Supersymmetric vortex defects in two dimensions Takuya Okuda - - PowerPoint PPT Presentation
Supersymmetric vortex defects in two dimensions Takuya Okuda University of Tokyo, Komaba 1 Plan Part I: Supersymmetric vortex defects [1705.10623 with K. Hosomichi and S. Lee] Part II: SUSY renormalization (Pauli-Villars and counterterms)
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[1705.06118 TO]
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Motivations and the set-up Three inequivalent definitions of defects Relations among definitions Applications
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A ∼ ηdϕ
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A ∼ ηdϕ
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Sometimes, defects characterized by the gauge field singularity are also described by the insertion of local degrees of freedom. (3d: Assel-Gomis,…, 4d: Gukov-Witten, Gaiotto, Nawata, …) What is the mechanism that guarantees the equivalence of the two descriptions? Will give an answer in the 2d abelian case.
A ∼ ηdϕ
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z(φ, ψ+), (ψ−, F) : finite
(ψ−, F) = O(rγ) , −1 < γ ≤ 0 (φ, ψ+) = O(rγ)
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Start with the smearing definition. For some values of vorticity 𝜃, the 2d bulk fields develop localized modes. The localized modes form 0d multiplets. The non-localized modes obey normal/flipped boundary conditions.
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Recall SUSY condition D=2𝜌iρ. We get . Expand in eigenmodes of . Zero-modes, if present, are annihilated by and are localized. Expand in eigenmodes of . Zero-modes, if present, are annihilated by and are localized.
S ∼ Z ¯ φ(−DzD¯
z + ¯
ΣΣ)φ + ¯ ψ ✓ ¯ Σ Dz D¯
z
Σ ◆ ψ + ¯ FF ψ = ✓ψ+ ψ− ◆ −DzD¯
z
φ, ψ+ ψ−, F −D¯
zDz
D¯
z
Dz
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zΨ = 0 for Ψ = φ, ψ+
m = 0, 1, . . . , bηc 1 Ψ = ˆ Ψ(r)eimϕ ˆ Ψ ⇠ ⇢ rm for r ⌧ ✏ rm−η for r ✏
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ˆ Ψ ⇠ ⇢ r−m for r ⌧ ✏ rm−η for r ✏
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z ˆ
Ψ = λˆ Ψ for ˆ Ψ = φ, ψ+ −D¯
zDz ˆ
Ψ = λˆ Ψ for ˆ Ψ = ψ−, F
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D(2d chiral)V smeared
η
= 8 > > > > < > > > > : D(2d chiral)V flipped
η
×
bηc1
Y
a=0
d(0d chiral)a (η > 0) D(2d chiral)V normal
η
×
bηc1
Y
α=0
d(0d Fermi)α (η < 0)
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η
= V flipped
η
✓Z D(0d chiral)e−S ◆N V flipped
η
= ✓Z D(0d Fermi)e−S ◆N = ΣN
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η−1
= 1 V flipped
η
= e−tV flipped
η−1
ΣN = e−t
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When the gauge field is non-dynamical, the smearing regularization gives a non-trivial
Flavor vortex defect V𝜃flavor realizes the twisted chiral operator e𝜃Y in the Hori-Vafa mirror theory. For discrete symmetries, vortex defects are nothing but twist fields.
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Level h-2 minimal model with h=2,3,4,… Its mirror is the Zh orbifold of itself. N=2 Landau-Ginzburg model with superpotential . Twist fields are vortex defects with vorticity η=-p/h, p=0,1,…,h-1.
c = 3(h − 2) h
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⌦ V−p/h(N)V−p/h(S) ↵
S2 = 1
h Γ( 1+p
h )
Γ(1 − 1+p
h )
= Γ( 1+p
h )2
hπ sin (1 + p)π h
Two-point functions of twist fields can be computed by localization. Agree with known results and mirror symmetry expectations. Explicit renormalization by Pauli-Villars and supergravity counterterms. [TO] Coincides with the known and mirror results.
