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



E&M Tensor field

3 41



E&M Tensor field

− +

Dipole:

Two endpoints interacting

with a 1-form gauge field A

with opposite charges.

3 41



E&M Tensor field

− +

Dipole:




C

Σ

Curvepole:

3 41



E&M Tensor field

− +

Dipole:




C

Σ

Curvepole:

Has closed curve boundary C = ∂Σ

charged under 2-form gauge field B.

3 41



E&M Tensor field

− +

Dipole:




C

Σ

Curvepole:



Bulk of Σ is inert.

3 41



E&M Tensor field

− +

Dipole:




C

Σ

Curvepole:



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


dB = H = −?H interacts with a field Φ whose quanta are:

4 41

The Goal



NOTE: The shape C of these curvepoles is fixed!

C is an external parameter of the theory.

4 41

The Goal



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


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


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



− I’ll present explicit formulas for up to quartic interactions.

5 41


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




− There is a restriction on C , imposed by SUSY.

5 41


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




− There is a restriction on C , imposed by SUSY. (sufficient condition)

5 41


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





− There is another restriction on C , imposed by Hollowness.

5 41


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






♦ If C is planar, both SUSY and Hollowness hold? (up to 4th order).

5 41


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






♦ 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


T 3

Vol(T 3)→ 0

M2

U-duality

6 41


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


T 3

Vol(T 3)→ 0

M2

U-duality

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


T 3

Vol(T 3)→ 0

M2

U-duality

U-dual T 3


Wrapped M5:

freehyper ’plet

& tensor ’plet

(1,0) multipletsΩ



curvepole

6 41


T 3

Vol(T 3)→ 0

M2

U-duality

U-dual T 3


Wrapped M5:

freehyper ’plet

& tensor ’plet

(1,0) multipletsΩ



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


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


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


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



⊂ SO(5)6,...,10

Momentum quantization: P5R5 ∈ Z + θJ2π

7 41


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



⊂ SO(5)6,...,10


Dual to : M2-area ∈ Z(2π)2R3R4 + (2πθR3R4)︸︷︷︸keep finite

J

7 41


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



⊂ SO(5)6,...,10


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




M5x7 x6

0, . . . , 5

8 41




Strong G1267 flux

M5x7 x6

0, . . . , 5

8 41




Strong G1267 flux

M5x7 x6

0, . . . , 5

M2

8 41




Strong G1267 flux

M5x7 x6

0, . . . , 5

M2

Σ

8 41




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


Free 6D (2, 0) tensor ’plet

9 41



(1, 0) tensor(B, χi, ϕ) i = 1, 2

SU(2)R index: (1, 0) hyper(φi, ψ)

9 41



(1, 0) tensor(B, χi, ϕ)

H(+) := 12 (dB + ∗dB) = 0

i = 1, 2SU(2)R index: (1, 0) hyper

(φi, ψ)

9 41




H(+) := 12 (dB + ∗dB) = 0


(φi, ψ)

want their quanta

to become curvepoles:

9 41




H(+) := 12 (dB + ∗dB) = 0


(φi, ψ)

want their quanta

to become curvepoles:

want their quanta

to stay pointlike.

9 41

Recall: 2nd quantized dipolesDipole:

− +

10 41


− +x

10 41


− +x

L

L is a fixed vector.(Counterpart of Σ)

10 41


− +x

L


Φ is 2nd quantized field of dipole.

10 41


− +x

L



Modified covariant derivative:

DµΦ(x) = ∂µΦ(x) + iAµ(x + L2 )Φ(x)− iAµ(x − L

2 )Φ(x)

10 41


− +x

L





2 )Φ(x)

[Bergman & OG, Bergman & Dasgupta & Karczmarek & Rajesh & OG]

10 41


− +x

L





2 )Φ(x)


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


− +x

L





2 )Φ(x)


Attach Wilson line:

− +

Px

P is a fixed path from −L2 to L

2 .

Φ(x) := e−i∫Px


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




Σ – fixed shape (open surface) around origin.

