Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Here the star product, also called the Moyal

product, is defined by

ðf  gÞð xÞ

¼ exp i











f ðx

Þgðx

Þj

¼x

½12

The star product is associative and noncommutative,

and satisfies

f  g =

g 

f under complex conjugation.

Also, for functions that vanish rapidly enough at

infinity, there holds

f  g ¼

g  f ¼

fg ½13

An interesting consequence of the nonlocality

as expressed by the noncommutative geometry [7]

is the existence of a dipole excitation whose extent is

proportional to its momentum, x = k. This rela-

tion is at the heart of the ‘‘IR/UV mixing phenom-

enon’’ (see below) of noncommutative field theory .

At one- (and higher-) loop level, the different

noncommutativities for the opposite boundaries of

the open string become essential and give rise to new

effects. In this case nonplanar diagrams require one

to put vertex operators at the two different

boundaries  = 0, . A more complicated phase

factor, whi ch involves internal as well as external

momentum, results. This leads to IR/UV mixing in

the noncommutative quantum field theory. The

different noncommutativity for the opposite bound-

aries of the open string [9] is the basic reason for the

IR/UV mixing in the noncommutative quantum field

theory. The commutation relations [5] are valid at all

loops; therefore, one can use them to construct the

higher-loop string amplitudes from first principles.

The effect of the B-field on the string interaction can

easily be implemented into the Reggeon vertex and

the complete higher loop amplitudes in the presence

of the B-field have been constructed.

Low-Energy Limit – The Seiberg–Witten Limit

and the NCOS Limit

The full open-string system is still quite complicated.

One may try to decouple the infinite number of

massive string modes to obtain a low-energy field-

theoretic description by taking the limit 

! 0.

Since open strings are sensitive to G and , one

should take the limit such that G an d  are fixed.

For the magnetic case B

= 0, Seiberg and Witten

showed that this can be achieved with the following

double scaling limit:



 

1=2

; g

  ! 0 ½14

with B

and everything else kept fixed. Assuming B

is of rank r, then [6] becomes

¼ð2

ðBg

1

BÞ

;

¼ðB

1

;

for i; j ¼ 1; ...; r ½15

Otherwise G

= g

, 

= 0. One may also argue that

the closed string decouples in this limit. As a result,

in the low-energy limit a greatly simplified non-

commutative Yang–Mills action F F is obtained

(see below for more discussion of this field theory).

For the case of a constant electric field back-

ground, say B

6¼ 0, there is a critical electric field

beyond which the open string becomes unstable and

the theory does not make sense. Due to the presence

of this upper bound of the electric field, one can

show that there is no decoupling limit where one

can reduce the string theory to a field theory on a

noncommutative spacetime. However, one can con-

sider a different scaling limit where one takes the

closed-string metric scale to infinity appropriately as

the electric field approaches the critical value. In this

limit, all closed-string modes decouple. One obtains

a novel noncritical string theory living on a

noncommutative spacetime known as the noncom-

mutative open string (NCOS).

Noncommutative Quantum Field Theory

Field theories on noncommutative spacetime are

defined by using the star product instead of the

ordinary product of the fields. To illustrate the

general ideas, let us consider a single real scalar field

theory with the action

S ¼



  @



 

    VðÞ



VðÞ¼



4

½16

Due to the property [13], free noncommutative field

theory is the same as an ordinary field theory.

Treating the interaction term as a perturbation, one

can perform the usual quantization and obtain the

Feynman rules: the propagator is unchanged and the

interaction vertex in the momentum space is given

by g times the phase factor

exp 

1a<b4

 p

½17

Here p  q  p









. The theory is nonlocal due to

the infinite order of derivatives that appear in the

interaction.

Noncommutative Geometry from Strings 517

Planar and Nonplanar Diagrams

The factor [17] is cyclically symmetric but not

permutation symmetric. This is analogous to the

situation of an M-field theory. Using the same

double-line notation as introduced by ’t Hooft, one

can similarly classify the Feynman diagrams of

noncommutative field theory according to its

genus. In particular, the total phase factor of a

planar diagram behaves quite differently from that

of a nonplanar diagram. It is easy to show that a

planar diagram will have the phase factor

ðp

; ...; p

Þ¼exp 

1a<bn

 p

½18

where p

, ..., p

are the (cyclically ordered) external

momenta of the graph. Note that the phase factor

[18] is independent of the internal momenta. This is

not the case for a nonplanar diagram. One can easily

show that a nonplanar diagram carries an additional

phase factor

¼ V

exp 

1a<bn

 p

½19

where C

is the signed intersection matrix of the

graph, whose ab matrix element counts the number

of times the ath (internal or external) line crosses the

bth line. The matrix C

is not uniquely determined

by the diagram as different ways of drawing the

graph could lead to different intersectio ns. However,

the phase factor [19] is unique due to momentum

conservation.

The different behavi ors of the planar and non-

planar phase factors have important consequences.

