Surface operators and duality relations in N=2 gauge theories - - PowerPoint PPT Presentation

Surface operators and duality relations in N=2 gauge theories

Alberto Lerda

Università del Piemonte Orientale and INFN – Torino, Italy

Trieste, 10 July 2018

This talk is mainly based on:

• S.K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, V. Gupta, R.R. John and A. L.,

“Surface operators, dual quivers and residues”, arXiv:1807.xxxxx

• S.K. Ashok, S. Ballav, M. Billò, E. Dell’Aquila, M. Frau, R.R. John and A. L.,

“Surface operators in conformal SQCD”, arXiv:180y.yyyyy

• S.K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, V. Gupta, R.R. John and A. L.,

“Surface operators in 5d gauge theories and duality relations”, JHEP 1805 (2018) 046, arXiv:1712.06946

• S.K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, V. Gupta, R.R. John and A. L.,

“Surface operators, chiral rings and localization in N=2 theories”, JHEP 1711 (2017) 137, arXiv:1707.08922

• S.K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, R.R. John and A. L., “Modular and

duality properties of surface operators in N=2* gauge theories”, JHEP 1707 (2017) 068, arXiv:1702.02833

but it builds on a very vast literature …

• 1. Introduction
• 2. Surface operators in N=2 gauge theories
• 3. Duality relations
• 4. Conclusions

Plan of the talk

§ The study of how a QFT responds to the presence of defects has recently received a lot of attention. § Defects are disturbances supported on sub-manifolds of the space-time that can be used to probe the theory in the bulk:

• 1. Local operators (d=0), e.g. chiral correlators

Defects

⌦ Trφn1(x1) · · · Trφnk(xk) ↵

§ The study of how a QFT responds to the presence of defects has recently received a lot of attention. § Defects are disturbances supported on sub-manifolds of the space-time that can be used to probe the theory in the bulk:

• 1. Local operators (d=0), e.g. chiral correlators
• 2. Line operators (d=1), e.g. Wilson lines

Defects

⌦ Trφn1(x1) · · · Trφnk(xk) ↵

D TrR P e

H

C iAE

§ The study of how a QFT responds to the presence of defects has recently received a lot of attention. § Defects are disturbances supported on sub-manifolds of the space-time that can be used to probe the theory in the bulk:

• 1. Local operators (d=0), e.g. chiral correlators
• 2. Line operators (d=1), e.g. Wilson lines
• 3. Surface operators (d=2)

Defects

⌦ Trφn1(x1) · · · Trφnk(xk) ↵

D TrR P e

H

C iAE

§ The study of how a QFT responds to the presence of defects has recently received a lot of attention. § Defects are disturbances supported on sub-manifolds of the space-time that can be used to probe the theory in the bulk:

• 1. Local operators (d=0), e.g. chiral correlators
• 2. Line operators (d=1), e.g. Wilson lines
• 3. Surface operators (d=2)
• 4. Domain walls (d=3)

Defects

⌦ Trφn1(x1) · · · Trφnk(xk) ↵

D TrR P e

H

C iAE

§ The study of how a QFT responds to the presence of defects has recently received a lot of attention. § Defects are disturbances supported on sub-manifolds of the space-time that can be used to probe the theory in the bulk:

• 1. Local operators (d=0), e.g. chiral correlators
• 2. Line operators (d=1), e.g. Wilson lines
• 3. Surface operators (d=2)
• 4. Domain walls (d=3)

Defects

⌦ Trφn1(x1) · · · Trφnk(xk) ↵

D TrR P e

H

C iAE

§ There are different ways of describing defects in a 4d QFT:

• 1. Modifying the QFT path-integral by imposing a singular

behavior of the fields on the defect.

• 2. Coupling the QFT to additional degrees of freedom localized
• n the defect.
• 3. Using a String Theory realization of the QFT and introducing

extra D-branes or M-branes to represent the defect.

• 4. …

Defects

§ There are different ways of describing defects in a 4d QFT:

• 1. Modifying the QFT path-integral by imposing a singular

behavior of the fields on the defect.

• 2. Coupling the QFT to additional degrees of freedom localized
• n the defect.
• 3. Using a String Theory realization of the QFT and introducing

extra D-branes or M-branes to represent the defect.

• 4. …

Defects

Surface operators

§ Surface operators in a 4d theory are operators supported on a 2d submanifold:

• They are particular examples of non-local operators that play the

role of “thermometers” for the QFT: when introduced in the path- integral, they provide us with valuable information (phases, non- perturbative features, ...) on the QFT.

Surface operators

D ⊂ R4

R4 ' C2 : (w1, w2) D ' C : (w1, 0) w1 w2

D

R4

(Gukov+Kapustin ’13)

§ Surface operators can be described:

• as singularities (or monodromy defects) for the gauge field Aμ along

the surface D in the 4d space-time

Surface operators

monodromy defects

§ Surface operators can be described:

• as singularities (or monodromy defects) for the gauge field Aμ along

the surface D in the 4d space-time

• as coupled 2d-4d systems, namely as a 2d theory supported on D

with a flavor symmetry group G that is gauged in 4d.

Surface operators

monodromy defects coupled 2d-4d systems

§ Surface operators can be described:

• as singularities (or monodromy defects) for the gauge field Aμ along

the surface D in the 4d space-time

• as coupled 2d-4d systems, namely as a 2d theory supported on D

with a flavor symmetry group G that is gauged in 4d.

