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

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finite products n

...n

(a multiplicity which may be

either finite or infinite) . Call such a formal infinite

product a generalized integer – or, perhaps, a

supernatural number! Two (countably) infinite

tensor products of matrix algebras are isomorphic

(just as in the finite tensor product case) if and only

if the corresponding supernatural numbers are

equal.

In formulating Glimm’s classification of infinite

tensor products of matrix algebras in this way,

Dixmier pointed out that each supernatural number

determines a subgroup of the rational numbers

(those with denominator dividing the supernatural

number) and that every subgroup of the rational

numbers containing the integers arises in this way.

He then gave an alternative derivation of Glimm’s

theorem by recovering this subgroup of the rational

numbers as a natural invariant of the algebra,

namely, as the subgroup generated by the values

on projections of the unique normalized trace. (By a

trace is meant here a unitarily invariant positive

linear functional.) This could even be interpreted as

an alternative statement of Glimm’s theorem.

Soon afterwards, Bratteli considered an extension

of Glimm’s class of C



-algebras, namely, the

inductive limits of arbitrary sequences of finite-

dimensional C



-algebras, and gave a classification of

these algebras in terms of the embedding multiplicity

data in the sequences. This was exactly analogous to

the original classification of Glimm, but now vastly

more complex, with the multiplicity data of the

sequence encoded in what is now called a Bratteli

diagram. (Note that a finite-dimensional C



-algebra

is just a direct sum of matrix algebras over the

complex numbers.) Bratteli diagrams have proved to

be very important, and in particular have been shown

by Putnam an d others to be useful for the study of

minimal homeomorphisms of the Cantor set.

Bratteli’s extension of Glimm’s tensor product

classification was followed by a corresponding

extension by the present author of Dixmier’s

approach to Glimm’s result. It was no longer

possible to express the appropriate data in terms of

traces (even in the case of a unique normalized

trace). Instead, the present author recalled the

concept of equivalence of projections introduced

by Murray and von Neumann forty years earlier,

together with the fact, proved by Murray and von

Neumann, that equivalence is compatible with

addition of orthogonal projections. (Two projec-

tions in a



-algebra are equivalent if they are equal

to x



x and xx



for some element x.) The resulting

elementary invariant – the set of equivalence classes

of projections with the operation of addition

whenever defined (whenever the equivalence classes

to be added have orthogonal representatives) – one

might refer to this as a local abelian semigroup –

which was used by Murray and von Neumann to

divide von Neumann algebras into what they called

types I, II, and III – was shown by the author to

determine Bratteli’s algebras up to isomorphism.

Bratteli called his algebras approximately finite-

dimensional C



-algebras, or AF algebras. The author

referred to his invariant simply as the range of the

(abstract) dimension, and pointed out that this

structure determined an enveloping ordered abelian

group, which he called the dimension group. It was

soon noticed that the dimension group was related

to the K-group introduced by Grothendieck in

algebraic geometry (see K-Theory), and by Atiyah

and Hirzebruch (see K-Theory) in topology.

Grothendieck’s K-group was defined for an arbi-

trary ring with unit, and Atiyah and Hirzebruch in

effect considered the special case of the ring of

continuous functions on a compact Hausdorff space –

in other words, a commutative C



-algebra – in the

process showing that the deep phenomenon of Bott

periodicity could be expressed in terms of this

invariant. The invariant itself (see below) is essen-

tially the same as that of Murray and von Neumann.

In the special case that the ring is an AF algebra, the

K-group coincides with the dimension group. (The

K-group has a natural ordered, or pre-ordered,

structure, although this was often suppressed.)

Let us consider the definition of the K-group of a

not necessarily unital C



-algebra; it is in this setting

that the statement of Bot t periodicity attains its

simplest form.

First, in the unital case, one constructs the abelian

local semigroup (addition just partially defined) of

Murray–von Neumann equivalence classes of pro-

jections, as described above in the case of an AF

algebra. Let us call this the dimension range. As

stated above, for AF algebras this is all that needs to

be done – the enveloping group of the dimension

range is already the K-group. In the general case,

one must repeat the construction for the algebra of

2  2 matrices over the given algebra, with the given

algebra considered as embedded as the upper left-

hand corner of the matrix algebra. The dimension

range of the given algebra then maps naturally into

(but not necessarily onto) the dimension range of the

matrix algebra. One should then repeat this con-

struction, doubling the order of the matrix algebra

at every stage (or, alternatively, increasing it just by

one). The enveloping group of the (algebraic)

inductive limit of this sequence of local semigroups

is then the K-group of the given algebra. (Alterna-

tively, one may just consider immediately the



-algebra of all infinite matrices over the given



-Algebras and their Classification 395



-algebra with only finitely many nonzero entries,

and form the dimension range of this



-algebra – and

the enveloping group of this abelian local semi-

group, now in fact a semigroup.)

