euro-strings,2011 padova · observablesin4d and2d theories (reviewof the agtrelation) nadav drukker...

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Observables in 4d and 2d theories

(review of the AGT relation)

Nadav Drukker

Euro-strings, 2011

Padova

September 5, 2011

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Introduction to the introduction

• Almost four years ago Pestun published his calculation of the expectation value of

BPS Wilson loop operators in N = 2 gauge theories on S4 using localization.

• Two years ago Alday, Gaiotto and Tachikawa reinterpreted his results

Nadav Drukker 2 AGT review

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The partition function of theories with

SU(2) gauge symmetry on S4 is the same

as a correlation function in Liouville CFT

Nadav Drukker 2-a AGT review

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The partition function of theories with

SU(2) gauge symmetry on S4 is the same

as a correlation function in Liouville CFT

• This realization spawned several new lines of study.

• In this talk I will review the original papers and explain some of the new

developments.

Nadav Drukker 2-b AGT review

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Outline

• Localization on S4

– Partition functions

• Statement of the AGT relation

• Generalizations:

– Gauge groups

– Observables I:

∗ Wilson and ’t Hooft loops operators

∗ Domain walls (and 3d theories)

– Observables II:

∗ Surface operators

– Yet more

• Discussion


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Localization on S4

4d theories on S4

• Consider a 4d gauge theory with N = 2 supersymmetry:

– Vector multiplets.

– Hyper multiplets.

• If the theory is conformal, there is a canonical definition of this theory on S4.

• For non-conformal theories, it is still possible to define them on S4 with symmetry

OSp(2|4).


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Localization on S4

4d theories on S4






OSp(2|4).

• We would like to calculate the partition function for such theories.

• More generally, the expectation values of observables.


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Localization on S4

4d theories on S4






OSp(2|4).

• We would like to calculate the partition function for such theories.

• More generally, the expectation values of observables.

• Crucial point: We require globally preserved SUSY: BPS observables

• The formalism needs the closure of SUSY off-shell. That allows us to calculate the

exact path integral.


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• Write S4 in stereographic coordinates

ds2 =R2 dx2

(R2 + x2)2x1 + ix2 = r1 e

iϕ1 x3 + ix4 = r2 eiϕ2

• We choose one supercharge Q which annihilates the vacuum and all the observables

we study.

• Near the north/south pole

Q = Q± S/R


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ds2 =R2 dx2

(R2 + x2)2x1 + ix2 = r1 e



we study.


Q = Q± S/R

• The specific choice has

Q2 = Lv +R+ gauge

with

v =∂

∂ϕ1+

∂

∂ϕ2


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ds2 =R2 dx2

(R2 + x2)2x1 + ix2 = r1 e



we study.


Q = Q± S/R

• The specific choice has

Q2 = Lv +R+ gauge

with

v =∂

∂ϕ1+

∂

∂ϕ2

• Fixed loci:

– 3d surfaces: r2 = const are invariant.

– 2d surfaces r2 = 0 are invariant

– Curves like r1 = const, r2 = 0 are fixed lines.

– Points at r1 = r2 = 0 and r1 = r2 = ∞.


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Localization

• Choose a fermionic potential V and deform the action

S → S + tQV

• The t dependance of the partition function is

∂t Z =

∫

QV eS+tQV


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Localization


S → S + tQV


∂t Z =

∫

QV eS+tQV

• Integrating by parts: Q commutes with the action, measure (and extra possible

insertions), but acts on tQV .

• It gives and insertion of tV LvV , so we need to require this to vanish.


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Localization


S → S + tQV


∂t Z =

∫

QV eS+tQV

• Integrating by parts: Q commutes with the action, measure (and extra possible

insertions), but acts on tQV .

• It gives and insertion of tV LvV , so we need to require this to vanish.

• Consider the limit t→ ∞. If QV is positive definite, then the path integral localizes

to the saddle points of QV .

• If we chose V well, the result is a finite dimensional integral.


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• For the vector multiplet we take V = ψQψ.

• the bosonic part of the localizing action is |Qψ|2.

• Need to solve Qψ = 0.


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• Scalar fields: Dφ = 0 leads to a constant real (or in Pestun conventions imaginary)

field a.

• Vector fields: Bi +Ei cos θ = 0:

⋆ F+ = 0 at north pole: Anti-instantons.

⋆ F− = 0 at south pole: Instantons.


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• Scalar fields: Dφ = 0 leads to a constant real (or in Pestun conventions imaginary)

field a.