Surface operators

monodromy defects coupled 2d-4d systems clarify & equate

§ The presence of the defect can be encoded in a singular behavior

• f the SU(N) gauge field on a loop around the defect:

where

Surface operators as monodromy defects

A = − diag (γ1, · · · , γ1, γ2, · · · , γ2, · · · , γM, · · · , γM) dθ | {z }

n1

| {z }

n2

| {z }

nM

(Gukov+Witten ’06, ‘08)

D

θ

w2

X

i

ni = N , w2 = ρ eiθ

§ The presence of the defect can be encoded in a singular behavior

• f the SU(N) gauge field on a loop around the defect:

where § This monodromy, i.e. the vector , breaks the gauge symmetry SU(N) to a (Levi) subgroup , which characterizes the defect

Surface operators as monodromy defects

A = − diag (γ1, · · · , γ1, γ2, · · · , γ2, · · · , γM, · · · , γM) dθ | {z }

n1

| {z }

n2

| {z }

nM

(Gukov+Witten ’06, ‘08)

~ n = {n1, n2, . . . , nM} L

SU(N) → S ⇥ U(n1) × U(n2) × · · · × U(nM) ⇤

X

i

ni = N , w2 = ρ eiθ

§ In presence of the defect, quantized magnetic fluxes are allowed for each group factor: § So besides the usual instanton factor in the path-integral we can insert also a new type of factor

Surface operators as monodromy defects

1 2π Z

D

Tr FU(ni) = mi ∈ Z , X

i

mi = 0

U(ni)

exp  2πi τ ⇣ 1 8π2 Z

R4 Tr F ∧ F

⌘

exp  2πi X

i

ηi ⇣ 1 2π Z

D

Tr FU(ni) ⌘

complexified gauge coupling constant “electric” parameters

§ In presence of the defect, quantized magnetic fluxes are allowed for each group factor: § It can be shown that this implies that

Surface operators as monodromy defects

instanton number new contribution due to the defect

1 2π Z

D

Tr FU(ni) = mi ∈ Z , X

i

mi = 0

1 8π2 Z

R4 Tr F ∧ F = k +

X

i

γi mi

U(ni)

§ In presence of the defect, quantized magnetic fluxes are allowed for each group factor: § It can be shown that this implies that § Thus

Surface operators as monodromy defects

1 2π Z

D

Tr FU(ni) = mi ∈ Z , X

i

mi = 0

1 8π2 Z

R4 Tr F ∧ F = k +

X

i

γi mi

−Sinst = 2πi τ ⇣ 1 8π2 Z

R4 Tr F ∧ F

⌘ + 2πi X

i

ηi ⇣ 1 2π Z

D

Tr FU(ni) ⌘

U(ni)

1 2π Z

D

Tr FU(ni) = mi ∈ Z , X

i

mi = 0

§ In presence of the defect, quantized magnetic fluxes are allowed for each group factor: § It can be shown that this implies that § Thus

Surface operators as monodromy defects

1 8π2 Z

R4 Tr F ∧ F = k +

X

i

γi mi

U(ni)

−Sinst = 2πi τ ⇣ k + X

i

γi mi ⌘ + 2πi X

i

ηi mi = 2πi τ k + 2πi X

i

(ηi + τ γi) mi

§ The non-perturbative weight in the path-integral is then: § So that the instanton partition function becomes

Surface operators as monodromy defects

with Zinst = X

{k,mi}

⇣ e2πi τ k Y

i

e2πi ti mi⌘ Z{k,mi}

e2πi τ k Y

i

e2πi ti mi ti = ηi + τ γi

§ The non-perturbative weight in the path-integral is then: § So that the instanton partition function becomes

