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M5-branes on a Circle

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M5-branes on a Circle

Kuo-Wei Huang (NTU)

Taiwan String Group Seminar Sep.16 2011

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Based on the paper:

P.-M. Ho, K.-W. Huang, and Y. Matsuo,

[A Non-Abelian Self-dual Gauge Theory in 5+1 Dimensions], JHEP 07 (2011) 021, [1104.4040].

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Outline Outline



Motivation : a short review



Gauge symmetry and Field strengths



Non-Abelianizing the Abelian theory



Action



Discussions



Summary and Outlook

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Motivation

 The precise formulation of “M-theory” is still unknown.

--- Its low-energy limit is 11D Supergravity.

 Fundamental objects: M2-brane and M5-brane.

--- M5-brane is the magnetic partner of M2-brane

 D-branes are hyperplanes with open strings ending on.

--- This picture can be shifted to M2-brane ends on M5- brane in 11D.

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Motivation

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Motivation

 There is a 2-form potential living in M5 that couples to string boundary of the M2.

 A nontrivial feature: in order to match the fermionic

degrees of freedom (8 modes) with Bosonic degrees of freedom.

 However, a 2-form in 6D has 6 polarizations, and

together with the 5 scalars: the degrees of freedom do not match..

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Motivation



The ‘’2-form’’ potential in fact should be self- dual:

Satisfies the self-dual condition:

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Motivation



No way to derive this system from an standard action:

 Also note: Self-duality condition is 1st order differential equation, which can not be obtained from an ordinary Kinetic term.

If

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Motivation

 The way out: Treat one direction differently (lost the covariance)

-- M.Perry and J.Schwarz (1996). ’’5+1’’ formulation

 ’’3+3’’ formulation: found by deriving M5-brane theory from the BLG model -- P. M. Ho and Y. Matsuo (2008).

’’4+2’’ formulation.-- W.-M. Chen and P.-M. Ho (2010).

 The manifestly covariant formulation by introducing the auxiliary field.—P.Pasti, D.Sorokin, M.Tonin (1996,2010).

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Motivation

 We have the nice formulation of a single M5-brane.

 But, how about N coinciding M5-branes?

 Recall that the U(1) theory of a D-brane get enhanced to the U(N) theory when N D-branes coincide.

 We expect that : N coinciding M5-branes is a kind of non-Abelian 2-form gauge theory:

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Motivation

 No-Go theorem: No non-trivial deformation of 2-form potential.—M. Henneaux, X. Bekaert, A. Sevrin (1999)

 Another subtle point: since M5-branes should reduce to D4-branes after the dimensional reduction

In contrast with the expected one:

by integrating over the reduced dimension from 6D..

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Motivation

 Here we will only focus on the gauge field part, which brings all the subtleties.

 We will build a non-Abelian chiral 2-form theory satisfying two major criteria:

(1) Reduce to a single M5 when the gauge symmetry is Abelian.

(2) Reduce to multiple D4 under dimensional reduction.

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Gauge Symmetry

 Consider M5-branes compactified on a circle of radius R, including all the Kaluza-Klein (KK) modes.

 Decomposition:

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Gauge Symmetry



A key point:

 Notice that previous no-go theorem assume the local deformation.

This non-local operator is well defined.

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Gauge Symmetry

 Define the covariant derivative:

and the field strength

:

 The should correspond to the YM gauge potential

‘’A’’ in D4-branes.

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Gauge Symmetry



Define the non-Abelian gauge transformation of 2-form (decomposed into zero/KK modes):

(1) Lie-algebra, and reduce to abelian case in abelian phase.

(2) Commutators involve at most one KK mode.

(3) Treat zero modes and KK modes as independent fields.

(we discuss this term later)

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Gauge Symmetry



Gauge transformations is closed by

where

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Gauge Symmetry

Why we need the non-locality?

It is crucial that in the 2-form theory, one has the “redundant symmetry”

is invariant under

Thus, only five of six gauge parameters are independent.

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Gauge Symmetry



Non-Abelian redundant symmetry:

 The non-locality insures the redundant symmetry can still exist in the non-Abelian theory.

Plug into

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Field Strengths

In the gauge , we have This motivate us to define which transforms covariantly

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Generalized

Generalized Jacobi Jacobi identities identities

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Field Strengths

The field strength transforms as

has an anomalous transformation !