In the case of a nonunital C



-algebra, one adjoins

a unit (as may be done, for instance, by representing

the C



-algebra faithfully on a Hilbert space, and

showing that the C



-algebra obtained by adjoining

the identity operator is independent of the representa-

tion – actually, one need only check that the



-algebra

structure is unique, as the C



-algebra norm on a



-algebra is always determined by the



-algebra

structure). The K-group of the resulting unital



-algebra then maps naturally into the K-group of

the natural one-dimensional quotient, and the kernel

of this map is, for reasons that will become clearer

later, defined to be the K-group of the nonunital

algebra.

Atiyah and Hirzebruch in fact referred to the

K-group of the C



-algebra as K

– the reason being

that there is another very natural group to consider,

namely, the K-group of the suspension of the



-algebra. (The suspension, SA,ofaC



-algebra A

is defined as the C



-algebra of all continuous

functions from the real line R into A which converge

to zero at 1, with the pointwise



-algebra

operations and the supremum norm. It may also be

defined as the (unique) C



-algebra tensor product

A  C

(R), where C

(R) denotes the suspension of

the C



-algebra C of complex number s.) Denoting

the K

-group of the suspension of a given C



-algebra

by K

, one might expect this process to continue,

but in fact it is periodic (K

, K

, ...). Bott

periodicity states that there is a natural isomorphism

of K

with K

.(C



-algebras can also be defined with

the field of real numbers as scalars, and in this case

the period of Bott periodicity is eight.)

Another way of stating Bott period icity, or, more

precisely, of embe dding it into the K-theory of



-algebras, is as follows. Given a short exact

sequence of C



-algebras,

0 ! J ! A ! A=J ! 0 ½3

i.e., given a C



-algebra A and a clos ed two-sided

ideal J (the quotient



-algebra is then a C



-algebra

with the quotient norm) – A is sometimes referred to

as an extension of J by A=J – consider the natural

short (not necessarily exact) sequences

ðJÞ!K

ðAÞ!K

ðA=JÞ½4

and

ðJÞ!K

ðAÞ!K

ðA=JÞ½5

and K

are functors!). There exist natural connect-

ing maps K

(A=J) !K

(J)andK

(A=J) !K

(J)–the

first referred to as the index map, and the second

(sometimes referred to as the odd-order index map)

obtained from this immediately from Bott periodicity

(as stated above) – such that the periodic six-term

sequence

ðJÞ!K

ðAÞ!K

ðA=JÞ

ðA=JÞ K

ðAÞ K

ðJÞ

is exact. (The perio dicity stated above can also be

recovered from this.)

Given that the functor K

classifies AF algebras,

one might expect the functor K

to be useful for

classification purposes also. In fact, this is the case.

(Indeed, as shown by Brown, the K

-functor is

already important for the theory of AF algebras – in

spite of, or even because of (!), the fact that the

-group of an AF algebra is zero.) Using the six-

term exact sequence of Bott periodicity described

above, corresponding to an extension of C



-algebras,

together with results of the present author, Brown

showed that any extension of one AF algebra by

another is again an AF algebra.

A rather large class of simple unital C



-algebras

has by now been classified by means of the

invariants K

and K

– together with the class of

the unit in K

, and the order (or pre-order) structure

on K

– and also taking into account the compact

convex set of tracial states on the C



-algebra

(a positive linear functional on a C



-algebra is called

a trace if it has the same value on x



x and xx



for

every element x, and a tracial state if it is a state,

that is, has norm 1, or has value 1 on the unit in the

case the algebra has a unit). In addition to the set of

tracial states, together with its natural topology and

convex structure, one should also keep track of the

natural pairing between traces and K

(any trace on

a unital C



-algebra has the same value on two

equivalent projections – equal to x



x and xx



for

some element x – and hence gives rise to an additive

real-valued functional on K

In terms of these invariants (which might, broadly

speaking, be called K-theoretical), it has been

possible to classify the simple unital C



-algebras

(not of type I) arising as inductive limits (i.e., as the

completions of increasing unions) of sequences of

finite direct sums of matrix algebras over separable

commutative C



-algebras, these assumed to have

spectra of dimension at most three, on the one hand

(work of the present author together with Guihua

Gong and Liangqing Li, a culmination of earlier

work of these authors together with a number of

others), and, on the other hand, it has been possible

(work of Kirchberg and Phillips, also based on

earlier work by a number of authors) to classify the

396 C



-Algebras and their Classification



-algebra tensor products (in a natural sense) of

these C



-algebras with what is called the Cuntz



-algebra O

(see below). In the first of these two

cases, the compact convex set of tracial states –

always a Choquet simplex – is an arbitrary (metriz-

able) such space.