• Vector fields: Bi +Ei cos θ = 0:

⋆ F+ = 0 at north pole: Anti-instantons.

⋆ F− = 0 at south pole: Instantons.

• Outcome:

Z =

∫

da |Zinst(a,mi)|2∆(a)2Z1-loop(a,mi)Zcl(a)

W =

∫

da |Zinst(a,mi)|2∆(a)2Z1-loop(a,mi)Zcl(a) TrR(e

a)


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Zs:

• The classical contribution comes from the scalar field

Zcl(a) = e−a2/g2

• The one loop part comes from evaluating the determinant of the localizing action QV .

– Vector multiplets: Off diagonal entries will give terms dependent on ai − aj :

∏

roots α

1

ΓB(α · a)2

– Fundamental hyper multiplets:∏

weights h

ΓB(h(a) +m)2

– Bi-fundamental hyper multiplets:∏

weights h, h′

ΓB(h(a)− h′(b) +m)2


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Zs:


Zcl(a) = e−a2/g2



∏

roots α

1

ΓB(α · a)2


weights h

ΓB(h(a) +m)2


weights h, h′

ΓB(h(a)− h′(b) +m)2

• Each term depends on one or two Coulomb branch parameter and one mass.

• They are “local” on a quiver diagram (see later).


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Zs:


Zcl(a) = e−a2/g2



∏

roots α

1

ΓB(α · a)2


weights h

ΓB(h(a) +m)2


weights h, h′

ΓB(h(a)− h′(b) +m)2

• Each term depends on one or two Coulomb branch parameter and one mass.

• They are “local” on a quiver diagram (see later).

• Same functions appear in the Liouville 3-point function!


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Instantons

• The instanton part is more complicated and depends on everything.

• We need to sum over all possible zero-size (anti)instantons at the fixed points: North

and south poles.


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Instantons



and south poles.

• Need a regularization

• Would like to use results for instantons in flat R4.


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Instantons



and south poles.

• Need a regularization

• Would like to use results for instantons in flat R4.

• Consider 5d Yang-Mills on S1 of radius β.

• Identify the R4 up to a rotation by Ω = eβε1L12+βε2L34 ,

Do a gauge rotation g = diag(eβa1 , · · · eβaN ).

• Take β → 0.

• Results in a Gaussian potential, restricts to fields excited around the origin.

• Gives a natural compactification of the instanton moduli space.


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Localize that

• Still need to calculate in the Omega background

Zinst(a, ...) =∑

k

qk∫

MN,k

something q = e2πiτ , τ =4πi

g2+

θ

2π

• Impose SUSY, so sum only over instantons invariant under Q (in a topologically

twisted theory).

• In particular invariant under Q2 = Lv + · · ·, so zero size instantons at the origin

Zinst(a, ...) =∑

k

qk∑

p∈MN,k

something

• The gauge moduli localize too, leading to a sum of N Young diagmars with a total of

k boxes.

Zinst(a, ...) =∑

~Y

q|~Y | × something


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Localize that

• Still need to calculate in the Omega background

Zinst(a, ...) =∑

k

qk∫

MN,k

something q = e2πiτ , τ =4πi

g2+

θ

2π

• Impose SUSY, so sum only over instantons invariant under Q (in a topologically

twisted theory).

• In particular invariant under Q2 = Lv + · · ·, so zero size instantons at the origin

Zinst(a, ...) =∑

k

qk∑

p∈MN,k

something

• The gauge moduli localize too, leading to a sum of N Young diagmars with a total of

k boxes.

Zinst(a, ...) =∑

~Y

q|~Y | × something

• Main point: A bit complicated, but very explicit and algorithmical.


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Back to S4

• Need instantons from the south pole and anti-instantons from the north pole.

• Should be invariant under Q = Q+ S/R.

• This is not a symmetry of massive N = 2 theories on R4.


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Back to S4




• Observation: After a field redefinition the theory and symmetry on S4 near the pole is

equivalent to the topological theory on the Omega-background (with ǫ=ǫ2 = 1/R).

• Can use the results for the topologically twisted N = 2 theories on the Omega

background.


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Back to S4




• Observation: After a field redefinition the theory and symmetry on S4 near the pole is

equivalent to the topological theory on the Omega-background (with ǫ=ǫ2 = 1/R).

• Can use the results for the topologically twisted N = 2 theories on the Omega

background.

upshot:

• Localization led to an explicit finite dimensional expression for the partition function

and Wilson loop in quite general N = 2 gauge theories.

