mtheory beijing v9

8/14/2019 Mtheory Beijing v9

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The tension of the branes we discussed is is just theirmassper unit worldvolume with units of (length) p 1


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per unit worldvolume, with units of (length) p .

http://find/http://goback/


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per unit worldvolume, with units of (length) p .

Since M-theory has only one dimensional parameter p, wecan predict that:

T M 213 p

, T M 516 p



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per unit worldvolume, with units of (length) .

Since M-theory has only one dimensional parameter p, wecan predict that:

T M 213 p

, T M 516 p

We will shortly argue that the correct answers are:

T M 2 =2

(2 p)3, T M 5 =

2(2 p)6

Thus we have argued that there is some supersymmetric


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g p y

theory dened in 11 at spacetime dimensions which hasmassless elds including agraviton, as well as stable 2-branesand 5-branes.



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g p y

theory dened in 11 at spacetime dimensions which hasmassless elds including agraviton, as well as stable 2-branesand 5-branes.We refer to this as M-theory.



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g p y

theory dened in 11 at spacetime dimensions which hasmassless elds including agraviton, as well as stable 2-branesand 5-branes.We refer to this as M-theory.There are two kinds of limitations in our knowledge of M-theory:



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(i) We have formulated it in a xed spacetime background and itis not clear how to study it in a background-independent way .



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(i) We have formulated it in a xed spacetime background and itis not clear how to study it in a background-independent way .

(ii) It is not obvious that it has a consistent ultraviolet completion .

The rst issue is also a problem instring theory.


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The rst issue is also a problem instring theory.However the second one is new. In string theory, using the

b l l b


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perturbative expansion, ultraviolet niteness can be quiteconvincingly demonstrated. The stringy nature cuts off UVinnities.


b i i l i l i b i


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perturbative expansion, ultraviolet niteness can be quiteconvincingly demonstrated. The stringy nature cuts off UVinnities.We may suspect that something similar holds in M-theory. Itsbrane excitations could perhaps provide an ultraviolet cutoff .


b i i l i l i b i


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So far, we dont know whether this is true and if so, whichbrane is responsible. But the most logical possibility is thatthe M2-brane, or membrane, governs the consistency of M-theory.


t b ti i lt i l t it b it


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So far, we dont know whether this is true and if so, whichbrane is responsible. But the most logical possibility is thatthe M2-brane, or membrane, governs the consistency of M-theory.

This is why the last few lectures of this series will bededicated to the M2-brane.

Outline

M-theory: Motivation and background

11d Supergravity


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M-branes as black branesCompactication to 10d

Branes and dualities from M-theory

M2-brane eld theory: MotivationSingle M2 brane

Bagger-Lambert theory

Interpretation of BL theory

Lorentzian 3-algebras

ABJM theory

Conclusions

Compactication to 10d


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In contrast to 11d, where supergravity (in at spacetime) isunique, in 10d there are three distinct classical supergravityactions: type IIA, type IIB and type I.



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In contrast to 11d, where supergravity (in at spacetime) isunique, in 10d there are three distinct classical supergravityactions: type IIA, type IIB and type I.Each of them is associated to a superstring theory: type IIA,

type IIB and type I/heterotic .



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type IIB and type I/heterotic .Now, compactifying 11d supergravity on a circle must lead to10d supergravity, therefore to one of the theories listed above.



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type IIB and type I/heterotic .Now, compactifying 11d supergravity on a circle must lead to10d supergravity, therefore to one of the theories listed above.This is very interesting and suggests a close relation betweenM-theory in 11d and superstring theory in 10d.

To compactify 11d supergravity, we have to split the 11dmetric into components:


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GMN G , G 10 , G1010where , = 0 , 1, , 9.



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GMN G , G 10 , G1010where , = 0 , 1, , 9.Thus the 11d metric gives rise to a metric, a vector eld and

a scalar in 10d.



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GMN G , G 10 , G1010where , = 0 , 1, , 9.Thus the 11d metric gives rise to a metric, a vector eld and

a scalar in 10d.Similarly the 3-form gives:

C MNP C , C 10and gives rise to a 3-form as well as a 2-form in 10d.


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This bosonic spectrum is identical to that of type IIAsupergravity.


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This bosonic spectrum is identical to that of type IIAsupergravity.Since circle compactication does not break supersymmetry,we can be quite sure that the 10d theory thus obtained isindeed going to be type IIA supergravity.


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This bosonic spectrum is identical to that of type IIAsupergravity.Since circle compactication does not break supersymmetry,we can be quite sure that the 10d theory thus obtained isindeed going to be type IIA supergravity.To see this more explicitly, we must parametrise the 11dmetric properly.

Let us rst look at the action of type IIA supergravity in acanonical normalisation:

S IIA =2

8 d10x g e 2 R + 4 12 H 3

2


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(2 s ) || || | |12 d10x ||g|| |F 2|2 + |F 4 + AH 3|2 12 B2F 4F 4where:

: dilaton, e = gsA : A dx , Ramond-Ramond 1-formB2 : B dxdx

, NS-NS 2-form

A3 : A dx

dx

dx, Ramond-Ramond 3-form

F 2 = dA, H 3 = dB2, F 4 = dA3

A useful trick is to instead work with the11d vielbein E AM .Let us start by parametrising it as:


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E AM =ea 0

0 e

where ea is the 10d vielbein, is the 10d scalar and we are

temporarily setting the 10d 1-form A to 0.