In the second case, this space is empty (as it is for

in particular). In both cases, K

and K

are

arbitrary countable abelian groups, with the proviso

that K

is not the sum of a torsion group and a

cyclic group. In the first case, the order structure on

, the class of the unit element, and the pairing of

with the space of traces have certain special

properties; as it turns out, these can be expressed in

a simple way. (The class of the unit need only be

positive and nonzero.) In the second case, the order

structure on K

is degenerate – every element is

positive – and the class of the unit can be arbitrary

(including zero!).

Let us just note that the Cuntz C



-algebra O

the unital C



-algebra generated by an infinite

sequence s

, s

, ... of isometries with orthogonal

ranges (in other words, elements s

such that s



the unit and s



= 0ifj 6¼ i). One need not require

the C



-algebra to have the universal property with

respect to these generators and relations as it is in

fact unique (up to an isomorphism preserving these

generators). In particular, this C



-algebra is simple.

(If one considers a finite sequence of isometries with

orthogonal ranges, and assumes in addition that the

sum of these is the unit, one also obtains a simple



-algebra, the Cuntz C



-algebra O

, n = 2, 3, ...).

The K

-group and K

-group of O

are, respectively,

Z and 0. (The K

-group and K

-groups of O

for

n = 2, 3, ... are, respectively, Z=(n  1)Z and 0.)

Both classes of C



-algebras considered in the

classification result stated above, although des-

cribed in rather a concrete way (in terms of

inductive limits and tensor products), can also be

characterized axiomatically, in a way that makes it

clear that they are, in fact, much more general than

they seem. (These axiomatizations are due to

Lin and to Kirchberg and Phillips. Typically, the

abstract axioms are easier to establish in a

given case than the inductive limit form described

above.)

In view of this, and the fact that one of the axioms

is a notion of amenability (the analogous property

for C



-algebras of a notion that has also been

considered for von Neumann algebras) and since

amenable von Neumann algebras (on a separable

Hilbert space) have been classified completely (in

remarkable work of Connes, together with many

others, starting with Murray and von Neumann –

and, one must also mention, ending with Haagerup,

who settled a particularly stubborn case), it is

natural to ask whether the K-theoretical invariants

described above might be sufficient to classify all

amenable separable C



-algebras, say, those which

are simple and unital.

The work of Villadsen has shown that additional

invariants must in fact be considered, if one is to

deal with arbitrary amenable simple C



-algebras,

and this has been confirmed in subsequent work of

Rørdam and of Toms. (Villadsen’s examples were

obtained by removing the condition of low dimen-

sion on the spectra of the commutative C



-algebras

appearing in the inductive limit decomposition

considered above.) The very nature of these authors’

work, however, has been to introduce additional

invariants, all of which it seems natural to consider

as, broadly speaking, K-theoretical. (And all of

which, as it happens, are already familiar.)

The question of the classifiability, in terms of

simple invariants (K-theoretical in nature, at least in

the broad sense, and including the spectrum which is

indispensable in the nonsimple case), of all (separ-

able) amenable C



-algebras would therefore still

appear to be on the agenda.

Already, in any case, just like the analogous

question for von Neumann algebras (now settled),

this question would appear to have had a noticeable

influence on the development of the subject – not

least in underlining the importance of K-theoretical

methods, which have proved to be pertinent both in

connection with the index theory of differential

operators on geometrical structures – from foliations

to fractals – and in connection with questions in

physics, related to quantum statistical mechanics

(see e.g., Quantum Hall Effect), to quantum field

theory (e.g., the standard model), and even to string

theory and M-theory.

See also: Axiomatic Quantum Field Theory; Bosons and

Fermions in External Fields; The Jones Polynomial;

K-Theory; Positive Maps on C *-Algebras; Quantum Hall

Effect; von Neumann Algebras: Introduction, Modular

Theory, and Classification Theory; von Neumann

Algebras: Subfactor Theory.

Further Reading

Davidson KR (1996) C



-Algebras by Example. Fields Institute

Monographs, 6. Providence, RI: American Mathematical

Society.

Dixmier J (1969) Les C



-Alge`bres et leurs Re´presentations,

2nd edn. Paris: Gauthier–Villars.

Elliott GA (1995) The classification problem for amenable



-algebras. In: Chatterji SD (ed.) Proceedings of the Interna-

tional Congress of Mathematicians, vols. 1, 2, pp. 922–932.

(Zu¨ rich, 1994). Basel: Birkha¨user.



-Algebras and their Classification 397

Evans DE and Kawahigashi Y (1998) Quantum Symmetries on

Operator Algebras. Oxford: Oxford University Press.

Fillmore PA (1996) A User’s Guide to Operator Algebras.

New York: Wiley.

Kadison RV and Ringrose J (1983–92) Fundamentals of the Theory

of Operator Algebras (4 volumes). New York: Academic Press.

Lin H (2001) An Introduction to the Classification of Amenable



-Algebras. Singapore: World Scientific.

Pedersen GK (1979) C



-Algebras and their Automorphism

Groups, London Math. Soc. Monographs. London: Academic

Press.