• Easy to solve only for N = 4, where Zinst = Z1-loop = 1.


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Liouville theory

• The action is

S =1

4π

∫

d2z(

gab∂aφ∂bφ+QRφ+ 4πµe2bφ)

,

• Q = b+ 1/b and the Liouville central charge is c = 1 + 6Q2.

• The vertex operators are labeled by α = Q/2 + ia

Vα(z, z) ≃ e2αφ(z,z)

They have conformal dimension ∆(α) = α(Q− α).


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Liouville theory

• The action is

S =1

4π

∫

d2z(

gab∂aφ∂bφ+QRφ+ 4πµe2bφ)

,

• Q = b+ 1/b and the Liouville central charge is c = 1 + 6Q2.

• The vertex operators are labeled by α = Q/2 + ia

Vα(z, z) ≃ e2αφ(z,z)

They have conformal dimension ∆(α) = α(Q− α).

• The three point function is given by the DOZZ formula

C(α1, α2, α3) =

(

πµΓ(b2)

Γ(1− b2)b2−2b2

)

1

b(Q−α1−α2−α3)

×Υ′(0)Υ(2α1)Υ(2α2)Υ(2α3)

Υ(α1 + α2 + α3 −Q)Υ(α1 + α2 − α3)Υ(α1 − α2 + α3)Υ(−α1 + α2 + α3)

Υ(x) =1

ΓB(x)ΓB(Q− x)


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• The four point function on the sphere is given by all intermediate states.

• We can sum over all primaries and account for the descendants by the conformal

blocks

µ1

µ2 µ3

µ4

α

〈Vµ1Vµ2

Vµ3Vµ4

〉 =

∫

dα∑

~m,~n

q|~n|〈Vµ1Vµ2

L~nVα〉K−1〈L~nVαVµ3

Vµ4〉

=

∫

dα |F(α, µ1, µ2, µ3, µ4, q)|2C(µ1, µ2, α)C(α, µ, µ4)

• This can be done since〈Vµ1

Vµ2L~nVα〉

〈Vµ1Vµ2

Vα〉is a known function of the dimensions and central

charge.


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The AGT relation

• For SU(2) gauge theory with NF = 4, the S4 partition function is the Liouville

4-point function.

• In particular, up to simple rearrangements:

Zinst = F

Z1-loop = C(µ1, µ2, µ3)

• α = Q/2 + ia, similar relation between µ and m.


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The AGT relation

• For SU(2) gauge theory with NF = 4, the S4 partition function is the Liouville

4-point function.

• In particular, up to simple rearrangements:

Zinst = F

Z1-loop = C(µ1, µ2, µ3)

• α = Q/2 + ia, similar relation between µ and m.

• more generally, can take any quiver:

2 2 2 2 2

µ1

µ2µ3 µ4

µ5

µ6α1 α2 α3


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Status

• In original paper checked perturbatively in q to high order.[

Alday,Gaiotto,Tachikawa

]

• More checkes[

Marshakov,Mironov2,Morozov2]

,. . .

• For certain examples the recursion relations for the conformal blocks were shown to

apply to the instanton partition functions.[

Fateev,Litvinov

][

Zamolochikov

][

Poghossian

][

Hadasz,Jaskolski,Suchanek

]

• Taking an extension of the Virasoro algebra led to a structure very similar to the

Young-diagrams for the instanton functions.[

Alba,Fateev,Litvinov,Tarnopolsky

]


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Status

• In original paper checked perturbatively in q to high order.[

Alday,Gaiotto,Tachikawa

]

• More checkes[

Marshakov,Mironov2,Morozov2]

,. . .

• For certain examples the recursion relations for the conformal blocks were shown to

apply to the instanton partition functions.[

Fateev,Litvinov

][

Zamolochikov

][

Poghossian

][

Hadasz,Jaskolski,Suchanek

]

• Taking an extension of the Virasoro algebra led to a structure very similar to the

Young-diagrams for the instanton functions.[

Alba,Fateev,Litvinov,Tarnopolsky

]

• Full proof exists (not published yet)[

Maulik,Okounkov

]


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Generalizations

Higher rank groups

• A similar story exists for quivers of SU(N). Liouville is replaced by AN−1 Toda CFT.[

Wyllard

]

• Generic physical states in these theories have N − 1 degrees of freedom matching the

Coulomb branch parameters.

• In the simplest case there are two punctures on the sphere with generic states and all

the rest have semi-degenerate fields with only one parameter (corresponding to the

mass).

• General punctures are quite complicated both in gauge theory and CFT.