A useful trick is to instead work with the11d vielbein E AM .Let us start by parametrising it as:


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E AM =ea 0

0 e

where ea is the 10d vielbein, is the 10d scalar and we are

temporarily setting the 10d 1-form A to 0.With the above parametrisation:

||E ||R(E ) e||e|| R(e) + 4

Comparing with the type IIA action we see that this is notwhat we want, so we perform a Weyl rescaling:


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E AM e E AM which has the effect:

||E || e11 ||e||, R e 2 Rso the RHS gets multiplied by e9 .



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||E || e11 ||e||, R e 2 Rso the RHS gets multiplied by e9 .Thus we require:

9 + 1 = 2



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||E || e11 ||e||, R e 2 Rso the RHS gets multiplied by e9 .Thus we require:

9 + 1 = 2

It follows that = 1/ 3.

Thus the correct decomposition is:

E AM = e / 3 e

a 0

eA e


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where we have now included the10d 1-form as well.


E AM = e / 3 e

a 0

eA e

h h l d d h d f ll


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where we have now included the10d 1-form as well.From this we easily nd that:

GMN = e 2/ 3g + e2AA e2A

e2

A e2


E AM = e / 3 e

a 0

eA e

h h i l d d h 10d 1 f ll


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where we have now included the10d 1-form as well.From this we easily nd that:

GMN = e 2/ 3g + e2AA e2A

e2

A e2

On the other hand, the M-theory 3-form becomes, ondimensional reduction:

C MNP C = AC 10 = B

With these identications, we can compactify 11d

i d i h h L i f 10d IIA


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supergravity and compare with the Lagrangian of 10d type IIAsupergravity.


it d ith th L i f 10d t IIA


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supergravity and compare with the Lagrangian of 10d type IIAsupergravity.Notice that the former after compactifying has twoparameters, p and R10 . On the other hand, the latter hastwo parameters, s and e = gs .


g it d ith th L g gi f 10d t IIA


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supergravity and compare with the Lagrangian of 10d type IIAsupergravity.Notice that the former after compactifying has twoparameters, p and R10 . On the other hand, the latter hastwo parameters, s and e = gs .Thus we should nd a relation between the two pairs of parameters. This will provide us the physical interpretation of the result.

Comparing Lagrangians, we right away nd:

2R 10

(2 p)9 =

1

g2s

1

(2 s )8


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(2 p) gs (2 s )


2R 10

(2 p)9 =

1

g2s

1

(2 s )8


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(2 p) gs (2 s )Additionally the relation between metrics tells us that:

p = g1/ 3s s


2R 10

(2 p)9 =

1

g2s

1

(2 s )8


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p = g1/ 3s s

Inserting the latter in the former, we nd:

R10 = gs s


2R 10

(2 p)9 =

1

g2s

1

(2 s )8


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p = g1/ 3s s

Inserting the latter in the former, we nd:

R10 = gs s

This is a truly striking result! It relates the radius of acompact dimension to the string length and coupling.

The proposed interpretation is as follows.


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When we compactify M-theory on a circle of radiusR10 it


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When we compactify M-theory on a circle of radiusR10 , itbecomes type IIA string theory in the limit R10 0.


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When we compactify M-theory on a circle of radiusR10 it


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When we compactify M theory on a circle of radiusR , itbecomes type IIA string theory in the limit R10 0.Conversely, if we start with 10d type IIA string theory andtake the coupling gs , the string description breaks down.In this limit, a new space dimension opens up and we getM-theory.This is a duality.

OutlineM-theory: Motivation and background

11d Supergravity

M-branes as black branes


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M branes as black branesCompactication to 10d


M2-brane eld theory: Motivation

Single M2 brane



Lorentzian 3-algebrasABJM theory

Conclusions

Branes and dualities from M-theoryLet us now try to justify the proposal that M-theory and typeIIA string theory are related as claimed:

M hcompactication

IIA i h


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M-theoryp strong coupling

type IIA string theory


M thcompactication t IIA t i th


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First we review the spectrum of branes in string theory.


M thcompactication t IIA t i g th


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First we review the spectrum of branes in string theory.In a classic calculation of the force between D-branes usingopen string theory, it was shown that:

T Dp =1gs

2(2 s ) p+1


M theory compactication type IIA string theory


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M-theory strong coupling


First we review the spectrum of branes in string theory.In a classic calculation of the force between D-branes usingopen string theory, it was shown that:

T Dp =1gs

2(2 s ) p+1

We also have:

T F 1 =2

(2 s )2, T NS 5 =

1g2s

2(2 s )6


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Let us now try to derive these results starting with M-branes.In principle this might not be possible at all!


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The tensions of string theory branes were calculated at weak


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coupling. One might expect them to be renormalised.




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coupling. One might expect them to be renormalised.However, the fact that these are supersymmetric branes savesus.




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It can be argued that the tension of supersymmetric branes isexact. This is an example of anon-renormalisation theorem .


The tensions of string theory branes were calculated at weakl h h b l d


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It can be argued that the tension of supersymmetric branes isexact. This is an example of anon-renormalisation theorem .Therefore we can compare the tensions of M-theory braneswith type IIA branes, and we will now do this.

When we compactify on a circle, the M2-brane can be eitherwrapped on the circle or transverse to the circle.


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When we compactify on a circle, the M2-brane can be eitherwrapped on the circle or transverse to the circle.In the rst case it looks (as R10

0) like a string or 1-brane.

