Instanton counting for 5d SYMs - KIAShome.kias.re.kr/MKG/upload/Joint workshop/Chiung Hwang.pdf ·...
Transcript of Instanton counting for 5d SYMs - KIAShome.kias.re.kr/MKG/upload/Joint workshop/Chiung Hwang.pdf ·...
Instanton counting for 5d SYMs
Chiung Hwang (POSTECH)High1, Jan 28 2015
based on arXiv:1406.6793 with Joonho Kim, Seok Kim and Jaemo Park
Outline
• Motivation
• Zinst
• Sp(1) theories w/o antisym
• Sp(1) theory for 6d SCFT
• Summary
Motivation
N=1 Sp(N) gauge theories
5d N=1 Sp(N) gauge theories
5-dimensional N=1 Sp(N) gauge theory with one antisym and Nf fund hypermultiplets
• nonrenormalizable -> effective field theory
• Nf=0,…,7: 5d SCFT at UV fixed point -> exhibits enhanced ENf+1 global symmetry
• Nf=8: the circle compactification of 6d SCFT -> E8 global symmetry
• engineered by type IIA string theory on R8,1×R+O8 D8
D4
R+
Seiberg 96
Instanton partition function
Instanton partition function
Instantons
• described by a nonlinear sigma model
• Small instanton singularities -> inherit the nonrenormalizability of the 5d theory
• UV completion -> ADHM gauged quantum mechanics
Fmn = ?4Fmn =1
2✏mnpqFpq
Instanton moduli space
Instanton partition function -> ADHM QM index
k =1
8⇡2
Z
R4
tr(F ^ F ) 2 Z+
• self-dual
• instanton charge
• preserve half SUSY
• form marginal bound states with W-bosons
Atiyah, Hitchin, Drinfeld, Manin 78 Nekrasov 04
Instanton partition function
ZkQM(✏1, ✏2,↵i, z) = Tr
h(�1)F e��{Q,Q†}e�✏1(J1+JR)e�✏2(J2+JR)e�↵i⇧ie�zF
i
Z =1
|W |
IZ1-loop
=1
|W |
IZV
Y
�
Z�
Y
Z
The ADHM QM index
Matrix integral:
Questions?• Which integration contour?
-> Jeffrey-Kirwan residue.
• Zinst=ZQM? -> No. Extra UV degrees of freedom might be captured.
Instanton partition function
ZkQM(✏1, ✏2,↵i, z) = Tr
h(�1)F e��{Q,Q†}e�✏1(J1+JR)e�✏2(J2+JR)e�↵i⇧ie�zF
i
The ADHM QM index
Matrix integral:
Questions?• Which integration contour?
-> Jeffrey-Kirwan residue.
• Zinst=ZQM? -> No. Extra UV degrees of freedom might be captured.
Z =1
|W |
IZ1-loop
=1
|W |
IZV
Y
�
Z�
Y
Z
Instanton partition function
ZkQM(✏1, ✏2,↵i, z) = Tr
h(�1)F e��{Q,Q†}e�✏1(J1+JR)e�✏2(J2+JR)e�↵i⇧ie�zF
i
The ADHM QM index
Matrix integral:
Questions?• Which integration contour?
-> Jeffrey-Kirwan residue.
• Zinst=ZQM? -> No. Extra UV degrees of freedom might be captured.
cf. avoiding contour issue by discarding antisym hyper for Sp(1) H. -C. Kim, S. Kim, K. Lee 12
Z =1
|W |
IZ1-loop
=1
|W |
IZV
Y
�
Z�
Y
Z
Instanton partition function
ZkQM(✏1, ✏2,↵i, z) = Tr
h(�1)F e��{Q,Q†}e�✏1(J1+JR)e�✏2(J2+JR)e�↵i⇧ie�zF
i
The ADHM QM index
Matrix integral:
Questions?• Which integration contour?
-> Jeffrey-Kirwan residue. Benini, Eager, Hori, Tachikawa 13
• Zinst=ZQM? -> No. Extra UV degrees of freedom might be captured.
Z =1
|W |
IZ1-loop
=1
|W |
IZV
Y
�
Z�
Y
Z
Instanton partition function
ZkQM(✏1, ✏2,↵i, z) = Tr
h(�1)F e��{Q,Q†}e�✏1(J1+JR)e�✏2(J2+JR)e�↵i⇧ie�zF
i
The ADHM QM index
Matrix integral:
Questions?• Which integration contour?
-> Jeffrey-Kirwan residue.
• Zinst=ZQM? -> No. Extra UV degrees of freedom might be captured.
=Zinst?Z =1
|W |
IZ1-loop
=1
|W |
IZV
Y
�
Z�
Y
Z
Instanton partition function
ZkQM(✏1, ✏2,↵i, z) = Tr
h(�1)F e��{Q,Q†}e�✏1(J1+JR)e�✏2(J2+JR)e�↵i⇧ie�zF
i
The ADHM QM index
Matrix integral:
Questions?• Which integration contour?