• Further generalization to linear quiver tails with decreasing rank also exists within

Toda CFT[

Kanno,Matsuo,Shiba

][

Drukker,Passerini

]


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Loop operators

• Wilson loops were included in the original calculation by[

Pestun

]

.

• They live on the equator of S4.

• The classification of general Wilson and ’t Hooft loop operators is given by curves on

the Riemann surface[

Drukker,Morrison,Okuda

]

• They are calculated by a Verlinde loop operator, or a topological defect in the 2d CFT[

Drukker,GomisOkuda,Teschner

][

Alday,Gaiotto,GukovTachikawa,Verlinde

] [

Petkova

][

Drukker,GaiottoGomis

]

• In the simplest case a Wilosn loop inserts cosh a into the Liouville bootstrap.

• ’t Hooft loops are non-diagonal operators on the conformal blocks leading to sums of

terms of the form F(a)F(a+ 1).


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• Works also for SU(N) and Toda.[

Passerini

][

Gomis,Le Floch

]

• Recently ’t Hooft loops were calculated by localization on S4.[

Gomis,Okuda,Pestun

]

• Result agrees with Liouville/Toda and with S-duality!


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Domain walls

• One can place domain walls on the S3 equator.

• Then they can be invariant under the bosonic and fermionic symmetries.

• We can guess some of them from Liouville/Toda.[


]


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Domain walls





]

• Consider a Janus wall, with one coupling on the north pole and one on the south.

Z =

∫

dν(α) F(α, µ, q′)F(α, µ, q)

• If q and q′ are related by S-duality, we can act on one hemisphere and get a duality

wall

Z =

∫

dν(α′) dν(α) F(α′, µ, q)ZS3

(α′, α)F(α, µ, q)


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Domain walls





]

• Consider a Janus wall, with one coupling on the north pole and one on the south.

Z =

∫

dν(α) F(α, µ, q′)F(α, µ, q)

• If q and q′ are related by S-duality, we can act on one hemisphere and get a duality

wall

Z =

∫

dν(α′) dν(α) F(α′, µ, q)ZS3

(α′, α)F(α, µ, q)

• For T-transformation the resulting 3d theory is Chern-Simons and adds a2 to the

action.

• For S-transformation of N = 2∗ this is T (SU(2)), which indeed gives the correct

Moore-Seiberg kernel.[

Hosomishi,Lee,Park

]


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Aside: 3d theories

• By the same tools one can calculate the partition function and Wilson loop in 3d

theories.[

Kapustin,Willett,Yaakov

]

• The result is much simpler, since there are no instantons. For vectors with a CS term

we get

Z3d =

∫

dµi∏

i<j

sinh2(

µi − µj2

)

e−1

2gs

∑i µ

2

i ,

• matter fields insert1

cosh(

µi−µj

2

)


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Aside: 3d theories


theories.[


]


we get

Z3d =

∫

dµi∏

i<j

sinh2(

µi − µj2

)

e−1

2gs

∑i µ

2

i ,


cosh(

µi−µj

2

)

• In particular for ABJM theory

ZABJM =1

N1!N2!

∫ N1∏

i=1

dµi2π

N2∏

j=1

dνj2π

∏

i<j 4 sinh(

µi−µj

2

)

4 sinh2(

νi−νj

2

)

∏

i,j

(

2 cosh(

µi−νj

2

))2 e−1

2gs(∑

i µ2

i−∑

j ν2

j )

• This matrix model can be solved![

Drukker,Marino,Putrov

]


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Aside: 3d theories


theories.[


]


we get

Z3d =

∫

dµi∏

i<j

sinh2(

µi − µj2

)

e−1

2gs

∑i µ

2

i ,


cosh(

µi−µj

2

)


ZABJM =1

N1!N2!

∫ N1∏

i=1

dµi2π

N2∏

j=1

dνj2π

∏

i<j 4 sinh(

µi−µj

2

)

4 sinh2(

νi−νj

2

)

∏

i,j

(

2 cosh(

µi−νj

2

))2 e−1

2gs(∑

i µ2

i−∑

j ν2

j )



]

• Other ideas:

– Correct R-charge extremizes Z.[

Jafferis

]

– Using the logZ, the S3 free energy as a generalization of the a theorem.[

Jafferis,KlebanovSilviu,Pufu,Safdi

]


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Aside: 3d theories


theories.[


]


we get

Z3d =

∫

dµi∏

i<j

sinh2(

µi − µj2

)

e−1

2gs

∑i µ

2

i ,


cosh(

µi−µj

2

)


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Aside: 3d theories


theories.[


]


we get

Z3d =

∫

dµi∏

i<j

sinh2(

µi − µj2

)

e−1

2gs

∑i µ

2

i ,


cosh(

µi−µj

2

)


ZABJM =1

N1!N2!