In the second case it is a 2-brane.


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In the second case it is a 2-brane.


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Doing the same thing for an M5-brane we get a 4-brane or a5-brane.



In the second case it is a 2-brane.f


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Doing the same thing for an M5-brane we get a 4-brane or a5-brane.To match with the branes in string theory, the only

possibilities are:

wrapped M2 F1, transverse M2 D2wrapped M5 D4, transverse M5 NS5

This is a denite set of predictions!


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This is a denite set of predictions!Start with the M2-brane. We had proposed that its tension is:

T M 2 =1

42 3 p


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T M 2 =1

42 3 p


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Wrapping on the circle, the tension of the resulting brane is:

T M 2 wrapped=

T M 2

2R 10

=1

2 2swhich is correct.


T M 2 =1

42 3 p


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T M 2 wrapped=

T M 2

2R 10

=1

2 2swhich is correct.

But this result really serves to x the tension of the M2-brane,which we had not determined previously.

Now consider the transverse M2-brane. Its tension is:

T M 2 =1

42 3 p1


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=1

42(g1/ 3s s )3

=1

gs1

42 3s= T D 2

Now consider the transverse M2-brane. Its tension is:

T M 2 =1

42 3 p1


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=1

42(g1/ 3s s )3

=1

gs1

42 3s= T D 2

This is a truly remarkable agreement!

For the M5-brane, things work out as follows.


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For the M5-brane, things work out as follows.We have proposed that its tension is:

T M 5 =1

325 6 p


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For the M5-brane, things work out as follows.We have proposed that its tension is:

T M 5 =1

325 6 p

Wrapping on the circle the tension of the resulting brane is:


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T M 5 wrapped = T M 5 2R 10= gs s

164g2s 6s

=1gs

2(2 s )5

=

T D 4

which is correct, but again can be thought of as adetermination of T M 5.

Finally, the transverse M5-brane gives:

T M 5 =

1

325 6

p

1 1


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=g2s 325 6s

=1

g2s2

(2 s )6

= T NS 5which is again a remarkable conrmation of the equivalencebetween M-theory and type IIA string theory.

This still leaves the D0 and D6 branes.


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Note that the mass of a D0 brane is:1 1


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T 0 =1

gs s=

1R10




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T 0 = gs s = R10What mode of M-theory can have this mass?




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T 0 = gs s = R10What mode of M-theory can have this mass?We will argue that it is the mode with one unit of momentumalong the compact direction .

Indeed, on a compact dimension of length L, the momentumis quantised in integers as:

p =2n

L


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p =2n

LFor massless particles in 11d, we have:


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E 2 = p21 + p29 + p210


p =2n



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E 2 = p21 + p29 + p210After compactication, a xed value of p10 will appear as amass.


p =2n



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E 2 = p21 + p29 + p210After compactication, a xed value of p10 will appear as amass.Since L = 2 R 10 , we have:

mass in 10d =

| p10

|=

n

R10

Thus a single D0-brane (n = 1 ) is a single unit of momentum

along x10 .


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along x10 .But we have a new prediction. For every integern, there


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should be a bound state of n D0-branes!.


along x10 .But we have a new prediction. For every integern, thereh ld b b d f D0 b !


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should be a bound state of n D0-branes!.This is a statement about string theory that we did not know

before the discovery of M-theory! And it has now beenveried directly within string theory.

Let us see how the charge of the D0-brane works out.


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Let us see how the charge of the D0-brane works out.Note that in 10 dimensions, a 0-brane is surrounded by S 8.Thus its charge is the integral of the spatial components of an8-form.


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The Poincare dual of this 8-form in 10d is a 2-form whichmust be the eld strength of the Ramond-Ramond 1-form A .



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The Poincare dual of this 8-form in 10d is a 2-form whichmust be the eld strength of the Ramond-Ramond 1-form A .

Therefore the D0-brane is electrically charged under A . Fromthe M-theory point of view, the latter is a Kaluza-Klein gaugeeld.

As we have seen, it is possible to have adual object which issurrounded by S 2 and is a magnetic source for the same eldstrength.


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As we have seen, it is possible to have adual object which issurrounded by S 2 and is a magnetic source for the same eldstrength.Such an object must be a 6-brane. Indeed it is known that int IIA t i th th 6 b i th ti d l f th


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type IIA string theory, the 6-brane is the magnetic dual of the0-brane.

As we have seen, it is possible to have adual object which issurrounded by S 2 and is a magnetic source for the same eldstrength.Such an object must be a 6-brane. Indeed it is known that intype IIA string theory the 6 brane is the magnetic dual of the


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type IIA string theory, the 6-brane is the magnetic dual of the0-brane.

In M-theory, A is a Kaluza-Klein gauge eld. Therefore amagnetically charged object must be a Kaluza-Kleinmonopole.

Let us rst discuss Kaluza-Klein monopoles abstractly.


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Let us rst discuss Kaluza-Klein monopoles abstractly.Consider the metric in 4 Euclidean dimensions:

ds2Taub-NUT = V ( x) dx

dx +

1

V ( x)dy + A

dx

2

where A is the vector potential for a magnetic monopole in 3di i


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dimensions: B = A

and V ( x) is a harmonic function in 3d determined by:

V = B

Let us rst discuss Kaluza-Klein monopoles abstractly.Consider the metric in 4 Euclidean dimensions:

ds2Taub-NUT = V ( x) dx

dx +

1

V ( x)dy + A

dx

2

where A is the vector potential for a magnetic monopole in 3dimensions:


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dimensions: B = A

and V ( x) is a harmonic function in 3d determined by:

V = BThis metric solves the 4d Euclidean Einstein equation withoutsources.

