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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
reminder: accessible N=1 models of class S
p = 4 (black) TN’s
N = 2
g = 3(p = 2g − 2)
reminder: accessible N=1 models of class S
p = 4 (black) TN’s
g = 3
N = 2
g = 3
N = 1 N = 1
N = 1
(p = 2g − 2)
reminder: accessible N=1 models of class S
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)
reminder: accessible N=1 models of class S
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 )
N = 1
N = 2
N = 2N = 2N = 1
N = 1
(p = 2g − 2)
reminder: accessible N=1 models of class S
why accessible?
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 )
N = 1
N = 2
N = 2N = 2N = 1
N = 1
(p = 2g − 2)
reminder: accessible N=1 models of class S
only ingredients are: (ℤ2 colored) TN’s & N=1 or N=2 tubes
why accessible?
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 )
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
clearly, this construction holds only for p,q ≥ 0 and g > 1
what about p,q < 0, or g ≤ 1?
clearly, this construction holds only for p,q ≥ 0 and g > 1
what about p,q < 0, or g ≤ 1?
inaccessible
clearly, this construction holds only for p,q ≥ 0 and g > 1
what about p,q < 0, or g ≤ 1?
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
• construct inaccessible models: p or q < 0, g ≤ 1
this talk
• how? deform TN to obtain 3 new building blocks
• construct inaccessible models: p or q < 0, g ≤ 1
• 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)
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
glue many together by gauging (diagonal combination of) flavor symmetries:
(only maximal punctures in this talk)
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
glue many together by gauging (diagonal combination of) flavor symmetries:
p = 2 black q = 0 red
T+N
N=2T+N
N = 2N=2 way
(only maximal punctures in this talk)
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
glue many together by gauging (diagonal combination of) flavor symmetries:
TrΦ(µ1 − µ2) ⊂ W
p = 2 black q = 0 red
T+N
N=2T+N
N = 2N=2 way
(only maximal punctures in this talk)
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
glue many together by gauging (diagonal combination of) flavor symmetries:
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
(only maximal punctures in this talk)
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
glue many together by gauging (diagonal combination of) flavor symmetries:
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
(only maximal punctures in this talk)
…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
CY3 = L1 ⊕ L2 → CgM-theory on
c1(L1) + c1(L2) = 2g − 2
Calabi-Yau condition reads
CC
CgM5's
p qa particular twist preserves only N=1 in 4d
on
[Bah-Beem-Bobev-Wecht]
…and in 6d
CY3 = L1 ⊕ L2 → CgM-theory on
c1(L1) + c1(L2) = 2g − 2
Calabi-Yau condition reads
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
[Bah-Beem-Bobev-Wecht]
…and in 6d
CY3 = L1 ⊕ L2 → CgM-theory on
c1(L1) + c1(L2) = 2g − 2
Calabi-Yau condition reads
CC
Cg
U(1)1
U(1)2
U(1)1 x U(1)2 global symmetry
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
[Bah-Beem-Bobev-Wecht]
[Intriligator-Wecht]
…and in 6d
4d model: low-energy dynamics of M5’s on Riemann surface Cg inside CY3
CY3 = L1 ⊕ L2 → CgM-theory on
c1(L1) + c1(L2) = 2g − 2
Calabi-Yau condition reads
CC
Cg
U(1)1
U(1)2
U(1)1 x U(1)2 global symmetry
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
[Bah-Beem-Bobev-Wecht]
[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
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
all red
back to 4d: enter flipping
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
all red
back to 4d: enter flipping
T−N
µA
µB
µC
can “flip” puncture’s color wrt parent TN’s color
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
all red
back to 4d: enter flipping
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
[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 · · ·...
......
. . .
⎤
⎥⎥⎥⎦
[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
superpotential term Tr MX 𝞵X. ⟨MX⟩ triggers relevant deformation
& closes puncture X⟨MX⟩nilpotent =
⎡
⎢⎢⎢⎣
0 1 0 · · ·0 0 1 · · ·0 0 0 · · ·...