which, through a change of variables, takes the form where and

Surface operators as monodromy defects

with Zinst = X

{k,mi}

⇣ e2πi τ k Y

i

e2πi ti mi⌘ Z{k,mi}

e2πi τ k Y

i

e2πi ti mi ti = ηi + τ γi

Zinst = X

{di}

q di

i

Z{di}

di ∈ N

• <latexit sha1_base64="x8KT6ZjhG6hk1HS1HF3zJ5UaquI=">AB+XicdVDLSgNBEJz1GeMr6tHLYBA8hZkgJrkFvHiMYB6QDWF20olDZh/M9AphyUd41ZM38erXePBfnI0RVLRORVU3XV1BopVFxt68ldW19Y3NwlZxe2d3b790cNixcWoktGWsY9MLhAWtImijQg29xIAIAw3dYHqZ+907MFbF0Q3OEhiEYhKpsZICndT1AzWZ+PNhqcwqjDHOc0Jr10wRxqNepXKc8thzJZojUsvfujWKYhRCi1sLbPWYKDTBhUsO86KcWEiGnYgJ9RyMRgh1ki7hzepagTFNwFCl6UKE7xuZCK2dhYGbDAXe2t9eLv7l9VMc1weZipIUIZL5IVQaFoesNMr1AHSkDCKPDlQFVEpjEAEo6iQ0ompK6bo+vh6mv5POtUKZxV+fV5uNpfNFMgxOSFnhJMaZIr0iJtIsmU3JMH8uhl3pP37L18jq54y50j8gPe6wdfWZRH</latexit><latexit sha1_base64="x8KT6ZjhG6hk1HS1HF3zJ5UaquI=">AB+XicdVDLSgNBEJz1GeMr6tHLYBA8hZkgJrkFvHiMYB6QDWF20olDZh/M9AphyUd41ZM38erXePBfnI0RVLRORVU3XV1BopVFxt68ldW19Y3NwlZxe2d3b790cNixcWoktGWsY9MLhAWtImijQg29xIAIAw3dYHqZ+907MFbF0Q3OEhiEYhKpsZICndT1AzWZ+PNhqcwqjDHOc0Jr10wRxqNepXKc8thzJZojUsvfujWKYhRCi1sLbPWYKDTBhUsO86KcWEiGnYgJ9RyMRgh1ki7hzepagTFNwFCl6UKE7xuZCK2dhYGbDAXe2t9eLv7l9VMc1weZipIUIZL5IVQaFoesNMr1AHSkDCKPDlQFVEpjEAEo6iQ0ompK6bo+vh6mv5POtUKZxV+fV5uNpfNFMgxOSFnhJMaZIr0iJtIsmU3JMH8uhl3pP37L18jq54y50j8gPe6wdfWZRH</latexit><latexit sha1_base64="x8KT6ZjhG6hk1HS1HF3zJ5UaquI=">AB+XicdVDLSgNBEJz1GeMr6tHLYBA8hZkgJrkFvHiMYB6QDWF20olDZh/M9AphyUd41ZM38erXePBfnI0RVLRORVU3XV1BopVFxt68ldW19Y3NwlZxe2d3b790cNixcWoktGWsY9MLhAWtImijQg29xIAIAw3dYHqZ+907MFbF0Q3OEhiEYhKpsZICndT1AzWZ+PNhqcwqjDHOc0Jr10wRxqNepXKc8thzJZojUsvfujWKYhRCi1sLbPWYKDTBhUsO86KcWEiGnYgJ9RyMRgh1ki7hzepagTFNwFCl6UKE7xuZCK2dhYGbDAXe2t9eLv7l9VMc1weZipIUIZL5IVQaFoesNMr1AHSkDCKPDlQFVEpjEAEo6iQ0ompK6bo+vh6mv5POtUKZxV+fV5uNpfNFMgxOSFnhJMaZIr0iJtIsmU3JMH8uhl3pP37L18jq54y50j8gPe6wdfWZRH</latexit><latexit sha1_base64="x8KT6ZjhG6hk1HS1HF3zJ5UaquI=">AB+XicdVDLSgNBEJz1GeMr6tHLYBA8hZkgJrkFvHiMYB6QDWF20olDZh/M9AphyUd41ZM38erXePBfnI0RVLRORVU3XV1BopVFxt68ldW19Y3NwlZxe2d3b790cNixcWoktGWsY9MLhAWtImijQg29xIAIAw3dYHqZ+907MFbF0Q3OEhiEYhKpsZICndT1AzWZ+PNhqcwqjDHOc0Jr10wRxqNepXKc8thzJZojUsvfujWKYhRCi1sLbPWYKDTBhUsO86KcWEiGnYgJ9RyMRgh1ki7hzepagTFNwFCl6UKE7xuZCK2dhYGbDAXe2t9eLv7l9VMc1weZipIUIZL5IVQaFoesNMr1AHSkDCKPDlQFVEpjEAEo6iQ0ompK6bo+vh6mv5POtUKZxV+fV5uNpfNFMgxOSFnhJMaZIr0iJtIsmU3JMH8uhl3pP37L18jq54y50j8gPe6wdfWZRH</latexit>

q1 ∼ e2πi t1 , qi ∼ e2πi(ti−ti−1) , i = 2, · · · , M − 1 , qM ∼ e2πi τ e−2πi tM−1 .

⇒ q ∼ e2πi τ = q1 · · · qM

§ The non-perturbative weight in the path-integral is then: § So that the instanton partition function becomes