∫ N1∏

i=1

dµi2π

N2∏

j=1

dνj2π

∏

i<j 4 sinh(

µi−µj

2

)

4 sinh2(

νi−νj

2

)

∏

i,j

(

2 cosh(

µi−νj

2

))2 e−1

2gs(∑

i µ2

i−∑

j ν2

j )



]


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Aside: 3d theories


theories.[


]


we get

Z3d =

∫

dµi∏

i<j

sinh2(

µi − µj2

)

e−1

2gs

∑i µ

2

i ,


cosh(

µi−µj

2

)


ZABJM =1

N1!N2!

∫ N1∏

i=1

dµi2π

N2∏

j=1

dνj2π

∏

i<j 4 sinh(

µi−µj

2

)

4 sinh2(

νi−νj

2

)

∏

i,j

(

2 cosh(

µi−νj

2

))2 e−1

2gs(∑

i µ2

i−∑

j ν2

j )



]

• Other ideas:

– Correct R-charge extremizes Z (?)[

Jafferis

]

– F = logZ, the S3 free energy is minimized along RG-flow (?)[

Jafferis,KlebanovSilviu,Pufu,Safdi

]


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Other generalizations

b 6= 1

• All the components of the calculation exist also for b 6= 1 (i.e. ǫ1 6= ǫ2).

• The conformal blocks in that case are still instanton partition functions.

• It is not known what the Liouville correlator calculates


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Other generalizations

b 6= 1

• All the components of the calculation exist also for b 6= 1 (i.e. ǫ1 6= ǫ2).

• The conformal blocks in that case are still instanton partition functions.

• It is not known what the Liouville correlator calculates

• In 3d one can deform the measure in the matrix model to

sinh

(

b(µi − µj)

2

)

sinh

(

µi − µj2b

)

And the matter contributions to a double sine function.

• This arises from localizing the 3d theory on a deformed S3.[

Hama,Hosimichi,Lee

]


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Surface operators

• 4d theories have a rich structure of surface operators.[

gukov,Witten

]

• If they wrap an S2 through the poles, they preserve the symmety v and can be BPS.

• Since they pass through the poles, they modify the instanton contribution.


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Surface operators


gukov,Witten

]



• For certain surface operators (with generic holonomies around the singularity) the

conformal blocks are closely related to those of affine SU(N) WZW theories.[

Alday,Tachikawa

]

• Rich generalization to Toda exists.


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Surface operators


gukov,Witten

]



• For certain surface operators (with generic holonomies around the singularity) the

conformal blocks are closely related to those of affine SU(N) WZW theories.[

Alday,Tachikawa

]

• Rich generalization to Toda exists.

• A complete calculation on S4 has not been done.

• Some “fudge factors” needed to get agreement are not understood from the CFT side.


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More

• One can study more modifications of the Instanton partition functions.

• On R2 × (R2/Zk) the instanton partition function is the same as for a surface

operator.[

Kanno, Tachikawa

]

• On R4/Zk the instanton partition function is the same as for supergroup Toda.

[

Nishioka,Tachikawa

]

• On R4/Z2 it’s super Liouville.

[

Bonelli,Maruyoshi,Tanzini

][

Belavin2,Bershtein

]

• Localization on S3 × R using a specific supercharge gives the index of the 4d theory.

This is related to a topological theory in 2d.[

Gadde,PomoniRastelli,Razamat

]


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Summary

• AGT relate the conformal blocks of CFTs and instanton partition functions.

• Moreover in certain cases, S4 partition functions have these as their ingredients to

produce complete CFT correlators.

• Extensive new tests of S-duality.

• Explicit results for Wilson loops, ’t Hooft loops and some domain walls on S4

• Interesting generalizations to 3d.


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Summary

• AGT relate the conformal blocks of CFTs and instanton partition functions.

• Moreover in certain cases, S4 partition functions have these as their ingredients to

produce complete CFT correlators.

• Extensive new tests of S-duality.

• Explicit results for Wilson loops, ’t Hooft loops and some domain walls on S4

• Interesting generalizations to 3d.

• What else can be localized?


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euro-strings,2011 padova · observablesin4d and2d theories (reviewof the agtrelation) nadav drukker...

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