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String and M-Theory:
The New Geometry of the 21st Century
National Univ. of Singapore, 2018
Supersymmetric Curvepoles in 6D
Ori Ganor
Berkeley Center for Theoretical Physics, UC Berkeley
December 14, 2018
arXiv:1710.06880, and thanks to Kevin Schaeffer for adiscussion in 2012 that provided motivation for this project.
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String and M-Theory:
The New Geometry of the 21st Century
National Univ. of Singapore, 2018
Supersymmetric Curvepoles in 6D
(New effects on M5-branes in strong flux?)
Ori Ganor
Berkeley Center for Theoretical Physics, UC Berkeley
December 14, 2018
arXiv:1710.06880, and thanks to Kevin Schaeffer for adiscussion in 2012 that provided motivation for this project.
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Dipoles and Curvepoles
Curvepole – counterpart of E&M dipole, interacting with a 2-form.
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Dipoles and Curvepoles
Curvepole – counterpart of E&M dipole, interacting with a 2-form.
E&M Tensor field
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Dipoles and Curvepoles
Curvepole – counterpart of E&M dipole, interacting with a 2-form.
E&M Tensor field
− +
Dipole:
Two endpoints interacting
with a 1-form gauge field A
with opposite charges.
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Dipoles and Curvepoles
Curvepole – counterpart of E&M dipole, interacting with a 2-form.
E&M Tensor field
− +
Dipole:
Two endpoints interacting
with a 1-form gauge field A
with opposite charges.
C
Σ
Curvepole:
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Dipoles and Curvepoles
Curvepole – counterpart of E&M dipole, interacting with a 2-form.
E&M Tensor field
− +
Dipole:
Two endpoints interacting
with a 1-form gauge field A
with opposite charges.
C
Σ
Curvepole:
Has closed curve boundary C = ∂Σ
charged under 2-form gauge field B.
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Dipoles and Curvepoles
Curvepole – counterpart of E&M dipole, interacting with a 2-form.
E&M Tensor field
− +
Dipole:
Two endpoints interacting
with a 1-form gauge field A
with opposite charges.
C
Σ
Curvepole:
Has closed curve boundary C = ∂Σ
charged under 2-form gauge field B.
Bulk of Σ is inert.
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Dipoles and Curvepoles
Curvepole – counterpart of E&M dipole, interacting with a 2-form.
E&M Tensor field
− +
Dipole:
Two endpoints interacting
with a 1-form gauge field A
with opposite charges.
C
Σ
Curvepole:
Has closed curve boundary C = ∂Σ
charged under 2-form gauge field B.
Bulk of Σ is inert.
Focus on 6D where (anti-)selfduality can impose restrictions.
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The Goal
Construct a supersymmetric second quantized theory of curvepolesinteracting with a 6D tensor multiplet of (1, 0) SUSY.
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The Goal
Construct a supersymmetric second quantized theory of curvepolesinteracting with a 6D tensor multiplet of (1, 0) SUSY.
dB = H = −?H interacts with a field Φ whose quanta are:
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The Goal
Construct a supersymmetric second quantized theory of curvepolesinteracting with a 6D tensor multiplet of (1, 0) SUSY.
dB = H = −?H interacts with a field Φ whose quanta are:
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The Goal
Construct a supersymmetric second quantized theory of curvepolesinteracting with a 6D tensor multiplet of (1, 0) SUSY.
dB = H = −?H interacts with a field Φ whose quanta are:
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The Goal
Construct a supersymmetric second quantized theory of curvepolesinteracting with a 6D tensor multiplet of (1, 0) SUSY.
dB = H = −?H interacts with a field Φ whose quanta are:
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The Goal
Construct a supersymmetric second quantized theory of curvepolesinteracting with a 6D tensor multiplet of (1, 0) SUSY.
dB = H = −?H interacts with a field Φ whose quanta are:
NOTE: The shape C of these curvepoles is fixed!
C is an external parameter of the theory.
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The Goal
Construct a supersymmetric second quantized theory of curvepolesinteracting with a 6D tensor multiplet of (1, 0) SUSY.
dB = H = −?H interacts with a field Φ whose quanta are:
NOTE: The shape C of these curvepoles is fixed!
C is an external parameter of the theory.
Does SUSY impose any restrictions on C?
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Preview of resultsConstruct action order by order:
L = L2 + L3 + L4 + · · ·δSUSY = δ0 + δ1 + δ2 + · · ·
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Preview of resultsConstruct action order by order:
L = L2 + L3 + L4 + · · ·δSUSY = δ0 + δ1 + δ2 + · · ·
Require SUSY and Hollowness.
Hollowness – theory depends only on C = ∂Σ, but not on Σ.
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Preview of resultsConstruct action order by order:
L = L2 + L3 + L4 + · · ·δSUSY = δ0 + δ1 + δ2 + · · ·
Require SUSY and Hollowness.
Hollowness – theory depends only on C = ∂Σ, but not on Σ.
− I’ll present explicit formulas for up to quartic interactions.
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Preview of resultsConstruct action order by order:
L = L2 + L3 + L4 + · · ·δSUSY = δ0 + δ1 + δ2 + · · ·
Require SUSY and Hollowness.
Hollowness – theory depends only on C = ∂Σ, but not on Σ.
− I’ll present explicit formulas for up to quartic interactions.
− There is a restriction on C , imposed by SUSY.
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Preview of resultsConstruct action order by order:
L = L2 + L3 + L4 + · · ·δSUSY = δ0 + δ1 + δ2 + · · ·
Require SUSY and Hollowness.
Hollowness – theory depends only on C = ∂Σ, but not on Σ.
− I’ll present explicit formulas for up to quartic interactions.