We choose a specic harmonic function V depending on areal number R, namely:

V ( x) = 1 +R

2rwhere r = |x|.


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V ( x) = 1 +R

2rwhere r = |x|.Thus the magnetic eld is:


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g

B =R

2

x

r 3


V ( x) = 1 +R

2rwhere r = |x|.Thus the magnetic eld is:


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g

B =R

2

x

r 3As r 0 this metric is apparently singular due to the terms:

R2r

dr 2 +2rR

dy2

The singularity can be avoided as follows. Dene:

r =

2rR


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r =

2rRThe dangerous terms then become:


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dr 2 +r 2

R2 dy

2


r =

2rRThe dangerous terms then become:


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dr 2 +r 2

R2 dy

2

Now the second term is non-singular only if y is an angle withperiodicity 2R .

Being a non-singular metric with a monopole charge, this iscalled a Kaluza-Klein monopole (if we add dt2 to make it aparticle).


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The monopole is located at the core near r 0, where theKaluza-Klein circleshrinks to zero size.


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The monopole is located at the core near r 0, where theKaluza-Klein circleshrinks to zero size.Let us now embed this solution in M-theory by taking the x

7 8 9 10


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directions to be x7, x8, x9 and the KK direction y to be x10

with periodicity 2R 10 .


The monopole is located at the core near r 0, where theKaluza-Klein circleshrinks to zero size.Let us now embed this solution in M-theory by taking the xd b 7 8 9 d h d b 10


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with periodicity 2R 10 .The resulting object is translational invariant alongx1, x2, , x6 so it is a 6-brane.


The monopole is located at the core near r 0, where theKaluza-Klein circleshrinks to zero size.Let us now embed this solution in M-theory by taking the xdi i b 7 8 9 d h KK di i b 10


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with periodicity 2R 10 .The resulting object is translational invariant alongx1, x2, , x6 so it is a 6-brane.And it is magnetically charged under the Kaluza-Klein gaugeeld arising from compactication of x10 .

So we have a candidate for the D6-brane of type IIA stringtheory.


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So we have a candidate for the D6-brane of type IIA stringtheory.To compute the tension, we just integrate the energy density

2V along the four dimensions in which the monopole is

embedded.


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embedded.Since V is independent of the compact direction, we get:

22 R d3 2V


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T KK 6 = (2 p)9 2R 10 d3x 2V = 2

(2 p)9 (2R 10)2

=1gs

2(2 s )7

= T D 6



embedded.Since V is independent of the compact direction, we get:

T2

2 R d3 2V


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T KK 6 = (2 p)9 2R 10 d3x 2V = 2

(2 p)9 (2R 10)2

=1gs

2(2 s )7

= T D 6

Success!!

We know that branes in type IIB string theory can beobtained from those of type IIA bycircle compactication andT-duality .


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We know that branes in type IIB string theory can beobtained from those of type IIA bycircle compactication andT-duality .

It is easy to check that this reproduces the tensions of all thebranes of type IIB: D1,D3,D5,D7 and NS5.


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We know that branes in type IIB string theory can beobtained from those of type IIA bycircle compactication andT-duality .It is easy to check that this reproduces the tensions of all thebranes of type IIB: D1,D3,D5,D7 and NS5.However it gives us some more information.Recall that in type IIB there are two types of strings:


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Recall that in type IIB there are two types of strings:

F-strings of tension: 12 2s

D-strings of tension:1gs

12 2s


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It has been argued that type IIB string theory has S-duality:

gs 1gs

, s gs s

Under this symmetry, the F-string and D-string areinterchanged . One can easily check that their tensions getinterchanged.


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gs 1gs

, s gs sUnder this symmetry, the F-string and D-string areinterchanged . One can easily check that their tensions getinterchanged.It has also been shown that p F-strings and q D-strings form


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p g q gstable bound states called ( p,q) strings, if p, q are co-prime.


gs 1gs



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stable bound states called ( p,q) strings, if p, q are co-prime.

These have tension:

T p,q = p2 + q2g2s 12 2s


gs 1gs



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stable bound states called ( p,q) strings, if p, q are co-prime.

These have tension:

T p,q = p2 + q2g2s 12 2sWe will now see that M-theory explains both these facts in abeautiful way.

Suppose we compactify M-theory on two circles x10 , x9 of radii R10 , R 9 to get type IIA string theory in 9 dimensions.


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M2-brane wrapped on x10 type IIA F-string


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M2-brane wrapped on x10 type IIA F-stringM2-brane wrapped on x

9

D2-brane wrapped on x9


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9


Now let us perform a T-duality on x9:


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9


Now let us perform a T-duality on x9:type IIA F-string type IIB F-stringD2 brane wrapped on x9 type IIB D string


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D2-brane wrapped on x type IIB D-string



9




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D2-brane wrapped on x type IIB D-stringIt follows that:



9




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D2-brane wrapped on x type IIB D-stringIt follows that:

F-string D-string (IIB) x9 x10 (M-theory)


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In fact we easily nd that:

gs (IIB) =R10R9

, s (IIB) = 3 pR10


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gs (IIB) =R10R9

, s (IIB) = 3 pR10Next, suppose in the same compactication, we wrap an

M2-brane p times along x10

and q times along x9.


