N=1 SCFT’s with DN blocks - UCFileSpace Tools - Homehomepages.uc.edu/~argyrepc/SCFT_Aspen/slides/Fazzi.pdf · N=1 SCFT’s with DN blocks Marco Fazzi based on 1609.08156 with Simone

N=1 SCFT’s with DN blocks

Marco Fazzi

based on 1609.08156 with Simone Giacomelli

related work by [Maruyoshi-Song,Nardoni]

builds on earlier proposal by [Agarwal-Intriligator-Song]

motivation:

want to explore corner of landscape of 4d N=1 theories.

N=1 class S: compactify A-type (2,0) on Riemann surface inside CY3

[Bah-Beem-Bobev-Wecht & several earlier and later works, both in field theory & holography]

motivation:

want to explore corner of landscape of 4d N=1 theories.

N=1 class S: compactify A-type (2,0) on Riemann surface inside CY3

which corner? inaccessible models

[Bah-Beem-Bobev-Wecht & several earlier and later works, both in field theory & holography]

index and terminology proposed by [Beem-Gadde]

reminder: accessible N=1 models of class S

p = 4 (black) TN’s

N = 2



N = 2

g = 3(p = 2g − 2)



g = 3

N = 2

g = 3

N = 1 N = 1

N = 1

(p = 2g − 2)


p = 4 (black) TN’s p = 3 black TN’s &

q = 1 red TN

g = 3

N = 2

g = 3

N = 1 N = 1

N = 1

(T+N )

(T−N )

(p = 2g − 2)



q = 1 red TN

g = 3

N = 2

g = 3

N = 1 N = 1

N = 1

(T+N )

(T−N )

N = 1

N = 2

N = 2N = 2N = 1

N = 1

(p = 2g − 2)


why accessible?


q = 1 red TN

g = 3

N = 2

g = 3

N = 1 N = 1

N = 1

(T+N )

(T−N )

N = 1

N = 2

N = 2N = 2N = 1

N = 1

(p = 2g − 2)


only ingredients are: (ℤ2 colored) TN’s & N=1 or N=2 tubes

why accessible?


q = 1 red TN

g = 3

N = 2

g = 3

N = 1 N = 1

N = 1

(T+N )

(T−N )

N = 1

N = 2

N = 2N = 2N = 1

N = 1

(p = 2g − 2)

clearly, this construction holds only for p,q ≥ 0 and g > 1


what about p,q < 0, or g ≤ 1?



inaccessible



constructing these latter cases gives field theory duals to all holographic solutions found by BBBW!

inaccessible

[Bah-Beem-Bobev-Wecht]

this talk

• construct inaccessible models: p or q < 0, g ≤ 1

this talk

• how? deform TN to obtain 3 new building blocks


this talk

• how? deform TN to obtain 3 new building blocks


• chiral ring relations for new blocks & a puzzle

BBBW engineering of accessible N=1 models: in 4d…

start from N=2 trinion w/ 3 maximal punctures N

µB

µA

µC

flavor symmetry SU(N)A x SU(N)B x SU(N)C

(only maximal punctures in this talk)



µB

µA

µC


glue many together by gauging (diagonal combination of) flavor symmetries:




µB

µA

µC



p = 2 black q = 0 red

T+N

N=2T+N

N = 2N=2 way




µB

µA

µC



TrΦ(µ1 − µ2) ⊂ W

p = 2 black q = 0 red

T+N

N=2T+N

N = 2N=2 way




µB

µA

µC



TrΦ(µ1 − µ2) ⊂ W

p = 2 black q = 0 red p = 1 black q = 1 red

T+N

N=2T+N

N = 2T+N

N=1T�NT−N

N = 1N=1 wayN=2 way




µB

µA

µC



TrΦ(µ1 − µ2) ⊂ W

p = 2 black q = 0 red p = 1 black q = 1 red

Trµ1µ2 ⊂ W

T+N

N=2T+N

N = 2T+N

N=1T�NT−N

N = 1N=1 wayN=2 way


…and in 6d

CY3 = L1 ⊕ L2 → CgM-theory on

c1(L1) + c1(L2) = 2g − 2

Calabi-Yau condition reads

p qa particular twist preserves only N=1 in 4d[Bah-Beem-Bobev-Wecht]

