Monopoles and Vorticesin 3d N=4 gauge theories

Hee-Cheol Kim

Harvard University

Based on arXiv :1609.xxxxx with M. Bullimore, T. Dimofte, D. Gaiotto, J. Hilburn

1

3d supersymmetric gauge theories with gauge group .• 8 real SUSY with R-symmetry• Vector multiplet

vector multiplet + Adjoint chiral multiplet

• Hypermultiplet Two chiral multiplets with

Rich structure of supersymmetric vacua parametrized by vacuum expectation value (VEV) of gauge invariant scalar operators.

Introduction

N = 4 GSU(2)H ⇥ SU(2)CQaa

↵

N = 2 Aµ, � '

N = 2 X,Y W = Tr'XY

Higgs branch Coulomb branchX,Y �,', V±

Higgs branch

• VEV of scalars in hypermultiplets (and ).• Superpotential constraint (F-term)

• D-term constraint

• Hyper-Kahler quotient

• No quantum correction. So classical geometry is exact.

X,Y � = ' = 0

MM = {X,Y |µC = µR = 0}/G

µC ⌘ @W

@'= X · Y = 0

µR ⌘ X ·X† � Y †Y = 0

MH

Coulomb branch

• VEV of three real scalars in vectormultiplets together with monopole operators .

• Monopole operator

• Hyper-Kahler with

• Quantum corrections on the classical geometry.

• Sometime we can study quantum-corrected Coulomb branch geometry indirectly using mirror symmetry.

�,'Vm

A± =

m

2

(±1� cos ✓)d�+ · · · (m 2 �G_weight lattice of G_

)

.

dimCMC = 2 rank G

MC

Z

S2

F = m

In Higgs branch, there are vortex solitons carrying magnetic flux.

Monopoles become interfaces (or kinks, or domain walls) in 1d vortex quantum mechanics (QM) living on the vortex world line (blue line).

We shall find UV gauge theory description for the 1d vortex system with the monopole interface.

vortex with n+A

nvortex with

VAx

3

n+A+A0

n

VA

VA0

Monopoles in Higgs branch

[Tong 03]

We will compute correlation functions of monopole operators using supersymmetric localization computation in 1d vortex QM.

Main goal is to extract exact chiral rings and Coulomb branch algebra from the correlation functions.

I will also show that the Coulomb branch algebra is naturally quantized in our setting.

|n+Ai

|ni

VA

Monopole correlation functions

Partition function of = hn+A |VA |n i

• Introduction

• Moduli space of classical Coulomb branch.

• Vortex quantum mechanics with monopole operators.

• Correlation functions and Coulomb branch algebra.

• Conclusion

Outline

We consider unitary gauge groups and fundamental (and also bi-fundamental) hypermultiplets.

Global symmetry :• Flavor symmetry acting on : • Topological symmetry acting on :

Mass deformations for

Fayet-Iliopoulos (FI) deformations for

-deformation for Lorentz rotation

Symmetries and deformationsG = ⌦iU(Ni)

GH = ⌦iSU(Nfi)

GC = (U(1)T )#of U(1)

0s inG

MH

MC

� ! � +mR , ' ! '+mC

µR = 0 ! µR = tR , µC = 0 ! µC = tC

GH

GC

⌦ U(1)z

' ! '� i✏Lz + ✏QH

Abelian gauge field can be dualized into a periodic scalar field , called dual photon.

Monopole operators are defined as

Classical Coulomb branch is

Quantum corrections modify this classical geometry.- For fundamental hypers, quantum-corrected Coulomb branch is

Abelian Coulomb branchAµ

and

a

da = ⇤dA a ⇠ a+ 2⇡g2

V± = e± 1

g2(�+ia)

MC = {', V±|V� = V �1+ } = R3 ⇥ S1

MC = {', V±}/(⇤) , (⇤) : V+V� =

NfY

i=1

('+miC)

Nf

singularity deformed by complex mass parametersANf�1

[Borokhov, Kapustin, Wu 02]

At generic points on , scalar fields take non-degenerate values in Cartan subalgebra of and the non-abelian gauge group will be broken to maximal abelian subgroup .