Rørdam M (2002) Classification of Nuclear, Simple C



-Algebras,

Encyclopaedia of Mathematical Sciences, vol. 126, pp. 1–145.

Berlin: Springer.

Sakai S (1971) C



-Algebras and W



-Algebras. Berlin: Springer.

Calibrated Geometry and Special Lagrangian Submanifolds

D D Joyce, University of Oxford, Oxford, UK

Calibrated Geometry

‘‘Calibrated geometry,’’ introduced by Harvey and

Lawson (1982), is the study of special classes of

‘‘minimal submanifolds’’ N of a Riemannian mani-

fold (M, g), defined using a closed form ’ on M

called a calibrat ion. For example, if (M, J, g)isa

Ka¨ hler manifold with Ka¨hler form !, then complex

k-submanifolds of M are calibrated with respect to

’ = !

=k!. Another important class of calibrated

submanifolds are special Lagrangian submanifolds

in Calabi–Yau manifolds, which is the focus of the

section ‘‘Special Lagrangian geometry.’’

Calibrations and Calibrated Submanifolds

We begin by defining ‘‘calibrations’’ and ‘‘calibrated

submanifolds.’’

Definition 1 Let (M, g) be a Riemannian manifold.

An ‘‘oriented tangent k-plane’’ V on M is a vector

subspace V of some tangent space T

M to M with

dimV = k, equipped with an orientation. If V is an

oriented tangent k-plane on M then gj

is a

Euclidean metric on V; so, combining gj

with the

orientation on V gives a natural volume form vol

on V, which is a k-form on V.

Now let ’ be a closed k-form on M. ’ is said to

be a calibration on M, if for every oriented k-plane

V on M, ’j

 vol

. Here, ’j

=   vol

for some

 2 R, and ’j

 vol

if   1. Let N be an

oriented submanifold of M with dimension k. Then

each tangent space T

N for x 2 N is an oriented

tangent k-plane. We say that N is a calibrated

submanifold if ’j

= vol

for all x 2 N.

It is easy to show that calibrated submanifolds

are automati cally ‘‘minimal submanifolds.’’ We

prove this in the compact case, but noncompact

calibrated submanifolds are locally volume-minimizing

as well.

Proposition 2 Let (M, g) be a Riemannian mani-

fold, ’ a calibration on M, and N a compact

’-submanifold in M. Then N is volume-minimizing

in its homology class.

Proof Let dim N = k, and let [N] 2 H

(M, R) and

[’] 2 H

(M, R) be the homology and cohomology

classes of N and ’. Then

½’½N¼

x2N

’j

x2N

vol

¼ VolðNÞ

since ’j

= vol

for each x 2 N,asN is a

calibrated submanifol d. If N

is any other compact

k-submanifold of M with [N

] = [N]inH

(M, R),

then

½’½N¼½’½N

¼

x2N

’j



x2N

vol

¼ VolðN

since ’j

 vol

because ’ is a calibration. The

last two equations give Vol(N)  Vol(N

). Thus, N

is volume-minimizing in its homology class.

Now let (M, g) be a Riemannian manifold with a

calibration ’, and let  : N ! M be an immersed

submanifold. Whether N is a ’-submanifold

depends upon the tangent spaces of N. That is, it

depends on  and its first derivative. So, for N to be

calibrated with respect to ’ is a first-order partial

differential equation on . But if N is calibrated then

N is minimal, and for N to be minimal is a second-

order partial differential equation on .

One moral is that the calibrated equations, being

first order, are often easier to solve than the minimal

submanifold equations, which are second order. So

calibrated geometry is a fertile source of examples of

minimal submanifolds.

Calibrated Submanifolds and Special Holonomy

A calibration ’ on (M, g) is only interesting if there

exist plenty of ’-submanifolds N in M, locally

or globally. Since ’j

= vol

for each x 2 N,

’-submanifolds will be abundant only if the family

’

of calibrated tangent k-planes V with ’j

= vol

398 Calibrated Geometry and Special Lagrangian Submanifolds

is ‘‘reasonably large’’ – say, if F

’

has small

codimension in the family of all tangent k-planes V

on M. A maximally boring example is the k-form

’ = 0, which is a calibration but has no calibrated

tangent k-planes, so no ’-submanifolds.

Thus, most calibrations ’ will have few or no

’-submanifolds, and only special calibrations ’ with

’

large will have interesting calibrated geometries.

Now the field of Riemannian holonomy groups is a

natural companion for calibrated geometry, because

it gives a simple way to generate interesting

calibrations ’ which automatically have F

’

large.

