Exact Results for 't Hooft Loops in Gauge Theories ... - Solvay Institutes

Introduction Problem Definitions Method Solution Result
Exact Results for ’t Hooft Loops in Gauge
Theories on S 4
Vasily Pestun
based on arXiv:1105.2568 with Jaume Gomis, Takuya Okuda
May 19, 2011
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Introduction Problem Definitions Method Solution Result
Supersymmetry is a powerful tool to address dynamics of
gauge theories
• The N = 2 effective low energy Seiberg-Witten theory
• Maldacena’s conjecture on the AdS/CFT correspondence
for N = 4 SYM
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Introduction Problem Definitions Method Solution Result
Loop operators are important gauge theory observables.
Wilson loop operator measures effective forces between
quarks.
(locally) supersymmetric Wilson loop operator was the key
object in several AdS/CFT studies for N = 4 SYM
• [Erickson-Semenoff-Zarembo] and [Drukker-Gross] circular
1/2 BPS Wilson loops in N = 4 SYM
• [Alday-Maldacena] polygonal Wilson loops and amplitudes
in N = 4 SYM
At λ → ∞, Wilson loop C ⊂ R 4 is dual to the minimal surface in
AdS landing on C ⊂ R 4
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Introduction Problem Definitions Method Solution Result
Motivation
Why we look on supersymmetric loops in N = 2 theories?
• What from N = 4 SYM can be pushed to N = 2?
• How S-duality acts? Can we compute vevs of Wilson and ’t
Hooft and test S-duality precisely?
• The loop observables in N = 2 theories are novel - not
computable from Seiberg-Witten theory - hence we’ll learn
something new about N = 2 dynamics if compute them
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Introduction Problem Definitions Method Solution Result
Wilson operators
1. Define N = 2 theory on S 4 of radius r.
2. Localize the path integral and get [V.P.’07]
〈W R (C)〉 S 4 = 1 ∫
[da]|Z Ω (ia, ɛ 1 , ɛ 2 , m, τ)| 2 tr R e 2πiar
Z S 4
Z Ω – Nekrasov’s partition function of N = 2 twisted theory on
R 4 ɛ 1 ,ɛ 2
[Losev-Moore-Nekrasov-Shatashvili’95,Nekrasov’02]
ɛ 1 = ɛ 2 = 1/r
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Introduction Problem Definitions Method Solution Result
Problem
Compute exact vev of supersymmetric ’t Hooft operator in
N = 2 gauge theories
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Introduction Problem Definitions Method Solution Result
Definitions
Define the N = 2 supersymmetry on S 4 : OSp(2|4) ⊂ SL(1|2, H)
— subgroup of the N = 2 superconformal group with
• 8 fermionic generators
• Sp(4) ≃ SO(5) bosonic subgroup – isometry of S 4
• SO(2) R symmetry
All zero modes in the OSp(2|4) theory on S 4 are lifted. We
integrate over all fields in the path integral. No moduli on S 4 .
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Introduction Problem Definitions Method Solution Result
To introduce hypermultiplet masses in OSp(2|4) theory on S 4 :
1 gauge the flavor symmetry
2 give expectation value to Φ 0 in the flavor vector multiplet
Remark on our conventions
In the OSp(2|4) theory on S 4
• m is real for positivity of the action
• in conformal theory m = 0
• under localization m relates to Ω-background m Ω of
[Nekrasov’02] as
m Ω = ɛ 1 + ɛ 2
2
+ im
The usual complex scalar field in N = 2 vector multiplet on R 4
corresponds to a pair of real fields on S 4 : Φ 0 and Φ 9 .
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Introduction Problem Definitions Method Solution Result
Supersymmetric Wilson loop
∮
W R (C) = tr R Pexp (Adx + iΦ 0 ds)
Supersymmetric ’t Hooft loop
Specify asymptotics in local R 3 normal to the loop (x = 0):
F A → − ⋆ B dx
2 x 2 , x → 0
Φ 9 → B 1
2 x , x → 0
The parameter is B ∈ g (weight of ∨ G) defining homomorphism
U(1) → G.
