory) ・Corresponds to generic point on tensor branch. [Bershadsky, Johansen ‘96] Tensor branch structure su k su k G = SU ( k ) : [ SU ( k )] 2 2 [ SU ( k )] . . . usp 2 k − 8 so 2 k G = SO (2 k ) : [ SO (2 k )] 1 4 . . . [ SO (2 k )] su 3 e 6 G = E 6 : [ E 6 ] 1 3 1 6 . . . [ E 6 ] so 7 e 7 su 2 su 2 G = E 7 : [ E 7 ] 1 2 3 2 1 8 . . . [ E 7 ] f 4 g 2 g 2 su 2 su 2 e 8 G = E 8 : [ E 8 ] 1 2 2 3 1 5 1 3 2 2 1 12 . . . [ E 8 ]

M5 fractionation 1/3 NS5 frozen E6 singularity 3-form charge of fractional M5 ・Number of fractions: 1/3 NS5 ・M-theory interpretation of quiver gauge theory on 1/6 NS5 1/6 NS5 SU(3) φ [E6] φ [E6] Fractional M5-brane tensor branch: [del Zotto, Heckman, Tomasiello, Vafa ’14] f ( SU ( k )) = 1 , f ( SO (2 k )) = 2 , f ( E 6 ) = 4 , f ( E 7 ) = 6 , f ( E 8 ) = 12

Frozen conformal matter theories ・ Frozen conformal matter theories: taking some Asymmetric flavor symmetry. matter: a wider class of 6d theories. ・Tensor branch flow from “unfrozen” conformal Non-simply laced flavor symmetry. outermost M5 fractions to infinity. so 7 e 7 e 7 so 7 su 2 su 2 su 2 su 2 [ E 7 ] 1 2 3 2 1 8 . . . 8 2 3 2 1 [ E 7 ] so 7 e 7 e 7 su 2 su 2 [ SU (2)] 3 2 1 8 . . . 8 1 2 [ SO (7)]

Another examples

Properties of frozen conformal matter theories

1. Higgsability

Frozen theories with pure Higgs branch ・Moduli space structure are important. ・In particular, we focus on the problem which 6d frozen conformal matter has pure Higgs branch . Pure Higgs branch: only hypers, no tensors/vectors. In some cases, we can eliminate all the tensors by 1 tensor -> 29 hypers “small instanton transition” In general, we can’t eliminate tensors and no pure Higgs branch.

Anomaly matching constraint ・6d theories with pure Higgs branch have the flow: 6d SCFT -> free hypers ・Gravitational anomaly matching requires: ・We have solved this constraint. 6d theory has pure Higgs branch only if its endpoint is = d H I hyper I SCFT | grav 8 8 φ , 4 , 52 , 352 , 622 , 7222 , 82222

Endpoint of 6d theories Repeated blowdown of -1 curves, until no more -1 curves. [Heckman, Morrison, Vafa ’13] ・For frozen conformal matters: ・ Endpoint is a specific point on tensor branch. 232…23 Blowdown formula: m, 1 , n → ( m − 1) , ( n − 1) f 4 f 4 g 2 g 2 g 2 g 2 su 2 su 2 e 8 su 2 su 2 [ E 8 ] 1 2 2 3 1 5 1 3 2 2 1 12 1 2 2 3 1 5 1 3 2 2 1 [ E 8 ] e 8 [ E 8 ] 2 [ E 8 ] e ( a 1 )2 n − 2 e ( a 2 ) t a 1 ( G ) − ( G ) − . . . − ( G ) − ( G ) a t 2 so 7 e 7 e 7 su 2 su 2 [ SU (2)] 3 2 1 8 . . . 8 1 2 [ SO (7)]

Endpoint of frozen theories ・For frozen theories, we realize all the linear endpoints classified in [HMV ’13].

M-theory interpretation ・Example of frozen theories with pure Higgs branch: ・M-theoretically, transition is recombination of M5 fractions and leaving off from the singularity. (endpoint:25) (endpoint:4) su (3) e 6 e 6 [1] 3 1 6 [1] , [ SU (3)] 1 6 1 [ SU (3)]

M-theory interpretation

2. Chiral anomalies

Chiral anomalies of frozen conformal matter ・Anomaly polynomial of conformal matters. Field theoretical method in [Ohmori, HS, Tachikawa, Yonekura ’14]. ・The result can be rewritten as follows: Add Green-Schwarz contribution. 1-loop contribution of massless multiplets of quiver gauge theory. 6d chiral anomalies cancel by bulk Chern-Simons term Anomaly inflow M-theoretic interepretaion: ( G fr , G fr ) [ G fr ] − ( G fr ) − . . . − ( G fr ) − [ G fr ] G → G fr I tot = 1 24 Q 3 | Γ G | 2 c 2 ( R ) 2 − QI 8 − 1 2 Q | Γ G | ( J 4 ,L + J 4 ,R ) − 1 − 1 2 I vec 2 I vec L R

Chiral anomalies of frozen conformal matter 11d CS term ・Chern-Simons term interpretation of anomaly 7d CS term on singularity Explicitly polynomial. I tot = 1 24 Q 3 | Γ G | 2 c 2 ( R ) 2 − QI 8 − 1 2 Q | Γ G | ( J 4 ,L + J 4 ,R ) − 1 − 1 2 I vec 2 I vec L R 2 π 2 π C ∧ J 4 ,L/R 6 C ∧ G ∧ G − 2 π C ∧ I 8 J 4 ,L/R = 1 1 tr F 2 48(4 c 2 ( R ) + p 1 ( T )) χ G → G fr + L/R 4 d G → G fr 12 1 χ G → G fr = r G − 11 + − | Γ G | d G → G fr

Chiral anomalies of frozen conformal matter ・For unfrozen conformal matters, we have derived 7d CS term in [Ohmori, HS, Tachikawa, Yonekura ’14]. ・We obtained generalization to frozen singularity. New results about M-theory! S Γ = 2 π C ∧ J 4 J 4 = 1 48(4 c 2 ( R ) + p 1 ( T )) χ G + 1 4tr F 2 G 1 Z χ G = c 2 ( L ) = r G + 1 − | Γ G | C 2 / Γ G

3. T^2 comapatification

T^2 compactification of frozen conformal matter ・T^2 compactification of (G,G) conformal matter. G=SU(k),SO(2k),E_k G-type (2,0) theory on ★ full puncture sphere [Ohmori, HS, Tachikawa, Yonekura ’15][delZotto, Vafa, Xie ‘15] full M5s = R 1 , 3 × T 2 × ● simple puncture

T^2 compactification of frozen conformal matter sphere See also [Tachikawa ’15] automorphism outer- ・We have generalized the result to frozen -type (2,0) theory on frozen singularity = ★ maximal twisted puncture conformal matter theories with . full M5s ( G fr , G fr ) G fr = F 4 , G 2 , USp (2 k ) ˆ G R 1 , 3 × T 2 × ● simple puncture ˆ G G fr SO (8) G 2 F 4 E 6 SO (2 k + 2) USp (2 k )

Conclusions

Conclusions ・We started a study of frozen variant of 6d conformal matters. What we have obtained: ・Anomaly polynomial formula. ・T^2 compactification of some theories. etc Thank you very much! ・Higgsability.