…and in 6d


c1(L1) + c1(L2) = 2g − 2


CC

CgM5's

p qa particular twist preserves only N=1 in 4d

on


…and in 6d


c1(L1) + c1(L2) = 2g − 2


CC

Cg

U(1)1

U(1)2

U(1)1 x U(1)2 global symmetry

M5's

p qa particular twist preserves only N=1 in 4d

on


…and in 6d


c1(L1) + c1(L2) = 2g − 2


CC

Cg

U(1)1

U(1)2


4d U(1)R is a combination: RSCFT(ϵ) = U(1)diag + ϵ U(1)anti-diag

M5's

p q

4d a(ϵ) & c(ϵ) from M5 anomaly polynomial. ϵ from a-maximization

a particular twist preserves only N=1 in 4d

on


[Intriligator-Wecht]

…and in 6d

4d model: low-energy dynamics of M5’s on Riemann surface Cg inside CY3


c1(L1) + c1(L2) = 2g − 2


CC

Cg

U(1)1

U(1)2


4d U(1)R is a combination: RSCFT(ϵ) = U(1)diag + ϵ U(1)anti-diag

M5's

p q

4d a(ϵ) & c(ϵ) from M5 anomaly polynomial. ϵ from a-maximization

a particular twist preserves only N=1 in 4d

on


[Intriligator-Wecht]

can assign a ℤ2 color (i.e. + or −) to the punctures as well

(locally the same as N=2 punctures)T−N

µA

µB

µC

back to 4d: enter flipping



µA

µB

µC

all red




µA

µB

µC

all red


T−N

µA

µB

µC

can “flip” puncture’s color wrt parent TN’s color



µA

µB

µC

all red


T−N

µA

µB

µC

can “flip” puncture’s color wrt parent TN’s color

TrMµX ⊂ WT−NµB

µC

M=

equivalent to introducing flipping field M: extra chiral in adjoint of flavor group SU(N)X

[Gadde-Maruyoshi-Tachikawa-Yan,Xie,Yonekura,Giacomelli]

[Heckman-Tachikawa-Vafa-Wecht, Gadde-Maruyoshi-Tachikawa-Yan, Agarwal-Bah-Maruyoshi-Song, Agarwal-Intriligator-Song, Maruyoshi-Song, Nardoni, …]

trick to construct new blocks: give M a maximal nilpotent vev



⟨MX⟩nilpotent =

⎡

⎢⎢⎢⎣

0 1 0 · · ·0 0 1 · · ·0 0 0 · · ·...

......

. . .

⎤

⎥⎥⎥⎦



superpotential term Tr MX 𝞵X. ⟨MX⟩ triggers relevant deformation

& closes puncture X⟨MX⟩nilpotent =

⎡

⎢⎢⎢⎣

0 1 0 · · ·0 0 1 · · ·0 0 0 · · ·...

......

. . .

⎤

⎥⎥⎥⎦





⎡

⎢⎢⎢⎣

0 1 0 · · ·0 0 1 · · ·0 0 0 · · ·...

......

. . .

⎤

⎥⎥⎥⎦

careful with a(ϵ) & c(ϵ): must shift R-symmetry RSCFT(ϵ)

to find unbroken combination along RG





⎡

⎢⎢⎢⎣

0 1 0 · · ·0 0 1 · · ·0 0 0 · · ·...

......

. . .

⎤

⎥⎥⎥⎦



N

µB µC

X=A

DNflavor:

SU(N)B x SU(N)C





⎡

⎢⎢⎢⎣

0 1 0 · · ·0 0 1 · · ·0 0 0 · · ·...

......

. . .

⎤

⎥⎥⎥⎦



N

µB µC eNN = 2

eNµC

X=A X=A,B

DNflavor:

SU(N)B x SU(N)C!DN flavor: SU(N)C





⎡

⎢⎢⎢⎣

0 1 0 · · ·0 0 1 · · ·0 0 0 · · ·...