− There is a restriction on C , imposed by SUSY. (sufficient condition)
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Preview of resultsConstruct action order by order:
L = L2 + L3 + L4 + · · ·δSUSY = δ0 + δ1 + δ2 + · · ·
Require SUSY and Hollowness.
Hollowness – theory depends only on C = ∂Σ, but not on Σ.
− I’ll present explicit formulas for up to quartic interactions.
− There is a restriction on C , imposed by SUSY. (sufficient condition)
− There is another restriction on C , imposed by Hollowness.
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Preview of resultsConstruct action order by order:
L = L2 + L3 + L4 + · · ·δSUSY = δ0 + δ1 + δ2 + · · ·
Require SUSY and Hollowness.
Hollowness – theory depends only on C = ∂Σ, but not on Σ.
− I’ll present explicit formulas for up to quartic interactions.
− There is a restriction on C , imposed by SUSY. (sufficient condition)
− There is another restriction on C , imposed by Hollowness.
♦ If C is planar, both SUSY and Hollowness hold? (up to 4th order).
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Preview of resultsConstruct action order by order:
L = L2 + L3 + L4 + · · ·δSUSY = δ0 + δ1 + δ2 + · · ·
Require SUSY and Hollowness.
Hollowness – theory depends only on C = ∂Σ, but not on Σ.
− I’ll present explicit formulas for up to quartic interactions.
− There is a restriction on C , imposed by SUSY. (sufficient condition)
− There is another restriction on C , imposed by Hollowness.
♦ If C is planar, both SUSY and Hollowness hold? (up to 4th order).
All of this is at the “classical” level.I’ll briefly mention issues regarding anomalies at the end.
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A digression: Connection to M-theory
T 3
Vol(T 3)→ 0
M2
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A digression: Connection to M-theory
T 3
Vol(T 3)→ 0
M2
U-duality
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A digression: Connection to M-theory
T 3
Vol(T 3)→ 0
M2
U-duality
U-dual T 3
Vol(U-dual T 3)→∞
Wrapped M5:
freehyper ’plet
& tensor ’plet
(1,0) multiplets
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A digression: Connection to M-theory
T 3
Vol(T 3)→ 0
M2
U-duality
U-dual T 3
Vol(U-dual T 3)→∞
Wrapped M5:
freehyper ’plet
& tensor ’plet
(1,0) multipletsΩ
Insert Ω ∈ Spin(5) twist
Scaling limit: Ω→ I with (Ω− I )/M4pVol2/3 → finite
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A digression: Connection to M-theory
T 3
Vol(T 3)→ 0
M2
U-duality
U-dual T 3
Vol(U-dual T 3)→∞
Wrapped M5:
freehyper ’plet
& tensor ’plet
(1,0) multipletsΩ
Insert Ω ∈ Spin(5) twist
Scaling limit: Ω→ I with (Ω− I )/M4pVol2/3 → finite
curvepole
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A digression: Connection to M-theory
T 3
Vol(T 3)→ 0
M2
U-duality
U-dual T 3
Vol(U-dual T 3)→∞
Wrapped M5:
freehyper ’plet
& tensor ’plet
(1,0) multipletsΩ
Insert Ω ∈ Spin(5) twist
Scaling limit: Ω→ I with (Ω− I )/M4pVol2/3 → finite
curvepole
[Bergman & OG, 2000; Bergman & Dasgupta & Karczmarek & Rajesh & OG, 2001; Alishahiha & OG 2003]c.f. [Douglas & Hull, 1997; Connes & Douglas & Schwarz, 1997]
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M-theory construction (more details)
T 3 T 3
M2
U
M5
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M-theory construction (more details)
T 3 T 3
M2
U
M5x5
x3
x4x5
x3
x4
0 ≤ xi ≤ 2πRi 0 ≤ xi ≤ 2πRi
V = (2π)3R3R4R5U
V = (2π)3R3R4R5 = (2π)6
M6pV
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M-theory construction (more details)
T 3 T 3
M2
U
M5x5
x3
x4x5
x3
x4
0 ≤ xi ≤ 2πRi 0 ≤ xi ≤ 2πRi
V = (2π)3R3R4R5U
V = (2π)3R3R4R5 = (2π)6
M6pV
Twist: x5 → x5 + 2πR5
accompanied by e iθ ∈ U(1)︸︷︷︸J
⊂ SO(5)6,...,10
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M-theory construction (more details)
T 3 T 3
M2
U
M5x5
x3
x4x5
x3
x4
0 ≤ xi ≤ 2πRi 0 ≤ xi ≤ 2πRi
V = (2π)3R3R4R5U
V = (2π)3R3R4R5 = (2π)6
M6pV
Twist: x5 → x5 + 2πR5
accompanied by e iθ ∈ U(1)︸︷︷︸J
⊂ SO(5)6,...,10
Momentum quantization: P5R5 ∈ Z + θJ2π
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M-theory construction (more details)
T 3 T 3
M2
U
M5x5
x3
x4x5
x3
x4
0 ≤ xi ≤ 2πRi 0 ≤ xi ≤ 2πRi
V = (2π)3R3R4R5U
V = (2π)3R3R4R5 = (2π)6
M6pV
Twist: x5 → x5 + 2πR5
accompanied by e iθ ∈ U(1)︸︷︷︸J
⊂ SO(5)6,...,10
Momentum quantization: P5R5 ∈ Z + θJ2π
Dual to : M2-area ∈ Z(2π)2R3R4 + (2πθR3R4)︸ ︷︷ ︸keep finite
J
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M-theory construction (more details)
T 3 T 3
M2
U
M5x5
x3
x4x5
x3
x4
0 ≤ xi ≤ 2πRi 0 ≤ xi ≤ 2πRi
V = (2π)3R3R4R5U
V = (2π)3R3R4R5 = (2π)6
M6pV
Twist: x5 → x5 + 2πR5
accompanied by e iθ ∈ U(1)︸︷︷︸J
⊂ SO(5)6,...,10
Momentum quantization: P5R5 ∈ Z + θJ2π
Dual to : M2-area ∈ Z(2π)2R3R4 + (2πθR3R4)︸ ︷︷ ︸keep finite
J
Area(Σ)7 41
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M-theory construction - what are the curvepoles?
