N=1 SCFT’s with DN blocks - UCFileSpace Tools -...
Transcript of N=1 SCFT’s with DN blocks - UCFileSpace Tools -...
![Page 1: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/1.jpg)
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]
![Page 2: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/2.jpg)
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]
![Page 3: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/3.jpg)
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]
![Page 4: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/4.jpg)
reminder: accessible N=1 models of class S
p = 4 (black) TN’s
N = 2
![Page 5: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/5.jpg)
reminder: accessible N=1 models of class S
p = 4 (black) TN’s
N = 2
g = 3(p = 2g − 2)
![Page 6: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/6.jpg)
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)
![Page 7: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/7.jpg)
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)
![Page 8: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/8.jpg)
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)
![Page 9: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/9.jpg)
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)
![Page 10: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/10.jpg)
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)
![Page 11: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/11.jpg)
clearly, this construction holds only for p,q ≥ 0 and g > 1
![Page 12: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/12.jpg)
clearly, this construction holds only for p,q ≥ 0 and g > 1
what about p,q < 0, or g ≤ 1?
![Page 13: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/13.jpg)
clearly, this construction holds only for p,q ≥ 0 and g > 1
what about p,q < 0, or g ≤ 1?
inaccessible
![Page 14: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/14.jpg)
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]
![Page 15: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/15.jpg)
this talk
• construct inaccessible models: p or q < 0, g ≤ 1
![Page 16: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/16.jpg)
this talk
• how? deform TN to obtain 3 new building blocks
• construct inaccessible models: p or q < 0, g ≤ 1
![Page 17: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/18.jpg)
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)
![Page 19: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/19.jpg)
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)
![Page 20: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/20.jpg)
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)
![Page 21: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/21.jpg)
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)
![Page 22: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/22.jpg)
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)
![Page 23: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/23.jpg)
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)
![Page 24: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/24.jpg)
…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]
![Page 25: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/25.jpg)
…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]
![Page 26: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/26.jpg)
…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]
![Page 27: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/27.jpg)
…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]
![Page 28: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/28.jpg)
…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]
![Page 29: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/29.jpg)
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
![Page 30: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/30.jpg)
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
![Page 31: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/31.jpg)
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
![Page 32: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/32.jpg)
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]
![Page 33: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/33.jpg)
[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
![Page 34: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/34.jpg)
[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 · · ·...
......
. . .
⎤
⎥⎥⎥⎦
![Page 35: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/35.jpg)
[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 · · ·...
......
. . .
⎤
⎥⎥⎥⎦
![Page 36: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/36.jpg)
[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
![Page 37: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/37.jpg)
[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
![Page 38: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/38.jpg)
[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
![Page 39: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/39.jpg)
[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: ∅
![Page 40: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/40.jpg)
T+N
DN
high-genus: many TN’s & DN’s
![Page 41: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/41.jpg)
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)
![Page 42: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/42.jpg)
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)
"
![Page 43: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/43.jpg)
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
![Page 44: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/44.jpg)
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
![Page 45: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/45.jpg)
torus
N
N
N
N
N
N
many DN’s glued together: e.g. g = 1, p = −q = 6
![Page 46: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/46.jpg)
torus
N
N
N
N
N
N
many DN’s glued together: e.g. g = 1, p = −q = 6
sphere(s)&
![Page 47: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/47.jpg)
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)&
![Page 48: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/48.jpg)
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)&
![Page 49: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/49.jpg)
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)
![Page 50: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/50.jpg)
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]
![Page 51: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/51.jpg)
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
![Page 52: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/52.jpg)
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
![Page 53: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/53.jpg)
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
![Page 54: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/54.jpg)
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
![Page 55: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/55.jpg)
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]
![Page 56: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/56.jpg)
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
![Page 57: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/57.jpg)
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
![Page 58: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/58.jpg)
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
![Page 59: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/59.jpg)
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
![Page 60: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/60.jpg)
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( )
![Page 61: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/61.jpg)
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
( )
![Page 62: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/62.jpg)
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
( )
![Page 63: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/63.jpg)
recap:
![Page 64: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/64.jpg)
recap:
• constructed inaccessible BBBW models in 4d by deforming TN
![Page 65: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/65.jpg)
recap:
• computed a & c central charges exactly in 4d
• constructed inaccessible BBBW models in 4d by deforming TN
![Page 66: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/66.jpg)
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
![Page 67: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/67.jpg)
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]
![Page 68: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/68.jpg)
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]
![Page 69: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/69.jpg)
Thanks
![Page 70: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/70.jpg)
![Page 71: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/71.jpg)
Cg
refined pants-decomposition of Cg
p = 2 black TN’s p = 1 black TN & q = 1 red TN
![Page 72: 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](https://reader031.fdocuments.in/reader031/viewer/2022030509/5ab8f7cf7f8b9a684c8d5ab0/html5/thumbnails/72.jpg)
remember: for N=2 class S
CY3 = O ⊕KCg → Cg ∼= C× T ∗Cg