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gs (IIB) =R10R9




The result, after T-dualising on x9, is a string in type IIBtheory that has p units of F-string charge as well as q units of D-string charge.


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gs (IIB) =R10R9




The result, after T-dualising on x9, is a string in type IIBtheory that has p units of F-string charge as well as q units of D-string charge.


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Its tension will be:T M 2 wrapped = T M 2 p(2R 10)2 + q(2R 9)2= p2 + q2g2s 12 2s = T p,q

reducing the existence of ( p,q) strings to Pythagoras theorem !

In these three lectures we have presented arguments that

11-dimensional M-theory exists.


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In these three lectures we have presented arguments that

11-dimensional M-theory exists.Its existence can be used to explain many features of stringtheory, including D-branes and dualities.


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It is likely that it holds the key to string theory, but no onehas yet been able to unlock this Forbidden City!


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Outline

M-theory: Motivation and background

11d Supergravity


Compactication to 10dBranes and dualities from M-theory



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Single M2 braneBagger-Lambert theory



ABJM theory

Conclusions


In string theory, the discovery of D-branes brought about arevolution in our understanding of quantum eld theory .


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In string theory, the discovery of D-branes brought about arevolution in our understanding of quantum eld theory .Like any soliton (e.g. monopole, cosmic string), a D-branepossesses degrees of freedom that are bound to it.


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In string theory, the discovery of D-branes brought about arevolution in our understanding of quantum eld theory .Like any soliton (e.g. monopole, cosmic string), a D-branepossesses degrees of freedom that are bound to it.These can be found by considering the light modes of the bulk


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theory expanded about the brane solution.


In string theory, the discovery of D-branes brought about arevolution in our understanding of quantum eld theory .Like any soliton (e.g. monopole, cosmic string), a D-branepossesses degrees of freedom that are bound to it.These can be found by considering the light modes of the bulk


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theory expanded about the brane solution.They are usually non-gravitational degrees of freedom, namelygauge elds, scalar elds and fermions.

The Dirichlet description of branes, in terms of open stringendpoints, provides an explicit construction of these degrees

of freedom - by quantising open strings.


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of freedom - by quantising open strings.It also provides information about their interactions .


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of freedom - by quantising open strings.It also provides information about their interactions .Thus we have a worldvolume eld theory on D-branes.


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of freedom - by quantising open strings.It also provides information about their interactions .Thus we have a worldvolume eld theory on D-branes.This is generically a non-renormalisable eld theory with


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arbitrarily high-derivative operators suppressed by the stringscale s .

For D-branes, this eld theory is in principlecompletelycalculable in string perturbation theory.


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On taking the string length ls =

0 it reduces to aconventional Yang-Mills type eld theory.


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On taking the string length ls =

0 it reduces to aconventional Yang-Mills type eld theory.Since the D-branes are supersymmetric, the eld theory willalso be supersymmetric.


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For D-branes, this eld theory is in principlecompletelycalculable in string perturbation theory.On taking the string length l

s=

0 it reduces to a

conventional Yang-Mills type eld theory.Since the D-branes are supersymmetric, the eld theory willalso be supersymmetric.By taking different brane congurations in different


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backgrounds, a variety of eld theories can be engineered inthis way.

The simplest special case arises when we takesuperstringtheory in at 10d spacetime with a stack of N parallelDp-branes.


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The simplest special case arises when we takesuperstringtheory in at 10d spacetime with a stack of N parallelDp-branes.The result (as ls 0) is maximally supersymmetricYang-Mills theory:

L = tr 14g2YM F F 12 DX iD X i g2YM

4 [X i , X j ]2

+ fermions


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where i = 1 , 2, , 9 p.



4 [X i , X j ]2

+ fermions


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where i = 1 , 2, , 9 p.Here A , X i , are all in the adjoint of U (N ).



4 [X i , X j ]2

+ fermions


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where i = 1 , 2, , 9 p.Here A , X i , are all in the adjoint of U (N ).The diagonal components of the scalars parametrise thetransverse directions to the brane.


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The group-theory structure has a nice pictorial representationin terms of open strings, shown here for the case of U (3) .


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This picture has it all: Cartan subalgebra, positive roots,negative roots, simple roots...



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This picture has it all: Cartan subalgebra, positive roots,negative roots, simple roots...Using orientifolds (orientation-reversing hyperplanes) one canget the other classical gauge groups, SO(N ), Sp(N ).



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This picture has it all: Cartan subalgebra, positive roots,negative roots, simple roots...Using orientifolds (orientation-reversing hyperplanes) one canget the other classical gauge groups, SO(N ), Sp(N ).

Using orbifolds (which do not reverse orientation) one can getdirect product gauge groups and bi-fundamental matter .


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Many features of the eld theory can be understood usingbranes along with the underlying superstring theory:

Nonabelian gauge symmetry (from stretched open strings).


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Nonabelian gauge symmetry (from stretched open strings).Higgs mechanism (from transverse motions of the branes).


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Nonabelian gauge symmetry (from stretched open strings).Higgs mechanism (from transverse motions of the branes).Supersymmetry (from spatial alignment).


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Nonabelian gauge symmetry (from stretched open strings).Higgs mechanism (from transverse motions of the branes).Supersymmetry (from spatial alignment).Duality (from duality of string theory).