The asymptotics locally satisfies Bogomolny equations
C
D A Φ 9 = ⋆F A
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Introduction Problem Definitions Method Solution Result
Method: Localization Principle
Localize the path integral:
∫
Z 0 = [Dφ] e −S 0[φ]
S 0 – the physical action
Q – global fermionic symmetry, QS 0 = 0
Q 2 = R – global bosonic symmetry
Deform the action
assuming {Q 2 , V } = 0
S t [φ] = S 0 [φ] + t{Q, V }
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Introduction Problem Definitions Method Solution Result
Now consider
Integrating by parts we get
∫
Z t = [Dφ] e −S 0[φ]−t{Q,V [φ]}
d
dt Z t = 0 =⇒ Z 0 = Z ∞
At t = ∞ the one-loop approximation to Z t is exact. But
Z t = Z 0 . Hence we get exact Z 0 .
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Introduction Problem Definitions Method Solution Result
We often take V = (Ψ, QΨ), where Ψ are the fermions of the
theory.
The path integral with action
S t = S 0 + tQV
and V = (Ψ, QΨ) localizes to configurations
QΨ = 0
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Introduction Problem Definitions Method Solution Result
∫
Z =
X
Localization summary
e −S = ∑ ∫
e −S| Yα Z1−loop [N Yα ]
α Y α⊂X
where:
Q is a fermionic symmetry
Q 2 = R is a bosonic symmetry
QS = 0
X = {Φ} is a space of fields
Y α = {Φ|QΨ = 0} are components of the localization locus
N Yα - normal bundle to Y α ⊂ X.
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Introduction Problem Definitions Method Solution Result
Solution strategy
We need to localize the path integral of N = 2 gauge theory on
S 4 in the presence of ’t Hooft singularity.
0 Define the action S, the t’ Hooft operator T and the
fermionic operator Q compatible with S and T
1 Find the localization loci Y α (solve QΨ = 0).
2 Compute exp(−S| Yα )
3 Compute the determinant Z 1−loop [N Yα ]
4 Integrate over Y α and sum over α
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Introduction Problem Definitions Method Solution Result
Step 0. The Q.
The Q is defined by a conformal Killing spinor ε on S 4
∇ µ ε = Γ µ˜ε
In the R 4 conformal frame generic ε is
ε = ˆε s + x µ Γ µˆε c
We choose certain ε s (chiral right) and ε c (chiral right) and fix
Q = Q ε such that
Q 2 = J + R
where J is a self-dual rotation of S 4 and R is R-symmery in
OSp(2|4)
North pole x = 0, ε is chiral right
South pole x = ∞, ε is chiral left
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Introduction Problem Definitions Method Solution Result
Step 1
The equations are QΨ = 0, where
QΨ = 1 2 F mnΓ mn ε − 1 2 φ aΓ aµ ∇ µ ε + iK i Γ 8i+4 ε
For our Q, the equations interpolate between
• north pole: instanton equations F + = 0 and DΦ 9 = 0
• equator: Bogomolny equations DΦ 9 = ∗F in the space
transversal to the S 1 orbits and invariance along S 1
• south pole: anti-instanton equations F − = 0 and DΦ 9 = 0.
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Introduction Problem Definitions Method Solution Result
Technical details:
Convenient coordinates
S 4 = S 1 × B 3
where B 3 is a 3d solid ball with a boundary B 3 : ∑ 3
i=1 x2 i < 1
and S 1 is a circle τ
The round S 4 metric is
ds 2 =
dx 2 i
(1 + x 2 ) 2 + (1 − x2 ) 2
(1 + x 2 ) 2 dτ 2
B 3 -flat rescaled metric
ds 2 = dx 2 i + (1 − x 2 ) 2 dτ 2
The ’t Hooft loop is the S 1 fiber at x i = 0.