Curvepoles can also be explained (heuristically) by a Myers effect
(dielectric open M2’s)[Bergman & OG, 2000; Bergman & Dasgupta & Karczmarek & Rajesh & OG, 2001]
M5
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M-theory construction - what are the curvepoles?
Curvepoles can also be explained (heuristically) by a Myers effect
(dielectric open M2’s)[Bergman & OG, 2000; Bergman & Dasgupta & Karczmarek & Rajesh & OG, 2001]
M5x7 x6
0, . . . , 5
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M-theory construction - what are the curvepoles?
Curvepoles can also be explained (heuristically) by a Myers effect
(dielectric open M2’s)[Bergman & OG, 2000; Bergman & Dasgupta & Karczmarek & Rajesh & OG, 2001]
Strong G1267 flux
M5x7 x6
0, . . . , 5
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M-theory construction - what are the curvepoles?
Curvepoles can also be explained (heuristically) by a Myers effect
(dielectric open M2’s)[Bergman & OG, 2000; Bergman & Dasgupta & Karczmarek & Rajesh & OG, 2001]
Strong G1267 flux
M5x7 x6
0, . . . , 5
M2
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M-theory construction - what are the curvepoles?
Curvepoles can also be explained (heuristically) by a Myers effect
(dielectric open M2’s)[Bergman & OG, 2000; Bergman & Dasgupta & Karczmarek & Rajesh & OG, 2001]
Strong G1267 flux
M5x7 x6
0, . . . , 5
M2
Σ
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M-theory construction - what are the curvepoles?
Curvepoles can also be explained (heuristically) by a Myers effect
(dielectric open M2’s)[Bergman & OG, 2000; Bergman & Dasgupta & Karczmarek & Rajesh & OG, 2001]
Strong G1267 flux
M5x7 x6
0, . . . , 5
M2
Σ
See also [Witten, 1997; Aganagic & Park & Popscu & Schwarz, 1997;
Bergshoeff & Berman, & van der Schaar & Sundell, 2000;
Berman & Tadrowski, 2007; Lambert & Orlando & Reffert, 2014; Lambert & Orlando & Reffert & Sekiguchi, 2018]
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Deforming the free theory to Curvepole Theory
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Deforming the free theory to Curvepole Theory
Free 6D (2, 0) tensor ’plet
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Deforming the free theory to Curvepole Theory
Free 6D (2, 0) tensor ’plet
(1, 0) tensor(B, χi, ϕ) i = 1, 2
SU(2)R index: (1, 0) hyper(φi, ψ)
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Deforming the free theory to Curvepole Theory
Free 6D (2, 0) tensor ’plet
(1, 0) tensor(B, χi, ϕ)
H(+) := 12 (dB + ∗dB) = 0
i = 1, 2SU(2)R index: (1, 0) hyper
(φi, ψ)
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Deforming the free theory to Curvepole Theory
Free 6D (2, 0) tensor ’plet
(1, 0) tensor(B, χi, ϕ)
H(+) := 12 (dB + ∗dB) = 0
i = 1, 2SU(2)R index: (1, 0) hyper
(φi, ψ)
want their quanta
to become curvepoles:
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Deforming the free theory to Curvepole Theory
Free 6D (2, 0) tensor ’plet
(1, 0) tensor(B, χi, ϕ)
H(+) := 12 (dB + ∗dB) = 0
i = 1, 2SU(2)R index: (1, 0) hyper
(φi, ψ)
want their quanta
to become curvepoles:
want their quanta
to stay pointlike.
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Recall: 2nd quantized dipolesDipole:
− +
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Recall: 2nd quantized dipolesDipole:
− +x
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Recall: 2nd quantized dipolesDipole:
− +x
L
L is a fixed vector.(Counterpart of Σ)
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Recall: 2nd quantized dipolesDipole:
− +x
L
L is a fixed vector.(Counterpart of Σ)
Φ is 2nd quantized field of dipole.
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Recall: 2nd quantized dipolesDipole:
− +x
L
L is a fixed vector.(Counterpart of Σ)
Φ is 2nd quantized field of dipole.
Modified covariant derivative:
DµΦ(x) = ∂µΦ(x) + iAµ(x + L2 )Φ(x)− iAµ(x − L
2 )Φ(x)
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Recall: 2nd quantized dipolesDipole:
− +x
L
L is a fixed vector.(Counterpart of Σ)
Φ is 2nd quantized field of dipole.
Modified covariant derivative:
DµΦ(x) = ∂µΦ(x) + iAµ(x + L2 )Φ(x)− iAµ(x − L
2 )Φ(x)
[Bergman & OG, Bergman & Dasgupta & Karczmarek & Rajesh & OG]
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Recall: 2nd quantized dipolesDipole:
− +x
L
L is a fixed vector.(Counterpart of Σ)
Φ is 2nd quantized field of dipole.
Modified covariant derivative:
DµΦ(x) = ∂µΦ(x) + iAµ(x + L2 )Φ(x)− iAµ(x − L
2 )Φ(x)
[Bergman & OG, Bergman & Dasgupta & Karczmarek & Rajesh & OG]
Attach Wilson line:
− +
Px
P is a fixed path from −L2 to L
2 .