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Nonabelian gauge symmetry (from stretched open strings).Higgs mechanism (from transverse motions of the branes).Supersymmetry (from spatial alignment).Duality (from duality of string theory).Monopoles (from D-strings ending on D3-branes).


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Nonabelian gauge symmetry (from stretched open strings).Higgs mechanism (from transverse motions of the branes).Supersymmetry (from spatial alignment).Duality (from duality of string theory).Monopoles (from D-strings ending on D3-branes).

Conformal invariance for D3 branes (from constancy of


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Conformal invariance for D3 branes (from constancy of dilaton).


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Conversely the eld theory explains many aspects of the

underlying string theory:M(atrix) theory .


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Conversely the eld theory explains many aspects of the

underlying string theory:M(atrix) theory .AdS/CFT correspondence (here the eld theory is the entirestring theory!).


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We would like to have a similar understanding for theworldvolume theory on M-branes.


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We would like to have a similar understanding for theworldvolume theory on M-branes.As we have seen, M-theory has two kinds of stable branes:


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M2-branes (membranes)


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M2-branes (membranes)M5-branes


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Besides the above motivations, one additional motivation isthat one may be able to use the M-brane eld theory to give aprecise denition of M-theory.


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Besides the above motivations, one additional motivation isthat one may be able to use the M-brane eld theory to give aprecise denition of M-theory.In the following lectures I will describe some recent progress in


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In the following lectures I will describe some recent progress inunderstanding the eld theory on multiple M2-branes, whichremained unknown for a decade.

We have seen that type IIA string theory lifts to M-theory asgs .


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We have seen that type IIA string theory lifts to M-theory asgs .In the process D2-branes lift to M2-branes.


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We have seen that type IIA string theory lifts to M-theory asgs .In the process D2-branes lift to M2-branes.The eld theory on N D2-branes is just maximallysupersymmetric or N = 8 Yang-Mills theory in (2 + 1) d,which has 7 transverse scalars.


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We have seen that type IIA string theory lifts to M-theory asgs .In the process D2-branes lift to M2-branes.The eld theory on N D2-branes is just maximallysupersymmetric or N = 8 Yang-Mills theory in (2 + 1) d,which has 7 transverse scalars.This is a super-renormalisable theory that inherits its couplingfrom the string coupling gs :

gs


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gYM = gsls

Therefore in the M-theory limit, gYM .


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Therefore in the M-theory limit, gYM .For a super-renormalisable theory, this is the infrared limit.


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Therefore in the M-theory limit, gYM .For a super-renormalisable theory, this is the infrared limit.Thus we may dene:

LM 2 = limgYM 1

g2YM LD 2and the problem is to nd an explicit form for this limitingtheory.


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LM 2 = limgYM 1

g2YM LD 2and the problem is to nd an explicit form for this limitingtheory.The limiting theory, if interacting, must be an infrared xedpoint and therefore conformal invariant.


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LM 2 = limgYM 1

g2YM LD 2and the problem is to nd an explicit form for this limitingtheory.The limiting theory, if interacting, must be an infrared xedpoint and therefore conformal invariant.


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Also, if the brane interpretation is to make sense, the eldtheory should have 8 scalars with an SO(8) global symmetry,describing transverse motion.

Thus we are looking for a2 + 1 d eld theory with:


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N = 8 supersymmetry.


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SO(8) global symmetry.


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SO(8) global symmetry.Superconformal invariance.


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SO(8) global symmetry.Superconformal invariance.

Possible application also to quantum critical points inCondensed Matter physics!


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Outline

M-theory: Motivation and background11d Supergravity





Single M2 brane



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ABJM theoryConclusions

Single M2 brane

For a single D2-brane the theory is relatively simple.


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Single M2 brane

For a single D2-brane the theory is relatively simple.It has been known for a long time how to relate it to asingleM2-brane.


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Single M2 brane

For a single D2-brane the theory is relatively simple.It has been known for a long time how to relate it to asingleM2-brane.In the limit s 0 (the Yang-Mills limit) the D2-brane isdescribed by a free Abelian gauge theory:

Lsingle D 2 = 1

4g2YM F F

12 X

i

X i

+ fermionsh X i i 1 2 7 t i th 7 t


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where X i , i = 1 , 2, , 7 parametrise the 7 transversedirections to the D2-brane.

The M2-brane action has to be different, because the branenow has 8 transverse directions so there should be 8 scalarelds.


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The M2-brane action has to be different, because the branenow has 8 transverse directions so there should be 8 scalarelds.The 8th scalar arises by a duality transformation as follows:

14g2YM F F 12 BF 12 g2YM BB


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The M2-brane action has to be different, because the branenow has 8 transverse directions so there should be 8 scalarelds.The 8th scalar arises by a duality transformation as follows:

14g2YM F F 12 BF 12 g2YM BB

Integrating out the auxiliary eld B gives back the originalaction. But integrating out A

instead gives:

B B = 0 =B =1 X


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B B 0 B gYM X where X 8 is a new scalar eld.

Flux quantisation of the original U (1) gauge eld implies thatX 8 is a periodic scalar:

X 8X 8 + 2 gYM


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X 8X 8 + 2 gYM

The action is therefore:

12

X i X i 12

X 8 X 8 + fermions

which still depends on gYM via the periodicity of X 8.


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X 8X 8 + 2 gYM

The action is therefore:

12

X i X i 12

X 8 X 8 + fermions

which still depends on gYM via the periodicity of X 8.In the limit gYM

the scalar X decompacties, and we

can relabel it X 8 to nd the theory for a single M2 brane:


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Lsingle M 2 = 12

X I X I + fermions

where I = 1 , 2,

, 8.