1. Since the phase factor [18] is independent of the

internal momenta, the divergences and renorma-

lizability of the planar diagrams will be (simply)

the same as in the commutative theory and can

be handled with standard renormalization tech-

niques. This is sharply different for the nonplanar

diagrams. In fact, due to the extra oscillatory

internal-momenta-dependent phase factor, one

can expect the nonplanar diagrams to have an

improved ultraviolet (UV) behavior. It turns out

that planar and nonplanar diagrams also differ

sharply in their infr ared (IR) behavior due to the

‘‘IR/UV mixing effect’’ (see below).

2. Moreover, at high energies one can expect that

noncommutative field theory will generically

become planar since the nonplanar diagrams will

be suppressed due to the oscillatory phase factor.

3. In the limit  !1, the nonplanar sector will be

totally suppressed since the rapidly oscillating

phase factor will cause the nonplanar diagram to

vanish upon integrating out the momenta. Thus,

generically the large  limit is analogous to the

large-N limit where only the planar diagrams

contribute. However, these expectations do not

apply for noncommutative gauge theory since

one needs to include ‘‘open Wilson lines’’ (see

below) in the construction of gauge invariant

observables, and the open Wilson line grows in

extent with energy and .

IR/UV Mixing

Due to the nonlocal nature of noncommutative field

theory, there is generally a mixing of the UV and

IR scales. The reason is roughly the following.

Nonplanar diagrams generally have phase factors

like exp (ikp) with k a loop momentum, p an

external momentum. Consider a nonplanar diagram

which is UV divergent when  = 0; one can expect

that for very high loop momenta the phase factor

will oscillate rapidly and render the integral finite.

However, this is only valid for a nonvanishing

external momentum p; the infinity will come back

as p ! 0. However, this time it appears as an IR

singularity. Thus, an IR divergence arises whose

origin is from the UV region of the momentum

integration and this is known as the IR/UV mixing

phenomenon.

To be more specific, consider the 

scalar theory

in D = 4 dimensions. The one-loop self-energy has a

nonplanar contribution given by



6ð2Þ

þ m

ikp



3ð4

ð

eff

þÞ ½20

where 

eff

= (1=

þ (p)

)

1

. One can see clearly

the IR/UV mixing: 

is UV finite as long as p 6¼ 0;

when p = 0, the quadratic UV divergence is

recovered, 

 

. For supersymmetric theory,

one has at most logarithmic IR singularities from

IR/UV mixing.

IR/UV mixing has a number of interesting

consequences.

1. Due to the IR/UV mixing, noncommutative

theory does not appear to have a consistent

Wilsonian description since it requires that

correlation functions computed at finite  differ

from their limiting values by terms of order 1=

for all values of mom enta. However, this is not

true for theory with IR/UV mixing. For example,

the two-point function [20] at finite value of 

differs from its value at =1 by the amount

518 Noncommutative Geometry from Strings





 

=1

/ 1=(p)

, for the range of momenta

(p)

 1=

. It has been argu ed that the IR

singularity may be associated with missing light

degrees of freedom in the theory. With new

degrees of freedom appropriately added, one may

recover a conventional Wilsonian description.

Moreover, it has been suggested to identify

these degrees of freedom with the closed-string

modes. However, the precise nature and origin of

these degrees of freedom is not known.

2. The renormalization of the planar diagrams is

straightforward; however, the situation is more

subtle for the nonplanar diagrams since the IR/

UV-mixed IR singularities may mix with other

divergences at higher loops and render the proof

of renormalizability much more difficult. IR/UV

mixing renders certain large N noncommutative

field theory nonrenormalizable. However, for

theories with a fixed set of degrees of freedom

to start with, it is believed that one can have

sufficiently good control of the IR divergences

and pro ve renormalizability. An example of

renormalizable noncommutative quantum field

theory is the noncommutative Wess–Zumino

model where IR/UV mixing is absent. However,

a general proof is still lacking.

3. One can show that IR/UV mixing in timelike

noncommutative theory (

6¼ 0) leads to break-

down of perturbative unitarity. For a theory

without IR/UV mixing, unitarity will be respected

even if the theory has a timelike noncommuta-

tivity. Theory with lightlike noncommutativity is

unitary.

Noncommutative Gauge Theory

Gauge theory on noncommutative space is defined

by the action

S ¼

dx tr F

ðxÞF

ðxÞ



½21

where the gauge fields A

are N N Hermitian

matrices, F

is the noncommutative field strength

= @

 @

 i[A

, A

]



, and tr is the ordinary

trace over N  N matrices. The theory is invariant

under the star-gauge transformation

! g  A

 g

 ig  @

½22

where the N  N matrix function g(x) is unitary

with respect to the star product g  g

= g

 g = I.

The solution is g = e

i



, where  is Hermitian. In

infinitesimal form, 



= @

 þ i[, A

]



. The non-

commutative gauge theory has N

Hermitian gauge

fields. Because of the star product, the U(1) sector of

the theory is not free and does not decouple from

the SU(N) factor as in the commutative case. Note

that this way of defining noncommutative gauge

theory does not work for other Lie groups since the

star commutator generally involves the commutator

as well as the anticommutator of the Lie algebra;

hence, the expressions above generally involve the

enveloping algebra of the underlying Lie group.