Φ(x) := e−i∫Px
AΦ(x) is gauge invariant.
Px := x + P is “P translated by x .”
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Recall: 2nd quantized dipolesDipole:
− +x
L
L is a fixed vector.(Counterpart of Σ)
Φ is 2nd quantized field of dipole.
Modified covariant derivative:
DµΦ(x) = ∂µΦ(x) + iAµ(x + L2 )Φ(x)− iAµ(x − L
2 )Φ(x)
[Bergman & OG, Bergman & Dasgupta & Karczmarek & Rajesh & OG]
Attach Wilson line:
− +
Px
P is a fixed path from −L2 to L
2 .
Φ(x) := e−i∫Px
AΦ(x) is gauge invariant.
Px := x + P is “P translated by x .”
DµΦ(x) = ∂µΦ(x) + iΦ(x)
∫Px
Fνµdxν
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Curvepole Theory (first try)
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
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Curvepole Theory (first try)
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
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Curvepole Theory (first try)
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
Step 1: “covariant” curvepole derivative
∂µφ→ Dµφ(x) := ∂µφ(x) + iφ(x)∫∂Σ Bµν(x + y)dyν
[Schaeffer, 2012 (unpublished)], (Implicit in [Alishahiha & OG, 2003])
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Curvepole Theory (first try)
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
Step 1: “covariant” curvepole derivative
∂µφ→ Dµφ(x) := ∂µφ(x) + iφ(x)∫∂Σ Bµν(x + y)dyν
[Schaeffer, 2012 (unpublished)], (Implicit in [Alishahiha & OG, 2003])
But action should depend only on H(−) := 12 (dB − ∗dB).
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Curvepole Theory (first try)
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
Step 1: “covariant” curvepole derivative
∂µφ→ Dµφ(x) := ∂µφ(x) + iφ(x)∫∂Σ Bµν(x + y)dyν
[Schaeffer, 2012 (unpublished)], (Implicit in [Alishahiha & OG, 2003])
But action should depend only on H(−) := 12 (dB − ∗dB).
For free tensor: H(+) := 12 (dB + ∗dB) = 0⇒ H(−) = dB,
a field redefinition φ(x) := e i∫
Σ Bµν(x+y)dyµ∧dyνφ(x) does it:
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Curvepole Theory (first try)
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
Step 1: “covariant” curvepole derivative
∂µφ→ Dµφ(x) := ∂µφ(x) + iφ(x)∫∂Σ Bµν(x + y)dyν
[Schaeffer, 2012 (unpublished)], (Implicit in [Alishahiha & OG, 2003])
But action should depend only on H(−) := 12 (dB − ∗dB).
For free tensor: H(+) := 12 (dB + ∗dB) = 0⇒ H(−) = dB,
a field redefinition φ(x) := e i∫
Σ Bµν(x+y)dyµ∧dyνφ(x) does it:
⇒ Dµφ(x) := ∂µφ(x) + i φ(x)∫
Σ(dB)µνσ(x + y)dyν ∧ dyσ
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Curvepole Theory (first try)
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
Step 1: “covariant” curvepole derivative
∂µφ→ Dµφ(x) := ∂µφ(x) + iφ(x)∫∂Σ Bµν(x + y)dyν
[Schaeffer, 2012 (unpublished)], (Implicit in [Alishahiha & OG, 2003])
But action should depend only on H(−) := 12 (dB − ∗dB).
For free tensor: H(+) := 12 (dB + ∗dB) = 0⇒ H(−) = dB,
a field redefinition φ(x) := e i∫
Σ Bµν(x+y)dyµ∧dyνφ(x) does it:
⇒ Dµφ(x) := ∂µφ(x) + i φ(x)∫
Σ(dB)µνσ(x + y)dyν ∧ dyσ
But interactions ⇒ H(+) := 12 (dB + ∗dB) = K (+) = (φ-contributions)
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Curvepole Theory
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
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Curvepole Theory
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
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Curvepole Theory
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
Step 1’: Modified “covariant” curvepole derivative
∂µφ→ Dµφ(x) := ∂µφ(x) + iVµ(x)φ(x)
Vµ(x) := 12
∫Σ dyν ∧ dyσH
(−)µνσ(x + y)
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Curvepole Theory
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
Step 1’: Modified “covariant” curvepole derivative
∂µφ→ Dµφ(x) := ∂µφ(x) + iVµ(x)φ(x)
Vµ(x) := 12
∫Σ dyν ∧ dyσH
(−)µνσ(x + y) −1
2
∫∂Σ dyµϕ(x + y)
required for SUSY
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Curvepole Theory
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
Step 1’: Modified “covariant” curvepole derivative
∂µφ→ Dµφ(x) := ∂µφ(x) + iVµ(x)φ(x)
Vµ(x) := 12
∫Σ dyν ∧ dyσH
(−)µνσ(x + y) −1
2
∫∂Σ dyµϕ(x + y)
required for SUSYAdditional terms in L
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Curvepole Theory
We want quanta of φ to be curvepoles:
[interacting with (H(−), χi, ϕ)]
Σ – fixed shape (open surface) around origin.
Σ
Step 1’: Modified “covariant” curvepole derivative
∂µφ→ Dµφ(x) := ∂µφ(x) + iVµ(x)φ(x)
Vµ(x) := 12
∫Σ dyν ∧ dyσH
(−)µνσ(x + y) −1
2
∫∂Σ dyµϕ(x + y)
required for SUSYAdditional terms in L
A condition on Σ
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What is the condition on Σ?
Define the curvepole integrals: Σ
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What is the condition on Σ?