Let G  O(n) be a possible holonomy group of a

Riemannian metric. In particular, we can take G to be

one of the holonomy groups U(m), SU(m), Sp(m), G

or Spin(7) from Berger’s classification. Then G acts

on the k-forms 

)



on R

,sowecanlookfor

G-invariant k-forms on R

. Suppose ’

is a nonzero,

G-invariant k-form on R

By rescaling ’

we can be arrange that for each

oriented k-plane U  R

, we have ’

 vol

,and

that ’

= vol

for at least one such U.LetH be the

stabilizer subgroup of this U in G.Then’

U

vol

U

by G-invariance, so   U is a calibrated

k-plane for all  2 G.Thus,thefamilyF

’

-calibrated k-planes in R

contains G=H,soitis

‘‘reasonably large,’’ and it is likely that the calibrated

submanifolds will have an interesting geometry.

Now let M be a manifold of dimension n,andg

a metric on M with Levi-Civita connection r and

holonomy group G. Then there is a k-form ’ on M

with r’ = 0, corresponding to ’

. Hence d’ = 0,

and ’ is closed. Also, the condition ’

 vol

for

all oriented k-planes U in R

implies that ’j



vol

for all oriented tangent k-planes V in M. Thus,

’ is a calibration on M. The family F

’

of calibrated

tangent k-planes on M fibers over M with fiber F

;

so, it is ‘‘reasonably large.’’

This gives a general method for finding interesting

calibrations on manifolds with reduced holonomy.

Here are the most significant examples.



Let G = U(m)  O(2m). Then G preserves a

2-form !

on R

.Ifg is a metric on M with

holonomy U(m), then g is Ka¨ hler with complex

structure J, and the 2-form ! on M associated to

is the Ka¨ hler form of g.

One can show that ! is a calibration on (M, g),

and the calibrated submanifolds are exactly the

‘‘holomorphic curves’’ in (M, J). More generally,

=k! is a calibration on M for 1  k  m, and

the corresponding calibrated submanifolds are the

complex k-dimensional sub manifolds of (M, J).



Let G = SU(m)  O(2m). Then G preserves a

complex volume form 

= dz

^^ dz

. Thus, a Calabi–Yau m-fold (M, g) with

Hol(g) = SU(m) has a holomorphic volume form

. The real part Re  is a calibration on M, and

the corresponding calibrated submanifolds are

called special Lagrangian submanifolds.



The group G

 O(7) prese rves a 3-form ’

and a

4-form ’

on R

. Thus , a Riemannian 7-ma nifold

(M, g) with holonomy G

comes with a 3-form ’

and 4-form ’, which are both calibrations. The

corresponding calibrated submanifolds are called

associative 3-folds and coassociative 4-folds.



The group Spin(7)  O(8) preserves a 4-form 

on R

. Thus a Riemannian 8-manifold (M, g) with

holonomy Spin(7) has a 4-form , which is a

calibration. The -submanifolds are called Cayley

4-folds.

It is an important general principle that to each

calibration ’ on an n-manifold (M, g) with special

holonomy constructed in this way, there corre-

sponds a constant calibration ’

on R

. Locally, ’-

submanifolds in M resemble the ’

-submanifolds in

, and have many of the same properties. Thus, to

understand the calibrated submanifolds in a mani-

fold with special holonomy, it is often a good idea to

start by studying the corresponding calibrated

submanifolds of R

In particular, singularities of ’-submanifolds in M

will be locally modeled on singularities of ’

submanifolds in R

. (In the sense of geometric

measure theory, the tangent cone at a singular point

of a ’-submanifold in M is a conical ’

-submanifold

in R

.) So by studying singular ’

-submanifolds in

, we may understand the singular behavior of

’-submanifolds in M.

Special Lagrangian Geometry

We now focus on one class of calibrated submani-

folds, special Lagrangian submanifolds in Calabi–

Yau manifolds. Calabi–Yau 3-folds are used to

make the spacetime vacuum in string theory, and

special Lagrangian 3-folds are the classical versions

of A-branes, or supersymmetric 3-cycles, in Calabi–

Yau 3-folds. Special Lagrangian geometry aroused

great interest amongst string theorists because of its

roˆ le in the SYZ conjecture, providing a geometric

basis for ‘‘mirror symmetry’’ of Calabi–Yau 3-folds.

Calabi–Yau Manifolds

Here is our definition of Calabi–Yau manifold.

Readers are warned that there are several differe nt

definitions of Calabi–Yau manifolds in use in the

literature. Ours is unusual in regarding  as part of

the given structure.

Calibrated Geometry and Special Lagrangian Submanifolds 399

Definition 3 Let m  2. A Cal abi–Yau m-fold is a

quadruple (M, J, g, ) such that (M, J) is a compact

m-dimensional complex manifold, g aKa¨hler metric

on (M, J) with Ka¨hler form !,and a holomorphic

(m, 0)-form on M called the holomorphic volume

form, which satisfies

=m! ¼ð1Þ

mðm1Þ=2

ði=2Þ

 ^



 ½1

The constant factor in [1] is chosen to make Re  a

calibration. It follows from [1] that g is Ricci-flat, 

is constant under the Levi-Civita connection, and

the holonomy group of g has Hol(g)  SU(m).