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Found a mechanism for the equivalence of the vortex defect defined by boundary condition and the defect defined by 0d-2d coupling. Gave a precise path-integral formulation of twist fields in Landau-Ginzburg realization of the minimal model. (In the paper) gave prescriptions for computing two-point functions of vortex defects.
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J = 0 J = j ∈ {1, . . . , 2NPV − 1} physical unphysical (PV ghosts) statistics ✏0 = +1 ✏j = ±1 U(1)PV-charge a0 = 0 aj ∈ R − {0} flavor/gauge charge b0 = +1 bj ∈ Z twisted mass
vector R-charge q0 = q R-charge cjq c0 = 1
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J
J
J
J
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X
J
✏Ja2
J =
X
J
✏Jb2
J =
X
J
✏Jc2
J =
X
J
✏JaJbJ = X
J
✏JbJcJ = X
J
✏JcJaJ = 0
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C0 := Y
j
|aj|−✏j , C1 := X
j
✏jbj log |aj| , C2 := X
j
✏jaj log |aj| , C3 := X
j
✏jcj log |aj| , Ξ1 := X
j
✏jbjsgn(aj) , Ξ2 := X
j
✏j|aj| , Ξ3 := X
j
✏jcjsgn(aj) , Ξ4 := X
j
✏jsgn(aj) .
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1-loop “=” ∞
Y
n=0
n + 1 + 1
2|B| − ˆ
σ n + 1
2|B| + ˆ
σ ⇣ ˆ = i`1 + q 2 ⌘
ZSUSY
1-loop, reg = ∞
Y
n=0
n + 1 + 1
2|B| − ˆ
σ n + 1
2|B| + ˆ
σ Y
j
✓n + 1 + 1
2|bjB| − Mj
n + 1
2|bjB| + Mj
◆✏j
Mj ≡ cj q 2 + i`(ajΛ + bjRe())
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ZSUSY
1-loop, reg =
Γ(ˆ + B
2 )
Γ(1 − ˆ + B
2 )
Y
j
✓ Γ(Mj + bj B
2 )
Γ(1 − Mj + bj B
2 )
◆✏j = C0ei π
2 Ξ1Be(C3−C1)qe2C1ˆ
e2iC2`Λ(`Λ)1−2ˆ
+ B
2 )
Γ(1 − ˆ + B
2 )
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Claim: the counterterms given by the following twisted superpotential renormalize the one-loop partition functions in arbitrary backgrounds. (μ: renormalization scale) :twisted chiral field constructed from the gravity multiplet/R-symmetry gauge multiplet in N=(2,2) U(1)V SUGRA. Similar to Witten’ s effective twisted superpotential.
f Wct(, b H, Λ) = − b H 8⇡ X
j
✏j log iajΛ µ + 1 4⇡ X
j
✏j(ajΛ + bj + cjq 2 b H) log ajΛ + bj + cjq
2 b
H µ e .
b H
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f Wct(σ, b H, Λ) = 1 − q 8π b H log Λ µ + 1 4π ✓ C1 − log Λ µ + iπ 2 Ξ1 ◆ σ + 1 4π ⇣ C2 + iπ 2 Ξ2 ⌘ Λ + q 4π ⇣ C3 + iπ 2 Ξ3 ⌘ b H 2 + 1 4π ⇣ log C0 − iπ 2 Ξ4 ⌘ b H 2 + O(Λ−1) .
ˆ cUV 8π Z d2x√g R log Λ µ
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log Λ µ − C1 + r(µ) = r0(Λ) , θ + π 2 Ξ1 = θ0 t = r − iθ
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ZSUSY = lim
Λ→∞ e−Sren−SctZSUSY 1-loop, reg
= e−Sren(`µ)1−2ˆ
+ B
2 )
Γ(1 − ˆ + B
2 )
= e−iB✓e4⇡i[r(µ)− 1
2π log(`µ)]`Re (`µ)1−q
Γ(ˆ + B
2 )
Γ(1 − ˆ + B
2 ) .
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2 )
Γ(1 − ˆ σ + B
2 )
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