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Field Strengths



In fact, we will not use the variables explicitly, so the anomalous covariant transformation law of

will never be used.



We will simply define to be the Hodge dual of so that its gauge transformation is

covariant as the same as other components.

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Non-Abelianizing the Abelian theory

 Start form the Lorentz-covariant action for an Abelian chiral 2-form (PST formulation)

where is the Hodge dual of and ‘’ ’’ is an auxiliary field.

 It is invariant under extra gauge transformations:

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Non-Abelianizing the Abelian theory

 By imposing the gauge fixing condition:

 Consider the compactification of the Abelian theory on a circle of radius R

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Non-Abelianizing the Abelian theory

The equation of motion of zero modes :

The equation of motion of KK modes :

Now, we non-abelianize them by

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Non-Abelianizing the Abelian theory

How to achieve the Non-abelain self-duality?

(1) For zero modes, we simply define as the Hodge dual of

(2) For KK modes, start form the equation of motion

which implies

for some tensor satisfying

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Non-Abelianizing the Abelian theory

 Then notice that we can shift

such that

(by using( ) )

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Non-Abelianizing the Abelian theory

to arrive the self-duality condition of KK modes We can absorb in

The non-locality makes the nonabelian self-duality possible.

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Action Action

 The action a straightforward generalization of the The action abelian action

 By identifying with the Yang-Mills theory for multiple D4-branes, we find .

 Recall the definition , the mentioned issue of the dependence of R in the 5D action is settled by allowing the coupling to depend on the compactified radius R.

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Action Action

 Normally the coupling constant is independent of whether the space is compactified.

 In the decompactified limit, the g is not really the

‘’coupling’’, since it should be a conformal field theory without free parameter.

 The viewpoint here: We define the 6D theory as the decompactified limit of this theory.

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Action Action

 Full equation of motion for the zero modes that modifies the Yang-Mills equation by extra commutators:

 On the other hand, we notice that via defining the useful variable , we can rewrite

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Action Action

 The dependence of and in S (KK) only

through the new variable . Consider its variation, we obtain the expected equation of motion

which implies the self-dual condition for the KK modes.

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Discussions

 No commutator involves two KK modes. No self-

interaction. All interactions are mediated by zero modes.

 In large R limit, KK modes represents the 2-form potential in 6D, while zero modes approach to zero. This does not imply no interaction, since the coupling becomes:

The product of the zero modes with the coupling can keep finite.

 In large R limit, a new field “A” appear as the connection that will not affect the degree of freedom.

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Discussions

 By matching the KK spectrum for string states of the M5- branes on a circle with spectrum of instanton states of 5D SYM.

--- Lambert, Papageorgakis and Sommerfelda (2011) They conjecture :

D4-branes (SYM) = M5-branes on a circle for arbitrary R.

Also suggested by M.Doglas independently (2011)

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Discussions

 Potential problems:

(1) They based on the broken phase.

No instanton solution, while it should still have nonzero momentum in the compactified direction.

(2) The instanton number gives the total value of the 5-th momentum and one could not specify the distributions over different states.

What prevent us to distinguish the distribution?

(perhaps it is the strong interaction…)

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Discussions

 Contrary to these proposals:

We have the explicit appearance of the KK modes through an action.

We interpret the instanton number as the 5-th

momentum of the A-field. While the B-field has its 5-th momentum explicitly kept in this formulation. No

ambiguity in the momentum carrier.

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Summary

We construct a non-Abelian chiral 2-forms theory on a circle times five dimensional spacetime.

(1) Reduces to the YM theory in 5D.

(2) Reduces to the Abelian case.

 Nonlocal in the compactified direction.

 Asymmetry between zero modes and KK modes.

 The interaction is mediated via the zero mode of B.

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Outlooks Outlooks



Supersymmetric version? Covariant version?



Hidden Lorentz symmetry?



Decompactified 6D theory?



Understanding N^3 problem?



Connection with lower dimensional physics?

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Outlooks Outlooks



Supersymmetric version? Covariant version?



Hidden Lorentz symmetry?



Decompactified 6D theory?



Understanding N^3 problem?



Connection with lower dimensional physics?

END Thank you!

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