This theory satises all the desired criteria. However that wasrelatively easy, since its a free eld theory.


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As we know, multiple D-branes form aninteracting Yang-Millstheory.


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As we know, multiple D-branes form aninteracting Yang-Millstheory.In this case the problem of nding the corresponding multipleM2-brane theory is much more difficult.


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Outline






Single M2 brane

Bagger-Lambert theoryInterpretation of BL theory


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ABJM theoryConclusions


Let us rst give arguments for some general properties of themultiple membrane theory.


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Let us rst give arguments for some general properties of themultiple membrane theory.Since membranes have 8 transverse directions, the eld theory

needs to have 8 scalar elds. By supersymmetry, it also musthave 4 two-component fermions.


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needs to have 8 scalar elds. By supersymmetry, it also musthave 4 two-component fermions.In (2 + 1) d the canonical dimensions of elds are as follows:

[X ] =1

2, [] = 1


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needs to have 8 scalar elds. By supersymmetry, it also musthave 4 two-component fermions.In (2 + 1) d the canonical dimensions of elds are as follows:

[X ] =1

2, [] = 1

Then scale invariance restricts the interactions to be of thef


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form:(X )6 and (X )2

A key insight was that the bosonic eld content can, andshould, include a nondynamical (Chern-Simons) gauge eldwith a Lagrangian:

S CS =k

2tr

A A +

2

3A A A


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S CS =k

2tr

A A +

2

3A A A

This is special to (2 + 1) d. This action is a topologicalinvariant and does not have any local gauge-invariant degreesof freedom.


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S CS =k

2tr

A A +

2

3A A A

This is special to (2 + 1) d. This action is a topologicalinvariant and does not have any local gauge-invariant degreesof freedom.Therefore the gauge eld does not contribute to thedynamical degrees of freedom, though it can help to close the


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y g , g psupersymmetry algebra.

The rst successful attempt to nd an interacting CFT

satisfying all the requirements was made by Bagger andLambert (a crucial step was provided independently byGustavsson). This is called the Bagger-Lambert A4 theory.


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satisfying all the requirements was made by Bagger andLambert (a crucial step was provided independently byGustavsson). This is called the Bagger-Lambert A4 theory.It relies on a mathematical structure called a Euclidean3-algebra.


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satisfying all the requirements was made by Bagger andLambert (a crucial step was provided independently byGustavsson). This is called the Bagger-Lambert A4 theory.It relies on a mathematical structure called a Euclidean3-algebra.This involves generators T A , a three-bracket, and a totallyantisymmetric 4-index structure constant f ABCD satisfying:

[T A , T B , T C ] = f ABC D T D


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satisfying all the requirements was made by Bagger andLambert (a crucial step was provided independently byGustavsson). This is called the Bagger-Lambert A4 theory.It relies on a mathematical structure called a Euclidean3-algebra.This involves generators T A , a three-bracket, and a totallyantisymmetric 4-index structure constant f ABCD satisfying:

[T A , T B , T C ] = f ABC D T D

A generalised trace over the three-algebra indices provides ametric on the 3-algebra:


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hAB = tr( T A , T B )

The structure constants satisfy the fundamental identity:

f AEF G f BCDG f BEF G f ACDG + f CEF G f ABDG f DEF G f ABCG = 0


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The scalars X I and fermions are three-algebra valued andthe interactions are:

Tr [X A , X B , X C ]2 and Tr [

A , X B , C ]X D


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The scalars X I and fermions are three-algebra valued andthe interactions are:

Tr [X A , X B , X C ]2 and Tr [

A , X B , C ]X D

And there is a gauge eld AAB with minimal couplings to thescalars and fermions, and a Chern-Simons interaction:

k f ABCD A AB ACD

+23 f

GAEF f BCDG A

AB A

CD A

EF


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where k is the quantised level.


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Here f ABCD

has been left abstract, but it was later shownthat there is only one consistent solution of the fundamentalidentity:

f ABCD = ABCD , A, B, C, D = 1 , , 4By taking suitable linear combinations of AAB one nds a pairof SU (2) gauge elds A , A .


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Here f ABCD

has been left abstract, but it was later shownthat there is only one consistent solution of the fundamentalidentity:

f ABCD = ABCD , A, B, C, D = 1 , , 4By taking suitable linear combinations of AAB one nds a pairof SU (2) gauge elds A , A .The Chern-Simons term reduces to the difference of twoactions:

k tr A dA +23 A A A A dA 23 A A A


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The scalars and fermions are bi-fundamentals :

X I a a and a a

and, e.g. DX = X A X + X A


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X I a a and a a


In this way the theory reduces to a conventional gauge theory.


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X I a a and a a


In this way the theory reduces to a conventional gauge theory.

Importantly we see that parity is preserved if we requireA A under parity.


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To complete this discussion let us write down the action of Bagger-Lambert theory in the bi-fundamental notation:

S A 4 =k

2 d3x Tr (D X I )D X I + i D 83 X IJK X IJK

13 i IJ [X I , X J , ] + 13 i IJ [X I , X J , ]+ 12

A A + 23 AA A

A A 23 AA A

where:

X IJK = X [I X J X K ]


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X IJK = X [I X J X K ]

[X I , X J , ] = X [I X J ]

X [I X J ] + X [I X J ]

Outline






Single M2 brane

Bagger-Lambert theoryInterpretation of BL theory

L t i 3 lg b


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ABJM theory

Conclusions


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One should now check if this theory really describesM2-branes.It should be noted that there is no coupling constant in thetheory. This is as expected since M-theory also has nocoupling constant.


