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Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
A Proposal for the Worldvolume Action ofMultiple M5-Branes
Chong-Sun Chu
seminar at IAS, HKUST4/2/2013
based on
1. A Theory of Non-Abelian Tensor Gauge Field with Non-Abelian GaugeSymmetry G x G, Chong-Sun Chu, arXiv:1108.5131.
2. Non-abelian Action for Multiple Five-Branes with Self-Dual Tensors, Chong-SunChu, Sheng-Lan Ko, arXiv:1203.4224.
3. Non-Abelian Self-Dual String Solutions, Chong-Sun Chu, Sheng-Lan Ko, PichetVanichchapongjaroen, arXiv:1207.1095.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Outline
1 Introduction
2 Non-abelian action for multiple M5-branesPerry-Schwarz action for a single M5-brane
3 Non-abelian self-dual string solution
4 Discussions
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Outline
1 Introduction
2 Non-abelian action for multiple M5-branesPerry-Schwarz action for a single M5-brane
3 Non-abelian self-dual string solution
4 Discussions
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Mysteries of M5-branes
What we know:
The low energy worldvolume dynamics is given by a 6d (2,0) SCFTwith SO(5) R-symmetry.
(Strominger, Witten)
The (2,0) tensor multiples contains 5 scalars and a selfdualantisymmetric 3-form field strength + fermions.
(Gibbons, Townsend; Strominger; Kaplan, Michelson)
What we don’t know:
What is the form of the gauge symmetry for multiple M5-branes ?
Interacting self-dual dynamics on M5-branes worldvolume?
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Self-dual dynamics for multiple M5-branes (?)
Generally, it is well known to be difficult to write down a Lorentzinvariant action for self-dual dynamics.
(Siegel 84; Floreanini, Jackiw 87)
For a single M5 case, problem solved by sacrificing manifest 6dLorentz symmetry.
(Henneaux-Teitelboim 88; Perry-Schwarz 97)
The action was later generalized to include kappa symmetry(Aganagic, Park, Popescu, Schwarz, 97)
Covariant construction was given by PST(Pasti-Sorokin-Tonin)
Not clear how to do this for N > 1 due to the other problem that anappropriate generalization of the tensor gauge symmetry was notknown.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Enhanced gauge symmetry of multiple M5-branes (?)
For multiple D-branes, symmetry is enhanced from U(1) to U(N):
δAaµ = ∂µΛa+[Aµ,Λ]a, F a
µν = ∂µAaν − ∂νAa
µ+[Aµ,Aν ]a.
For multiple M5-branes, it is not known how to non-Abelianize2-form (or higher form) gauge fields:
δBaµν = ∂µΛa
ν−∂νΛaµ+(?), Ha
µνλ = ∂µBaνλ+∂νB
aλµ+∂λB
aµν +(?).
to have nontrivial self interaction.
Moreover, exists no-go theorems: there is no nontrivial deformationof the Abelian 2-form gauge theory if locality of the action and thetransformation laws are assumed.
(Henneaux; Bekaert; Sevrin; Nepomechie)
These no-go theorems, however, suggest an important direction ofgiven up locality.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
The need of nonlocality for M5-branes should not be surprising:ABJM and BLG theory for multiple M2-branes are also non-local.
We will get around the no-go theorem by similarly introducing a setof auxillary fields: the theory become nonlocal when these fields areeliminated using their equations of motion.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
In this talk, I will explain a 6d proposal of the low energyworldvolume theory (Ko and Chu).
There is also a 5d proposal of the (2,0) theory as strong couplinglimit of 5d SYM. (Douglas; Lambert, Papageorgakis and
Schmidt-Sommerfeld)
analogy:low energy SUGRA ← quantum mechanical M-theory (proposed tobe defined by D0s’ SQM)low energy theory of M5-branes ← quantum M5-branes theory(proposed to be defined by 5d SYM)
Other related works/approaches: Smith, Ho, Huang, Masuo, Samtleben,
Sezgin, Wimmer, Wulff, Saemann, Wolf, Czech, Huang, Rozali, Tachikawa,
H.C. Kim, S. Kim, Koh, K. Lee, S. Lee, Bolognesi, Maxfield, Sethi, Bak,
Gustavsson, Hosomichi, Seong, Nosaka, Terashima, Kallen, Minahan, Nedelin,
Zabzine, Palmer, Sorokin, Bandos, ...