The North pole, the first fixed point of Q 2 is x = (0, 0, 1)
he South pole, the second fixed of Q 2 is x = (0, 0, −1)
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Introduction Problem Definitions Method Solution Result
In coordinates (x i , τ) the equations for vector multiplet are
[
F 1τ + D 1 , i (
1 + |⃗x|
2 ) ]
Φ 0 − x 3 Φ 9 + x 1 F 12 = 0
2
F 2τ +
[
F 3τ + D 3 , i 2
[
[
D 2 , i (
1 + |⃗x|
2 ) ]
Φ 0 − x 3 Φ 9 − x 2 F 21 = 0
2
(
1 + |⃗x|
2 ) Φ 0 − x 3 Φ 9
]
+ x 1 F 32 − x 2 F 31 = 0
(
1 + |⃗x|
2 ) Φ 0 − x 3 Φ 9
]
+ x 1 F τ2 − x 2 F τ1 = 0
D τ , i 2
[
[Φ 9 , D τ ] + Φ 9 , i (
1 + |⃗x|
2 ) ]
Φ 0 + x 1 [Φ 9 , D 2 ] − x 2 [Φ 9 , D 1 ] = 0
2
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Introduction Problem Definitions Method Solution Result
−(1+x 2 1−x 2 2−x 2 3)[D 1 Φ 9 ]−2x 1 x 2 [D 2 Φ 9 ]−2x 1 x 3 [D 3 Φ 9 ]−2x 2 [Dˆ4 Φ 9]
− 2x 1 Φ 9 − 2x 1 x 3 F 12 + 2x 1 x 2 F 13 + (1 − x 2 1 + x 2 2 + x 2 3)F 23 − 2x 3 F 1ˆ4
+ i(1 + |⃗x| 2 )K 1 + 2x 1 F 3ˆ4 = 0
−(1−x 2 1+x 2 2−x 2 3)[D 2 Φ 9 ]−2x 1 x 2 [D 1 Φ 9 ]−2x 2 x 3 [D 3 Φ 9 ]+2x 1 [Dˆ4 Φ 9]
− 2x 2 Φ 9 − 2x 2 x 3 F 12 − 2x 1 x 2 F 23 − (2 2 + x 2 1 − x 2 2 + x 2 3)F 13 − 2x 3 F 2ˆ4
+ i(1 + |⃗x| 2 )K 2 + 2x 2 F 3ˆ4 = 0
2x 1 x 3 [D 1 Φ 9 ] + 2x 2 x 3 [D 2 Φ 9 ] − (1 + x 2 1 + x 2 2 − x 2 3)[D 3 Φ 9 ] + 2x 3 Φ 9
+ 4irΦ 0 + (1 − x 2 1 − x 2 2 + x 2 3)F 12 + 2x 1 x 3 F 23 − 2x 2 x 3 F 13
− 2x 2 F 2ˆ4 − 2x 3F 3ˆ4 − 2x 1F 1ˆ4 + i ( 1 + |⃗x| 2) K 3 = 0
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Introduction Problem Definitions Method Solution Result
More compact form of equations
Let
a i = 1 2 (D iΦ 9 + 1 2 ɛ ijkF jk ), b i = 1 2 (D iΦ 9 − 1 2 ɛ ijkF jk ).
Then the 3d equations on B 3 are
b i − δ i x i Φ 9 + δ i x 2 T ij a j = 0
where
δ 1 = δ 2 = −δ 3 = 1
T ij = δ ij − 2x ix j
x 2
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Introduction Problem Definitions Method Solution Result
Smooth solutions
The only topologically trivial solution and smooth away from
singularity is
x i
F jk = − B 2 ɛ ijk
|⃗x| 3 , F iˆ4 = −ig2 θ B x i
16π 2 |⃗x| 3 ,
Φ 9 =
B
2|⃗x| , Φ 0 = −g 2 θ B 1
16π 2 |⃗x| + a
,
1 + |⃗x|2
4r 2
a/r
K 3 = −(
) 2
,
1 + |⃗x|2
4r 2
The moduli space of solutions is parametrized by a ∈ g.
