Noncommutative Geometries in M-theory

David Berman (Queen Mary, London)Neil Copland (DAMTP, Cambridge)Boris Pioline (LPTHE, Paris)Eric Bergshoeff (RUG, Groningen)

Introduction

Noncommutative geometries have a natural realisation in string theory.

M-theory is the nonperturbative description of string theory.

How does noncommutative geometry arise in M-theory?

Outline

Review how noncommutative theories arise in string theory: a physical perspective.

M-theory as a theory of membranes and fivebranes. The boundary term of membrane. Its quantisation. A physical perspective. M2-M5 system and fuzzy three-spheres. The degrees of freedom of the membrane.N

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Noncommutative geometry in string theory Simplest approach: the coupling of a string to a

background two form B:

For constant B field this is a boundary term:

This is the action of the interaction of a charged particle in a magnetic field.

S =R§ dX

M ^dX N BM N

S =R@§ dtBM N X M @tX N

Noncommutative geometry in string theory We quantise this action (1st order) and we

obtain:

where

Including the neglected kinetic terms

[X M ;X N ]D =µM N

µ= B ¡ 1

µ= B(1+B 2)

Noncommutative geometry in string theory Therefore, the open strings see a

noncommutative space, in fact the Moyal plane. The field theory description of the low energy dynamics of open strings will then be modelled by field theory on a Moyal plane and hence the usual product will be replaced with the Moyal *product.

Lets view this another way (usefull for later) .

Noncommutative geometry in string theory Instead of quantising the boundary term of

the open string consider the classical dynamics of an open string.

The boundary condition of the string in a background B field is:

This can solved to give a zero mode solution:

@¾X i +B i j@¿X j =0

X i =pi0¿ +B i j pj0¾

Noncommutative geometry in string theory The string is stretched into a length:

The canonical momentum is given by:

The elogation of the string is proportional to the momentum:

¢ i =B i j pj0

P i = (@¿X i ¡ B i j@¾X j ) = (1+B2)pi0

¢ i = µi j Pj µ= B1+B 2

Noncommutative geometry in string theory The interactions will be via their end points

thus in the effective field theory there will be a nonlocal interaction:

Á(x+ 12µP )Á(x ¡

12µP ) = Á¤Á

L int =Á(x+ 12¢ )Á(x ¡

12¢ )

Noncommutative geometry in string theory The effective metric arises from considering

the Hamiltonian:

E =p(1+B2)p20

E =pGi j PiPj Gi j = ±i j

1+B 2

M-theory

For the purposes of this talk M-theory will be a theory of Membranes and Five-branes in eleven dimensional spacetime.

A membrane may end on a five-brane just as an open string may end on a D-brane.

The background fields of eleven dimensional supergravity are C3 , a three form potential and the metric.

What happens at the boundary of a membrane when there is a constant C field present?

What is the effective theory of the five-brane?

Boundary of a membrane

The membrane couples to the background three form via a pull back to the membrane world volume.

Constant C field, this becomes a boundary term:

1st order action, quantise a la Dirac(This sort of action occurs in the effective theory of

vortices; see eg. Regge, Lunde on He3 vortices).

S =R§ dX

i ^dX j ^dX kCi j k

S =R@§ X

idX j ^dX kCi j k

Boundary of a membrane

Resulting bracket is for loops; the boundary of a membrane being a loop as opposed to the boundary of string being a point:

[X (¾)i ;X (~¾)j ]D = 1C±(¾¡ ~¾)@¾X k²i j k

Strings to Ribbons

Look at the classical analysis of membranes in background fields.

The boundary condition of the membrane is

This can be solved by:

@½X i ¡ Ci j k@¾X j@¿X k = 0 ½=0;¼

X i =pi0¿ +ui¾+¢ i½

Strings to ribbons

Where after calculating the canonical momentum

One can as before express the elongation of the boundary string as

With

Pi =Ci j k@¾X j@½X k

¢ i = µ² i j k uj P k

kuk2

µ= C1+C 2

Strings to Ribbons

Thus the string opens up into a ribbon whose width is proportional to its momentum.

For thin ribbons one may model this at low energies as a string.

The membrane Hamiltonian in light cone formulation is given by:

With g being the determinant of the spatial metric, for ribbon this becomes

P ¡ =Rd¾d½1

P + (p20+g)

g= C 2

(1+C 2)2 (P2 ¡ (ui P i )2

kuk2 )

Ribbons to strings

After expressing p0 in terms of P and integrating over rho the Hamiltonian becomes:

The Lagrangian density becomes

This is the Schild action of a string with tension C!