Let (M, J) be a compact, complex manifold, and g

aKa¨hler metric on M, with Ricci curvature R

.Define

the Ricci form  of g by 

= J

. Then  is a closed

real (1, 1)-form on M,withdeRhamcohomologyclass

[] = 2c

(M) 2 H

(M, R), where c

(M) is the first

Chern class of M in H

(M, Z). The Calabi conjecture

specifies which closed (1, 1)-forms can be the Ricci

forms of a Ka¨hler metric on M.

The Calabi conjecture Let (M, J) be a compact,

complex manifold, and g

aKa¨hler metric on M,

with Ka¨hler form !

. Suppose that  is a real, closed

(1, 1)-form on M with [] = 2c

(M). Then there

exists a unique Ka¨ hler metric g on M with Ka¨hler

form !, such that [!] = [!

] 2 H

(M, R), and the

Ricci form of g is .

Note that [!] = [!

] says that g and g

are in the

same Ka¨hler class. The conjecture was posed by Calabi

in 1954, and was eventually proved by Yau in 1976.

Its importance to us is that when the canonical bundle

is trivial, so that c

(M) = 0, we can take   0, and

then g is Ricci-flat. Since K

is trivial, it has a nonzero

holomorphic section, a holomorphic (m, 0)-form .As

g is Ricci-flat, it follows that r=0, where r is the

Levi-Civita connection of g.Rescaling by a complex

constant makes [1] hold, and then (M, J, g, )isa

Calabi–Yau m-fold. This proves:

Theorem 4 Let (M, J) be a compact complex m-

manifold with K

trivial. Then every Ka¨ hler class

on M contains a unique Ricci-flat Ka¨hler metric g.

There exists a holomorphic (m, 0)-form , unique

up to change of phase  7!e

i

, such that

(M, J, g, ) is a Calabi–Yau m-fold.

Using algebraic geometry, one can produce many

examples of complex m-folds ( M, J) satisfying these

conditions, such as the Fermat (m þ 2)-tic

½z

; ...; z

mþ1



2 CP

mþ1

: z

mþ2

þþz

mþ2

mþ1

¼ 0



½2

Therefore, Calabi–Yau m-folds are very abundant.

Special Lagrangian Submanifolds

Definition 5 Let (M, J, g, ) be a Calabi–Yau m-fold.

Then Re  is a calibration on the Riemannian

manifold (M, g). An oriented real m-dimensional

submanifold N in M is called a special Lagrangian

submanifold (SL m-fold) if it is calibrated with respect

to Re .

Here is an alternative definition of SL m-folds. It

is often more useful than Definition 5.

Proposition 6 Let (M, J, g, ) be a Calabi–Yau

m-fold, with Ka¨ hler form ! , and N a real m-dimen-

sional submanifold in M. Then N admits an

orientation making it into an SL m-fold in M if

and only if !j

 0 and Im j

 0.

Regard N as an immersed submanifold, with

immersion  : N ! M. Then [! j

] and [ Im j

] are

unchanged under continuous variations of the

immersion . Thus, [!j

] = [Im j

] = 0 is a neces-

sary condition not just for N to be special

Lagrangian, but also for any isotopic submanifold

in M to be special Lagrangian. This proves:

Corollary 7 Let (M, J, g, ) be a Calabi–Yau m-

fold,andNacompactrealm-submanifoldinM.

Then a necessary condition for N to be isotopic

to a special Lagrangian submanifold N

in M

is that [!j

] = 0 in H

(N, R) and [Im j

] = 0 in

(N, R).

Deformations of Compact SL m-Folds

The deformation theory of compact special Lagran-

gian manifolds was studied by McLean (1998), who

proved the following result:

Theorem 8 Let (M, J, g, ) be a Calabi–Yau

m-fold, and N a compact special Lagrangian

m-fold in M. Then the moduli space M

of special

Lagrangian deformations of N is a smooth manifold

of dimension b

(N), the first Betti number of N.

Sketch proof. Suppose for simplicity that N is an

embedded submanifold. There is a natural orthogo-

nal decomposition TMj

= TN , where  ! N is

the normal bundle of N in M.AsN is Lagrangian,

the complex structure J : TM ! TM gives an iso-

morphism J :  ! TN. But the metric g gives an

isomorphism TN ﬃ T



N. Composing these two

gives an isomorphism  ﬃ T



Let T be a small tubular neighborhood of N in M.

Then we can identify T with a neighborhood of the

zero section in . Using the isomorphism  ﬃ T



N,we

have an identification between T and a neighborhood of

the zero section in T



N. This can be chosen to identify

the Ka¨hler form ! on T with the natural symplectic

400 Calibrated Geometry and Special Lagrangian Submanifolds

structure on T



N.Let : T ! N be the obvious

projection.

