euro-strings,2011 padova · observablesin4d and2d theories (reviewof the agtrelation) nadav drukker...
TRANSCRIPT
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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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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
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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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
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
Nadav Drukker 3 AGT review
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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).
Nadav Drukker 4 AGT review
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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).
• We would like to calculate the partition function for such theories.
• More generally, the expectation values of observables.
Nadav Drukker 4-a AGT review
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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).
• 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.
Nadav Drukker 4-b AGT review
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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
Nadav Drukker 5 AGT review
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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
• The specific choice has
Q2 = Lv +R+ gauge
with
v =∂
∂ϕ1+
∂
∂ϕ2
Nadav Drukker 5-a AGT review
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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
• 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 = ∞.
Nadav Drukker 5-b AGT review
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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
Nadav Drukker 6 AGT review
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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
• 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.
Nadav Drukker 6-a AGT review
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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
• 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.
Nadav Drukker 6-b AGT review
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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.
Nadav Drukker 7 AGT review
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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.
• 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.
Nadav Drukker 7-a AGT review
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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.
• 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)
Nadav Drukker 7-b AGT review
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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
Nadav Drukker 8 AGT review
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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
• Each term depends on one or two Coulomb branch parameter and one mass.
• They are “local” on a quiver diagram (see later).
Nadav Drukker 8-a AGT review
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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
• 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!
Nadav Drukker 8-b AGT review
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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.
Nadav Drukker 9 AGT review
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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.
• Need a regularization
• Would like to use results for instantons in flat R4.
Nadav Drukker 9-a AGT review
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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.
• 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.
Nadav Drukker 9-b AGT review
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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
Nadav Drukker 10 AGT review
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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.
Nadav Drukker 10-a AGT review
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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.
Nadav Drukker 11 AGT review
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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.
• 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.
Nadav Drukker 11-a AGT review
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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.
• 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.
Nadav Drukker 11-b AGT review
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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).
Nadav Drukker 12 AGT review
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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)
Nadav Drukker 12-a AGT review
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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.
Nadav Drukker 13 AGT review
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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.
Nadav Drukker 14 AGT review
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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
Nadav Drukker 14-a AGT review
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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
]
Nadav Drukker 15 AGT review
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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
]
Nadav Drukker 15-a AGT review
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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
]
Nadav Drukker 16 AGT review
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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).
Nadav Drukker 17 AGT review
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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!
Nadav Drukker 18 AGT review
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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.[
Drukker,GaiottoGomis
]
Nadav Drukker 19 AGT review
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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.[
Drukker,GaiottoGomis
]
• 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)
Nadav Drukker 19-a AGT review
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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.[
Drukker,GaiottoGomis
]
• 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
]
Nadav Drukker 19-b AGT review
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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
)
Nadav Drukker 20 AGT review
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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
)
• 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
]
Nadav Drukker 20-a AGT review
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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
)
• 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
]
• 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
]
Nadav Drukker 20-b AGT review
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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
)
Nadav Drukker 21 AGT review
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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
)
• 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
]
Nadav Drukker 21-a AGT review
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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
)
• 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
]
• Other ideas:
– Correct R-charge extremizes Z (?)[
Jafferis
]
– F = logZ, the S3 free energy is minimized along RG-flow (?)[
Jafferis,KlebanovSilviu,Pufu,Safdi
]
Nadav Drukker 21-b AGT review
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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
Nadav Drukker 22 AGT review
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&
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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
]
Nadav Drukker 22-a AGT review
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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.
Nadav Drukker 23 AGT review
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&
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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.
• 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.
Nadav Drukker 23-a AGT review
'
&
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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.
• 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.
Nadav Drukker 23-b AGT review
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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
]
Nadav Drukker 24 AGT review
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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.
Nadav Drukker 25 AGT review
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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?
Nadav Drukker 25-a AGT review
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The end
Nadav Drukker 26 AGT review