string and m-theory: the new geometry of the 21st century ... · string and m-theory: the new...

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.

1 41

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.

2 41

Dipoles and Curvepoles

Curvepole – counterpart of E&M dipole, interacting with a 2-form.

3 41

Dipoles and Curvepoles

Curvepole – counterpart of E&M dipole, interacting with a 2-form.

E&M Tensor field

3 41

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.

3 41

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:

3 41

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.

3 41

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.

3 41

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.

3 41

The Goal

Construct a supersymmetric second quantized theory of curvepolesinteracting with a 6D tensor multiplet of (1, 0) SUSY.

4 41

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:

4 41

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:

4 41

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:

4 41

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:

4 41

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.

4 41

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?

4 41

Preview of resultsConstruct action order by order:

L = L2 + L3 + L4 + · · ·δSUSY = δ0 + δ1 + δ2 + · · ·

5 41

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.

5 41

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.

5 41

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.

5 41

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)

5 41

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.

5 41

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).

5 41

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.

5 41

A digression: Connection to M-theory

T 3

Vol(T 3)→ 0

M2

6 41

A digression: Connection to M-theory

T 3

Vol(T 3)→ 0

M2

U-duality

6 41

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

6 41

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

6 41

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

6 41

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]

6 41

M-theory construction (more details)

T 3 T 3

M2

U

M5

7 41

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

7 41

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

7 41

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

7 41

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

7 41

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

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

8 41

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

8 41

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

8 41

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

8 41

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

8 41

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]

8 41

Deforming the free theory to Curvepole Theory

9 41

Deforming the free theory to Curvepole Theory

Free 6D (2, 0) tensor ’plet

9 41

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)

9 41

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)

9 41

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:

9 41

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.

9 41

Recall: 2nd quantized dipolesDipole:

− +

10 41

Recall: 2nd quantized dipolesDipole:

− +x

10 41

Recall: 2nd quantized dipolesDipole:

− +x

L

L is a fixed vector.(Counterpart of Σ)

10 41

Recall: 2nd quantized dipolesDipole:

− +x

L

L is a fixed vector.(Counterpart of Σ)

Φ is 2nd quantized field of dipole.

10 41

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)

10 41

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]

10 41

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 .”

10 41

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

10 41

Curvepole Theory (first try)

We want quanta of φ to be curvepoles:

[interacting with (H(−), χi, ϕ)]

11 41

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

11 41

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])

11 41

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).

11 41

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:

11 41

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

11 41

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)

11 41

Curvepole Theory

We want quanta of φ to be curvepoles:

[interacting with (H(−), χi, ϕ)]

12 41

Curvepole Theory

We want quanta of φ to be curvepoles:

[interacting with (H(−), χi, ϕ)]

Σ – fixed shape (open surface) around origin.

Σ

12 41

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)

12 41

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

12 41

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

12 41

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

12 41

What is the condition on Σ?

Define the curvepole integrals: Σ

13 41

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

13 41

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.

13 41

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)

13 41

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

13 41

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.

13 41

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

14 41

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

14 41

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

14 41

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)

14 41

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

15 41

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)

15 41

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)

15 41

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)

15 41

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

16 41

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

17 41

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

18 41

Hollowness

C = ∂Σ

Σ

19 41

Hollowness

C = ∂Σ

Σ

It would be nice if the theory depended only on C ; not on Σ.

19 41

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.

19 41

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.

19 41

HollownessRecall dipole case:

− +

20 41

HollownessRecall dipole case:

− +x

20 41

HollownessRecall dipole case:

− +x

L

20 41

HollownessRecall dipole case:

− +x

L

DµΦ(x) =

∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L

2 )]Φ(x)

20 41

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.

20 41

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

20 41

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

20 41

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.

20 41

Hollowness for cuvepoles

21 41

Hollowness for cuvepoles

Σ

21 41

Hollowness for cuvepoles

Σ

Σ′

21 41

Hollowness for cuvepoles

Σ

Σ′

ξ ξ : Σ→ R6

is the deformation vector

δξ Φµν

= ξσ∂σΦµν

+ ξµ∂σΦνσ− ξν∂σΦ

µσ

21 41

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

21 41

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.

21 41

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

21 41

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].

21 41

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 . . .

21 41

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

22 41

Hollowness (continued)

Σ

Σ′

ξ ξ : Σ→ R6

is the deformation vector

δξ Φµν

= ξσ∂σΦµν

+ ξµ∂σΦνσ− ξν∂σΦ

µσ

23 41

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.),

23 41

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

)=

23 41

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 .

23 41

Hollowness (continued . . . )

Σ

Σ′

ξ ξ : Σ→ R6

is the deformation vector

24 41

Hollowness (continued . . . )

Σ

Σ′

ξ ξ : Σ→ R6

is the deformation vector

δξ∫ (· · · · · · · · ·

)d6x = πi

32

∫εαβγµνσ Jα

βγξτ∂τJ

µ νσd6x .

24 41

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)

24 41

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)

24 41

Applications of hollowness

Insisting on Hollowness is useful:

25 41

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

25 41

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:

25 41

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);

25 41

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.

25 41

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 . . .

25 41

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

.

26 41

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

26 41

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

26 41

Open Questions

27 41

Open Questions

Higher than quartic orders? Additional restrictions on Σ?

27 41

Open Questions

Higher than quartic orders? Additional restrictions on Σ?

Hollowness anomaly cancellation? (Green-Schwarz-like?)

Quantum corrections? (protected by anti-selfduality?)

27 41

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.)

27 41

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.

27 41

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]

27 41

More Questions

28 41

More Questions

Can pure-spinor techniques simplify the equations?

cf. [Cederwall & Karlsson, 2011] for BI.

28 41

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]

28 41

Summary

Constructed a SUSY action of interacting curvepoles

(up to quartic terms)

29 41

Summary

Constructed a SUSY action of interacting curvepoles

(up to quartic terms)

There is a geometric constraint on curvepole shape

29 41

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

29 41

Thanks!

30 41

MORE DETAILS

31 41

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

32 41

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]

33 41

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

34 41

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

35 41

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

36 41

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

37 41

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

38 41

Summary (repeat)

Constructed a SUSY action of interacting curvepoles

(up to quartic terms)

39 41

Summary (repeat)

Constructed a SUSY action of interacting curvepoles

(up to quartic terms)

There is a geometric constraint on curvepole shape

39 41

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

39 41

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

39 41

The bigger picture?

This is another example of new nonlocal theories that appear

when probing regions of strong flux with branes.

Other examples include:

40 41

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]

40 41

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]

40 41

Thanks!

41 41