Under this identification, submanifolds N

in T 

M which are C

close to N are identified with the

graphs of small smooth sections  of T



N. That is,

submanifolds N

of M close to N are identified with

1-forms  on N. We need to know: which 1-forms 

are identified with SL m-folds N

Now, N

is special Lagrangian if !j

 Im j

 0.

But j

: N

! N is a diffeomorphism, so we can

push !j

and Im j

down to N, and regard them

as functions of . Calculation shows that





ðÞ¼d and 



Im j

ðÞ¼Fð; rÞ

where F is a nonlinear function of its arguments.

Thus, the moduli space M

is locally isomorphic to

the set of small 1-forms  on N such that d  0

and F(, r)  0.

Now it turns out that F satisfies F(, r) 

d()when is small. Therefore, M

is locally

approximately isomorphic to t he vector space of 1-

forms  with d = d() = 0. But by Hodge theory,

this is isomorphic to the de Rham cohomology

group H

(N, R), and is a manifold with dimension

(N).

To carry out this last step rigorously requires

some technical machinery: one must work with

certain Banach spaces of sections of T



N, 



and 



N, use elliptic regularity results to prove

that the map  7!(d, F(, r)) has closed image in

these Banach spaces, and then use the implicit

function theorem for Banach spaces to show that

the kernel of the map is what is expected.

Obstructions to Existence of Compact SL m-Folds

Let {(M, J

, g

, 

):t 2 (, )} be a smooth one-

parameter family of Calabi–Yau m-folds. Suppose

is an SL m-fold in (M, J

, g

, 

). When can we

extend N

to a smooth family of SL m-folds N

(M, J

, g

, 

) for t 2 (, )?

By Corollary 7, a necessary condition is that

] = [Im 

] = 0 for all t. Our next result

shows that locally, this is also a sufficient condition.

Theorem 9 Let {( M, J

, g

, 

):t 2 (, )} be a

smooth one-parameter family of Calabi–Yau m-folds,

with Ka¨hler forms !

. Let N

be a compact SL m-fold

in (M, J

, g

, 

), and suppose that [!

] = 0

in H

, R) and [Im 

] = 0 in H

, R) for all

t 2 (, ). Then N

extends to a smooth one-

parameter family {N

:t2 (, )}, where 0 < 

and N

is a compact SL m-fold in (M, J

, g

, 

This can be proved using similar techniques to

Theorem 8. Note that the condition [Im 

] = 0

for all t can be satisfied by choosing the phases of

the 

appropriately, and if the image of H

(N, Z)in

(M, R) is zero, then the condition [! j

] = 0 holds

automatically.

Thus, the obstructions [!

] = [Im 

] = 0in

Theorem 9 are actually fairly mild restrictions, and

SL m-folds should be considered as pretty stable

under small deformations of the Calabi–Yau

structure.

Remark The deformation and obstruction theory

of compact SL m-folds are extremely well behaved

compared to many other moduli space problems in

differential geometry. In other geometric problems

(such as the deformations of complex structures on a

complex manifold, or pseudoholomorphic curves in

an almost-complex manifold, or instantons on a

Riemannian 4-manifold), the deformation theory

often has the following general structure.

There are vector bundles E, F over a compact

manifold M, and an elliptic operator P : C

(E) !

(F), usually first order. The kernel Ker P is the

set of infinitesimal deformations, and the cokernel

Coker P the set of obstructions. The actual moduli

space M is locally the zeros of a nonli near map

 : Ker P ! Coker P.

In a generic case, Coker P = 0, and then the

moduli space M is locall y isomorphic to Ker P,

and so is locally a manifold with dimension ind(P).

However, in nongeneric situations Coker P may be

nonzero, and then the moduli space M may be

nonsingular, or have an unexpected dimension.

However, SL m-folds do not follow this pattern.

Instead, the obstructions are topologically determined,

and the moduli space is always smooth, with dimen-

sion given by a topological formula. This should be

regarded as a minor mathematical miracle.

Mirror Symmetry and the SYZ Conjecture

Mirror symmetry is a mysterious relationship

between pairs of Calabi–Yau 3-folds M,

M, arising

from a branch of phy sics known as string theory,

and leading to some very strange and exciting

conjectures about Calabi–Yau 3-folds, many of

which have been proved in special cases.

In the beginning (the 1980s), mirror symmetry

seemed mathematically completely mysterious. But

there are now two complementary conjectural

theories, due to Kontsevich and Strominger–Yau–

Zaslow, which explain mirror symmetry in a fairly

mathematical way. Probably both are true, at some

level. The second proposal, due to Strominger, Yau,

and Zas low (1996), is known as the SYZ conjecture.

Here is an attempt to state it.