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Outline
1 Introduction
2 Non-abelian action for multiple M5-branesPerry-Schwarz action for a single M5-brane
3 Non-abelian self-dual string solution
4 Discussions
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Review of the Perry-Schwarz formulation
Perry-Schwarz formulation:1. A direction, say x5, is treated differently: denote the 5d and 6dcoord. by xµ and xM = (xµ, x5).2. the tensor gauge field potential is represented by a 5× 5antisymmetric tensor field Bµν . It can be thought of as a (tensor)gauge fixed formulation in which Bµ5 never appear.
Perry and Schwarz considered the action:
S0(B) =1
2
∫d6x
(−HµνHµν + Hµν∂5Bµν
)where
Hµν :=1
6εµνρλσHρλσ.
Note that manifest Lorentz symmetry is lost.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
EOM:εµνρλσ∂ρ(Hλσ − ∂5Bλσ) = 0
has the general solution
Hλσ − ∂5Bλσ = ∂µαν − ∂ναµ, for arbitrary αµ.
The action is invariant under the gauge symmetry
δBµν = ∂µϕν − ∂νϕµ, for arbitrary ϕµ.
This allows one to reduce the general solution to the EOM to thefirst order form
Hµν = ∂5Bµν .
This is the self-duality equation in this theory.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Modified Lorentz symmetry
The action has manifest 5d Lorentz invariance and a non-manifestLorentz symmetry mixing µ with the 5 direction.
Lorentz transformation (active view)
Standard Lorentz transformation
δBµν = (Λ · L)Bµν + δspinBµν
has a orbital part:Λ · L = (Λ · x)∂5 − x5(Λ · ∂)
and a spin part:δspinBµν = ΛνBµ5 − ΛµBν5.
PS proposed to consider the following transformation
δBµν = (Λ · x)Hµν − x5(Λ · ∂)Bµν ,
where Λµ = Λ5µ denote the corresponding infinitesimal transformationparameters.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
1. On shell, it is equal to the standard Lorentz transformation
δBµν = (Λ · x)Hµν − x5(Λ · ∂)Bµν = (Λ · x)∂5Bµν − x5(Λ · ∂)Bµν
2. Commutator:
[δΛ1 , δΛ2 ]Bµν = δ(5d)Λαβ
Bµν + EOM + gauge symmetry
whereδBµν = ∂µϕν − ∂νϕµ, ϕν = xαΛαλBν
λ
is a gauge symmetry of the PS theory. So the PS transformation canbe considered as a (modified) Lorentz transformation.
Note that modified Lorentz symmetry is typical of action of self-dualdynamics
(Siegel 84)
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Non-abelian action
The idea is to try to generalize the Perry-Schwarz approach:
1 Represent the self-dual tensor gauge field by a 5× 5 antisymmetricfield Ba
µν in the adjoint and giving up manifest 6d Lorentz symmetry
2 Moreover we’ll introduce a set of YM gauge fields Aaµ for gauge
group G .
Comments
1. Motivation for 2:- Multiple M2-branes (ABJM) was used to probe the M5-branes system.The gauge non-invariance of the boundary Chern-Simons action wasshown to imply the existence of a Kac-Moody current algebra on theworldsheet of multiple self-dual strings
- This was argued to induces a U(N)× U(N) gauge symmetry in the
theory of N coincident M5-branes. (Chu, Smith; Chu)
2. YM gauge field Aaµ was also introduced in many other approaches,
e.g. Ho, Huang, Masuo; Samtleben, Sezgin, Wimmer, Wulff;Sorokin’s talk; ...) The main difference in our apporach is that theyare not propagating and so there is no extra degrees of freedom.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
The Action
Our proposed action S = S0 + SE consists of two pieces:
S0 =1
2
∫d6x tr
(−HµνHµν + Hµν∂5Bµν
),
is the non-abelian generalization of the Perry-Schwarz action, where
Hµνλ = D[µBνλ], Dµ = ∂µ + Aµ.