With the help of the ‘‘Seiberg–Witten map’’ (see

below), one can construct an enveloping-algebra-

valued gauge theory which has the same number of

independent gauge fields and gauge parameters as

the ordinary Lie-algebra-valued gauge theory. How-

ever, the quantum properties of these theories are

much less understood. One may also introduce

certain automorphisms in the noncommutative

U(N) theory to restrict the dependence of the

noncommutative space coordinates of the field

configurations and obtain a notion of noncommu-

tative theory with orthogonal and symplectic star-

gauge group. However, the theory does not reduce

to the standard gauge theory in the commutative

limit  ! 0.

Open Wilson Line and Gauge-Invariant

Observables

One remarkable feature of noncommutative gauge

theory is the mixing of noncommutative gauge

transformations and spacetime translations, as can

be seen from the following identity:

ikx

 f ðxÞe

ikx

¼ f ðx þ kÞ½23

for any function f. This is analogous to the situation

in general relativity where translations are also

equivalent to gauge transformations (general coor-

dinate transformations). Thus , as in general relativ-

ity, there are no local gauge-invariant observables in

noncommutative gauge theory. The unification of

spacetime and gauge fields in noncommutative

gauge theory can also be seen from the fact

that derivatives can be realized as commutators,

f !i[

1

, f ], and get absorbed into the vector

potential in the covariant derivative

¼ @

þ iA

!i

1

þ iA

½24

Equation [24] clearly demonstrates the unification

of spacetime and gauge fields. Note that the field

strength takes the form F

= i[D

, D

] þ 

1

The Wilson line operator for a path C running

from x

to x

is defined by

WðCÞ¼P



exp i



½25

Noncommutative Geometry from Strings 519



denotes the path ordering with respect to the star

product, with A(x

) at the right. It transforms as

WðCÞ!gðx

ÞWðCÞgðx

½26

In commutative gauge theory, the Wilson line

operator for closed loop (or its Fourier transform)

is gauge invariant. In noncommutative gauge

theory, the closed Wilson loops are no longer

gauge invariant. Noncommutative generalization

of the gauge invariant Wilson loop operator can

be constructed most readily by deforming the

Fourier transform of the Wilson loop operator. It

turns o ut that the closed loop has t o open in a

specific way to form an open Wilson line in order

to be gauge invariant. To see this, let us consider a

path C connecting points x and x þ l.Using[23] ,it

is easy to see that the operator

WðkÞ

dx tr WðCÞe

ikx

; with l

¼ k



½27

is gauge invariant. Just like Wilson loops in ordinary

gauge theory, these operators also constitute an

overcomplete set of gauge-invariant operators para-

metrized by the set of curves C. When  = 0, C

becomes a closed loop and we reobtain the (Fourier

transformed) usual closed Wilson loop in commu-

tative gauge theory. Noncommutative version of the

loop equation for closed Wilson loop has been

constructed and involves open Wilson line. The

open Wilson line is instrumental in the construction

of gauge-invariant observables. An important appli-

cation is in the construction of various couplings of

the noncommutative D-brane to the bulk super-

gravity fields. The equivalence of the commutative

and noncommutative couplings to the RR fields

leads to the exact expression for the Seiberg–Witten

map. It is remarkable that the one-loop nonplanar

effective action for noncommutative scalar theory,

gauge theory, as well as the two-loop effective

action for scalar can be written compactly in terms

of open Wilson line. Based on this result, the

physical origin of the IR/UV mixing has been

elucidated. One may identify the open Wilson line

with the dipole excitation generically presents in

noncommutative field theory and hence ex plain the

presence of the IR/UV mixing. IR/UV mixing may

also be identified with the instability associated with

the closed-string exchange of the noncommutative

D-branes.

The Seiberg–Witten Map

The open string is coupled to the 1-form A

living on

the D-brane through the coupling

@

A. For slowly

varying fields, the effective action for this gauge

potential can be determined from the S-matrix and

is given by the Dirac– Born–Infeld (DBI) action. In

the presence of a B-field, the discussion above (see

eqn [11]) leads to the noncommutative DBI

Lagrangian

NCDBI

FÞ¼G

1



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detð G þ 2

FÞ

½28

where 

= (2)

p

(

)

(pþ1)=2

is the D-brane tension

and

F is the noncommutative field strength.

However, one may also exploit the tensor gauge

invariance on the D-brane (i.e., the string sigma

model is invariant under A ! A  , B ! B þ d )

and consider the combination F þ B as a whole. In

this case, it is like having the open string coupled

to the boundary gauge field strength F þB and

there is no B field. One has the usual DBI

Lagrangian

DBI

ðFÞ¼g

1



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detðG þ 2

ðF þ BÞÞ

½29

In [28] and [29], G

and g

are the effective open-

string couplings in the noncommutative and com-

mutative descriptions. Although they look quite

different, Seiberg and Witten showed that the

commutative and noncommutative DBI actions

are indeed equivalent if the open-string couplings

are related by g

= G

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det (g þ 2

B)= det G

and

there is a field redefinit ion that relates the

commutative and noncommutative gauge fields.

