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,
Hiroyuki Shimizu (Kavli IPMU)
Based on arXiv:1707.05785 with N.Mekareeya K.Ohmori, and A.Tomasiello
@YITP workshop Field Theory and String 2017
6d conformal matters
[delZotto, Heckman, Tomasiello, Vafa ’14]
・An important example of 6d N=(1,0) theories: N M5 branes on ALE singularity . C2/ΓG ・We have many example of 6d N=(1,0) theories.
[Heckman, Morrison, (Rudelius,) Vafa ’13, ‘15]
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
(review) (new)
N M5 branes x6 x7~x10
7d SYM 7d SYM
[del Zotto, Heckman, Tomasiello, Vafa ’14]
・Worldvolume theory of N M5-banes on top of the singularity. C2/ΓG C2/ΓG ・ flavor symmetry. G × G G = SU(k), SO(2k), Ek
・F-theory is useful to examine the tensor branch. ・Dual to F-theory on non-compact elliptic CY3: B2
Z_N singularity on B2. G-type 7-brane wrapping these two non-compact curves
B2 T 2 CY3
・We’d like to know the tensor structure of theory. → Repeated blowup on the base.
[E6] [E6] [E6] [E6] φ φ
SU(3) 1 3 1 minus of self-intersection # of CP^1
CP1 CP1 CP1
・Blowup rule: m, n → (m + 1), 1, (n + 1)
・Quiver gauge theory is hidden at intersection.
・We have the following sequence of CP^1s. (Quiver gauge theory) G = SU(k) : [SU(k)]
suk
2 . . .
suk
2 [SU(k)] G = SO(2k) : [SO(2k)]
usp2k−8
1
so2k
4 . . . [SO(2k)] G = E6 : [E6] 1
su3
3 1
e6
6 . . . [E6] G = E7 : [E7] 1
su2
2
so7
3
su2
2 1
e7
8 . . . [E7] G = E8 : [E8] 1 2
su2
2
g2
3 1
f4
5 1
g2
3
su2
2 2 1
e8
12 . . . [E8]
[Bershadsky, Johansen ‘96]
・Corresponds to generic point on tensor branch.
・M-theory interpretation of quiver gauge theory on tensor branch: Fractional M5-brane
[E6] φ [E6] φ SU(3) 1/3 NS5 1/6 NS5 1/6 NS5 1/3 NS5
・Number of fractions:
f(SU(k)) = 1 , f(SO(2k)) = 2 , f(E6) = 4 , f(E7) = 6 , f(E8) = 12
3-form charge of fractional M5 frozen E6 singularity
[del Zotto, Heckman, Tomasiello, Vafa ’14]
・Frozen conformal matter theories: taking some
[SU(2)]
so7
3
su2
2 1
e7
8 . . .
e7
8 1
su2
2 [SO(7)] [E7] 1
su2
2
so7
3
su2
2 1
e7
8 . . .
e7
8
su2
2
so7
3
su2
2 1 [E7] ・Tensor branch flow from “unfrozen” conformal matter: a wider class of 6d theories.
Asymmetric flavor symmetry. Non-simply laced flavor symmetry.
・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.
・6d theories with pure Higgs branch have the flow: 6d SCFT -> free hypers ・Gravitational anomaly matching requires: ISCFT
8
= dHIhyper
8
|grav ・We have solved this constraint. 6d theory has pure Higgs branch only if its endpoint is φ, 4, 52, 352, 622, 7222, 82222
Repeated blowdown of -1 curves, until no more -1 curves.
m, 1, n → (m − 1), (n − 1)
Blowdown formula:
[E8] 1 2
su2
2
g2
3 1
f4
5 1
g2
3
su2
2 2 1
e8
12 1 2
su2
2
g2
3 1
f4
5 1
g2
3
su2
2 2 1 [E8]
[E8]
e8
2 [E8]
・Endpoint is a specific point on tensor branch. ・For frozen conformal matters: e(a1)2n−2e(a2)t a1(G) − (G) − . . . − (G) − (G)at
2
[Heckman, Morrison, Vafa ’13]
[SU(2)]
so7
3
su2
2 1
e7
8 . . .
e7
8 1
su2
2 [SO(7)]
232…23
・For frozen theories, we realize all the linear endpoints classified in [HMV ’13].
・Example of frozen theories with pure Higgs branch: ・M-theoretically, transition is recombination of M5 fractions and leaving off from the singularity. [1]
su(3)
3 1
e6
6 [1] , [SU(3)] 1
e6
6 1 [SU(3)]
(endpoint:25) (endpoint:4)
Chiral anomalies of frozen conformal matter ・Anomaly polynomial of conformal matters. (Gfr, Gfr)
Field theoretical method in [Ohmori, HS, Tachikawa, Yonekura ’14].
・The result can be rewritten as follows:
Itot = 1 24Q3|ΓG|2c2(R)2 − QI8 − 1 2Q|ΓG|(J4,L + J4,R) − 1 2Ivec
L
− 1 2Ivec
R
G → Gfr M-theoretic interepretaion: Anomaly inflow
6d chiral anomalies cancel by bulk Chern-Simons term 1-loop contribution of massless multiplets of quiver gauge theory. Add Green-Schwarz contribution.
[Gfr] − (Gfr) − . . . − (Gfr) − [Gfr]
Chiral anomalies of frozen conformal matter ・Chern-Simons term interpretation of anomaly polynomial.
Itot = 1 24Q3|ΓG|2c2(R)2 − QI8 − 1 2Q|ΓG|(J4,L + J4,R) − 1 2Ivec
L
− 1 2Ivec
R
11d CS term
2π 6 C ∧ G ∧ G − 2πC ∧ I8
7d CS term on singularity J4,L/R = 1 48(4c2(R) + p1(T))χG→Gfr + 1 4dG→Gfr trF 2
L/R
2πC ∧ J4,L/R χG→Gfr = rG − 11 + 12 dG→Gfr − 1 |ΓG| Explicitly
Chiral anomalies of frozen conformal matter ・For unfrozen conformal matters, we have derived 7d CS term in [Ohmori, HS, Tachikawa, Yonekura ’14]. SΓ = 2πC ∧ J4 J4 = 1 48(4c2(R) + p1(T))χG + 1 4trF 2
G
χG = Z
C2/ΓG
c2(L) = rG + 1 − 1 |ΓG| ・We obtained generalization to frozen singularity. New results about M-theory!
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]
R1,3 × T 2×
full M5s
=
T^2 compactification of frozen conformal matter
・We have generalized the result to frozen conformal matter theories with .
R1,3 × T 2×
full M5s
Gfr = F4, G2, USp(2k) (Gfr, Gfr)
★ maximal twisted puncture
sphere
ˆ G
frozen singularity
ˆ G Gfr SO(8) G2 E6 F4 SO(2k + 2) USp(2k)
automorphism
See also [Tachikawa ’15]
=
・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.