......

. . .

⎤

⎥⎥⎥⎦



N

µB µCeeNeN

N = 2eN

µC

X=A X=A,B X=A,B,C

DNflavor:

SU(N)B x SU(N)C!DN flavor: SU(N)C !!DN no flavor: ∅

T+N

DN

high-genus: many TN’s & DN’s

T+N

DN


to find new trial a central charge, must add contribution from multiplets M

to TN’s central charges under new RSCFT(ϵ)

T+N

DN




a(ϵ)mod. tube =3

32

!ϵ3(3N3 − 3)− ϵ(3N3 − 2N − 1)

"

T+N

DN


central charges of theory w/ (p’,q’) = (10,-6)



a(ϵ)mod. tube =3

32

!ϵ3(3N3 − 3)− ϵ(3N3 − 2N − 1)

"

+ }a(ϵ)(p,q)

p = 4 → p′ = 4 + 6

q = 0 → q′ = 0− 6

T+N

DN


central charges of theory w/ (p’,q’) = (10,-6)

this reproduces results of BBBW obtained by integrating M5 anomaly polynomial on surface with g > 1 and generic (p,q) (one >0, one <0)



a(ϵ)mod. tube =3

32

!ϵ3(3N3 − 3)− ϵ(3N3 − 2N − 1)

"

+ }a(ϵ)(p,q)

p = 4 → p′ = 4 + 6

q = 0 → q′ = 0− 6

torus

N

N

N

N

N

N

many DN’s glued together: e.g. g = 1, p = −q = 6

torus

N

N

N

N

N

N


sphere(s)&

torus

N

N

N

N

N

N

eeN


only 1 : g = 0, p = 1, q = −3

!!DN

sphere(s)&

torus

N

N

N

N

N

N

eeN

eNN = 2 N N = 2 N

· · · eN


only 1 : g = 0, p = 1, q = −3

!!DN

n DN’s & 2 ’s at the tails: g = 0, p = 2+n, q = −(4+n)

!DN

sphere(s)&

torus

N

N

N

N

N

N

eeN

eNN = 2 N N = 2 N

· · · eN


only 1 : g = 0, p = 1, q = −3

!!DN

n DN’s & 2 ’s at the tails: g = 0, p = 2+n, q = −(4+n)

!DN

sphere(s)&

central charges for all new blocks also appeared in [Maruyoshi-Song,Nardoni]

purely 4d field theory computation of a & c reproduces all results by BBBW, for every genus and generic choice of (p,q)

we studied their chiral ring

in TNTrµk

A = TrµkB = Trµk

C

µAQ = µBQ = µCQ[Benini-Tachikawa-Wecht,

Gadde-Maruyoshi-Tachikawa-Yan Maruyoshi-Tachikawa-Yan-Yonekura,

Hayashi-Tachikawa-Yonekura, Lemos-Peelaers]


in TNTrµk

A = TrµkB = Trµk

C

µAQ = µBQ = µCQ[Benini-Tachikawa-Wecht,

Gadde-Maruyoshi-Tachikawa-Yan Maruyoshi-Tachikawa-Yan-Yonekura,

Hayashi-Tachikawa-Yonekura, Lemos-Peelaers]

flavor SU(N)A x SU(N)B x SU(N)C

Q i j k


in TNTrµk

A = TrµkB = Trµk

C

µAQ = µBQ = µCQ

in TN + MA

µAQ = µBQ = µCQ = 0

TrµkA = Trµk

B = TrµkC = 0

[Benini-Tachikawa-Wecht, Gadde-Maruyoshi-Tachikawa-Yan

Maruyoshi-Tachikawa-Yan-Yonekura, Hayashi-Tachikawa-Yonekura,

Lemos-Peelaers]


Q i j k

[MF-Giacomelli]

