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Toward a Proof of Montonen-Olive Duality via Multiple M2-branes

25th Sep 2008 @KIAS conference “Recent Developments in String/M Theory”. Toward a Proof of Montonen-Olive Duality via Multiple M2-branes. Koji Hashimoto (RIKEN). arXiv/0809.2137(hep-th) w/ T.S.Tai, S.Terashima. Why BLG/ABJM ?.  Integration of .  Integration of .

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Toward a Proof of Montonen-Olive Duality via Multiple M2-branes

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  1. 25th Sep 2008 @KIAS conference “Recent Developments in String/M Theory” Toward a Proof of Montonen-Olive Duality via Multiple M2-branes Koji Hashimoto (RIKEN) arXiv/0809.2137(hep-th) w/ T.S.Tai,S.Terashima

  2. Why BLG/ABJM ?

  3.  Integration of   Integration of  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 Let’s remind Abelian duality for elemag system 3 dimensions: Scalar – vector duality 4 dimensions: Elemag duality

  4. 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

  5. 4d U(N) super Yang-Mills has a non-local symmetry of the theory, Montonen-Olive duality conjecture Proof wanted !!!

  6. IIA string w/ S1 w/ S1 w/ IIB string w/ S1 w/ 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

  7. 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

  8. Plan 1. Why BLG/ABJM? 2. M2  D2 : Review 3. M2  D3 4. MO duality 4 slides 3 slides 6 slides 3 slides

  9. 2. M2  D2 : Review [Mukhi-Papageorgakis] [Distler-Mukhi-Papageorgakis-VanRaamsdonk]

  10. 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

  11. : fixed 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

  12. : fixed Scalar kinetic term for dissapears. 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)

  13. 3. M2  D3

  14. 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

  15. Torus compactification Moduli space : [Terashima,Yagi] [Imamura,Kimura] It’s already a double orbifold, so we take Turn on two vevs : M2 on a shrinking torus

  16. Diagonalize gauge fields Think of them as KK modes of a 4d YM 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

  17. 1 2 3 2n 1st step For simplicity in this talk I will explain the case Linear combination : Integrating out the auxiliary fields

  18. 2nd step  Diagonalize. In the limit , KK radius equivalent 4d YM

  19. For generic , cross terms remain  theta 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] Taylor’s T-duality. [Taylor]

  20. 4. MO duality

  21. 1 1 1 2 2 2 3 3 3 2n 2n 2n Choice of the combination Prime case Second case Third case

  22. Emergence of SL(2,Z) These couplings are related by SL(2,Z) transformation! Generators : Prime  Second Prime  Third Choice of combination of CS gauge fields SL(2,Z) transformation =

  23. Identified 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

  24. 5. Summary

  25. No. Our S is a parity, due to our special values of . 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.

  26. 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

  27. Discussions Toward a Proof ? Other vevs don’t help for our quiver…..  Different quivers necessary? IIB physics  M physics? Interpolation of AdS5 AdS4 ?

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