The map

A =

A(A) is called the Seiberg–Witten

map. Moreover, the noncommutative g auge sym-

metry is equivalent to the ordinary gauge symme-

try in the sense that t hey h ave the same set of

orbits under gauge transformation:

AðAÞþ





AðAÞ¼

AðA þ 



AÞ½30

Here

and

 are, respectively, the noncommu-

tative gauge field and noncommutative gauge

transformation parameter, and A

and  are,

respectively, the ordinary gauge field and ordinary

transformation paramete r. The map between

and A

is called the Seiberg–Witten map. Equation

[30] can be solved only if the transformation

parameter

 =

(, A) is field dependent. The

Seiberg–Witten map is characterized by the Seiberg–

Witten differential equation



ðÞ¼



ð@

þð@

Þ

½31

An exact solution for the Seiberg–Witten map can

be written down with the help of the open Wilson

520 Noncommutative Geometry from Strings

line. For the case of U(1) with constant F, we have

the exact solution

F = (1 þ F)

1

That there is a field redefinition that allo ws one to

write the effective action in terms of different fields

with different gauge symmetries may seem puzzling

at first sight. However, it has a clear physical origin

in terms of the string world sheet. In fact, there are

different possible schemes to regularize the short-

distance divergence on the world sheet. One can

show that the Pauli–Villars regularization gives the

commutative description, while the point-splitting

regularization gives the noncommutative descrip-

tion. Since theories defined by different regulariza-

tion schemes are related by a coupling-constant

redefinition, this implies that the commutative and

noncommutative descriptions are related by a field

redefinition, because the couplings on the world

sheet are just the spacetime fields.

Despite this formal equivalence, the physics of the

noncommutative theories is generally quite different

from the commutative case. First, it is clear that

generally the Seiberg–Witten map may take non-

singular configurations to singular configurations.

Second, the observables one is interested in are also

generally different. Moreover, the two descriptions

are generally good for different regimes: the con-

ventional gauge theory description is simpler for

small B and the noncommutative description is

simpler for large B.

Perturbative Gauge Theory Dynamics

The noncommutative gauge symmetry [22] can be

fixed as usual by employing the Faddeev–Po pov

procedure, resulting in Feynman rules that are

similar to the convent ional gauge theory. The

important difference is that now the structure

constants in the phase factors [18] and [19] should

be amended. It turns out that the nonplanar U(N)

diagrams contribute (only) to the U(1) part of the

theory. As a result, unlike the commutative case, the

U(1) part of the theory is no longer decoupled and

free. Noncommutative gauge theory is one-loop

renormalizable. The -function is determined solely

by the planar diagrams and, at one loop, is given by

ðgÞ¼

16

for N  1 ½32

Note that the -function is independent of ; the

noncommutative U(1) is asymptotically free and

does not reduce to the commutative theory when

 ! 0. Noncommutative theory beyond the tree

level is generally not smooth in the limit  ! 0.

Discontinuity of this kind was also noted for the

Chern–Simon system.

Gauge anomalies can be similarly discussed and

satisfy the noncommutative generalizations of the

Wess–Zumino consistency conditions. In d = 2n

dimensions, the anomaly involves the combination

tr(T

T

nþ1

) rather than the usual symme-

trized trace, since the phase factor is not permutation

symmetric. As a result, the usual cancellation of the

anomaly does not work and is the main obstacle to

the construction of noncommutat ive chiral gauge

theory.

There are a number of interesting features to

mention for the IR/UV mixing in noncomm utative

gauge theory.

1. IR/UV mixing generically yields pole-like IR

singularities. Despite the appearance of IR

poles, gauge invariance of the theory is not

endangered.

2. One can show that only the U(1) sector is

affected by IR/UV mixing.

3. As a result of IR/UV mixing, noncommutative

U(1) photons polarized in the noncommutative

plane will have different dispersion relations

from those which are not. Strange as it is, this

is consistent with gauge invariance.

Noncommutative Solitons, Instantons

and D-Branes

Solitons and instantons play important roles in the

nonperturbative aspects of field theory. The non-

locality of the star product gives noncommutative

field theory a stringy nature. It is remarkable that

this applies to the nonperturba tive sector as well.

Solitons and instantons in the noncommutative

gauge theory amazingly reproduce the properties of

D-branes in the string.

