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