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One should now check if this theory really describesM2-branes.It should be noted that there is no coupling constant in thetheory. This is as expected since M-theory also has nocoupling constant.However the levelk of the Chern-Simons actions acts as a

coupling.


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One should now check if this theory really describesM2-branes.It should be noted that there is no coupling constant in thetheory. This is as expected since M-theory also has nocoupling constant.However the levelk of the Chern-Simons actions acts as a

coupling.This is a puzzle: what isk doing here?


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The interpretation of the Bagger-Lambert theory becomesclearer when we consider the Higgs mechanism.


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The interpretation of the Bagger-Lambert theory becomesclearer when we consider the Higgs mechanism.Take k = 1 to start with. If we give a vevv to one componentof the bi-fundamental elds, then at energies below this vev,the Lagrangian becomes:

LBL vev v =1v2 L

U (2)SY M + O

1v3

and one SU (2) gauge eld has become dynamical!


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The interpretation of the Bagger-Lambert theory becomesclearer when we consider the Higgs mechanism.Take k = 1 to start with. If we give a vevv to one componentof the bi-fundamental elds, then at energies below this vev,the Lagrangian becomes:

LBL vev v =1v2 L

U (2)SY M + O

1v3

and one SU (2) gauge eld has become dynamical!This is an unusual result. In Yang-Mills with gauge group G,when we give a vev to one component of an adjoint scalar, atlow energy the Lagrangian becomes:

12 L

(G)SY M =

12 L

(G G)SY M


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g2YM SY M vev v g2YM SY M

where G is the subgroup that commutes with the vev.

Lets give a quick derivation of this novel Higgs mechanism:

LCS = tr A dA +

23 A

A

A Ad

A

23

A

A

A

= tr A F + +16 A A A

where A = A A , F + = dA + + 12 A + A + .


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LCS = tr A dA +

23 A

A

A Ad

A

23

A

A

A

= tr A F + +16 A A A

where A = A A , F + = dA + + 12 A + A + .Also the covariant derivative on a scalar eld is:

DX = X A X + X A


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LCS = tr A dA +

23 A

A

A Ad

A

23

A

A

A

= tr A F + +16 A A A


DX = X A X + X A If X = v 1 then:

(DX )2

v2(A )(A ) +


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LCS = tr A dA +

23 A

A

A Ad

A

23

A

A

A

= tr A F + +16 A A A


DX = X A X + X A If X = v 1 then:

(DX )2

v2(A )(A ) +

Thus, A is massive but not dynamical. Integrating it outgives us:

1

2 (F + ) (F + ) + O

13


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4v2 ( + ) ( + ) O v3so A + becomes dynamical.

One can check that the bi-fundamental X I

reduces to anadjoint under A + . The rest of N = 8 SYM assembles itself correctly.


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reduces to anadjoint under A + . The rest of N = 8 SYM assembles itself correctly.But how should we physically interpret this?

LBLvev v

= 1v2

LU (2)SY M + O 1v3


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LBLvev v

= 1v2

LU (2)SY M + O 1v3It seems like the M2 branes are becoming a pair of D2 braneswith gYM = v.


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LBLvev v

= 1v2

LU (2)SY M + O 1v3It seems like the M2 branes are becoming a pair of D2 braneswith gYM = v.

Have we somehow compactied the spacetime theory? This isnot really possible because we have not done anything to thebulk spacetime.


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The resolution is to note that for any nite v, there arecorrections to the Yang-Mills action. These decouple only asv . So at best we can say that:

LBL vev v = limv1v2 L

U (2)SY M


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U (2)SY M

The RHS is by denition the theory on M 2-branes! So this ismore like a proof that the original Chern-Simons theoryreally is the theory on two M 2-branes.


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U (2)SY M

The RHS is by denition the theory on M 2-branes! So this ismore like a proof that the original Chern-Simons theoryreally is the theory on two M 2-branes.

More precisely, this is the case far out on the moduli space (atlarge Higgs vevv).


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However once we introduce the Chern-Simons level k then the

analysis is different:

LBLvev v

=kv2

LU (2)SY M + Okv3


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LBLvev v

=kv2

LU (2)SY M + Okv3

If we take k , v with v2/k = gYM xed, then in thislimit the RHS actually becomes:1

g2YM LU (2)SY M

and this is denitely the Lagrangian for two D 2 branes atnite coupling.


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LBLvev v

=kv2

LU (2)SY M + Okv3

If we take k , v with v2/k = gYM xed, then in thislimit the RHS actually becomes:1

g2YM LU (2)SY M

and this is denitely the Lagrangian for two D 2 branes atnite coupling.So this time we have compactied the theory! How can thatbe?


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be?

It has been proposed that the level k corresponds to the orderof an orbifold group.


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It has been proposed that the level k corresponds to the orderof an orbifold group.In this proposal, the branes described by Bagger-Lamberttheory are not transverse to R 8, but to R 8/Z k for some actionof the group Z k .


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This is a potentially nice explanation since the levelk is aninteger which ts well with the fact that the order of a nitegroup is also an integer.


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This is a potentially nice explanation since the levelk is aninteger which ts well with the fact that the ord

mtheory beijing v9

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