P ¡ =Rd¾ 1

P + (1+C 2) (P2+ C 2(P i ui )2

kuk2 )

L = (@¿X )2+ C 2

kuk2 fXi ;X j gfX i ;X j g

Strings to matrices

For those who are familiar with matrix regularisation of the membrane one may do the same here to obtain the matrix model with light cone Hamiltonian:

P ¡ = 1P + ([A0;X i ]2+C2[X i ;X j ][X i ;X j ])

Interactions

The interactions would be nonlocal in that the membranes/ribbons would interact through their boundaries and so this would lead to a deformation from the point of view of closed string interactions.

Some loop space version of the Moyal product would be required.

Branes ending on branes

We have so far discussed the effective field theory on a brane in a background field.

Another interesting application of noncommutative geometry to string theory is in the description of how one brane may end on another.

Description of D-branes

When there are multiple D-branes, the low energy effective description is in terms of a non-abelian (susy) field theory. Branes ending on branes may be seen as solitonic configurations of the fields in the brane theory.

Branes ending on branes: k-D1, N-D3 D3 brane perspective ½ BPS solution of the

world volume theory N=1, BIon solution to

nonlinear theory, good approximation in large k limit

N>1, Monopole solution to the U(N) gauge theory

Spike geometry

D1 brane perspective ½ BPS solution of the

world volume theory Require k>1, good

approximation in large N limit.

Fuzzy funnel geometry

D1 ending on a D3

D3 brane perspective

Monopole equation

D1 brane perspective

Nahm Equation

DÁ= ¤F d©i

d¾ = § i2²i j k[©

j ;©k]

Nahm equation

Solution of the Nahm equation gives a fuzzy two sphere funnel:

Where

andR̂(¾) = § 1

2(¾¡ ¾1 )

©i = R̂(¾)®iN ; i = 1;2;3

[®iN ;®jN ]= 2i²i j k®kN

Fuzzy Funnel

The radius of the two sphere is given by

With

Which implies

P 3i=1(®

iN )

2 = (N 2 ¡ 1)1N £ N

R(¾)2 = (2¼ls )2

N

P 3i=1Tr[©

i (¾)2]

R(¾) = N ¼̀ s¾¡ ¾1

q1¡ 1

N 2

BIon Spike The BIon solution:

Agreement of the profile in the large N limit between BIon description and fuzzy funnel.

Also, agreement between spike energy per unit length; Chern Simons coupling; and fluctuations.

The Nahm Transform takes you between D1 and D3 brane descriptions of the system.

Á(r) = ¼lsNr

Trivial observation on fuzzy 2-spheres Consider harmonics on a 2-sphere with

cutoff, E. Number of modes: Where k is given by:

If the radius R is given by: Then the number of modes in the large N

limit scale as:

P kl (2l +1) = (k+1)2

k(k+1)R 2 =E 2

R2 =N 2 ¡ 1

N 2

M2 branes ending on M5 branes D1 ending on D3

branes BIon Spike Nahm Equation Fuzzy Funnel with a

two sphere blowing up into the D3

M2 branes ending on M5 branes

Self-dual string Basu-Harvey Equation Fuzzy Funnel with a

three sphere blowing up in to M5

Self-dual string

Solution to the ½ BPS equation on the M5 brane,

BIon like spike gives the membrane

H = ¤dÁ

Á(r) = cNr 2

Basu-Harvey equation

Where

And G5 is a certain constant matrix

Conjectured to be the equivalent to the Nahm equation for the M2-M5 system

dX i

ds + M 311

8¼p2N

14!²i j kl[G5;X j ;X k;X l ]= 0

[X 1;X 2;X 3;X 4]=P

perms ¾sign(¾)[X¾(1);X ¾(2);X ¾(3);X ¾(4)]

Fuzzy funnel Solution

Solution:

Where Gi obeys the equation of a fuzzy

3-sphere

X i (s) = ip2¼

M3211

1psG

i

Gi + 12(n+2) ²i j klG5GjGkGl =0

Properties of the solution

The physical radius is given by

Which yields

Agreeing with the self-dual string solution

R =

s ¯¯¯¯T rP

(X i )2

T r1

¯¯¯¯

s » NR 2

From a Hamiltonian

Consider the energy functional

Bogmolnyi type construction yields

E = T22

Rd2¾Tr

³X i0X i0 ¡ 1

3![Xj ;X k;X l][X j ;X k;X l]

´

E = T22

Rd2¾

½Tr

³X i0+gi j kl 14![H

¤;X j ;X k;X l]´2

+T¾

T = ¡ T2Rd2¾Tr

³gi j klX i0 1

4![H¤;X j ;X k;X l]

´

From a Hamiltonian

For more than 4 active scalars also require:

H must have the properties:

For four scalars one recovers B-H equation and H=G5

13!gi j klgipqrTr

¡[X j ;X k;X l][X p;X q;X r ]

¢= Tr

¡[X i ;X j ;X k][X i ;X j ;X k]

¢

fH ¤;X ig= 0 H ¤2 =1

Properties of this solution

Just as for the D1 D3 system the fluctuation spectra matches and the tension matches.