The path integral localizes to
Y 0 = g
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Introduction Problem Definitions Method Solution Result
Step 2. The action S| Y0
Evaluating S 0 at supersymmetric configurations we get
(
S cl [a] = − 8π2 2π
2
g 2 Tr a2 +
g 2 + g2 θ 2 )
32π 2 Tr B 2
In terms of
we can rewrite
τ = θ
2π + 4πi
g 2
where
S cl [a] = −πiτ Tr â(N) 2 + πi¯τ Tr â(S) 2
â(N) = iΦ 0 (N) − Φ 9 (N)
â(S) = iΦ 0 (S) + Φ 9 (S)
are the parameters of the gauge transformation generated by
Q 2 at fixed points N and S
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Introduction Problem Definitions Method Solution Result
Step 3. The one-loop determinant
Denoting the fields of even and odd statistics with a subindex e
and o respectively, the Q multiplets are
ˆQ · ϕ e,o = ˆϕ o,e
ˆQ · ˆϕ o,e = R · ϕ e,o .
and then
ˆQ 2 · ϕ e,o = R · ϕ e,o ,
We can show that the one-loop determinant is
det CokerD vmR| o
det KerD vmR| e
· det CokerD hmR| o
det KerD hmR| e
.
where D is a certain (tranversally elliptic) differential operator
defined from our tQV term.
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Introduction Problem Definitions Method Solution Result
To find the ratio of determinants (equivariant Euler character)
we consider the index of operator D (equivariant Chern
character)
ind D = tr KerD e R − tr CokerD e R
read the weights of R and combine them into determinant
using the rule
∑
c j e w j(ε 1 ,ε 2 ,â, ˆm f ) → ∏ w j (ε 1 , ε 2 , â, ˆm f ) c j
j
j
Recall that R = J + R + [Φ, ·]
J is the SD spatial rotation
R is the R-symmetry rotation
[Φ, ·] is the gauge transformation.
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Introduction Problem Definitions Method Solution Result
Excision property of the index/Atiya-Singer theory
To find index for transversally elliptic operator D on a manifold
M under equivariant U(1) action R we can cut M into three
pieces
• neighborhood of the north pole
• neighborhood of the equator
• neighborhood of the south pole
and apply Atiyah-Singer fixed point formula for the index
ind D(R) = ∑ p∈F
tr E0 (p) R − tr E1 (p) R
.
det T Mp (1 − R)
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Introduction Problem Definitions Method Solution Result
North (South) pole
Near the North pole the operator D, defining the equations, is
the self-dual operator
The index is then
D SD : Ω 1 d∗ ⊕d +
−→ Ω 0 ⊕ Ω 2+
ind(D SD,C )(t 1 , t 2 ) = (t 1t 2 + t −1
1 t−1 2 + 2) − (t 1 + t −1
1 + t 2 + t −1
2 )
(1 − t 1 )(1 − t −1
1 )(1 − t 2)(1 − t −1
2 )
1 + t 1 t 2
=
(1 − t 1 )(1 − t 2 ) . (5.1)
for the U(1) action z 1 → t 1 z 1 , z 2 → t 2 z 2 .
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Introduction Problem Definitions Method Solution Result
When we tensor product sections with the adjoint
representation of G the index becomes (at t i = e iε i
)
ind(D)(ε 1 , ε 2 , â) =
To read the R weights we expand
1 + e iε 1
e iε 2
(1 − e iε 1 )(1 − e iε 2)
= ∑
(1 + e iε 1+iε 2
) ∑
2(1 − e iε 1 )(1 − e iε 2)
n 1 ,n 2 ≥0
w∈adj
e iw·â . (5.2)
(1 + e iε 1
e iε 2
)e in 1ε 1
e in 2ε 2
(5.3)
Therefore, the one-loop contribution from the pole is
∏
[n 1 ε 1 + n 2 ε 2 + α · â] 1/2 [(n 1 + 1)ε 1 + (n 2 + 1)ε 2 + α · â] 1/2 .