µA = 0

MAQ = 0


in TNTrµk

A = TrµkB = Trµk

C

µAQ = µBQ = µCQ

in TN + MA


TrµkA = Trµk

B = TrµkC = 0



Lemos-Peelaers]


Q i j k

[MF-Giacomelli]

QN j k

QN-1 j k

Q2 j k

Q1 j k

…µA = 0

MAQ = 0


in TNTrµk

A = TrµkB = Trµk

C

µAQ = µBQ = µCQ

in TN + MA


TrµkA = Trµk

B = TrµkC = 0



Lemos-Peelaers]


Q i j k

[MF-Giacomelli]

plug in ⟨MA⟩ + fluctuations

chiral ring of DN has only 1 generator*: bifundamental of SU(N)B x SU(N)C Q1jk (Q’s w/ higher i index written in terms of Q1jk and components of MA)

QN j k

QN-1 j k

Q2 j k

Q1 j k

…µA = 0

MAQ = 0

*oversimplifying a bit

which operator corresponds to an M2 in inaccessible models?

wrapped M2Cg ⊂ CY3

supersymmetric cycleBPS “heavy operator” O:

scaling dim. ∆ = energy of M2[Gaiotto-Maldacena,BBBW]





accessible models

O =!

QTN

N = 1 N = 1

N = 1





accessible models inaccessible model (e.g. high-genus)

O =!

QTN O =!

QTNQ1jkDN

N = 1 N = 1

N = 1






O =!

QTN O =!

QTNQ1jkDN

∆(O) =3

4(N − 1) [(p+ q)− ϵ(p− q)]

in inaccessible models, this matches holographic computation provided we use the unique independent bifundamental for each DN block!

N = 1 N = 1

N = 1






O =!

QTN O =!

QTNQ1jkDN

∆(O) =3

4(N − 1) [(p+ q)− ϵ(p− q)]

in inaccessible models, this matches holographic computation provided we use the unique independent bifundamental for each DN block!

knowledge of chiral ring instrumental in identifying correct heavy operator

N = 1 N = 1

N = 1

a remark:

in Gabi’s & Shlomo’s talks M5’s probing Ak-1

global symmetry for (1,0): SU(k) x SU(k) x U(1)t

in our case k=1: (2,0), not (1,0)!

global symmetry for (2,0) seen as (1,0): just U(1)

in specific CY3 background, identified w/ combination of U(1)1 & U(1)2( )

a remark:



in our case k=1: (2,0), not (1,0)!


in specific CY3 background, identified w/ combination of U(1)1 & U(1)2

discrete choice: flux for U(1)t labels theory

choice of (p,q) labeling (in)accessible BBBW models

( )

a remark:



in our case k=1: (2,0), not (1,0)!


in specific CY3 background, identified w/ combination of U(1)1 & U(1)2

discrete choice: flux for U(1)t labels theory

choice of (p,q) labeling (in)accessible BBBW models

explicit 4d field theory operation equivalent to turning on fluxes for global symmetry in 6d

should be applicable to most (1,0)’s

( )

recap:

recap:

• constructed inaccessible BBBW models in 4d by deforming TN

recap:

• computed a & c central charges exactly in 4d


recap:



• derived chiral ring relations for 3 new blocks & new N=1 dualities

recap:




• discussed unitary bound violations in eeN eNN = 2

eN&

already in [Maruyoshi-Song] [MF-Giacomelli]

recap:




• counted relevant operators and matched against N=1 class S index

• discussed unitary bound violations in eeN eNN = 2

eN&

already in [Maruyoshi-Song] [MF-Giacomelli]

[Beem-Gadde]

Thanks

Cg

refined pants-decomposition of Cg

p = 2 black TN’s p = 1 black TN & q = 1 red TN

remember: for N=2 class S

CY3 = O ⊕KCg → Cg ∼= C× T ∗Cg

Documents

N=1 SCFT’s with DN blocks - UCFileSpace Tools - Homehomepages.uc.edu/~argyrepc/SCFT_Aspen/slides/Fazzi.pdf · N=1 SCFT’s with DN blocks Marco Fazzi based on 1609.08156 with Simone