There is no equivalent of the Nahm transform.

The membrane theory it is derived from is not understood.

Questions???

Can the B-H equation be used to describe more than the M2 ending on a single M5?

How do the properties of fuzzy spheres relate to the properties of nonabelian membranes?

What is the relation between the B-H equation and the Nahm equation?

Supersymmetry??? How many degrees of freedom are there on

the membrane?

M-theory Calibrations

Configurations with less supersymmetry that correspond to intersecting M5 and M2 branes

Classified by the calibration that may be used to prove that they are minimal surfaces

Goal: Have the M2 branes blow up into generic M-theory calibrations

M-theory Calibrations

Planar five-brane

M5: 1 2 3 4 5M 2: 1 #

g2345 = 1 º = 1=2

X 20= ¡ H ¤[X 3;X 4;X 5] ; X 30=H ¤[X 4;X 5;X 2] ;

X 40= ¡ H ¤[X 5;X 2;X 3] ; X 50=H ¤[X 2;X 3;X 4] :

M-theory CalibrationsIntersecting five branes

M5: 1 2 3 4 5M 5: 1 2 3 6 7M 2: 1 #

g2345 =g2367 = 1 º = 1=4

X 20= ¡ H ¤[X 3;X 4;X 5]¡ H ¤[X 3;X 6;X 7] ; X 30=H ¤[X 4;X 5;X 2]+H ¤[X 6;X 7;X 2] ;

X 40= ¡ H ¤[X 5;X 2;X 3] ; X 50=H ¤[X 2;X 3;X 4] ;

X 60= ¡ H ¤[X 7;X 2;X 3] ; X 70=H ¤[X 2;X 3;X 6] ;

[X 2;X 4;X 6]= [X 2;X 5;X 7] ; [X 2;X 5;X 6]= ¡ [X 2;X 4;X 7] ;

[X 3;X 4;X 6]= [X 3;X 5;X 7] ; [X 3;X 5;X 6]= ¡ [X 3;X 4;X 7];

[X 4;X 5;X 6]= [X 4;X 5;X 7] = [X 4;X 6;X 7]= [X 5;X 6;X 7]= 0:

M-theory Calibrations

Intersecting five branes

M5: 1 2 3 4 5M 5: 1 2 3 6 7M 5: 1 2 3 8 9M 2: 1 #

g2345 = g2367 =g2389 =1 º = 1=8

M-theory CalibrationsX 20= ¡ H ¤[X 3;X 4;X 5] ¡ H ¤[X 3;X 6;X 7]¡ H ¤[X 3;X 8;X 9];

X 30=H ¤[X 4;X 5;X 2] + H ¤[X 6;X 7;X 2]+H ¤[X 8;X 9;X 2] ;

X 40= ¡ H ¤[X 5;X 2;X 3] ; X 50=H ¤[X 2;X 3;X 4] ;

X 60= ¡ H ¤[X 7;X 2;X 3] ; X 70=H ¤[X 2;X 3;X 6] ;

X 80= ¡ H ¤[X 9;X 2;X 3] ; X 90=H ¤[X 2;X 3;X 8] ;

[X 2;X 4;X 6]= [X 2;X 5;X 7] ; [X 2;X 5;X 6]= ¡ [X 2;X 4;X 7];

[X 2;X 4;X 8]= [X 2;X 5;X 9] ; [X 2;X 5;X 8]= ¡ [X 2;X 4;X 9] ;

[X 2;X 6;X 8]= [X 2;X 7;X 9] ; [X 2;X 7;X 8]= ¡ [X 2;X 6;X 9];

[X 3;X 4;X 6]= [X 3;X 5;X 7] ; [X 3;X 5;X 6]= ¡ [X 3;X 4;X 7];

[X 3;X 4;X 8]= [X 3;X 5;X 9] ; [X 3;X 5;X 8]= ¡ [X 3;X 4;X 9] ;