SE =
∫d5x tr
((Fµν − c
∫dx5 Hµν)Eµν
).
where Eµν is a 5d auxiliary field implementing the constraint
Fµν = c
∫dx5 Hµν
- Note that there is no Bµ5 and A5. Aµ and Eµν live in 5-dimensions- Note also the presence of fields Aµ,Eµν that is not expected from (2,0)susy.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
The action is invariant under:
1. Yang-Mills gauge symmetry
δAµ = ∂µΛ + [Aµ,Λ], δBµν = [Bµν ,Λ], δEµν = [Eµν ,Λ].
2. Tensor gauge symmetry:
δTAµ = 0, δTBµν = D[µΛν], δTEµν = 0,
for arbitrary Λµ(xM) such that [F[µν ,Λλ]] = 0.
3. Moreover there is a gauge symmetry
δEµν = αµν
for arbitrary α(xλ) such that D[µανλ] = 0, Dµαµλ = 0.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Property 1: Self-Duality
EOM of Eµν gives the constraint
Fµν = c
∫dx5 Hµν .
EOM of BµνεµνρλσDρ(Hλσ − ∂5Bλσ) = 0
has the general solution
Hλσ − ∂5Bλσ = Φλσ,
where D[λΦµν] = 0 .
Again with an appropriate fixing of the tensor gauge symmetry, onecan reduce the second order EOM to the self-duality equation
Hµν = ∂5Bµν .
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Property 2: Degrees of freedom
After eliminating the aux field Eµν and substituting the F = Hconstraint, the resulting action becomes highly nonlinear andinteracting. To count the degrees of freedom, we use the linearizedtheory.
At the quadratic level, the non-abelian action is simply given bydimG copies of the Perry-Schwarz action. We obtain 3× dimGdegrees of freedom in Bµν .
Our theory contains 3× dimG degrees of freedom as required by(2,0) supersymmetry.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Property 3: Lorentz Symmetry
The action is invariant under the 5-µ Lorentz transformation:
δBµν = (Λ · x)Hµν + x5ΛκHκµν + Λκφµνκ,
where
φaµνκ =
∫dy G abµ′ν′
µν (x , y)Jbµ′ν′κ(y)
Jµνκ is some expression linear in Hand G ab
µν,µ′ν′(x , y) is the Green function satisfying
∂5Gabµ′ν′
µν −1
2εµν
αβγ(D(y)α )acG
cbµ′ν′
βγ = δµ′ν′
µν δabδ(6)(x − y)
with the BC: G abµ′ν′
µν (x , y) = 0, |x5| → ∞.
As before, the commutator closes to the standard 5d Lorentztransformation plus terms vanishing onshell, plus a gaugetransformation.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Note that our proposed Lorentz symmetry is nonlocal.
This is not unexpected since we are working in a gauge fixedformulation without Bµ5. The Lorentz symmetry is also nonlocal forQED in Coulomb gauge or string in lightcone gauge.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Property 4: Reduction to D4-branes
Consider a compactification of x5 on a circle of radius R. Thedimensional reduced action reads
S =2πR
2
∫d5x tr
(−H2
µν + (Fµν − 2πRcHµν)Eµν)
Integrate out Eµν , we obtain
Fµν = 2πRcHµν .
and eliminate Hµν , we obtain the 5d Yang-Mills action
SYM = − 1
4πRc2
∫d5x tr F 2
µν .
This gives the YM coupling and the gauge group to be
g2YM = Rc2, G = U(N)
for a system of N M5-branes.
However there is a subtilty.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Property 4: Reduction to D4-branes
Consider a compactification of x5 on a circle of radius R. Thedimensional reduced action reads
S =2πR
2
∫d5x tr
(−H2
µν + (Fµν − 2πRcHµν)Eµν)
Integrate out Eµν , we obtain
Fµν = 2πRcHµν .
and eliminate Hµν , we obtain the 5d Yang-Mills action
SYM = − 1
4πRc2
∫d5x tr F 2
µν .
This gives the YM coupling and the gauge group to be
g2YM = Rc2, G = U(N)
for a system of N M5-branes.