which, through a change of variables, takes the form where and

Surface operators as monodromy defects

with Zinst = X

{k,mi}

⇣ e2πi τ k Y

i

e2πi ti mi⌘ Z{k,mi}

e2πi τ k Y

i

e2πi ti mi ti = ηi + τ γi

ramified instanton partition function

Zinst = X

{di}

q di

i

Z{di}

• <latexit sha1_base64="x8KT6ZjhG6hk1HS1HF3zJ5UaquI=">AB+XicdVDLSgNBEJz1GeMr6tHLYBA8hZkgJrkFvHiMYB6QDWF20olDZh/M9AphyUd41ZM38erXePBfnI0RVLRORVU3XV1BopVFxt68ldW19Y3NwlZxe2d3b790cNixcWoktGWsY9MLhAWtImijQg29xIAIAw3dYHqZ+907MFbF0Q3OEhiEYhKpsZICndT1AzWZ+PNhqcwqjDHOc0Jr10wRxqNepXKc8thzJZojUsvfujWKYhRCi1sLbPWYKDTBhUsO86KcWEiGnYgJ9RyMRgh1ki7hzepagTFNwFCl6UKE7xuZCK2dhYGbDAXe2t9eLv7l9VMc1weZipIUIZL5IVQaFoesNMr1AHSkDCKPDlQFVEpjEAEo6iQ0ompK6bo+vh6mv5POtUKZxV+fV5uNpfNFMgxOSFnhJMaZIr0iJtIsmU3JMH8uhl3pP37L18jq54y50j8gPe6wdfWZRH</latexit><latexit sha1_base64="x8KT6ZjhG6hk1HS1HF3zJ5UaquI=">AB+XicdVDLSgNBEJz1GeMr6tHLYBA8hZkgJrkFvHiMYB6QDWF20olDZh/M9AphyUd41ZM38erXePBfnI0RVLRORVU3XV1BopVFxt68ldW19Y3NwlZxe2d3b790cNixcWoktGWsY9MLhAWtImijQg29xIAIAw3dYHqZ+907MFbF0Q3OEhiEYhKpsZICndT1AzWZ+PNhqcwqjDHOc0Jr10wRxqNepXKc8thzJZojUsvfujWKYhRCi1sLbPWYKDTBhUsO86KcWEiGnYgJ9RyMRgh1ki7hzepagTFNwFCl6UKE7xuZCK2dhYGbDAXe2t9eLv7l9VMc1weZipIUIZL5IVQaFoesNMr1AHSkDCKPDlQFVEpjEAEo6iQ0ompK6bo+vh6mv5POtUKZxV+fV5uNpfNFMgxOSFnhJMaZIr0iJtIsmU3JMH8uhl3pP37L18jq54y50j8gPe6wdfWZRH</latexit><latexit sha1_base64="x8KT6ZjhG6hk1HS1HF3zJ5UaquI=">AB+XicdVDLSgNBEJz1GeMr6tHLYBA8hZkgJrkFvHiMYB6QDWF20olDZh/M9AphyUd41ZM38erXePBfnI0RVLRORVU3XV1BopVFxt68ldW19Y3NwlZxe2d3b790cNixcWoktGWsY9MLhAWtImijQg29xIAIAw3dYHqZ+907MFbF0Q3OEhiEYhKpsZICndT1AzWZ+PNhqcwqjDHOc0Jr10wRxqNepXKc8thzJZojUsvfujWKYhRCi1sLbPWYKDTBhUsO86KcWEiGnYgJ9RyMRgh1ki7hzepagTFNwFCl6UKE7xuZCK2dhYGbDAXe2t9eLv7l9VMc1weZipIUIZL5IVQaFoesNMr1AHSkDCKPDlQFVEpjEAEo6iQ0ompK6bo+vh6mv5POtUKZxV+fV5uNpfNFMgxOSFnhJMaZIr0iJtIsmU3JMH8uhl3pP37L18jq54y50j8gPe6wdfWZRH</latexit><latexit sha1_base64="x8KT6ZjhG6hk1HS1HF3zJ5UaquI=">AB+XicdVDLSgNBEJz1GeMr6tHLYBA8hZkgJrkFvHiMYB6QDWF20olDZh/M9AphyUd41ZM38erXePBfnI0RVLRORVU3XV1BopVFxt68ldW19Y3NwlZxe2d3b790cNixcWoktGWsY9MLhAWtImijQg29xIAIAw3dYHqZ+907MFbF0Q3OEhiEYhKpsZICndT1AzWZ+PNhqcwqjDHOc0Jr10wRxqNepXKc8thzJZojUsvfujWKYhRCi1sLbPWYKDTBhUsO86KcWEiGnYgJ9RyMRgh1ki7hzepagTFNwFCl6UKE7xuZCK2dhYGbDAXe2t9eLv7l9VMc1weZipIUIZL5IVQaFoesNMr1AHSkDCKPDlQFVEpjEAEo6iQ0ompK6bo+vh6mv5POtUKZxV+fV5uNpfNFMgxOSFnhJMaZIr0iJtIsmU3JMH8uhl3pP37L18jq54y50j8gPe6wdfWZRH</latexit>

di ∈ N

q1 ∼ e2πi t1 , qi ∼ e2πi(ti−ti−1) , i = 2, · · · , M − 1 , qM ∼ e2πi τ e−2πi tM−1 .

⇒ q ∼ e2πi τ = q1 · · · qM

§ Like the usual instanton partition function, also the ramified instanton partition function can be computed using localization § The idea is to consider the orbifold

in the presence of an Omega-background with parameters

and to regulate the divergences and localize § Introducing v.e.v’s for the adjoint scalars and going to a generic point in the Coulomb branch, we have

Ramified instanton partition function

(Kanno, Tachikawa ‘11)

R4 D ✏1

✏2

C✏1 × C✏2 × C ZM × C × C

(Nekrasov ’02,…)

Zinst = X

{di}

q di

i

Z{di}(a, ✏1, ✏2)

explicitly calculable

a

§ From we obtain § The prepotential F describes the effective 4d dynamics § The (twisted) superpotential W accounts for the effective dynamics on the 2d defect

Ramified instanton partition function

(twisted) superpotential

Zinst

log(1 + Zinst) = −Finst ✏1✏2 + Winst ✏1 + · · ·

prepotential

§ From we obtain § The prepotential F describes the effective 4d dynamics § The (twisted) superpotential W accounts for the effective dynamics on the 2d defect § Indeed

