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.
3
The N 2 degrees of freedom of the membrane.
Noncommutative geometry in string
theory
Simplest approach: the coupling of a string to a
background two form B:
S=
R
M
N
dX ^ dX B M N
§
For constant B field this is a boundary term:
S=
R
@§
N
dtB M N X M @
X
t
This is the action of the interaction of a charged
particle in a magnetic field.
Noncommutative geometry in string
theory
We quantise this action (1st order) and we
obtain:
[X
M
N
MN
; X ]D = µ
where µ = B ¡ 1
Including the neglected kinetic terms
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:
i
j
@
¾X + B i j @
¿X = 0
This can solved to give a zero mode solution:
i
X =
pi0 ¿ +
j
B i j p0 ¾
Noncommutative geometry in string
theory

The string is stretched into a length:
¢i =

j
B i j p0
The canonical momentum is given by:
i
j
2 i
P i = (@
¿X ¡ Bi j @
¾X ) = (1 + B )p0
The elogation of the string is proportional to
the momentum:
i
ij
¢ = µ Pj
µ=
B
1+ 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:
L i n t = Á(x +
Á(x +
1
2 µP)Á(x
¡
1
2¢
)Á(x ¡
1
2 µP)
1
2¢
= Á¤Á
)
Noncommutative geometry in string
theory

The effective metric arises from considering
the Hamiltonian:
p
E = (1 + B 2 )p20
E=
p
Gi j Pi Pj
ij
G =
±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.
R
i
j
k
S = § dX ^ dX ^ dX Ci j k

Constant C field, this becomes a boundary term:
S=
R
@§
X i dX j ^ dX k Ci j k
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).

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:
i
j
[X (¾) ; X (~
¾) ]D =
1
C
k ij k
±(¾¡ ¾
~)@
¾X ²
Strings to Ribbons


Look at the classical analysis of membranes
in background fields.
The boundary condition of the membrane is
i
j
k
@
½X ¡ Ci j k @
¾X @
¿X = 0
This can be solved by:
X i = pi0¿ + ui ¾+ ¢ i ½
½= 0; ¼
Strings to ribbons

Where after calculating the canonical
momentum
j
k
Pi = Ci j k @
¾X @
½X

One can as before express the elongation of
the boundary string as
i
¢ =
With
µ=
µ² i j k u j P k
kuk 2
C
1+ 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
R
is given by:
1
¡
2
P
=
d¾d½P + (p0 + g)
With g being the determinant of the spatial metric,
for ribbon this becomes g = C 22 2 (P 2 ¡ ( u i P i2) 2 )
( 1+ C )
ku k
Ribbons to strings

After expressing p0 in terms of P and
integrating over rho the Hamiltonian
becomes:R
C 2 (P i ui )2
1
¡
2
P = d¾P + ( 1+ C 2 ) (P +
)
ku k 2

The Lagrangian density becomes
2
L = (@
¿X ) +

C2
i
j
f
X
;
X
2
kuk
gf X i ; X j g
This is the Schild action of a string with
tension C!
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
¡
=
1
P+
([A 0 ; X i ]2 + C 2 [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
DÁ = ¤F
Monopole equation

d© i
d¾
D1 brane perspective
= §
i
j
2 ² i j k [©
Nahm Equation
k
;© ]
Nahm equation

Solution of the Nahm equation gives a fuzzy
two sphere funnel:
^ )®i ;
©i = R(¾
N
i = 1; 2; 3
Where
i
[®N
j
; ®N
k
] = 2i ² i j k ®N
and
^ )= §
R(¾
1
2(¾¡ ¾1 )
Fuzzy Funnel

The radius of the two sphere is given by
2
R(¾) =
With P
( 2¼l s ) 2
N
3
i 2
(®
N)
i= 1
P
Tr [©i (¾) 2 ]
= (N 2 ¡ 1)1N £ N
Which implies
R(¾) =
3
i= 1
q
N ¼` s
¾¡ ¾1
1¡
1
N2
BIon Spike

The BIon solution:
Á(r ) =



¼l s N
r
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.
Trivial observation on fuzzy 2-spheres



Consider harmonics on a 2-sphere with
cutoff, E.
Pk
2
Number of modes:
(2l
+
1)
=
(k
+
1)
l
Where k is given by: k ( k + 1)
R2


= E
2
If the radius R is given by: R2 = N 2 ¡ 1
Then the number of modes in the large N
limit scale as:
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,
H = ¤dÁ
BIon like spike gives the membrane
Á(r ) =
cN
r2
Basu-Harvey equation
dX i
ds
+
M 131 1
j
p
²
[G
;
X
5
8¼ 2N 4! i j k l
;X k;X l] = 0
Where
1
2
3
4
[X ; X ; X ; X ] =
P
per m s ¾ sign(¾)[X
¾(1)
; X ¾( 2) ; X ¾( 3) ; X ¾(4) ]
And G5 is a certain constant matrix
Conjectured to be the equivalent to the Nahm
equation for the M2-M5 system
Fuzzy funnel Solution
Solution:
i
X (s) =
i
p
2¼ p1
i
G
3
s
2
M 11
Where Gi obeys the equation of a fuzzy
3-sphere
i
G +
1
j
²
G
G
i
j
k
l
5
2( n + 2)
k
l
G G = 0
Properties of the solution

The physical radius is given by
s ¯ P
¯
¯T r ( X i ) 2 ¯
¯
R= ¯
¯ Tr1
¯
Which yields
s»
N
R2
Agreeing with the self-dual string solution
From a Hamiltonian