However there is a subtilty.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
EOM gives DµFµν = 0 instead of
DµFµν = −πR
2εναβγδ[Fαβ ,Bγδ]?
Need to be more careful with the implementation of Delta function:∫[DA][DB][DE ]e−S =
∫[DA][DB]e−SYM δ(Fµν−2πRHµν) =
∫[DA]e−SYM−S′ ,
where consistency requires that
δS ′
δAν=
1
2εναβγδ[Fαβ ,Bγδ]
The 5d theory is thus given by the action S5d = SYM + S ′.S ′ describes a higher derivative correction term to the Yang-Mills theorysince [F ,B] ∼ DDB and B is of the order of F .It might be possible that S ′ captures the non-abelian DBI action ofD4-branes.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
We have constructed a non-abelian action of tensor fields with theproperties:
1. the action admits a self-duality equation of motion,2. the action has manifest 5d Lorentz symmetry and a modified 6d
Lorentz symmetry,3. on dimensional reduction, the action gives the 5d Yang-Mills action
plus corrections.
Based on these properties, we propose our action to be the bosonictheory describing the gauge sector of coincident M5-branes in flatspace.
We still need to include the scalar field and fermions, and to susycomplete the action.
Another interesting direction is to explore the dynamical content ofthe theory.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Outline
1 Introduction
2 Non-abelian action for multiple M5-branesPerry-Schwarz action for a single M5-brane
3 Non-abelian self-dual string solution
4 Discussions
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Self-dual string on M5-brane
M2-branes can end on M5-brane. The endpoint gives strings livingon the M5-brane.These self-dual strings appear as solitons of the M5-brane theory.
In the Abelian case, the self-dual string solition has been obtained byPerry-Schwarz (1996) and also by Howe-Lambert-West (1997).
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Self-Dual String Soliton in the Perry-Schwarz Theory
The Perry-Schwarz non-linear field equation is given by
Hµν =(1− y1)Hµν5 + Hµρ5H
ρσ5Hσν5√1− y1 + 1
2y21 − y2
,
where y1 := − 12Hµν5H
µν5, y2 := 14Hµν5H
νρ5Hρσ5Hσµ5.
A solution aligning in the x5 direction is solved by the ansatz:
B = α(ρ)dtdx5 +β
8(±1− cos θ)d φdψ,
where the 6d metric is
ds2 = −dt2 + (dx5)2 + dρ2 + ρ2dΩ23,
with the three-sphere given in Euler coordinates
dΩ23 =
1
4[(dψ + cos θd φ)2 + (d θ2 + sin2 θd φ2)],
Note that Bφψ = β(±1− cos θ) is the potential of a Dirac monopole!
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
For this ansatz, y1 = α′ 2, y2 = α′ 4/2 and the EOM reads
α′(ρ) =β√
β2 + ρ6.
The solution is regular everywhere:
α ∼ ρ as ρ→ 0,
α ∼ − β
2ρ2+ const. as ρ→∞.
The magnetic charge P and electric charge Q (per unit length) ofthe string:
P =
∫S3
H, Q =
∫S3
∗H,
givesP = Q = 2π2β
hence the string is self-dual.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
charge quantization condition for self-dual string in 6d
PQ ′ + QP ′ ∈ 2πZ
(Deser, Gomberoff, Henneaux, Teitelboim)
for the self-dual string gives
β = ±√
n
4π3,
i.e.P = Q = ±
√nπ.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
For the non-abelian theory, the equations of motion to be solved are:
Hµν = ∂5Bµν .
Fµν = c
∫dx5 Hµν
Need to do:
1. construct a string aligns in a different direction, say x4.The gauge field is auxiliary:
Fµν = c
∫dx5 Hµν = c(Bµν(x5 =∞)− Bµν(x5 = −∞))
Therefore if the non-Abelian solution is translationally invariant along x5,
then Fµν = 0 is trivial.