Ramified instanton partition function

(twisted) superpotential

Zinst

log(1 + Zinst) = −Finst ✏1✏2 + Winst ✏1 + · · ·

prepotential

1 ✏1✏2 ∼ Svol

• C2

✏1✏2

• ∼

Z d4x d2✓1 d2✓2 1 ✏1 ∼ Svol

• C✏1
• ∼

Z d2x d✓1 d¯ ✓1

§ From we obtain § The prepotential F describes the effective 4d dynamics § The (twisted) superpotential W accounts for the effective dynamics on the 2d defect § For example, in the case , we have

Ramified instanton partition function

(twisted) superpotential

Zinst

log(1 + Zinst) = −Finst ✏1✏2 + Winst ✏1 + · · ·

prepotential

SU(2) → S ⇥ U(1) × U(1) ⇤

hφi = diag(a, a)

Winst = q1 2a − q2 2a + · · ·

Zinst = 1 + q1 ✏1(2a + ✏1 + ✏2) + q2 ✏1(−2a + ✏1 + ✏2) + · · ·

§ From we obtain § The prepotential F describes the effective 4d dynamics § The (twisted) superpotential W accounts for the effective dynamics on the 2d defect

Ramified instanton partition function

(twisted) superpotential

Zinst

log(1 + Zinst) = −Finst ✏1✏2 + Winst ✏1 + · · ·

prepotential

Can we calculate W in a different way and obtain some physical intuition ?

§ One considers a 4d gauge theory with group G=SU(N) § One then couples it with a (2,2) 2d GLSM with G as a global symmetry § Simplest example: CP1 2d σ – model with SU(2) flavour symmetry

Surface defect as a coupled 2d/4d system

(Gukov+Witten ’06, … Gadde+Gukov ‘13, Gaiotto+Gukov+Seiberg ‘13, ...)

§ One considers a 4d gauge theory with group G=SU(N) § One then couples it with a (2,2) 2d GLSM with G as a global symmetry § Simplest example: CP1 2d σ – model with SU(2) flavour symmetry 2d U(1) gauge theory with a chiral field in the fundamental of SU(2)

Surface defect as a coupled 2d/4d system

(Gukov+Witten ’06, … Gadde+Gukov ‘13, Gaiotto+Gukov+Seiberg ‘13, ...)

§ One considers a 4d gauge theory with group G=SU(N) § One then couples it with a (2,2) 2d GLSM with G as a global symmetry § Simplest example: CP1 2d σ – model with SU(2) flavour symmetry 2d U(1) gauge theory with a chiral field in the fundamental of SU(2)

Surface defect as a coupled 2d/4d system

(Gukov+Witten ’06, … Gadde+Gukov ‘13, Gaiotto+Gukov+Seiberg ‘13, ...)

2d U(1) gauge theory

§ One considers a 4d gauge theory with group G=SU(N) § One then couples it with a (2,2) 2d GLSM with G as a global symmetry § Simplest example: CP1 2d σ – model with SU(2) flavour symmetry 2d U(1) gauge theory with a chiral field in the fundamental of SU(2)

Surface defect as a coupled 2d/4d system

(Gukov+Witten ’06, … Gadde+Gukov ‘13, Gaiotto+Gukov+Seiberg ‘13, ...)

2d U(1) gauge theory 4d SU(2) gauge theory

The coupled 2d/4d system

§ Understanding the SU(2) node just as a flavour symmetry, we can write the effective twisted superpotential W for the CP1 model as

(d’Adda+Di Vecchia+ ...’82)

W = 2πi t σ − 2 h σ ⇣ log σ µ − 1 ⌘i

§ Understanding the SU(2) node just as a flavour symmetry, we can write the effective twisted superpotential W for the CP1 model as This is a purely 2d point of view

The coupled 2d/4d system

(d’Adda+Di Vecchia+ ...’82)

W = 2πi t σ − 2 h σ ⇣ log σ µ − 1 ⌘i

FI coupling twisted U(1) chiral superfield 2 because of the SU(2) flavour symmetry UV scale

The coupled 2d/4d system

§ Understanding the SU(2) node just as a flavour symmetry, we can write the effective twisted superpotential W for the CP1 model as § Understanding the SU(2) node as a 4d theory, and giving a vev to the adjoint scalar φ, we introduce a twisted mass for σ and W becomes

(d’Adda+Di Vecchia+ ...’82)

W = 2πi t σ − 2 h σ ⇣ log σ µ − 1 ⌘i

W = 2πi t σ Tr h (σ hφi) ⇣ log σ hφi µ 1 ⌘i

(Hanany+Hori ‘97)

§ Understanding the SU(2) node just as a flavour symmetry, we can write the effective twisted superpotential W for the CP1 model as § Understanding the SU(2) node as a 4d theory, and giving a vev to the adjoint scalar φ, we introduce a twisted mass for σ and W becomes The 4d theory is treated only classically

The coupled 2d/4d system

(d’Adda+Di Vecchia+ ...’82)

W = 2πi t σ − 2 h σ ⇣ log σ µ − 1 ⌘i

W = 2πi t σ Tr h (σ hφi) ⇣ log σ hφi µ 1 ⌘i

(Hanany+Hori ‘97)

4d vev’s as twisted masses for σ

The coupled 2d/4d system

§ Understanding the SU(2) node just as a flavour symmetry, we can write the effective twisted superpotential W for the CP1 model as § Understanding the SU(2) node as a 4d theory, and giving a vev to the adjoint scalar φ, we introduce a twisted mass for σ and W becomes § Taking into account the quantum fluctuations of the 4d SU(2) theory, we finally get

(d’Adda+Di Vecchia+ ...’82)

W = 2πi t σ − 2 h σ ⇣ log σ µ − 1 ⌘i

W = 2πi t σ Tr h (σ hφi) ⇣ log σ hφi µ 1 ⌘i

(Hanany+Hori ‘97)

W = 2πi t σ − D Tr h (σ − φ) ⇣ log σ − φ µ − 1 ⌘iE

(Gaiotto+Gukov+Seiberg ‘13, ….)