[X 3;X 6;X 8]= [X 3;X 7;X 9] ; [X 3;X 7;X 8]= ¡ [X 3;X 6;X 9];

[X 4;X 5;X 6]+ [X 6;X 8;X 9]= 0 ; [X 4;X 5;X 7]+ [X 7;X 8;X 9]= 0 ;

[X 4;X 5;X 8]+ [X 6;X 7;X 8]= 0 ; [X 4;X 5;X 9]+ [X 6;X 7;X 9]= 0 ;

[X 4;X 6;X 7]+ [X 4;X 8;X 9]= 0 ; [X 5;X 6;X 7]+ [X 5;X 8;X 9]= 0 ;

[X 4;X 6;X 8]= [X 4;X 7;X 9] + [X 5;X 6;X 9]+ [X 5;X 7;X 8] ;

[X 5;X 7;X 9]= [X 5;X 6;X 8] + [X 4;X 7;X 8]+ [X 4;X 6;X 9] :

M-theory Calibrations

M5: 1 2 3 4 5M 5: 1 2 3 6 7M 5: 1 4 5 6 7M 2: 1 #

g2345 = g2367 =g4567 =1 º = 1=8

M-theory Calibrations

M5: 1 2 3 4 5M 5: 1 2 4 6 8¹M 5: 1 2 3 6 7M 5: 1 2 5 7 8M 2: 1 #

g2345 =g2468 = ¡ g2367 = g2578 =1 º = 1=8

The solutions For example, two intersecting 5-branes

This is a trivial superposition of the basic B-H solution.

There are more solutions to these equations corresponding to nonflat solutions.

Calibrations

It is the calibration form g that goes into the generalised B-H equation.

Fuzzy funnels can successfully described all sorts of five-brane configurations.

Interesting to search for and understand the non-diagonal solutions.

Fuzzy Funnel description of membranes We have seen a somewhat ad hoc

description of membranes ending on five-brane configurations. Is there any further indication that this approach may have more merit??

Back to the basic M2 ending on an M5. The basic equation is that of a fuzzy 3-sphere.

How many degrees of freedom are there on a fuzzy three sphere?

Fuzzy Three Sphere

Again consider the number of modes of a three sphere with a fixed UV cut-off

Number of modes scales as k^3 (large k limit) k is given by R is given by Number of Modes

k2 =E 2R2

R =pN

N 3=2

Non-Abelian Membranes

This recovers (surprisingly) the well known N dependence of the non-Abelian membrane theory (in the large N limit).

The matrices in the action were originally just any NxN matrices but the solutions yielded a representation of the fuzzy three sphere.

Other fuzzy three sphere properties:

1. The algebra of a fuzzy three sphere is nonassociative.

2. The associativity is recovered in the large N limit.

Relation to the Nahm Equation To relate the Basu-Harvey equation to the

Nahm equation we do this by introducing a projection.

Projection P should project out G4 and then the remaining projected matrices obey the Nahm equation.

Consider:

P = 1=2(1+ i¡ 4¡ 5)

Projecting to Nahm

Properties:

P 2 = P P ¡ 4P = P ¡ 5P = 0 P ¡ aP = ¡ a a= 1::3

P ¡ 4¡ 5P = incP

Apply to Basu-Harvey

Project the Basu-Harvey equation

Case i=4, the equation vanishes

Case i=1,2,3 then one recovers the Nahm equation

P (dXi

ds + M 311

8¼p2N

14!²i j kl[G5;X j ;X k;X l])P = 0

Projected Basu-Harvey equation Provided:

Giving (in the large N limit)

X 4 = 32¼R 11G4

3c

dX a

d¾ + i2®0²abc[X b;X c]= 0

Discussion

Ad hoc attempts to generalise the Nahm equation have lead to interesting conjectures for the non-Abelian membrane theory.

Successes include the incorporation of calibrations corresponding to various fivebrane intersections. The geometric profile, fluctuations and tensions match known results.

The relation to the Nahm equation is through a projection (a bit different to the usual dimensional reduction.

A key note of interest is the interpretation of the degrees of freedom of the membrane as coming from the fuzzy thee sphere.

N 3=2

Conclusions

Noncommutative geometry arises naturally in the effective theory of strings- Moyal plane, fuzzy 2-sphere etc.

M-theory is the nonperturbative version of string theory.

It seems to require generalisations of these ideas to more exotic geometries.

eg. Noncommutative loop spaces, deformed string interactions, fuzzy three spheres, the encoding nonabelian degrees of freedom.