Monopole operators in IR abelian theory can be explicitly constructed as

Therefore, the classical Coulomb branch takes the form

This description breaks down by quantum effects when W-bosons becomes massless and thus non-abelian gauge symmetry is restored.

We shall compute Coulomb branch chiral rings with full quantum corrections by studying monopole actions on vortices in Higgs branch.

Non-Abelian Coulomb branch (classical analysis)MC 'g

U(1)r ⇢ G

Vm = e1g2

m·(�+ia)

MclasC ⇠ {(R3 ⇥ S1)r}/WG

Focus on gauge theory with fundamental hypermultiplets and a real FI parameter .

Higgs branch which admits vortex solution is Grassmannian parametrized by (and ).

On a Higgs vacuum labelled by , we find BPS vortices satisfying

Moduli space of vortex solutions is a complex manifold with

which respects symmetry .

Higgs branch and vorticesU(N) Nf

G(N,Nf )

tR > 0

X Y = � = ' = 0

Fzz +X ·X† = tR , DzX = 0Z

R2

Fzz = n , Xz!1! hXi⌫

⌫

z = x

1+ix

2

dimCMn⌫ = nNf

U(1)z ⇥ S[U(N)⇥ U(Nf �N)]

Mn⌫

Complex masses and -deformation such asleaves isolated massive vacua which are fixed points w.r.t equivariant gauge and flavor rotations.

Vortex moduli space on a massive vacuum will also be lifted and localized to isolated fixed points labelled by

-deformation effectively compactifies 3d gauge theory to 1d quantum mechanics at .

Hilbert space of the 1d quantum mechanics is the equivariant cohomology of the moduli space of the 1/2 BPS vortices :

Vortex charge and

⌦ ' ! '+mC � i✏Lz + ✏QH

⌦z = 0

✓Nf

N

◆

n k = (k1, · · · , kN ) ,NX

i=1

ki = n

⌫

H⌫ = �fixed p

C|pi

= �n,k

C|n, ki

Brane construction of vortices.

D1-brane worldvolume theory is 1d gauged quantum mechanics of

• Global symmetry :

• Higgs branch agrees with vortex moduli space .

Vortex quantum mechanics

NS5

NS50

tRn D1

x

3,4,5

x

7,8,9

x

6

N D3

Nf�N D3

N = (2, 2)

N

Nf�N

Bq

q

n

Mn⌫

U(1)z ⇥ S[U(N)⇥ U(Nf �N)]

[Hanany, Tong 03]

NS5 NS50

N D3

Nf D3

We propose

• 1d vortex system with the interface preserves SUSY• Interface connects vortex QM with U(n) gauge group and

another vortex QM with U(n+A) gauge group.

Monopole as Interface

Monopole operator in 3d = 1/2 BPS interface in vortex QM

vortex with charge n+A

nvortex with charge

U(n)

U(n+A) gauge theory

gauge theory

VA Monopole Interface

1d

N = (0, 2)

Monopole interface between and vortices is constructed as follows:

1. Neumann boundary condition on 1d vector and chiral multiplets.

2. Additional 0-dimensional degrees of freedom and .

3. Superpotentials

This interface provides a (surjective) map from to , when .So, we claim that the interface realizes 3d monopole operator.In mathematics, this map is known as Hecke correspondence.

Induces vector and chiral multiplets at the interface. N = (0, 2)

N Nf�N

n n0⇠

�

⌘ ⌘

B B0

q

q0q0

q

: chiral multiplet: Fermi multiplet

⇠ �, ⌘, ⌘

J� = ⇠B0 �B⇠ , J⌘ = ⇠q0 � q , J⌘ = q0 � q⇠ (E� = E⌘ = E⌘ = 0)

Mn0

⌫ Mn⌫ n0 � n

n n0

[Nakajima 11], [Braverman, Feigin, Finkelberg, Rybnikov 11]

Partition function of the 1d vortex system with the interface on .