GMS Solitons

Derrick’s theorem says that commutative scalar field

theories in two or higher dimensions do not admit

any finite-energy classical solution. This follows

from a simple scaling argument, which will fail

when the theory becomes noncommutative since

noncommutativity introduces a fixed lengt h scale

ﬃﬃﬃ



. Noncommutative solitons in pure scalar theory

can be easily constructed in the limit  = 1. For

example, consider a (2 þ 1)-dimensional single sca-

lar theory with a potential V and noncommutativity



= . In the limit  = 1, the potential term

dominates and the noncommutative solitons are

determined by the equation

@V=@ ¼ 0 ½33

Noncommutative Geometry from Strings 521

Equation [33] can be easily solved in terms of

projectors. Assuming V has no linear term, the

general soliton (up to unitary equiva lence) is

 ¼



½34

where 

are the roots of V

() = 0 and P

is a set of

orthogonal projectors. For real scalar field theory,

the sum is restricted to real roots only. These

solutions are known as the Gopakumar–Minwalla–

Strominger (GMS) solitons. A simple example of a

projector is given by P = j0ih0j, which corresponds

to a Gaussian profile in the x

, x

plane with width

ﬃﬃﬃ



. The soliton continues to exist until  decreases

below a certain critical 

New solutions can be generated from known ones

using the so-ca lled solution-generating technique. If

 is a solution of [33], then



¼ T

T ½35

is also a solution provided that TT

= 1. In an

infinite-dimensional Hilbert space, T is not necessa-

rily unitary, that is, T

T 6¼ 1. In this case, T is said

to be a partial isometry. The new solution 

different from  since they are not related by a

global transformation of basis.

Tachyon Condensation and D-Branes

A beautiful application of the noncommutative

soliton is in the construction of D-branes as solitons

of the tachyon field in noncommutative open-string

theory. For the bosonic string theory, one may

consider it to be a space-filling D25 brane. Integrat-

ing out the massive-string modes leads to an

effective action for the tachyon and the massless

gauge field A



. It should be remarked that, contrary

to the pure scalar case, noncommutative solitons can

be constructed exactly for finite  in a system with

gauge and scalar fields. Although the detailed form

of the effective action is unknown, one has enough

confidence to say what the true vacuum configura-

tion is according to the Sen conjecture. One can then

apply the solution-generating technique to generate

new soliton solutions. In this manner, with a B-field

of rank 2k, one can construct solutions which are

localized in R

and represent a D(25 – 2k) brane.

This is supported by the matching of the tension

and the spectrum of fluctuations around the

soliton configuration. Similar ideas can also be

applied to construct D -branes in type II string

theory. Again the starting point is an unstable

brane configuration with tachyon field(s). There

are two types of unstable D-branes: non-BPS Dp

branes (p odd in IIA theory and p even in IIB

theory) and BPS branes–antibranes Dp–

systems. A similar analysis allows one to identify

the non commut ative soliton with the lowe r-

dimensional B PS D-branes which arises from

tachyon cond ensation.

One main motivation for studying tachyon

condensation in open-string theory is the hope

that open-string theory may provide a f undamental

nonperturbative formulation of string theory. It

may not be too surprising that D-branes can be

obtained in terms of open-string fields. However,

to describe closed strings and NS branes in terms

of open-string degrees of freedom remains an

obstacle.

Noncommutative Instanton and Monopoles

Instantons on noncommutative R



can be readily

constructed using the Atiyah–Drinfeld–Hitchin–

Manin (ADHM) formalism by modifying the

ADHM constraints with a constant addit ive

term. The result is that the self-dual (resp. anti-

self-dual) instanton moduli space depends only on

the anti-self-dual (resp. self-dual) part. The con-

struction goes through even in the U(1) case.

Consider a self-dual ; the ADHM constraints for

the self-dual instanton are t he same as in the

commutative case, and there is no nonsingular

solution. On the other hand, the ADHM c on-

straints for the anti-self-dual instanton get mod-

ified and admit nontrivial solutions. T his

noncommut ative instanton solution i s nonsingular

with size

ﬃﬃﬃ



. The noncommutative instanton

represents a D(p–4) brane within a Dp brane.

The ADHM constraints are just the D-flatness

condition for the D-brane world-volume gauge

theory. The additive constant to the ADHM

constraints also has a simple interpretation as a

Fayet–Iliopolous parameter which appears in the

presence of a B-field. Although the ADHM

method does not give a se lf-dual instant on, a

direct construction can be applied to obtain non-

ADHM self-dual instantons. Recall that the gauge

fieldstrengthcanbewrittenasF

= i[D

, D

] þ 

1

where D

is given by the function on the right-

hand side of [24]. Thus, a simple self-dual solution

can be constructed with

¼ i

1

T ½36

where T is a partial isometry which satisf ies

= 1, but T

T = 1  P is not necessarily the

identity. It is clear that P is a projector. The field

strength

¼ 

1

P ½37

522 Noncommutative Geometry from Strings

is self-dual and has instanton number n where n is

the rank of the projector.

On noncommutative R

(say 

= ), BPS mono-

poles satisfy the Bogomolny equation:

 ¼B

; i ¼ 1; 2; 3 ½38

and can be obtained by solving the Nahm

equation

¼ 

ijk

þ 

 ½39

are k  k Hermitian matrices depending on an

auxiliary vari able z and k gives the charge of the

monopole. Noncommutativity modifies the Nahm

equation with a constant term, which can be

absorbed by a constant shift of the generators.

Therefore, unlike the case of instanton, the mono-

pole moduli space is not mod ified by noncommu ta-

tive deformation. The Nahm construction has a

clear physical meaning in string theory. The mono-

pole (electric charge) can be interpreted as a D-string

(fundamental string) ending on a D3 brane. One can

also suspend k D-string between a collection of N

parallel D3 braness; this would correspond to a

charge k monopole in a Higgsed U(N) gauge theory.

