M5-branes on a Circle
Kuo-Wei Huang (NTU)
Taiwan String Group Seminar Sep.16 2011
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].
Outline Outline
Motivation : a short review
Gauge symmetry and Field strengths
Non-Abelianizing the Abelian theory
Action
Discussions
Summary and Outlook
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.
Motivation
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..
Motivation
The ‘’2-form’’ potential in fact should be self- dual:
Satisfies the self-dual condition:
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
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).
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:
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..
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.
Gauge Symmetry
Consider M5-branes compactified on a circle of radius R, including all the Kaluza-Klein (KK) modes.
Decomposition:
Gauge Symmetry
A key point:
Notice that previous no-go theorem assume the local deformation.
This non-local operator is well defined.
Gauge Symmetry
Define the covariant derivative:
and the field strength
:
The should correspond to the YM gauge potential
‘’A’’ in D4-branes.
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)
Gauge Symmetry
Gauge transformations is closed by
where
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.
Gauge Symmetry
Non-Abelian redundant symmetry:
The non-locality insures the redundant symmetry can still exist in the non-Abelian theory.
Plug into
Field Strengths
In the gauge , we have This motivate us to define which transforms covariantly
Generalized
Generalized Jacobi Jacobi identities identities
Field Strengths
The field strength transforms as
has an anomalous transformation !
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.
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:
Non-Abelianizing the Abelian theory
By imposing the gauge fixing condition:
Consider the compactification of the Abelian theory on a circle of radius R
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
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
Non-Abelianizing the Abelian theory
Then notice that we can shift
such that
(by using( ) )
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.
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.
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.
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
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.
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.
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)
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…)
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.
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.
Outlooks Outlooks
Supersymmetric version? Covariant version?
Hidden Lorentz symmetry?
Decompactified 6D theory?
Understanding N^3 problem?
Connection with lower dimensional physics?
Outlooks Outlooks
Supersymmetric version? Covariant version?
Hidden Lorentz symmetry?
Decompactified 6D theory?
Understanding N^3 problem?