The coupled 2d/4d system

§ This 2d/4d quiver describes a 4d SU(2) theory coupled with a 2d U(1) theory inside. § Thus, it should describe the defect corresponding to and to the partition . Can we make a quantitative check ?

SU(2) → S[U(1) × U(1)]

~ n = [1, 1]

The coupled 2d/4d system

§ This 2d/4d quiver describes a 4d SU(2) theory coupled with a 2d U(1) theory inside. § Thus, it should describe the defect corresponding to and to the partition . Can we make a quantitative check ?

YES !

SU(2) → S[U(1) × U(1)]

~ n = [1, 1]

§ Consider the twisted superpotential § The 2d vacuum is determined by

The coupled 2d/4d system

W = 2πi t σ − D Tr h (σ − φ) ⇣ log σ − φ µ − 1 ⌘iE = − D Tr h (σ − φ) ⇣ log σ − φ Λ1 − 1 ⌘iE

2d dynamically generated scale Λ2

1 = e2πi t µ2

exp ⇣∂W ∂σ ⌘ = 1 ⇐ ⇒ exp D Tr h log σ − φ Λ1 iE = 1

§ Consider the twisted superpotential § The 2d vacuum is determined by § This equation can be solved using the Seiberg-Witten curve of the 4d SU(2) theory

The coupled 2d/4d system

W = 2πi t σ − D Tr h (σ − φ) ⇣ log σ − φ µ − 1 ⌘iE = − D Tr h (σ − φ) ⇣ log σ − φ Λ1 − 1 ⌘iE

2d dynamically generated scale Λ2

1 = e2πi t µ2

y2 = P2(x)2 − 4Λ4

4d dynamically generated scale

P2(x) = x2 1 2hTrφ2i

Λ4 = e2πi τ µ4

characteristic polynomial of SW

exp ⇣∂W ∂σ ⌘ = 1 ⇐ ⇒ exp D Tr h log σ − φ Λ1 iE = 1

§ Consider the twisted superpotential § The 2d vacuum is determined by § This equation can be solved using the Seiberg-Witten curve of the 4d SU(2) theory

The coupled 2d/4d system

W = 2πi t σ − D Tr h (σ − φ) ⇣ log σ − φ µ − 1 ⌘iE = − D Tr h (σ − φ) ⇣ log σ − φ Λ1 − 1 ⌘iE

2d dynamically generated scale Λ2

1 = e2πi t µ2

y2 = P2(x)2 − 4Λ4

4d dynamically generated scale

P2(x) = x2 1 2hTrφ2i

Λ4 = e2πi τ µ4

characteristic polynomial of SW

exp ⇣∂W ∂σ ⌘ = 1 ⇐ ⇒ exp D Tr h log σ − φ Λ1 iE = 1

§ From the Seiberg-Witten relation , one gets § With simple manipulations, the vacuum equation becomes which is solved by

The coupled 2d/4d system

σ? = s 1 2 ⌦ Tr φ2↵ + Λ2

1 + Λ4

Λ2

1

= a + 1 2a ⇣ Λ2

1 + Λ4

Λ2

1

⌘ + · · ·

v.e.v. of φ instanton terms D Tr 1 x − φ E = P 0

2(x)

y

exp ⇣∂W ∂σ ⌘ = 1 ⇐ ⇒ P2(σ) = Λ2

1 + Λ4

Λ2

1

D Tr h log σ − φ Λ1 iE = Z dσ D Tr h 1 σ − φ iE = · · · = log ⇣P2(σ) + p P2(σ)2 − 4Λ4 2Λ2

1

⌘

§ Evaluating W on the vacuum solution σ* , we find § Notice that and thus

The coupled 2d/4d system

W(σ?)

• inst = 1

2a ⇣ Λ2

1 − Λ4

Λ2

1

⌘ + · · ·

Λ2

1 ∼ e2πi t → q1

, Λ4 Λ2

1

∼ e2πi τ e−2πi t → q q1 = q2

W(σ?)

• inst = q1

2a − q2 2a + · · ·

§ Evaluating W on the vacuum solution σ* , we find § Notice that and thus § Furthermore, we see that

The coupled 2d/4d system

W(σ?)

• inst = 1

2a ⇣ Λ2

1 − Λ4

Λ2

1

⌘ + · · ·

Λ2

1 ∼ e2πi t → q1

, Λ4 Λ2

1

∼ e2πi τ e−2πi t → q q1 = q2

W(σ?)

• inst = q1

2a − q2 2a + · · ·

Λ1 Λ

§ Evaluating W on the vacuum solution σ* , we find § Notice that and thus This exactly matches the twisted superpotential due to ramified instantons !!

The coupled 2d/4d system

W(σ?)

• inst = 1

2a ⇣ Λ2

1 − Λ4

Λ2

1

⌘ + · · ·

Λ2

1 ∼ e2πi t → q1

, Λ4 Λ2

1

∼ e2πi τ e−2πi t → q q1 = q2

W(σ?)