The matrices X

correspond to the matrix transverse

coordinates of the D-strings which lie within the D3

branes.

Further Topics

Finally, in the following some further topics of

interest are discussed briefly.

1. The noncommutative geome try discussed here is

of canonical type. Other deformations exist, for

example, kappa-deformation and fuzzy sphere

which are of the Lie-algebra type, and quantum

group deformation which is a quadratic-type

deformation: x

= q

1

, whose consis-

tency is guaranteed by the Yang–Baxter equation.

It is interesting to see whether these noncommu-

tative geometries arise from string theory.

Another natural generalization is to consider

noncommutative geo metry of superspace. A

simple example is to consider the fermionic

coordinates to be deformed with the nonvanish-

ing relation

f



;



g¼C



½40

where C



are constants. It has been shown that

[40] arises in certain Calabi–Yau compactification

of type IIB string theory in the presence of RR

background. The deformation [40] reduces the

number of supersymmetries by half. Therefore,

it is called N = 1=2 supersymmetry. The

noncommutativity [40] can be implemented on

the superspace (y

, 









) as a star product for the





’s. Unlike the bosonic deformation which

involves an infinite number of higher derivatives,

the star product for [40] stops at order C

due to

the Grassmannian nature of the fermionic coordi-

nates. Field theory with N = 1=2 supersymmetry

is local and differs from the ordinary N = 1theory

by only a small number of supersymmetry break-

ing terms. The N = 1=2 Wess–Zumino model is

renormalizable if extra F and F

terms are added

to the original Lagrangian, where F is the auxiliary

field. The N = 1=2 gauge theory is also

renormalizable.

2. Integrability of a theory provides valuable infor-

mation beyond the perturbative level. An integr-

able field theory is characterized by an infinite

number of conserved charges in involution. It is

natural to ask whether integrability is preserved

by noncommutative deformation. Noncommuta-

tive integrable field theories have been con-

structed. In the commutative case, Ward has

conjectured that all (1 þ 1)- and (2 þ 1)-dimen-

sional integrable systems can be obtained from

the four-dimensional self-dual Yang–Mills equa-

tion by reduction. Validity of the noncommuta-

tive version of the Ward conjecture has been

confirmed so far. It will be interesting to see

whether it is true in general.

3. Locality and Lorentz symmetry form the corner-

stones of quantum field theory and standard

model physics of particles. Noncommutative field

theory provides a theoretical framework where

one can discuss effects of nonlocality and Lorentz

symmetry violation. Possible phenome nological

signals have been investigated (mostly at the tree

level) and a bound has been placed on the extent

of noncommutativity. A proper understanding

and better control of the IR/UV mixing remains

the crux of the problem. Noncommutative

geometry may also be relevant for cosmology

and inflation.

4. Like the standard AdS/CFT correspondence, the

noncommutative gauge theory should also have

a gravity-dual description. The s upergravity

background can be determined by c onsidering

the decoupling limit of D-branes with an NS

B-field background. However, since the non-

commutative gauge theory does not permit any

conventional local gauge-invariant observable,

the usual AdS/CFT correspondence that relates

field theory correlators with bulk interaction

does not seem to apply. It has b een argued that

generic properties such as the relation between

length and momentum for open Wilson lines

Noncommutative Geometry from Strings 523

can be seen from the gravity side. A more precise

understanding of the duality map is called for.

See also: Brane Construction of Gauge Theories;

Deformation Quantization; Gauge Theories from Strings;

Noncommutative Tori, Yang–Mills, and String Theory;

Positive Maps on C



-Algebras; Solitons and Other

Extended Field Configurations; String Field Theory;

Superstring Theories.

Further Reading

Chu CS and Ho PM (1999) Noncommutative open string and

D-brane. Nuclear Physics B 550: 151.

Connes A (1994) Noncommutative Geometry. New York:

Academic Press Inc.

Connes A, Douglas MR, and Schwarz A (1998) Noncommutative

geometry and matrix theory: compactification on tori. JHEP

9802: 003.

Douglas MR and Hull CM (1998) D-branes and the noncommu-

tative torus. JHEP 9802: 008.

Douglas MR and Nekrasov NA (2001) Noncommutative field

theory. Reviews of Modern Physics 73: 977.

Harvey JA (2001) Komaba lectures on noncommutative solitons

and D-branes. arXiv:hep-th/0102076.

Konechny A and Schwarz A (2002) Introduction to M(atrix)

theory and noncommutative geometry. Physics Reports 360:

353.

Nekrasov NA (2000) Trieste lectures on solitons in noncommu-

tative gauge theories. arXiv:hep-th/0011095.

Polchinski J (1998) String Theory. Cambridge: Cambridge

University Press.

Schomerus V (1999) D-branes and deformation quantization.

JHEP 9906: 030.

Seiberg N and Witten E (1999) String theory and noncommuta-

tive geometry. JHEP 9909: 032.

Sen A (2004) Tachyon dynamics in open string theory. arXiv:hepth/

0410103.

Szabo RJ (2003) Quantum field theory on noncommutative spaces.