Σ

11 41





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





Σ




But action should depend only on H(−) := 12 (dB − ∗dB).

11 41





Σ





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





Σ








⇒ Dµφ(x) := ∂µφ(x) + i φ(x)∫

Σ(dB)µνσ(x + y)dyν ∧ dyσ

11 41





Σ








⇒ 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



12 41

Curvepole Theory




Σ

12 41

Curvepole Theory




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




Σ



Vµ(x) := 12


(−)µνσ(x + y) −1

2

∫∂Σ dyµϕ(x + y)

required for SUSY

12 41

Curvepole Theory




Σ



Vµ(x) := 12


(−)µνσ(x + y) −1

2


required for SUSYAdditional terms in L

12 41

Curvepole Theory




Σ



Vµ(x) := 12


(−)µνσ(x + y) −1

2


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



Ψµν

:=∫

Σ Ψ(x + y)dyµ ∧ dyν

Ψµν

:=∫

Σ Ψ(x − y)dyµ ∧ dyν

13 41



Ψµν

:=∫


Ψµν

:=∫


A condition on Σ appears by requiring SUSY at quartic order.

13 41



Ψµν

:=∫


Ψµν

:=∫



SUSY preserved if:

(up to quartic order)

13 41



Ψµν

:=∫


Ψµν

:=∫



SUSY preserved if:


Ψ αβγδ

= Ψ γδαβ

for arbitrary Ψ

13 41



Ψµν

:=∫


Ψµν

:=∫



SUSY preserved if:


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:=∫


Ψµν

:=∫


Balanced curvepole: Ψ αβγδ

= Ψ γδαβ

for arbitrary Ψ

14 41

Balanced Curvepoles

ΣΨ

µν:=∫


Ψµν

:=∫



= 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:=∫


Ψµν

:=∫



= Ψ γδαβ

for arbitrary Ψ



for arbitrary Ψ,Φ

(This is a stronger condition)

Ψ αβ Φ γδ = Ψ γδ Φ αβ

14 41

Balanced Curvepoles

ΣΨ

µν:=∫


Ψµν

:=∫



= Ψ γδαβ

for arbitrary Ψ



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

:=∫


Ψµν

:=∫



= Ψ γδαβ

for arbitrary Ψ

15 41


ΣΨµν

:=∫


Ψµν

:=∫



= Ψ γδαβ

for arbitrary Ψ

Parity invariant (Σ = −Σ) (balanced)

15 41


ΣΨµν

:=∫


Ψµν

:=∫



= Ψ γδαβ

for arbitrary Ψ


Planar (Σ ⊂ R2 ⊂ R6) (balanced)

15 41


ΣΨµν

:=∫


Ψµν

:=∫



= Ψ γδαβ

for arbitrary Ψ


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


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


If ∂Σ = ∂Σ′, we’d like theory (Σ) ' theory(Σ′)

up to a field redefinition.

19 41

Hollowness

C = ∂Σ

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


− +x

20 41


− +x

L

20 41


− +x

L

DµΦ(x) =

∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L

2 )]Φ(x)

20 41


− +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


20 41


− +x

L

DµΦ(x) =

∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L

2 )]Φ(x)

− +

Px


Φ(x) := e−i∫Px



∫Px

Fνµdxν

20 41


− +x

L

DµΦ(x) =

∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L

2 )]Φ(x)

− +

Px


Φ(x) := e−i∫Px



∫Px

Fνµdxν

P ′x

20 41


− +x

L

DµΦ(x) =

∂µΦ(x) + i [Aµ(x + L2 )− Aµ(x − L

2 )]Φ(x)

− +

Px


Φ(x) := e−i∫Px



∫Px

Fνµdxν

P ′x

Px → P ′x can be compensated by a field redefinition.