n 1 ,n 2 ≥0
(5.4)
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Introduction Problem Definitions Method Solution Result
Barnes G-function
The double product can be regularized
G(1 + z) = (2π) z/2 e −((1+γz2 )+z)/2
∞∏
n=1
(
1 + z ) n
e
−z+ z2
2n . (5.5)
n
and then the one-loop contribution from each pole is
Z 1-loop,pole (â) = ∏ ( ) (
α · â
G 1/2 G 1/2 2 + α · â )
ε
ε
α
and recall that the gauge parameter â at the poles is (at θ = 0)
â(N) = ia − B 2r
â(S) = ia + B 2r ,
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Introduction Problem Definitions Method Solution Result
Contribution to Z 1−loop from equatior
Skipping derivation, the answer (for vectormultiplet)
Z 1-loop,eq (â, B) = ∏ α>0
[ ( (â
sin πα ·
ε + B ))] −α·B
(5.6)
2
Schematically, this comes from extra modes that appear near
the equator in the singular monopole background.
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Introduction Problem Definitions Method Solution Result
Step 5. Integrate over Y 0
Combination of exp(−S| Y0 ) and Z 1−loop factorizes nicely into
〈
ZT (B)
〉
=
∫
where
and
Z 1-loop,pole =
∫
da Z north · Z south · Z equator =
da |Z north | 2 · Z equator
Z north =Z cl (â(N), q) Z 1-loop,pole (â(N), im f )
Z south =Z cl (â(S), ¯q) Z 1-loop,pole (â(S), im f )
Z equator =Z 1-loop,eq (â(E), im f , B) ,
[ ]
1
Z cl (â, q) = exp 2πiτ Tr â 2 .
2ε 1 ε 2
∏
∏ NF
∏ [
f=1 w∈R
G
[ (
G
α·â
) (
ε G 2 +
α·â
ε
(
)
1 + w·â
ε
− ˆm f
ε
G
α
)] 1/2
(
1 − w·â
ε
)]
+ ˆm 1/2
f
ε
and Z 1-loop,eq Vasily Pestun Localization for ’t Hooft operators on S 4 30/35

Introduction Problem Definitions Method Solution Result
Other components Y α : instanton corrections
In the limit x → 0, up to the terms O(x 2 ) our Q complex in the
neighborhood of the north pole is exactly as the one of the
gauge theory on R 4 ε,ε [Nekrasov’02] . Hence the other
components of the localization locus Y α are the point instantons
F + = 0 sitting at the north pole as in [Nekrasov’02] .
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Introduction Problem Definitions Method Solution Result
Result
Including the point instanton contributions at the north and
south poles, we finally get the result
〈
ZT (B)
〉
=
∫
da
∣ Z north(ia − B ∣ ∣∣∣
2
2 ) · Z equator (ia)
where Z north is Nekrasov’s partition function including classical,
(slightly modified) one-loop and the instanton factors.
Our gauge theory result agrees with the computation in the
Liouville/Toda theories by [Alday-Gaiotto-Tajikawa] conjecture
performed by [Drukker,Gomis,Okuda,Teschner’09] and [Alday,
Gaiotto, Gukov, Tachikawa, Verlinde’09] and [Gomis,Floch’10] .
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Introduction Problem Definitions Method Solution Result
At the minimal B the result is exact. At higher B there are
further corrections coming from the screening of the ’t Hooft
loop by the monopoles condensing on the loop.
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Introduction Problem Definitions Method Solution Result
S-duality
Finally, let us compare our formulae for Wilson and ’t Hooft
loops and
∫
〈 〉
ZW (R) τ = [da]|Z north (ia, τ)| ∑ 2 e 2πiaw
w
〈
ZT (B)
〉
τ = ∫
da
∣ Z north(ia − B ∣ ∣∣∣
2
2 , τ) · Z equator (ia)
Notice:
Wilson loop of weight w inserts operator exp(2πiwa)
t’ Hooft loop of coweight B inserts shift operator exp(B ∂
∂a )
Hence, the latter formula (magnetic) at τ ∨ = − 1 τ
is Fourier
transform of the former (electric).
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Introduction Problem Definitions Method Solution Result
The Nekrasov’s functions Z north (â) are too complicated to
explicitly integrate analytically over â, but the integrand is well
defined as a quickly convergent series and we were able to
perform numerical check of the S-duality invariance.
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