Physics Reports 378: 207.

Taylor W (2000) M(atrix) theory: matrix quantum mechanics as a

fundamental theory. Reviews of Modern Physics 73: 419.

Noncommutative Tori, Yang–Mills, and String Theory

A Konechny, Rutgers, The State University of New

Jersey, Piscataway, NJ, USA

Introduction

Noncommutative tori are historically among the

oldest an d by now the most developed examples

of noncommutative spaces. Noncommutative

Yang–Mills theory can be obtained from string

theory. This connection led to a cross-fertilization

of research in ph ysics and mathematics on Ya ng–

Mills theory on noncommutative tori. One

important result stemming from that work is the

link between T-duality in string theory and

Morita equivalence of associative algebras. In

this article, we give an overview of the basic

results in the differential geometry of noncommu-

tative tori. Yang–Mills theory on noncommuta-

tive tori, the duality induced by Morita

equivalence and its link w ith T-duality are

discussed. The noncommutative Nahm transform

for instantons is introduced.

Noncommutative Tori

The Algebra of Functions

The basic notions of noncommutative differential

geometry were introduced and illustrated on the

example of a two-dimensional noncommutative

torus by Connes (1980). To define an algebra of

functions on a d-dimensional noncommutative

torus, consider a s et of linear generators U

labeled

by n 2Z

–ad-dimensional vector with integral

entries. The multiplication is defined by the

formula

¼e

in



n þm

½1

where 

is an antisymmetric d d matrix, and

summation over repeated indices is assumed. We

further extend the multiplication from finite linear

combinations to formal infin ite series

C(n)U

where the coefficients C(n) tend to zero faster than

any power of knk. The resulting algebra constitutes

the algebra of smooth functions on a noncommuta-

tive torus and will be denoted by T



. Sometimes for

brevity we will omit the dimension label d in the

notation of the alge bra. We introduce an involution

 in T



by the rule U



= U

n

. The elements U

are

assumed to be unitary with respect to this involu-

tion, that is, U



= U

n

= 1 U

. One can

further introduce a norm and take an appropriate

completion of the involutive algebra T



to obtain

the C



-algebra of functions on a noncommutative

torus. For our purposes, the norm structure will not

be impo rtant. A canonically normalized trace on T



is introduced by specifying

tr U

¼

n;0

½2

Projective Modules

According to the general approach to noncommu ta-

tive geometry, finitely generated projective modules

524 Noncommutative Tori, Yang–Mills, and String Theory

over the algebra of functions are natural analogs of

vector bundles. Throu ghout this article, when speak-

ing of a projective module, we will assume a finitely

generated left projective module.

A free module (T



)

is equipped with a T



-valued

Hermitian inner product h.,.i



defined by the

formula

hða

; ...; a

Þ; ðb

; ...; b

Þi



i¼1



½3

A projective module E is by definition a direct

summand of a free module. Thus, it inherits the

inner product h.,.i



. Consider the endomorphi sms

of the module E, that is, linear mappings E !E

commuting with the action of T



. These endo-

morphisms form an associative unital algebra

denoted End



E. A decomposition (T



)

= E E

determines an endomorphism P :(T



)

!(T



)

that

projects (T



)

onto E. The algebra End



E can then

be identified with a subalgebra of Mat



) – the

endomorphisms of the free module (T



)

. The latter

has a canonical trace that is the composition of the

matrix trace with the trace specified in [2].By

restriction, it gives rise to a canonical trace tr on

End



E. The same embedding also provides a

canonical involution on End



E by a composition of

the matrix transposition and the involution  on T



A large class of examples of projective modules

over noncomm utative tori are furnished by the

so-called Heisenberg modules. They are constructed

as follows. Let G be the direct sum of R

and an

abelian finitely generated group, and let G



be its

dual group. In the most general situation

G = R

Z

F where F is a finite group. Then



ﬃR

T

F



Consider the linear space S(G) of functions on G

decreasing at infinity faster than any power. We

define operators U

(,

)

: S(G) !S(G) labeled by a

pair (,

) 2G G



acting as follows:

ðU

ð;~Þ

f ÞðxÞ¼

ðxÞf ðx þ Þ½4

One can check that the operators U

(,

)

satisfy the

commutation relations

ð;

Þ

ð;

Þ

ðÞ



1

ðÞU

ð;

Þ

ð;

Þ

½5

If ( ,

) run over a d-dimensional discrete subgroup

 G G



,  ﬃZ

, then formula [4] defines a

module over a d-dimensional noncomm utative

torus T



with

expð2i

Þ¼



ð



1

ð

Þ½6

for a given basis (



) of the lattice . This module is

projective if  is such that G G



= is compact.

If that is the case, then the projective T



-module at

hand is called a Heisenberg module and denoted

by E



Heisenberg modules play a special role. If the

matrix 

is irrational in the sense that at least one

of its entries is irrational, then any projective

module over T



can be represented as a direct sum

of Heisenberg modules. In that sense, Heisenberg

modules can be used as building blocks to construct

an arbitrary module.

