Toward a Proof of MontonenOlive Duality via Multiple M2branes

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Toward a Proof of Montonen-Olive Duality via
Multiple M2-branes
25th Sep 2008 _at_KIAS conference Recent
Developments in String/M Theory

• Koji Hashimoto (RIKEN)

arXiv/0809.2137(hep-th) w/ T.S.Tai, S.Terashima
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Why BLG/ABJM ?
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Use of ABJM?
Forget importance of M-theory.
Is BLG / ABJM useful for any other purpose?
Ans Surely it is very useful, for non-Abelian
duliaty
Lets remind Abelian duality for elemag system
3 dimensions Scalar vector duality
4 dimensions Elemag duality
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Non-Abelianization?
Q Can we non-Abelianize these classical
dualities?
3 dimensions Yes
Nonabelian Scalar-Chern-Simons ? 3d YM
Mukhi-Papageorgakis
Distler-Mukhi-Papageorgakis-VanRaamsdonk
Real help of string/M theory for field theory
duality !
4 dimensions Montonen-Olive duality
Renowned conjecture, no proof yet.
Our target Proof of MO duality via ABJM
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Montonen-Olive duality conjecture
Proof wanted !!!
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Non-Abelian dualities ? String/M-theory
M-theory
IIA string on S1
D2
M2
D2
T-dual
IIB string on S1
D3
D3
M ? IIA relations
M2 ? D2 Scalar-Vector duality
M ? IIB relations
Decompactified IIB M on shrinking torus
D3 ? D3 MO duality 9-11 flip in M-theory
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Strategy for a field theory proof of MO
ABJM theory of multiple M2-branes
Realize a torus compactification of transverse
space for ABJM
9-11 flip
Shrinking torus limit
...
...




4d YM
4d YM
4d YM
MO duality
Question 1 Torus in ABJM ?
Ans Quiver generalization of ABJM 2 vevs
? Sec.3
Question 2 Choice of the torus cycles ?
Ans Different pairings of CS gauge fields
? Sec.4
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Plan

• 1. Why BLG/ABJM?
• 2. M2 ? D2 Review
• 3. M2 ? D3
• 4. MO duality

4 slides
3 slides
6 slides
3 slides
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2. M2 ? D2 Review
Mukhi-Papageorgakis
Distler-Mukhi-Papageorgakis-VanRaamsdonk
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ABJM theory
Aharony,Bergman,Jafferis,Maldacena
U(N) x U(N) ABJM theory N M2s on C4 / Zk
U(1) gauge transf.
Boundary term remains,
devides the C4
? Moduli space C4 / Zk
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ABJM and circle compactification
Distler-Mukhi-Papageorgakis-VanRaamsdonk
M2
M2
Orbifold C4 / Zk
S1 compactification
Quiver diagram of ABJM
Linear combination
Vev gives the mass term for
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ABJM ? 3d YM
Mukhi-Papageorgakis
Distler-Mukhi-Papageorgakis-VanRaamsdonk
CS kinetic term is written as
The massive is an auxiliary field,
integrated out
3d YM
ABJM in the limit D2 action (3d U(N) YM)
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3. M2 ? D3
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Another circle ?
We need a torus Another orbifold is necessary
Orbifolded U(N)n x U(N)n ABJM
Bena,Klebanov,Klose,Smedback Terashima,Yagi
Imamura,Kimura Hosomichi, Lee3,Park
ABJM
Douglas-Moore
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Torus compactification
Moduli space
Terashima,Yagi Imamura,Kimura
Its already a double orbifold, so we take
Turn on two vevs
M2 on a shrinking torus
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2 steps for getting 4d YM
(i)
(ii)
Infinite of 3d massive YMs
Orbifolded ABJM
4d YM
Introduce two scalar vevs
Make a linear combination
Integrate out
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1st step
For simplicity in this talk I will explain the
case
Linear combination
Integrating out the auxiliary fields
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2nd step
? Diagonalize.
In the limit ,
KK radius
equivalent
4d YM
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Final 4d action
We obtain 4d YM action from the orbifolded ABJM,
Theta term appears.
3d ? 4d is .
Deconstruction.
Arkani-Hamed,Cohen,Georgi Hill,Pokorski,Wang
Taylors T-duality.
Taylor
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4. MO duality
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Choice of the combination
Prime case
Second case
Third case
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Emergence of SL(2,Z)
These couplings are related by SL(2,Z)
transformation!
Generators
Prime ? Second
Prime ? Third
SL(2,Z) transformation
Choice of combination of CS gauge fields

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MO duality proven ?
Is this a proof of MO duality ?!?!
Not really..
We find Our S transf. Parity transf.
Our coupling is a one-parameter family
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5. Summary
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Summary importance

Torus compactification of M2 transverse space is
realized.
Orbifolded ABJM 2 vevs, with a new scaling limit
4d YM on N D3s is derived from 3d ABJM.
Deconstruction (T-duality) with theta term
Infinitely many 4d YM are derived.
New geometric interpretation of quiver diagrams
Montonen-Olive duality is proven?
However, field theory realization of T transf. is
nontrivial
We got half the way to the goal.
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M-theory interpretation
2 vevs
? 2 1-cycles specified by vectors
Torus modurus
Due to M-IIB duality, this should become
axio-dilaton.
Our field theory result is identical with this
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Discussions

Toward a Proof ?
Other vevs dont help for our quiver..
? Different quivers necessary?
IIB physics ? M physics?
Interpolation of AdS5 ? AdS4 ?
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