Define the curvepole integrals: Σ
Ψµν
:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
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What is the condition on Σ?
Define the curvepole integrals: Σ
Ψµν
:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
A condition on Σ appears by requiring SUSY at quartic order.
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What is the condition on Σ?
Define the curvepole integrals: Σ
Ψµν
:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
A condition on Σ appears by requiring SUSY at quartic order.
SUSY preserved if:
(up to quartic order)
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What is the condition on Σ?
Define the curvepole integrals: Σ
Ψµν
:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
A condition on Σ appears by requiring SUSY at quartic order.
SUSY preserved if:
(up to quartic order)
Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
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What is the condition on Σ?
Define the curvepole integrals: Σ
Ψµν
:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
A condition on Σ appears by requiring SUSY at quartic order.
SUSY preserved if:
(up to quartic order)
Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
This is a geometric condition on Σ.
Let’s call such Σ’s balanced curvepoles.
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Balanced Curvepoles
ΣΨ
µν:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
Balanced curvepole: Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
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Balanced Curvepoles
ΣΨ
µν:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
Balanced curvepole: Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
Equivalent to:∫d6x Ψ αβ Φ γδ =
∫d6x Ψ γδ Φ αβ
for arbitrary Ψ,Φ
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Balanced Curvepoles
ΣΨ
µν:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
Balanced curvepole: Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
Equivalent to:∫d6x Ψ αβ Φ γδ =
∫d6x Ψ γδ Φ αβ
for arbitrary Ψ,Φ
(This is a stronger condition)
Ψ αβ Φ γδ = Ψ γδ Φ αβ
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Balanced Curvepoles
ΣΨ
µν:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
Balanced curvepole: Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
Equivalent to:∫d6x Ψ αβ Φ γδ =
∫d6x Ψ γδ Φ αβ
for arbitrary Ψ,Φ
(This is a stronger condition)
Ψ αβ Φ γδ = Ψ γδ Φ αβPlanar curvepole
(Σ ⊂ R2 ⊂ R6)
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Balanced and Unbalanced Curvepoles
ΣΨµν
:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
Balanced curvepole: Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
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Balanced and Unbalanced Curvepoles
ΣΨµν
:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
Balanced curvepole: Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
Parity invariant (Σ = −Σ) (balanced)
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Balanced and Unbalanced Curvepoles
ΣΨµν
:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
Balanced curvepole: Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
Parity invariant (Σ = −Σ) (balanced)
Planar (Σ ⊂ R2 ⊂ R6) (balanced)
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Balanced and Unbalanced Curvepoles
ΣΨµν
:=∫
Σ Ψ(x + y)dyµ ∧ dyν
Ψµν
:=∫
Σ Ψ(x − y)dyµ ∧ dyν
Balanced curvepole: Ψ αβγδ
= Ψ γδαβ
for arbitrary Ψ
Parity invariant (Σ = −Σ) (balanced)
Planar (Σ ⊂ R2 ⊂ R6) (balanced)
(a chair) (not balanced)
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The action - quadratic termsThe bosonic quadratic terms are:
L(bosonic)2 = 1
12πHµνσHµνσ + 1
4π∂µϕ∂µϕ+ ∂µφi∂
µφi
whereHµνσ := 3∂[µBνσ] 3-form field strength
H(±)µνσ := 1
2 (Hµνσ ± i6εµνσαβγH
αβγ)selfdual and anti-selfdual components
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The action - cubic termsThe bosonic cubic terms are:
L(bosonic)3 = −JµVµ
whereJµ := i(φi∂µφ
i − ∂µφiφi) U(1) current
Vµ := 12 H(−)
µνσ
νσ
− 12 ∂
νϕµν
effective gauge field defined above
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The action - quartic termsThe additional bosonic quartic terms are:
L(bosonic)4 = VµVµφiφ
i − 3π16 J [µ
σν]
J[µσν]
+ π2 ∂
µM ji µσ
∂νMij
νσ
whereJµ := i(φi∂µφ
i − ∂µφiφi) U(1) current
M ji := φiφ
j − 12δ
jiφkφ
k SU(2)R triplet
Vµ := 12 H(−)
µνσ
νσ
− 12 ∂
νϕµν
effective gauge field defined above
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Hollowness
C = ∂Σ
Σ
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Hollowness
C = ∂Σ
Σ
It would be nice if the theory depended only on C ; not on Σ.
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Hollowness
C = ∂Σ
Σ
It would be nice if the theory depended only on C ; not on Σ.
If ∂Σ = ∂Σ′, we’d like theory (Σ) ' theory(Σ′)
up to a field redefinition.
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Hollowness
C = ∂Σ
Σ
It would be nice if the theory depended only on C ; not on Σ.
If ∂Σ = ∂Σ′, we’d like theory (Σ) ' theory(Σ′)
up to a field redefinition.
That turns out to be true . . . well . . . sort of.
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HollownessRecall dipole case:
− +
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HollownessRecall dipole case:
− +x
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HollownessRecall dipole case:
− +x
L
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HollownessRecall dipole case:
− +x
L
DµΦ(x) =
∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L
2 )]Φ(x)
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HollownessRecall dipole case:
− +x
L
DµΦ(x) =
∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L
2 )]Φ(x)
− +
Px
Field redefinition (attach Wilson line):
Φ(x) := e−i∫Px
AΦ(x) is gauge invariant.
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HollownessRecall dipole case:
− +x
L
DµΦ(x) =
∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L
2 )]Φ(x)
− +
Px
Field redefinition (attach Wilson line):
Φ(x) := e−i∫Px
AΦ(x) is gauge invariant.