Consider the energy functional
E=

T2
2
R
³
0
d2 ¾T r X i X i
0
´
¡ 3!1 [X j ; X k ; X l ][X j ; X k ; X l ]
Bogmolnyi type construction yields
E=
T2
2
R
½ ³
¾
´
2
2
i0
d ¾ Tr X + gi j k l 4!1 [H ¤ ; X j ; X k ; X l ] + T
T = ¡ T2
R
³
´
d2 ¾T r gi j k l X i 4!1 [H ¤ ; X j ; X k ; X l ]
0
From a Hamiltonian

For more than 4 active scalars also require:
1
3! gi j k l gi pqr

¡
¢
¡ i
¢
j
k
i
j
k
Tr [X ; X ; X ][X ; X ; X ] = Tr [X ; X ; X ][X ; X ; X ]
j
k
l
p
q
r
H must have the properties:
¤
i
f H ;X g= 0 H

¤2
= 1
For four scalars one recovers B-H equation
and H=G5
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 5
M2: 1
g2345 = 1
X
20
¤
3
4
5
#
º = 1=2
30
= H ¤ [X 4 ; X 5 ; X 2 ] ;
= ¡ H [X ; X ; X ]
;
X
X 4 = ¡ H ¤ [X 5 ; X 2 ; X 3 ]
;
X 5 = H ¤ [X 2 ; X 3 ; X 4 ] :
0
0
M-theory Calibrations
Intersecting five branes
M5:
M5:
M2:
1 2 3 4 5
1 2 3
6 7
1
g2345 = g2367 = 1
0
X 2 = ¡ H ¤ [X 3 ; X 4 ; X 5 ] ¡ H ¤ [X 3 ; X 6 ; X 7 ]
#
º = 1=4
0
;
X 3 = H ¤ [X 4 ; X 5 ; X 2 ] + H ¤ [X 6 ; X 7 ; X 2 ] ;
;
X 5 = H ¤ [X 2 ; X 3 ; X 4 ] ;
X 6 = ¡ H ¤ [X 7 ; X 2 ; X 3 ]
;
X 7 = 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 4 ; X 5 ; X 6 ] = [X 4 ; X 5 ; X 7 ]
;
=
[X 3 ; X 5 ; X 6 ] = ¡ [X 3 ; X 4 ; X 7 ];
[X 4 ; X 6 ; X 7 ] = [X 5 ; X 6 ; X 7 ] = 0:
0
X 4 = ¡ H ¤ [X 5 ; X 2 ; X 3 ]
0
0
0
M-theory Calibrations
Intersecting five branes
M5:
M5:
M5:
M2:
1 2 3 4 5
1 2 3
6 7
1 2 3
8
1
#
g2345 = g2367 = g2389 = 1
º = 1=8
9
M-theory Calibrations
0
X 2 = ¡ H ¤ [X 3 ; X 4 ; X 5 ] ¡
0
X 3 = H ¤ [X 4 ; X 5 ; X 2 ] +
0
X 4 = ¡ H ¤ [X 5 ; X 2 ; X 3 ]
H ¤ [X 3 ; X 6 ; X 7 ] ¡ H ¤ [X 3 ; X 8 ; X 9 ];
H ¤ [X 6 ; X 7 ; X 2 ] + H ¤ [X 8 ; X 9 ; X 2 ] ;
0
;
X 5 = H ¤ [X 2 ; X 3 ; X 4 ] ;
;
X 7 = H ¤ [X 2 ; X 3 ; X 6 ] ;
X 8 = ¡ H ¤ [X 9 ; X 2 ; X 3 ]
;
X 9 = 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 3 ; X 4 ; X 6 ] = [X 3 ; X 5 ; X 7 ]
;
;
[X 2 ; X 7 ; X 8 ] = ¡ [X 2 ; X 6 ; X 9 ];
[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 4 ; X 6 ; X 8 ] = [X 4 ; X 7 ; X 9 ] +
[X 5 ; X 6 ; X 7 ] + [X 5 ; X 8 ; X 9 ] = 0 ;
[X 5 ; X 6 ; X 9 ] + [X 5 ; X 7 ; X 8 ] ;
0
X 6 = ¡ H ¤ [X 7 ; X 2 ; X 3 ]
0
[X 5 ; X 7 ; X 9 ] = [X 5 ; X 6 ; X 8 ] +
0
0
[X 4 ; X 7 ; X 8 ] + [X 4 ; X 6 ; X 9 ] :
M-theory Calibrations
M5:
M5:
M5:
M2:
1 2 3 4 5
1 2 3
6 7
1
4 5 6 7
1
#
g2345 = g2367 = g4567 = 1
º = 1=8
M-theory Calibrations
M5:
M5:
M¹ 5 :
M5:
M2:
1
1
1
1
1
2 3 4 5
2
4
6
8
2 3
6 7
2
5
7 8
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 fivebrane 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 k 2 = E 2 R2
p
R is given by R = N
Number of Modes
N
3=2
Non-Abelian Membranes



1.
2.
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:
The algebra of a fuzzy three sphere is
nonassociative.
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¡ 4 P = P¡ 5 P = 0 P¡ a P = ¡ a a = 1::3
P¡ 4 ¡ 5 P = i ncP
Apply to Basu-Harvey
Project the Basu-Harvey equation
dX i
P( ds
+
M 131 1
j
p
²
[G
;
X
5
8¼ 2N 4! i j k l
; X k ; X l ])P = 0
Case i=4, the equation vanishes
Case i=1,2,3 then one recovers the Nahm
equation
Projected Basu-Harvey equation

Provided:
4
X =

32¼R 11 G 4
3c
Giving (in the large N limit)
dX a
d¾
+
i
b
c
²
[X
;
X
]
2®0 abc
= 0
Discussion




Ad hoc attempts to generalise the Nahm equation
have lead to interesting conjectures for the nonAbelian 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 N 3=2
degrees of freedom of the membrane as coming
from the fuzzy thee sphere.
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.