......
. . .
⎤
⎥⎥⎥⎦
[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
superpotential term Tr MX 𝞵X. ⟨MX⟩ triggers relevant deformation
& closes puncture X⟨MX⟩nilpotent =
⎡
⎢⎢⎢⎣
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
[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
superpotential term Tr MX 𝞵X. ⟨MX⟩ triggers relevant deformation
& closes puncture X⟨MX⟩nilpotent =
⎡
⎢⎢⎢⎣
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
N
µB µC
X=A
DNflavor:
SU(N)B x SU(N)C
[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
superpotential term Tr MX 𝞵X. ⟨MX⟩ triggers relevant deformation
& closes puncture X⟨MX⟩nilpotent =
⎡
⎢⎢⎢⎣
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
N
µB µC eNN = 2
eNµC
X=A X=A,B
DNflavor:
SU(N)B x SU(N)C!DN flavor: SU(N)C
[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
superpotential term Tr MX 𝞵X. ⟨MX⟩ triggers relevant deformation
& closes puncture X⟨MX⟩nilpotent =
⎡
⎢⎢⎢⎣
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
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
high-genus: many TN’s & DN’s
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
high-genus: many TN’s & DN’s
to find new trial a central charge, must add contribution from multiplets M
to TN’s central charges under new RSCFT(ϵ)
a(ϵ)mod. tube =3
32
!ϵ3(3N3 − 3)− ϵ(3N3 − 2N − 1)
"
T+N
DN
high-genus: many TN’s & DN’s
central charges of theory w/ (p’,q’) = (10,-6)
to find new trial a central charge, must add contribution from multiplets M
to TN’s central charges under new RSCFT(ϵ)
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
high-genus: many TN’s & DN’s
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)
to find new trial a central charge, must add contribution from multiplets M
to TN’s central charges under new RSCFT(ϵ)
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
many DN’s glued together: e.g. g = 1, p = −q = 6
sphere(s)&
torus
N
N
N
N
N
N
eeN
many DN’s glued together: e.g. g = 1, p = −q = 6
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
many DN’s glued together: e.g. g = 1, p = −q = 6
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
many DN’s glued together: e.g. g = 1, p = −q = 6
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]
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]
flavor SU(N)A x SU(N)B x SU(N)C
Q i j k
we studied their chiral ring
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]
flavor SU(N)A x SU(N)B x SU(N)C
Q i j k
[MF-Giacomelli]
µA = 0
MAQ = 0
we studied their chiral ring
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]
flavor SU(N)A x SU(N)B x SU(N)C
Q i j k
[MF-Giacomelli]
QN j k
QN-1 j k
Q2 j k
Q1 j k
…µA = 0
MAQ = 0
we studied their chiral ring
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]
flavor SU(N)A x SU(N)B x SU(N)C
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]
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
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 inaccessible model (e.g. high-genus)
O =!
QTN O =!
QTNQ1jkDN
N = 1 N = 1
N = 1
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 inaccessible model (e.g. high-genus)
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
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 inaccessible model (e.g. high-genus)
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 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
discrete choice: flux for U(1)t labels theory
choice of (p,q) labeling (in)accessible BBBW models
( )
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
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
• constructed inaccessible BBBW models in 4d by deforming TN
recap:
• computed a & c central charges exactly in 4d
• constructed inaccessible BBBW models in 4d by deforming TN
• derived chiral ring relations for 3 new blocks & new N=1 dualities
recap:
• computed a & c central charges exactly in 4d
• constructed inaccessible BBBW models in 4d by deforming TN
• derived chiral ring relations for 3 new blocks & new N=1 dualities
• discussed unitary bound violations in eeN eNN = 2
eN&
already in [Maruyoshi-Song] [MF-Giacomelli]
recap:
• computed a & c central charges exactly in 4d
• constructed inaccessible BBBW models in 4d by deforming TN
• derived chiral ring relations for 3 new blocks & new N=1 dualities
• 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
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