• If , and if , .

Localization computation gives rise to the correlation function (at 1st vacuum)

where is the equivariant weight at fixed point

Correlation functions

t2 > t > t1

t1

t2|n0i

|ni

O hn0|O|ni (for )N = 1

n0 = n O = 1 n0 = n+ 1 O = V+

|ni

at

at

hn0|O|ni =Qn

s=1(n0 + 1� s)✏

QNf

i=2

Qn0

s=1(m1 �mi + s✏)

!n !n0

!n =nY

s=1

(n+ 1� s)✏

NfY

i=2

nY

s=1

(m1 �mi + s✏)

Identity interface when :

• This gives the correct normalization of the vortex state .• 3d vortex partition function on interval with Neumann b.c. is

Correlation functions of monopole operators when :

O = 1 n0 = n

hn|O|ni = hn|ni = 1

!n

|ni|N i

Zvortex

= hN|N i =X

n�0

qnhn|ni =X

n�0

qn1

!n

V± n0 = n+ 1

hn+1|V+|ni = hn|V�|n+1i = 1Qn

s=1

h(n+ 1� s)✏

QNf

i=2(m1 �mi + s✏)i

We can compute monopole actions and Coulomb branch algebra from the monopole correlation functions.

Monopoles act on the state as

In addition, the 3d complex vectormultiplet scalar acts as

Monopole action :

Monopole action and Coulomb branch algebra

V+|ni = C+|n+1i , V�|ni = C�|n� 1i

C+ =hn+1|V+|nihn+1|n+1i , C� =

hn�1|V�|nihn�1|n�1i

where

'

'|ni = (�mi � n✏� ✏2 )|ni (at i-th vacuum)

|ni

V+|ni = P ('+ ✏2 )|n+1i , V�|ni = |n�1i

P (x) =

NfY

a=1

(x+ma)with

Hilbert space is the highest-weight Verma module of the Coulomb branch algebra.

The Coulomb branch algebra is

The algebra is quantized by -parameter .

V+V� = P ('+ ✏2 ) , V�V+ = P ('� ✏

2 ) , [', V±] = ⌥✏V±

“Spherical rational Cherednik algebra"

⌦ ✏

VA

VB

Commutative

VA

VB

✏

Non-commutative

H⌫

|ni ⇠ (V+)n|0i 8 |ni 2 H⌫for

N = 1

⌦ -deformation

for

[Nekrasov, Shatashvili 09], [Yagi 14]

For general , monopole operators acts on the vortex states as

Quantized Coulomb branch algebra :

Therefore, Coulomb branch chiral ring is

V +a |n, ki =

P ('a +✏2 )QN

b 6=a('b � 'a)|n+1, k+�ai

V �a |n, ki = 1

QNb 6=a('b � 'a)

|n�1, k��ai

'a|n, ki = (�ma � ka✏� ✏2 )|n, ki

V +a V �

a =P ('a +

✏2 )Q

b 6=a('a � 'b)('b � 'a � ✏), V �

a V +a =

P ('a � ✏2 )Q

b 6=a('a � 'b)('b � 'a + ✏)

N

C[MC ] = {V ±a ,'a}/{⇤,WG}

⇤

[Bullimore, Dimofte, Gaiotto 15]

Exact Coulomb branch algebra can be computed using correlation functions of monopoles operators which are partition functions of 1d vortex QM with interfaces.

Generalization to “Triangular quiver gauge theories”.• Finite algebra and finite AGT correspondence

Moduli matrix approach for monopole operators.

Extensions :

• General unitary quiver theories

• Higher dimensional generalization

• N=3 or N=2 gauge theories

• Other gauge groups ?

Conclusion

[Braverman, Feigin, Rybnikov, Finkelberg 11], [Nakajima 11]

[Bullimore, Dimofte, Gaiotto 15]

WN