DµΦ(x) = ∂µΦ(x) + iΦ(x)
∫Px
Fνµdxν
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HollownessRecall dipole case:
− +x
L
DµΦ(x) =
∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L
2 )]Φ(x)
− +
Px
Field redefinition (attach Wilson line):
Φ(x) := e−i∫Px
AΦ(x) is gauge invariant.
DµΦ(x) = ∂µΦ(x) + iΦ(x)
∫Px
Fνµdxν
P ′x
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HollownessRecall dipole case:
− +x
L
DµΦ(x) =
∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L
2 )]Φ(x)
− +
Px
Field redefinition (attach Wilson line):
Φ(x) := e−i∫Px
AΦ(x) is gauge invariant.
DµΦ(x) = ∂µΦ(x) + iΦ(x)
∫Px
Fνµdxν
P ′x
Px → P ′x can be compensated by a field redefinition.
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Hollowness for cuvepoles
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Hollowness for cuvepoles
Σ
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Hollowness for cuvepoles
Σ
Σ′
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Hollowness for cuvepoles
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
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Hollowness for cuvepoles
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
Vµ := 12 H(−)
µνσ
νσ
− 12 ∂
νϕµν
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Hollowness for cuvepoles
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
Vµ := 12 H(−)
µνσ
νσ
− 12 ∂
νϕµν
∂νϕµν
=∫
Σ ∂νϕ(x + y)dyµ ∧ dyν =
∫∂Σ ϕ(x + y)dyµ is hollow.
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Hollowness for cuvepoles
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
Vµ := 12 H(−)
µνσ
νσ
− 12 ∂
νϕµν
∂νϕµν
=∫
Σ ∂νϕ(x + y)dyµ ∧ dyν =
∫∂Σ ϕ(x + y)dyµ is hollow.
H(−)µνσ
νσ
would have been hollow if dH(−) = 0 but
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Hollowness for cuvepoles
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
Vµ := 12 H(−)
µνσ
νσ
− 12 ∂
νϕµν
∂νϕµν
=∫
Σ ∂νϕ(x + y)dyµ ∧ dyν =
∫∂Σ ϕ(x + y)dyµ is hollow.
H(−)µνσ
νσ
would have been hollow if dH(−) = 0 but
∂[αH(−)µνσ] = −3πi
32 εαµνσγδ ∂τJ[γ
δτ ]
+ 3π4 ∂[αJµ
νσ].
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Hollowness for cuvepoles
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
Vµ := 12 H(−)
µνσ
νσ
− 12 ∂
νϕµν
∂νϕµν
=∫
Σ ∂νϕ(x + y)dyµ ∧ dyν =
∫∂Σ ϕ(x + y)dyµ is hollow.
H(−)µνσ
νσ
would have been hollow if dH(−) = 0 but
∂[αH(−)µνσ] = −3πi
32 εαµνσγδ ∂τJ[γ
δτ ]
+ 3π4 ∂[αJµ
νσ]. Nevertheless . . .
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The action (again)
L(bosonic) =
112πHµνσH
µνσ + 14π∂µϕ∂
µϕ+ ∂µφi∂µφi + π
2 ∂µM j
i µσ∂νM
ij
νσ
︸ ︷︷ ︸hollow
−JµVµ + VµVµφiφ
i − 3π16 J [µ
σν]
J[µσν]︸ ︷︷ ︸
not hollow
Jµ := i(φi∂µφi − ∂µφiφi) U(1) current
M ji := φiφ
j − 12δ
jiφkφ
k SU(2)R triplet
Vµ := 12 H(−)
µνσ
νσ
−12 ∂
νϕµν︸ ︷︷ ︸
hollow
effective gauge field defined above
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Hollowness (continued)
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
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Hollowness (continued)
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
But with a suitable field redefinition ( δξBµν = · · · , δξφi = · · · , etc.),
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Hollowness (continued)
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
But with a suitable field redefinition ( δξBµν = · · · , δξφi = · · · , etc.),
δξ∫d6x
(· · · −JµVµ + VµV
µφiφi − 3π
16 J [µσν]
J[µσν]︸ ︷︷ ︸
non-hollow terms
)=
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Hollowness (continued)
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ Φµν
= ξσ∂σΦµν
+ ξµ∂σΦνσ− ξν∂σΦ
µσ
But with a suitable field redefinition ( δξBµν = · · · , δξφi = · · · , etc.),
δξ∫d6x
(· · · −JµVµ + VµV
µφiφi − 3π
16 J [µσν]
J[µσν]︸ ︷︷ ︸
non-hollow terms
)=
πi32
∫εαβγµνσ Jα
βγξτ∂τJ
µ νσd6x .
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Hollowness (continued . . . )
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
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Hollowness (continued . . . )
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ∫ (· · · · · · · · ·
)d6x = πi
32
∫εαβγµνσ Jα
βγξτ∂τJ
µ νσd6x .
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Hollowness (continued . . . )
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ∫ (· · · · · · · · ·
)d6x = πi
32
∫εαβγµνσ Jα
βγξτ∂τJ
µ νσd6x .
εαβγµνσ · · · βγ · · · νσ = 0 for 3-planar Σ (i.e., Σ ⊂ R3 ⊂ R6)
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Hollowness (continued . . . )
Σ
Σ′
ξ ξ : Σ→ R6
is the deformation vector
δξ∫ (· · · · · · · · ·
)d6x = πi
32
∫εαβγµνσ Jα
βγξτ∂τJ
µ νσd6x .