2. supersymmetrize: Perry-Schwarz self-dual string solution is non-BPSas there is no other matter field turned on to cancel the tensor fieldforce.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Self-dual string in x4 direction
It is instructive to first obtain the PS solution in Cartesiancoordinates for a string in the x4 direction. The result is
H =α′
ρdtdw
(xdx + ydy + zdz + x5dx5) +
β
ρ4
(x5dxdydz − xdydzdx5 + zdydxdx5 − ydzdxdx5)
,
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
To get B, we integrate directly the self-duality equation
Hµν5 = ∂5Bµν
and get
Bij = −1
2
βεijkxkr3
(x5r
ρ2+ tan−1(x5/r)
), Btw = − β
2ρ2,
i , j = 1, 2, 3,w = x4.
Although the auxillary field does not appear in the PS construction,it is amazing that
Fij = −cβπ
2
εijkxkr3
, Ftw = 0
i.e. a Dirac monopole in the (x , y , z) subspace if cβ = −2/π !
The presence of a Dirac monopole was already apparent in theoriginal solution of PS. Here, we reveal that the monopoleconfiguration also appears (more directly) in the auxiliary gauge field.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
It turns out the use of an non-abelian monopole in place of theDirac monopole is precisely what is needed to construct thenon-abelian self-dual string solution.
Here we have two candidates of non-abelian monopole: theWu-Yang monopole and the ’t Hooft-Polyakov monopole.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Non-abelian Wu-Yang & ’t Hooft-Polyakov monopole
Wu-Yang
Consider SU(2) gauge group
[T a,T b] = iεabcT c , a, b, c = 1, 2, 3.
The non-abelian Wu-Yang monopole is given by
Aai = −εaik
xkr2, F a
ij = εijmxmxar4
, i , j = 1, 2, 3.
Note that the field strength for the Wu-Yang solution is related tothe field strength of the Dirac monopole by a simple relation:
F aij = F
(Dirac)ij
xa
r.
Note that the color magnetic charge vanishes∫S2
F a = 0
and the Wu-Yang solution is actually not a monopole. Neverthless itplays a key role in the construction of non-abelian monopole of ’tHooft-Polyakov.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
’t Hooft-Polyakov
In the BPS limit, the ’t Hooft-Polyakov monopole satisfies
1
2εijkFij = Dkφ, D2
kφ = 0
where φ is an adjoint Higgs scalar field.
The solution is given by
Aai = −εaik
xk
r2(1− kv (r)), φa =
vxa
rhv (r),
where
kv (r) :=vr
sinh(vr), hv (r) := coth(vr)− 1
vr.
Asymptotically r →∞, we have
Aai → −εaik
xk
r2, φa → |v |x
a
r:= φ∞,
i.e. precisely the Wu-Yang monopole at infinity.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Unbroken U(1) gauge symmetry at infinity may be identified as theelectromagnetic gauge group and the electromagnetic field strengthcan be obtained as a projection:
Fij = F aij
φa
|v |= εijk
xk
r3, for large r .
The magnetic charge is given by p =∫S2 F = 4π, which corresponds
to a magnetic monopole of unit charge.
Note that at the core r → 0, we have
Ai → 0, φ→ 0
and hence the SU(2) symmetry is restored at the monopole core.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Non-abelian self-dual string solution
Inspired by the relation of Dirac monopole to the Wu-Yang solution,try the ansatz
Haµνλ = H
(PS)µνλ
xa
r
Here r =√
x2 + y2 + z2 and H(PS)µνλ is the field strength for the
linearized Perry-Schwarz solution aligning in the x4 direction.Self-duality is automatically satisfied!
B can be obtained by integrating Hµν5 = ∂5Bµν and we obtain
Baµν = B(PS)
µν
xa
r,
It is amusing that the auxillary field configuration is given by
F aij = −cβπ
2
εijmxmxar4
, F atw = 0.
This is the Wu-Yang monopole if we take cβ = − 2π .
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Like the Wu-Yang monopole, the color magnetic charge of ourWu-Yang string solution vanishes.
However this is not a problem as we should include scalar fields(which are natural from the point of view of (2,0) supersymmetry).
Although we do not have the full (2,0) supersymmetric theory, onecan argue (a simple dimensional analysis) that the self-dualityequation of motion is not modified by the presence of the scalarfields.