• inst = q1

2a − q2 2a + · · ·

ramified instanton counting parameters qi

Recap

monodromy defects coupled 2d-4d systems

4d 2d 2d 4d Winst

Winst

from localization from vacuum equations dynamically generated scales Λi Winst

• loc = W(σ)
• vacuum

§ The quiver which describes the general defect in the SU(N) theory is § This quiver corresponds to the monodromy defect with

General case

bi-fundamental 2d chiral matter 2d U(ri) gauge theories 4d SU(N) flavor theory

~ n = [n1, · · · , nM]

ri =

i

X

j=1

nj

SU(N) → S ⇥ U(n1) × U(n2) × · · · × U(nM) ⇤

§ The quiver which describes the general defect in the SU(N) theory is § Given the structure of the quiver, the twisted superpotential is

General case

bi-fundamental 2d chiral matter 2d U(ri) gauge theories 4d SU(N) flavor theory

$(x) = x log x µ − 1

• <latexit sha1_base64="jBKiw8DbrfPmJVjwSQMqGf+AGfM=">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</latexit><latexit sha1_base64="jBKiw8DbrfPmJVjwSQMqGf+AGfM=">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</latexit><latexit sha1_base64="jBKiw8DbrfPmJVjwSQMqGf+AGfM=">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</latexit><latexit sha1_base64="jBKiw8DbrfPmJVjwSQMqGf+AGfM=">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</latexit>

W = 2⇡i

M−1

X

i=1 ri

X

s=1

ti (i)

s

−

M−2

X

i=1 ri

X

s=1 ri+1

X

t=1

$((i)

s

− (i+1)

t

) −

rM−1

X

s=1

D Tr $((M−1)

s

− ) E

§ The non-perturbative contributions to W are computed from the vacuum equations and exactly match those obtained from the localization approach!

General case

exp ⇣ ∂W ∂σ(i)

s

⌘ = 1

§ The non-perturbative contributions to W are computed from the vacuum equations and exactly match those obtained from the localization approach! Technical but important point § The ramified instanton partition function is given in terms of an multiple integral; for example at 1-instanton we have § Thus we have to specify the contour of integration, or equivalently specify which poles contribute to the integral

General case

Z1 = −

M

X

i=1

qi ✏1 Z di 2⇡i

ni

Y

s=1

1

• as − i + ✏1+✏2

2

• ni+1

Y

t=1

1

• i − at + ✏1+✏2

2

• <latexit sha1_base64="DUS02uORWX9QBYkudf4Lj/xoQ=">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</latexit><latexit sha1_base64="hgyfN9Iaj/oe2CpgvDN+hBejbJ0=">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</latexit><latexit sha1_base64="hgyfN9Iaj/oe2CpgvDN+hBejbJ0=">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</latexit><latexit sha1_base64="9OsjuY+YDbpvqsWRi0UrlTSl0s0=">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</latexit>

exp ⇣ ∂W ∂σ(i)

s

⌘ = 1

§ The precise match is obtained by giving a positive imaginary part to the ε parameters and selecting the contribution from the following poles:

Integration contour

i = as + ✏1 + ✏2 2 s = 1 · · · , ni , i = 1, · · · , M − 1

M = at − ✏1 + ✏2 2 t = 1 · · · , n1

(Gorsky+LeFloch+... ‘17 Ashok+Billo+... ‘17,’ 18)

§ The precise match is obtained by giving a positive imaginary part to the ε parameters and selecting the contribution from the following poles:

Integration contour

a2 − ✏1 + ✏2 2

a1 − ✏1 + ✏2 2

a2 + ✏1 + ✏2 2

a1 + ✏1 + ✏2 2

χ1, · · · , χM−1

i = as + ✏1 + ✏2 2 s = 1 · · · , ni , i = 1, · · · , M − 1

M = at − ✏1 + ✏2 2 t = 1 · · · , n1

(Gorsky+LeFloch+... ‘17 Ashok+Billo+... ‘17,’ 18)

§ The precise match is obtained by giving a positive imaginary part to the ε parameters and selecting the contribution from the following poles:

Integration contour

a2 − ✏1 + ✏2 2

a1 − ✏1 + ✏2 2

a2 + ✏1 + ✏2 2

a1 + ✏1 + ✏2 2

i = as + ✏1 + ✏2 2 s = 1 · · · , ni , i = 1, · · · , M − 1

M = at − ✏1 + ✏2 2 t = 1 · · · , n1

χM

(Gorsky+LeFloch+... ‘17 Ashok+Billo+... ‘17,’ 18)

χ1

χ2

χM−1

upper upper upper

Integration contour

χM

lower

§ This integration prescription can be elegantly specified with a Jeffrey-Kirwan vector η given by χM

χ1

χ2

χM−1

upper upper upper lower

Integration contour

+ ξM χM

η = − χ1 − χ2 · · · − χM−1

(ξM 1)

§ This integration prescription can be elegantly specified with a Jeffrey-Kirwan vector η given by χM

χ1

χ2

χM−1

upper upper upper lower

Integration contour

+ ξM χM

• sign of the beta-function for the FI couplings of the U(ri)

η = − χ1 − χ2 · · · − χM−1

(ξM 1)