εαβγµνσ · · · βγ · · · νσ = 0 for 3-planar Σ (i.e., Σ ⊂ R3 ⊂ R6)
=⇒ δξ∫Ld6x = 0 for 3-planar Σ. (up to quartic terms)
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Applications of hollowness
Insisting on Hollowness is useful:
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Applications of hollowness
Insisting on Hollowness is useful:
1. SUSY alone doesn’t rule out adding a quartic term
∆L = (#)
(12 Jσ µν
Jσµν− ∂σM
ijµν∂σM j
i
µν
+ fermions
)︸ ︷︷ ︸
not hollow
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Applications of hollowness
Insisting on Hollowness is useful:
1. SUSY alone doesn’t rule out adding a quartic term
∆L = (#)
(12 Jσ µν
Jσµν− ∂σM
ijµν∂σM j
i
µν
+ fermions
)︸ ︷︷ ︸
not hollow
2. Ruling out higher order terms:
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Applications of hollowness
Insisting on Hollowness is useful:
1. SUSY alone doesn’t rule out adding a quartic term
∆L = (#)
(12 Jσ µν
Jσµν− ∂σM
ijµν∂σM j
i
µν
+ fermions
)︸ ︷︷ ︸
not hollow
2. Ruling out higher order terms:
hollowness requires a ∂ for every · · · .
Terms of the form ∂n−2 · · · n−2φn have dimension (n + 2);
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Applications of hollowness
Insisting on Hollowness is useful:
1. SUSY alone doesn’t rule out adding a quartic term
∆L = (#)
(12 Jσ µν
Jσµν− ∂σM
ijµν∂σM j
i
µν
+ fermions
)︸ ︷︷ ︸
not hollow
2. Ruling out higher order terms:
hollowness requires a ∂ for every · · · .
Terms of the form ∂n−2 · · · n−2φn have dimension (n + 2);
n + 2 > 6 for n > 4.
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Applications of hollowness
Insisting on Hollowness is useful:
1. SUSY alone doesn’t rule out adding a quartic term
∆L = (#)
(12 Jσ µν
Jσµν− ∂σM
ijµν∂σM j
i
µν
+ fermions
)︸ ︷︷ ︸
not hollow
2. Ruling out higher order terms:
hollowness requires a ∂ for every · · · .
Terms of the form ∂n−2 · · · n−2φn have dimension (n + 2);
n + 2 > 6 for n > 4. But . . .
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Is hollowness anomalous?
The field redefinition is
δBγδ = πi4 εαµνσγδ ξ
αJµνσ
δφi = iεφi,
δφi
= −iεφi,
δψi = iεψi,
δψi
= −iεψi,
ε := 12 ξ
αH(−)αµν
µν
.
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Is hollowness anomalous?
The field redefinition is
δBγδ = πi4 εαµνσγδ ξ
αJµνσ
δφi = iεφi,
δφi
= −iεφi,
δψi = iεψi,
δψi
= −iεψi,
ε := 12 ξ
αH(−)αµν
µν
.
Anomaly ∼∫εdV ∧ dV ∧ dV at order · · · 4
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Is hollowness anomalous?
The field redefinition is
δBγδ = πi4 εαµνσγδ ξ
αJµνσ
δφi = iεφi,
δφi
= −iεφi,
δψi = iεψi,
δψi
= −iεψi,
ε := 12 ξ
αH(−)αµν
µν
.
Anomaly ∼∫εdV ∧ dV ∧ dV at order · · · 4
Reminder: Vµ := 12 H(−)
µνσ
νσ
− 12 ∂
νϕµν
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Open Questions
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Open Questions
Higher than quartic orders? Additional restrictions on Σ?
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Open Questions
Higher than quartic orders? Additional restrictions on Σ?
Hollowness anomaly cancellation? (Green-Schwarz-like?)
Quantum corrections? (protected by anti-selfduality?)
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Open Questions
Higher than quartic orders? Additional restrictions on Σ?
Hollowness anomaly cancellation? (Green-Schwarz-like?)
Quantum corrections? (protected by anti-selfduality?)
How are these theories related to the M-theory construction?
Which Σ are realized? (Probably disks.)
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Open Questions
Higher than quartic orders? Additional restrictions on Σ?
Hollowness anomaly cancellation? (Green-Schwarz-like?)
Quantum corrections? (protected by anti-selfduality?)
How are these theories related to the M-theory construction?
Which Σ are realized? (Probably disks.)
Perhaps technique of [Ho & Matsuo, 2008, and Ho & Huang & Matsuo, 2011],
applying Bagger-Lambert-Gustavsson theory to
Poisson triple-product, can help here.
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Open Questions
Higher than quartic orders? Additional restrictions on Σ?
Hollowness anomaly cancellation? (Green-Schwarz-like?)
Quantum corrections? (protected by anti-selfduality?)
How are these theories related to the M-theory construction?
Which Σ are realized? (Probably disks.)
Perhaps technique of [Ho & Matsuo, 2008, and Ho & Huang & Matsuo, 2011],
applying Bagger-Lambert-Gustavsson theory to
Poisson triple-product, can help here.
Applications to interactions induced on M5-brane by G -flux?
cf. [Perry & Schwarz, Aganagic & Park & Popescu & Schwarz, Bergshoeff & Berman & van der Schaar
& Sundell, Gopakumar & Minwalla & Seiberg & Strominger, Lambert & Orlando & Reffert & Sekiguchi]
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More Questions
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More Questions
Can pure-spinor techniques simplify the equations?
cf. [Cederwall & Karlsson, 2011] for BI.
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More Questions
Can pure-spinor techniques simplify the equations?
cf. [Cederwall & Karlsson, 2011] for BI.
Recast in projective superspace?[Galperin & Ivanov & Ogievetsky & Sokatchev; Karlhede & Lindstrom & Rocek]
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Summary
Constructed a SUSY action of interacting curvepoles
(up to quartic terms)
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Summary
Constructed a SUSY action of interacting curvepoles
(up to quartic terms)
There is a geometric constraint on curvepole shape
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Summary
Constructed a SUSY action of interacting curvepoles
(up to quartic terms)
There is a geometric constraint on curvepole shape
Explicit formulas for the quartic interactions
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Thanks!