As for the scalar field’s EOM, the self-interacting potential vanishesif there is only one scalar field turned on (R-symmetry). As a result,the equation of motion of the scalar field is
D2Mφ = 0.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
A reasonable form of the BPS equation is the non-abeliangeneralization of the BPS equation of Howe-Lambert-West:
Hijk = εijk∂5φ, Hij5 = −εijkDkφ.
This follows immediately from the supersymmetry transformation
δψ = (ΓMΓIDMφI +
1
3!2ΓMNPHMNP)ε
and the 1/2 BPS condition
Γ046ε = −ε.
Note: this is the most natural non-abelian generalization of the abelian
(2,0) supersymmetry transformation.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
The BPS equation can be solved with
φa = −(u +β
2ρ2)xa
r,
The transverse distance |φ| defined by |φ|2 = φaφa gives
|φ| = |u +β
2ρ2|.
This describes a system of M5-branes with a spike at ρ = 0 and leveloff to u as ρ→∞. Hence the physical interpretation of our self-dualstring is that two M5-branes are separating by a distance u and withan M2-brane ending on them.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Asymptotic U(1) B-field is Bµν ≡ φa∞Baµν = ±B(PS)
µν and we obtain
P = Q = −4π
|c |.
Charge quantization yields
c = ±4√π
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
One may also consider the compactified case with x5 compactifiedon a circle.
Fourier mode expansion
HMNP =∑n
e inx5/RH
(n)MNP(r).
Integrating Hµν5 = ∂5Bµν , we get
Bµν =x5
2πRcFµν(r) +
∞∑n=−∞
e inx5/RB(n)
µν (r),
where
H(n 6=0)µν5 (r) =
in
RB(n 6=0)µν (r).
Note the presence of winding modes.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Consider an ansatz with the only nonzero components Btw and Bij ,the self-duality condition reads
D[iB(0)jk] = εijk
Ftw
2πRc, DkB
(0)tw = − fk
2πRc
Dkb(n)k =
in
RB
(n)tw , DkB
(n)tw = −b(n)
k
in
R, n 6= 0,
where
fk(r) :=1
2εijkFij and b
(n)k (r) :=
1
2εijkB
(n)ij for n 6= 0.
Note the red equation takes exactly the same form as the BPS
equation for the HP monopole if we identify −2πRcB(0)tw := φ as the
scalar field there.
Hence it can be solved with Aai taken to be that of the HP monopole
Aai = −εaik
xk
r2(1− kv (r)).
This implies Ftw = 0.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
The whole system of equations can be solved exactly and we obtain
Batw = −hv (r)
2πRc
vxa
r+∑n 6=0
αneinx5/R e
−|n|r/R
vr
(1 +
vR
|n|coth(vr)
)vxa
r,
Baij =
x5
2πRcF aij (r) + c0F
aij (r) +
∑n 6=0
e inx5/RB
a (n)ij (r),
where
Ba (n)ij = εijk
R
in
(− v3(ra′n − kv (r)an)
xkxa
r− δakankv (r)
v
r
), n 6= 0.
As before, we can include a scalar field φ as needed in the (2,0)theory. The charge is found to be the same as the uncompactifiedstring (as it should be by consistency).
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Outline
1 Introduction
2 Non-abelian action for multiple M5-branesPerry-Schwarz action for a single M5-brane
3 Non-abelian self-dual string solution
4 Discussions
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
I. We have constructed a non-abelian action of tensor fields that wepropose to be the bosonic theory describing the gauge sector ofcoincident M5-branes in flat space.
II. We have also constructed the non-abelian string solutions of thetheory. By including a scalar field, we argue how the solution can bepromoted to become a solution of the (2,0) theory.This provides dynamical support to our proposed theory.
Introduction Non-abelian action for multiple M5-branes Non-abelian self-dual string solution Discussions
Further questions
Supersymmetry: (2,0)? (1,0)?Scalar potential and BPS equation?
Other solutions? BPS string junction?
Integrability?
Covariant PST extension of our model?
Connection with the 5d SYM proposal of Douglas andLambert-Papageorgakis-Schmidt-Summerfield?Where is the B-field in the SYM description? (similar to theproblem of extracting the gravity field in the BFSS matrix model?)
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