§ This integration prescription can be elegantly specified with a Jeffrey-Kirwan vector η given by χ1

χ2

χM−1

upper upper upper

Integration contour

+ ξM χM

FI couplings of the U(ri) nodes of the quiver

• sign of the beta-function for the FI couplings of the U(ri)

η = − ξ1 χ1 − ξ2 χ2 · · · − ξM−1 χM−1

ξM

ξ1 ⌧ ξ2 ⌧ · · · ⌧ ξM−1 ⌧

lower

χM

Λ1 Λ2 · · · ΛM−1

Λ

↔

Duality

§ Consider a 2d U(r) theory with NF fundamentals and NA ( < NF ) anti-fundamentals § This theory is dual to a 2d U(NF – r) theory with the roles of fundamentals and anti-fundamentals reversed

Duality

(Seiberg ‘94)

+ superpotential term

A) B)

W A

class = 2πi t Trσ

→ W B

class = −2πi t Trσ + 2πi t NF

X

f=1

mf

§ We can apply this duality rule to the quiver representing a surface defect

Duality

Wclass = · · · + 2πi

• t1 Tr σ1 − t Tr σ + (t2 + t) Tr σ2

+ · · ·

Wclass = · · · + 2πi

• t1 Tr σ1 + t Tr σ + t2 Tr σ2

+ · · ·

Wclass = · · · + 2πi

• (t1 + t) Tr σ1 − t Tr σ + (t2 + t) Tr σ2

+ · · ·

§ We can apply this duality rule to the quiver representing a surface defect

Duality

Wclass = · · · + 2πi

• t1 Tr σ1 + t Tr σ + t2 Tr σ2

+ · · ·

All these different quiver theories provide alternative descriptions of the same surface operator corresponding to

SU(N) ↓ S ⇥ U(n1) × U(n2) × U(n3) × U(n4) ⇤

Wclass = 2πi

• t1 Tr σ1 + t2 Tr σ2 + t3 Tr σ3

Wclass = 2πi

• − t1 Tr σ1 + (t1 + t2) Tr σ2 + t3 Tr σ3

Wclass = 2πi

• t1 Tr σ1 + t2 Tr σ2 + t3 Tr σ3

Wclass = 2πi

• − t1 Tr σ1 + (t1 + t2) Tr σ2 + t3 Tr σ3

Wclass = 2πi

• t1 Tr σ1 + t2 Tr σ2 + t3 Tr σ3

Wclass = 2πi

• t3 Tr σ1 − (t2 + t3) Tr σ2 − t1 Tr σ3

Wclass = 2πi

• − t3 Tr σ1 − t2 Tr σ2 − t1 Tr σ3

§ For any of these quiver theories we can work out the q vs Λ map just as we did for the simplest quiver § For example, after one duality step we find Note that the exponents are the (absolute values of the) beta-functions for

the FI couplings ξi , and 2N is the beta-function of the 4d SU(N) theory.

Duality

q1 ∼ Λn1+n2

1

, q2 ∼ Λn1+2n2+n3

2

Λn1+n2

1

, q3 ∼ Λn3+n4

3

, q4 ∼ Λ2N q1q2q3

§ For any of these quiver theories we can work out the q vs Λ map just as we did for the simplest quiver § For example, after one duality step we find § Moreover, the sign change in the FI term of the first node leads to consider a new Jeffrey-Kirwan vector

Duality

q1 ∼ Λn1+n2

1

, q2 ∼ Λn1+2n2+n3

2

Λn1+n2

1

, q3 ∼ Λn3+n4

3

, q4 ∼ Λ2N q1q2q3

Wclass = 2πi

• − t1 Tr σ1 + (t1 + t2) Tr σ2 + t3 Tr σ3

η = + ξ1 χ1 − ξ2 χ2 · · · − ξM−1 χM−1

+ ξM χM

χ1 is now integrated in the lower plane !

§ Solving the vacuum equations leads again to a perfect match (term by term) with the superpotential of ramified instantons with the new JK parameter! § The same is true for any other quiver diagram of the sequence we considered

Duality

exp ⇣∂W ∂σ ⌘ = 1

§ Solving the vacuum equations leads again to a perfect match (term by term) with the superpotential of ramified instantons with the new JK parameter! § The same is true for any other quiver diagram of the sequence we considered, and also in sequences with non-linear quivers

Duality

exp ⇣∂W ∂σ ⌘ = 1

§ Starting from the same localization integrand and choosing different integration contours (or different Jeffrey-Kirwan vectors) we obtain a perfect match with different quiver theories related to each other by duality transformations

Duality

M

Y

i=1

I di 2⇡i qdi

i

di! Zdi(a, ✏1, ✏2)

JK1 JK2 JK3

Conclusions

§ [n1 , n2 , … , nM ] § 4d v.e.v’s of SU(N) § Ramified instanton counting parameters qi § Winst from localization § Integration contour

Summary of results

monodromy defects coupled 2d-4d systems

§ Ranks of gauge groups § 2d twisted masses § Dynamically generates scales Λi § Winst from vacuum eqs. § Duality frame

§ Similar analysis can be done for other type of theories

• N=2* theories
• SQCD’s (with matter in fundamental representation)
• Chern-Simons like theories in 3d/5d
• Theories with other groups (orthogonal, symplectic,…)

§ Explore the connection with integrable models § ...

Conclusions

Thanks a lot for your attention!