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MORE DETAILS
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Antiselfduality with interactions
Action: S = 14π
∫dB ∧ ∗dB +
∫(dB − ∗dB) ∧ K
Field strength: H(±) := 12 (dB ± ∗dB)
EOMs: 0 = d[
12π∗dB + (K + ∗K )
]= d
[1πH
(+) + (K + ∗K )]
Corrections to free anti-selfduality condition:
H(+) = −π(K + ∗K )
Can define: H(+) := 12 (dB + ∗dB) + π(K + ∗K ) = 0
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Spinor notation in 6Da,b, c,d, . . . = 1, . . . 4 spinor indices
∂ab = 12εabcd∂
cd derivative
Bab 2-form (Ba
a = 0)
H(−)ab = H
(−)ba antiselfdual 3-form
H(+)ab = H(+)ba selfdual 3-form
H(−)ab = 2∂c(aBb)
c ⇐⇒H
(−)µνσ = 1
2 (Hµνσ − i3!εµνσαβγH
αβγ)
for Hµνσ = 3∂[σBµν]
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SUSY transformations at 0th order
δ0φi = ηaiψa
δ0φi = ηai ψa
δ0ψa = −2iηbi ∂abφi
δ0ψa = −2iηbi ∂abφi
δ0ϕ = ηai χia
δ0χia = iηbi∂abϕ+ iηbiHab
δ0Hab = ηci ∂caχib + ηci ∂cbχ
ia
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Effective vector multipletOut of H and χ we construct a vector multiplet:
Vab := Hc[ac
b]+ ∂c[a ϕ
c
b] Vector field
ρia := ∂ab χic
c
bGaugino field
Fba = ∂bcVac − 1
4δba∂
dcVdc Associated field strength
Their 0th order SUSY transformations:
δ0Vab = −ηai ρib + ηbi ρia − ∂abλ
δ0ρia = 2iηbiFa
b
λ := ηai χib
b
aField-dependent gauge parameter
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SUSY transformations at 1st order
δ1φi = −iηaj φi χ
jb
b
a
δ1φi
= iηaj φiχjb
b
a
δ1ψa = 2ηbi Hc[bc
a]φi + 2ηbi φ
i∂c[b ϕc
a]+ iηbi ψa χ
ic
c
b
δ1ψa = 2ηbi Hc[ac
b]φi+ 2ηbi φ
i∂c[a ϕ
c
b]− iηbi ψa χ
ic
c
b
δ1ϕ = 0
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SUSY transformations at 1st order (Continued)
δ1χia = πiηci ψbψ(a
b
c)+ πiηci ψbψ(a
b
c)
−πηcj12φ
j∂bcφ
i + 12φ
j∂bcφi+ 3
2φi∂bcφ
j + 32φ
i∂bcφj
b
a
−πηcj14φ
i∂abφj+ 1
4φi∂abφ
j + 34φ
j∂abφi+ 3
4φj∂abφ
ib
c
δ1Bab = − iπ
2 ηci ψcφ
i+ ψcφ
ia
b+ iπηci ψbφ
i+ ψbφ
ia
c
+iπηai ψcφi+ ψcφ
ic
b− iπ
2 δabη
ci ψdφ
i+ ψdφ
id
c
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Origin of Balanced Curvepole condition
We want (δ0 + δ1 + δ2 + · · · )∫
(L2 + L3 + L4 + · · · ) = 0.
Look at the φψψψ terms in δ1L3
+ηei ∂ab φiψa
c
bψcψd
d
e− ηei ∂ab φ
iψa
d
eψcψd
c
b
+ηei ∂ab φiψa
c
eψcψd
d
b− ηei ∂ab φ
iψa
d
bψcψd
c
e
+ 12η
ei ∂
ab φiψa
c
dψbψc
d
e− 1
2ηei ∂
ab φiψa
d
eψbψc
c
d
+ 12η
ei ∂
ab φiψa
c
eψdψb
d
c− 1
2ηei ∂
ab φiψa
d
cψdψb
c
e
Doesn’t seem to be possible to cancel with δ0L4 + δ2L2 unless
∫X
c
b Yd
e =∫
Xd
e Yc
b
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Summary (repeat)
Constructed a SUSY action of interacting curvepoles
(up to quartic terms)
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Summary (repeat)
Constructed a SUSY action of interacting curvepoles
(up to quartic terms)
There is a geometric constraint on curvepole shape
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Summary (repeat)
Constructed a SUSY action of interacting curvepoles
(up to quartic terms)
There is a geometric constraint on curvepole shape
Explicit formulas for the quartic interactions
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Summary (repeat)
Constructed a SUSY action of interacting curvepoles
(up to quartic terms)
There is a geometric constraint on curvepole shape
Explicit formulas for the quartic interactions
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The bigger picture?
This is another example of new nonlocal theories that appear
when probing regions of strong flux with branes.
Other examples include:
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The bigger picture?
This is another example of new nonlocal theories that appear
when probing regions of strong flux with branes.
Other examples include:
Noncommutative spacetime[Connes & Douglas & Schwarz, Douglas & Hull, Seiberg & Witten, 1998]
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The bigger picture?
This is another example of new nonlocal theories that appear
when probing regions of strong flux with branes.
Other examples include:
Noncommutative spacetime[Connes & Douglas & Schwarz, Douglas & Hull, Seiberg & Witten, 1998]
Dipole theories
Puff Field Theory [Hashimoto & Jue & Kim & Ndirango & OG, 2007]
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Thanks!
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