-> Jeffrey-Kirwan residue.
• Zinst=ZQM? -> Not always. UV completion might give rise to extra degrees of freedom.
Z =1
|W |
IZ1-loop
=1
|W |
IZV
Y
�
Z�
Y
Z
Zinst
ADHM quantum mechanics
The ADHM QM index:
ZQM
=1
|W |X
�⇤
JK-Res(Q(�⇤), ⌘) Z1-loop
(�, ✏+
, z)
D0-D0 : O(k) antisymmetric (At,') , (�̄A↵̇ ,�a↵)
O(k) symmetric (a↵�̇ ,'aA) , (�A↵ , �̄a↵̇)
D0-D4 : Sp(N)⇥O(k) bif. (q↵̇) , ( A, a)
D0-D8 : SO(2Nf )⇥O(k) bif. ( l)
Zinst=ZQM?
• noncompact Coulomb branch: lifted for Nf≤7 • part of Higgs branch: non 5-dimensional degrees of freedom
D8
D4D0
R+
O8 Douglas 96; Aharony, Hanany, Kol 97
Extra string theory states
D0-D8-O8 bound states (unbounded from D4)
ZNf=0 = PE
� t2q
(1� tu)(1� t/u)(1� tv)(1� t/v)
�
Z1Nf5 = PE
� t2
(1� tu)(1� t/u)(1� tv)(1� t/v)q�(yi)
SO(2Nf )
2Nf�1
�
ZNf=6 = PE
� t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
⇣q�(yi)
SO(12)32 + q2
⌘�
ZNf=7 = PE
� t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
⇣q�(yi)
SO(14)64 + q2�(yi)
SO(14)14
⌘�
• captured by D0 quantum mechanics on D8-O8
• Dropping bifundamentals from D0-D4,
ZQM
=1
|W |X
�⇤
JK-Res(Q(�⇤), ⌘) Z1-loop
(�, ✏+
, z)
-> exactly the 0th order term of ZQM expanded in the electric charge fugacity
Extra string theory states
D8-O8 system• 9d SO(2Nf) SYM theory
• enhanced to ENf+1 refering to string duality
• The perturbative index:
Qa↵ , QA
↵ , Q̄a↵̇ , Q̄A
↵̇
2 sinh✏12
· 2 sinh ✏22
· 2 sinh m+ ✏+2
· 2 sinh m� ✏+2
1⇣2 sinh ✏1
2 · 2 sinh ✏22 · 2 sinh m+✏+
2 · 2 sinh m�✏+2
⌘2 �SO(2Nf )adj (yi)
+× ×
the translations on R8 the electric charges
fpert =�SO(2Nf )adj (yi)+
2 sinh ✏12 · 2 sinh ✏2
2 · 2 sinh m+✏+2 · 2 sinh m�✏+
2
= �t2�
SO(2Nf )adj (yi)+
(1� tu)(1� t/u)(1� tv)(1� t/v)
broken
Extra string theory states
The perturbative index for D8-O8
The nonperturbative index for D0-D8-O8
E4 = SU(5) : 24 ! 10 + 150 + 41 + 4�1
E5 = SO(10) : 45 ! 10 + 280 + (8s)1 + (8s)�1
E6 : 78 ! 10 + 450 + 161 + 16�1
E7 : 133 ! 10 + 660 + 321 + 32�1 + 12 + 1�2
E8 : 248 ! 10 + 910 + 641 + 64�1 + 142 + 14�2
f0 = � t2
(1� tu)(1� t/u)(1� tv)(1� t/v)q
fNf = � t2
(1� tu)(1� t/u)(1� tv)(1� t/v)q�(yi)
SO(2Nf )
2Nf�1
f6 = � t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
hq�(yi)
SO(12)32 + q2
i
f7 = � t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
hq�(yi)
SO(14)64 + q2�(yi)
SO(14)14
i
• ENf+1 enhancement -> supports duality between I’ & heterotic
• UV completion additionally captures 9d spectrum.
f9d SYM = �t2�
SO(2Nf )adj (yi)+
(1� tu)(1� t/u)(1� tv)(1� t/v)
Zinst = ZQM/Zstring
Sp(1) theories w/o antisym
Sp(1) theories w/o antisym: revisited
Two classes of 5d rank N SCFTs -> Sp(N) SYMs w/ or w/o an antisym hyper
• 1st class: UV fixed points for Nf≤7, engineered by D4-D8-O8 or M-theory on CY3
• 2nd class: UV fixed points for Nf≤2N+4, engineered by M-theory on CY3
• For Sp(1), the two classes are expected to yield the same SCFTs.