Connections

Next we would like to define connections on a

projective module over T



. To this end, let us first

define a Lie algebra of shifts L



acting on T



specifying a basis consisting of derivations



: T



, j = 1, ..., d satisfying



ðU

Þ¼2in

½7

These derivations span a d-dimensional abelian Lie

algebra that we denote by L



A connection on a module E over T



is a set of

operators r

: E !E, X 2L



, depending linearly on

X and satisfying

½r

; U

¼

ðU

Þ½8

where U

are operators E !E representing the

corresponding generators of T



. In the standard

basis [7], this relation reads as

½r

; U

¼2in

½9

The curvature of the connection r

defined as the

commutator F

= [r

, r

] is an exterior 2-form

on the adjoint vector space L





with values in

End



K-Theory: Chern Character

The K-groups of a noncommutative torus coincide

with those for commutative tori:

ðT



ÞﬃZ

d1

ﬃK





The Chern character of a projective module E

over a noncommutative torus T



can be defined as

chðEÞ¼tr exp

2i



2

even







½10

where F is the curvature form of a connection on

E, 

even





) is the even part of the exterior algebra

of L





and tr is the canonical trace on End



E. This

Noncommutative Tori, Yang–Mills, and String Theory 525

mapping gives rise to a noncommutative Chern

character

ch : K





!

even







½11

The component ch

(E) = tr 1  dim(E) is called the

dimension of the module E.

A distinctive feature of the noncommutative

Chern character [11] is that its image does not

consist of integral elements, that is, there is no

lattice in L





that generates the image of the Chern

character. However, there is a different integrality

statement that replaces the commutative one. Con-

sider a basis in L





in which the derivations

corresponding to basis elements satisfy [7]. Denote

the exterior forms corresponding to the basis

elements by 

, ..., 

. Then an arbitrary element

of (L





) can be represented as a polynomial in the

anticommuting variables 

. Next let us consider the

subset 

even

) 

even





) that consists of poly-

nomials in 

having integer coefficients. It was

proved by Elliott that the Chern character is

injective and its range on K



) is given by the

image of 

even

) under the action of the operator

exp 

@



@



This fact implies that the K-group K



) can be

identified with the additive group 

even

The K-theory class (E) 2

even

) of a module E

can be computed from its Chern character by the

formula

ðEÞ¼exp

@



@



chðEÞ½12

Note that the anticommuting variables 

and the

derivatives @=@

satisfy the anti commutation rela-

tion {

, @=@

} = 

The coefficients of (E) standing in front of

monomials in 

are integers to which we will

refer as the topological numbers of the module E.

These numbers can also be interpreted as numbers

of D-branes of a definite kind although in non-

commutative geometry it is difficult to talk about

branes as geometrical objects wrapped on torus

cycles.

One can show that for noncomm utative tori T



with irrational matrix 

the set of elements of



) that represent a projective module (i.e., the

positive cone) consist exact ly of the elements of

positive dimension. Moreover, if 

is irrational, any

two projective modules which represent the same

element of K



) are isomorphic; that is, the

projective modul es are essentially specified in this

case by their topological numbers.

The complex differential geometry of noncommu-

tative tori and its relation with mirror symmetry is

discussed in Polishchuk and Schwarz (2003).

Yang–Mills Theory on Noncommutative

Tori

Let E be a projective module over T



. We call a

Yang–Mills field on E a connection r

-compatible

with the Hermitian structure, that is, a connection

satisfying

; 



þ ; r





¼

ð ; 



Þ½13

for any two elements ,  2E. Given a positive-

definite metric on the Lie algebra L



, we can define

a Yang–Mills functional

ðr

Þ¼

trðF

Þ½14

Here g

stands for the metric tensor in the canonical

basis [7], V =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jdet gj

, g

is the Yang–Mills

coupling constant, tr stands for the canonical trace

on End



E discussed above, and summ ation over

repeated indices is assumed. Compatibility with the

Hermitian structure [13] can be shown to imply

the positive definiteness of the functional S

. The

extrema of this functional are given by the solutions

to the Yang–Mills equations

½r

; F

¼0 ½15

A gauge transformation in the noncommutative

Yang–Mills theory is specified by a unitary endo-

morphism Z 2End



E, that is, an endomorphism

satisfying ZZ



= Z



Z = 1. The corresponding gauge

transformation acts on a Yang–Mills field as

7!Zr



½16

The Yang–Mills functional [14] a nd the Yang–

Mills equations [15] are invariant under these

transformations.

It is easy to see that Yang–Mills fields whose

curvature is a scalar operator, that is, [ r

, r

] =



1 with 

a real-number-valued tensor, solve the

Yang–Mills equations [15]. A characterization of

modules admitting a constant curvature connection

and a description of the moduli spaces of consta nt

curvature connections (i.e., the space of such

connections modulo gauge transformations)

is revi ewed in Konechny and Schwarz (2002).

Another interesting class of solutions to the Yang–

Mills equations is instantons (see below).

As in the ordinary field theory, one can construct

various extensions of the noncommutative Yang–

Mills theory [14] by adding other fields. To obtain a

526 Noncommutative Tori, Yang–Mills, and String Theory