20 41

Hollowness for cuvepoles

21 41


Σ

21 41


Σ

Σ′

21 41


Σ

Σ′

ξ ξ : Σ→ R6

is the deformation vector

δξ Φµν

= ξσ∂σΦµν

+ ξµ∂σΦνσ− ξν∂σΦ

µσ

21 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ Φµν

= ξσ∂σΦµν


µσ

Vµ := 12 H(−)

µνσ

νσ

− 12 ∂

νϕµν

21 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ Φµν

= ξσ∂σΦµν


µσ

Vµ := 12 H(−)

µνσ

νσ

− 12 ∂

νϕµν

∂νϕµν

=∫

Σ ∂νϕ(x + y)dyµ ∧ dyν =

∫∂Σ ϕ(x + y)dyµ is hollow.

21 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ Φµν

= ξσ∂σΦµν


µσ

Vµ := 12 H(−)

µνσ

νσ

− 12 ∂

νϕµν

∂νϕµν

=∫



H(−)µνσ

νσ

would have been hollow if dH(−) = 0 but

21 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ Φµν

= ξσ∂σΦµν


µσ

Vµ := 12 H(−)

µνσ

νσ

− 12 ∂

νϕµν

∂νϕµν

=∫



H(−)µνσ

νσ


∂[αH(−)µνσ] = −3πi

32 εαµνσγδ ∂τJ[γ

δτ ]

+ 3π4 ∂[αJµ

νσ].

21 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ Φµν

= ξσ∂σΦµν


µσ

Vµ := 12 H(−)

µνσ

νσ

− 12 ∂

νϕµν

∂νϕµν

=∫



H(−)µνσ

νσ


∂[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


22 41

Hollowness (continued)

Σ

Σ′

ξ ξ : Σ→ R6


δξ Φµν

= ξσ∂σΦµν


µσ

23 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ Φµν

= ξσ∂σΦµν


µσ

But with a suitable field redefinition ( δξBµν = · · · , δξφi = · · · , etc.),

23 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ Φµν

= ξσ∂σΦµν


µσ


δξ∫d6x

(· · · −JµVµ + VµV

µφiφi − 3π

16 J [µσν]


non-hollow terms

)=

23 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ Φµν

= ξσ∂σΦµν


µσ


δξ∫d6x

(· · · −JµVµ + VµV

µφiφi − 3π

16 J [µσν]


non-hollow terms

)=

πi32

∫εαβγµνσ Jα

βγξτ∂τJ

µ νσd6x .

23 41

Hollowness (continued . . . )

Σ

Σ′

ξ ξ : Σ→ R6


24 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ∫ (· · · · · · · · ·

)d6x = πi

32


βγξτ∂τJ

µ νσd6x .

24 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ∫ (· · · · · · · · ·

)d6x = πi

32


βγξτ∂τJ

µ νσd6x .

εαβγµνσ · · · βγ · · · νσ = 0 for 3-planar Σ (i.e., Σ ⊂ R3 ⊂ R6)

24 41


Σ

Σ′

ξ ξ : Σ→ R6


δξ∫ (· · · · · · · · ·

)d6x = πi

32


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



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




∆L = (#)

(12 Jσ µν

Jσµν− ∂σM

ijµν∂σM j

i

µν

+ fermions

)︸︷︷︸

not hollow

2. Ruling out higher order terms:

25 41




∆L = (#)

(12 Jσ µν

Jσµν− ∂σM

ijµν∂σM j

i

µν

+ fermions

)︸︷︷︸

not hollow


hollowness requires a ∂ for every · · · .

Terms of the form ∂n−2 · · · n−2φn have dimension (n + 2);

25 41




∆L = (#)

(12 Jσ µν

Jσµν− ∂σM

ijµν∂σM j

i

µν

+ fermions

)︸︷︷︸

not hollow




n + 2 > 6 for n > 4.

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∆L = (#)

(12 Jσ µν

Jσµν− ∂σM

ijµν∂σM j

i

µν

+ fermions

)︸︷︷︸

not hollow




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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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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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


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

Quantum corrections? (protected by anti-selfduality?)

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Open Questions




How are these theories related to the M-theory construction?

Which Σ are realized? (Probably disks.)

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Open Questions






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






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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String and M-Theory: The New Geometry of the 21st Century ... · String and M-Theory: The New...

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