The same SCFT but different string theory engineerings -> The same Zinst but different ZQM’s
Seiberg 96 Intriligator, Morrison, Seiberg 97
• M2-branes wrapping 2-cycles
• signal of noncompact modulus: Z1-loop approaches a const for Nf=6.
Sp(1) theories w/o antisym: revisited
Zstring = PE
� (1 + t2)q2
2(1� tu)(1� t/u)
�
Zstring
=Zw/o
QM
Zw/
inst
= 1
Instantons in the CY engineering
-> for Nf<6, no extra UV degrees of freedom
-> for Nf=6, M2 can escape to infinity
Sp(1) theory for 6d SCFT
Sp(1) theory for 6d SCFT
Sp(1) theory with Nf = 8 (& antisym) • 8 D8 + O8 -> zero D8-brane charge
• Uplift to M-theory on R8,1×R+×S1 -> M9-M5 system
• The circle compactification of 6d (1,0) SCFT
• E-string Klemm, Mayr, Vafa 96
O8 D8
D4
R+
M9M5
M2
Sp(1) theory for 6d SCFT
f =
t(v + v�1 � u� u�1)
(1� tu)(1� t/u)� (t+ t3)(u+ u�1 + v + v�1)
2(1� tu)(1� t/u)(1� tv)(1� t/v)
�q2
1� q2
� t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
�(yi)
SO(16)120
q2
1� q2+ �(yi)
SO(16)128
q
1� q2
�
-> Not manifest E8 due to nonzero Wilson lines; y8 -> y8q
Extra string states• zero electric charge sector:
Sp(1) theory for 6d SCFT
f =
t(v + v�1 � u� u�1)
(1� tu)(1� t/u)� (t+ t3)(u+ u�1 + v + v�1)
2(1� tu)(1� t/u)(1� tv)(1� t/v)
�q2
1� q2
� t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
�(yi)
SO(16)120
q2
1� q2+ �(yi)
SO(16)128
q
1� q2
�
-> Not manifest E8 due to nonzero Wilson lines; y8 -> y8q
Extra string states• zero electric charge sector:
fpert = �t2�
SO(2Nf )adj (yi)+
(1� tu)(1� t/u)(1� tv)(1� t/v)= � t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
h�+91 + y28�
SO(14)14
i• the second line +
� t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
�E8248(yi)
q2
1� q2+ �E8
248(yi)+�
-> the KK tower of 10d E8 SYM
Sp(1) theory for 6d SCFT
f =
t(v + v�1 � u� u�1)
(1� tu)(1� t/u)� (t+ t3)(u+ u�1 + v + v�1)
2(1� tu)(1� t/u)(1� tv)(1� t/v)
�q2
1� q2
� t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
�(yi)
SO(16)120
q2
1� q2+ �(yi)
SO(16)128
q
1� q2
�
Extra string states• zero electric charge sector:
• Type I SUGRA: dilaton φ, RR 2-form C2, graviton gµν, dilatino λ, gravitino ψµ
(1� 28� 35v)boson � (8s � 56s)fermion
= (8v ⌦ 8v)sym � (8v ⌦ 8c)� (8c ⌦ 8c)anti
� (t+ t3)(u+ u�1 + v + v�1)
(1� tu)(1� t/u)(1� tv)(1� t/v)
�(8v) = (t+ t�1)(u+ u�1 + v + v�1)
�(8c) = �t2 � 2� t�2 � (u+ u�1)(v + v�1)
�(8s) = �(t+ t�1)(u+ u�1 + v + v�1)
the translations on R8
+
Sp(1) theory for 6d SCFT
f =
t(v + v�1 � u� u�1)
(1� tu)(1� t/u)� (t+ t3)(u+ u�1 + v + v�1)
2(1� tu)(1� t/u)(1� tv)(1� t/v)
�q2
1� q2
� t2
(1� tu)(1� t/u)(1� tv)(1� t/v)
�(yi)
SO(16)120
q2
1� q2+ �(yi)
SO(16)128
q
1� q2
�
Extra string states• zero electric charge sector:
ZSCFT =ZQM
Zstring
6d (1,0) SCFT index
• compared with E-string index Klemm, Mayr, Vafa 96; Haghighat, Lockhart, Vafa 14; J. Kim, S. Kim, K. Lee, J. Park, Vafa 14; Cai, Huang, Sun 14
Summary
• Instanton partition functions of 5d SYMs -> good observables for 5d/6d SCFTs
• Nonrenormalizability of 5d gauge theories -> singularities in instanton moduli space
• UV completion -> ADHM gauged quantum mechanics
• Matrix integral for ADHM index -> JK-residue, generally applicable to 1d N=2 theories
• ADHM index may capture extra UV degrees of freedom
• Different UV completions -> different ZQM’s but the same Zinst
• String theory can be a guideline for handling nonrenoramlizable effective field theories