Calibrated Geometry and Special Lagrangian Submanifolds 401

The SYZ conjecture Suppose M and

M are mirror

Calabi–Yau 3-folds. Then (under some additional

conditions), there should exist a compact topologi-

cal 3-manifold B and surjective, continuous maps

f : M ! B and

f :

M ! B, such that

(i) There exists a dense open set B

 B, such that

for each b 2 B

, the fibers f

1

(b) and

1

(b) are

nonsingular special Lagrangian 3-tori T

in M

and

M. Furthermore, f

1

(b) and

1

(b) are in

some sense dual to one another.

(ii) For each b 2 =BnB

, the fibers f

1

(b) and

1

(b) are expected to be singular special

Lagrangian 3-folds in M and

The fibrations f and

f are called special Lagran-

gian fibrations, and the set of singular fibers  is

called the discriminant. In part (i), the nonsingular

fibers of f and

f are supposed to be dual tori. What

does this mean?

On the topological level, we can define duality

between two tori T,

T to be a choice of isomorph-

ism H

(T, Z) ﬃ H

(

T, Z). We can also define

duality between tori equipped with flat Riemannian

metrics. Write T = V=, where V is a Euclidean

vector space and  a lattice in V. Then the dual

torus

T is defined to be V



=



, where V



is the

dual vector space and 



the dual lattice. However,

there is no notion of duality between nonflat

metrics on dual tori.

Strominger, Yau, and Zaslow argue only that

their conjecture holds when M,

M are close to the

‘‘large complex structure limit.’’ In this case, the

diameters of the fibers f

1

(b),

1

(b) are expected to

be small compared to the diameter of the base space

B, and away from singularities of f ,

f, the metrics on

the nonsingular fibers are expected to be approxi-

mately flat. So, part (i) of the SYZ conjecture says

that for b 2 BnB

, f

1

(b) is approximately a flat

Riemannian 3-torus, and

1

(b) is approximately the

dual flat Riemannian torus.

Mathematical research on the SYZ conjecture has

followed two broad approaches. The first could be

described as symplectic topo logical. For this, we

treat M,

M just as symplectic manifolds and f ,

f just

as Lagrangian fibrations. We also suppose B is a

smooth 3-manifold and f ,

f are smooth maps. Under

these simplifying assumptions, Mark Gross, Wei-

Dong Ruan, and others have built up a beautiful,

detailed picture of how dual SYZ fibrations work at

the global topological level.

The second approach could be described as local

geometric. Here, we try to take the special Lagran-

gian condition seriously from the outset, and focus

on the local behavior of special Lagrangian

submanifolds, and especially their singularities,

rather than on global topological questions. In

addition, we are intrested in what fibrations of

generic Calabi–Yau 3-folds might look like.

There is now a well-developed theory of SL

m-folds with isolated singularities modeled on

cones (Joyce 2003a). This is applied to SL

fibrations and the SYZ conjecture in Joyce

(2003a, b), leading to the tentative conclusions

that for generic Calabi–Yau 3-folds M, special

Lagrangian fibrations f : M ! B will be only piece-

wise smooth, and have discriminants  of real

codimension 1 in B, in contrast to smooth fibra-

tions which have  of codimension 2. We also

argue that for generic mirrors M,

M and f ,

the discriminants ,

 cannot be homeomorphic

and so do not coincide. This contradicts part (ii)

above.

A better way to formulate the SYZ conjecture

may be in terms of families of mirror Calabi–Yau

3-folds M

and fibrations f

: M

! B,

B for t 2 (0, ) which approach the ‘‘large complex

structure limit’’ as t ! 0. Then we could require the

discriminants 



of f

to converge to some

common, codimension 2 limit 

as t ! 0.

It is an important, and difficult, open problem to

construct examples of special Lagrangian fibrations

of compact, holonomy SU(3) Calabi–Yau 3-folds.

None are currently known.

See also: Minimal submanifolds; Mirror Symmetry:

A Geometric Survey; Moduli Spaces: An Introduction;

Riemannian Holonomy Groups and Exceptional Holonomy.

Further Reading

Gross M, Huybrechts D, and Joyce D (2003) Calabi–Yau

Manifolds and Related Geometries, Universitext Series, Berlin:

Springer.

Harvey R and Lawson HB (1982) Calibrated geometries. Acta

Mathematica 148: 47–157.

Joyce DD (2000) Compact Manifolds with Special Holonomy.

Oxford: Oxford University Press.

Joyce DD (2003a) Special Lagrangian submanifolds with isolated

conical singularities. V. Survey and applications. Journal of

Differential Geometry 63: 279–347, math.DG/0303272.

Joyce DD (2003b) Singularities of special Lagrangian fibrations

and the SYZ conjecture. Communications in Analysis and

Geometry 11: 859–907, math.DG/0011179.

Joyce DD (2003c) U(1)-invariant special Lagrangian 3-folds in C

and special Lagrangian fibrations. Turkish Mathematical

Journal 27: 99–114, math.DG/0206016.

McLean RC (1998) Deformations of calibrated submanifolds.

Communications in Analysis and Geometry 6: 705–747.

Strominger A, Yau S-T, and Zaslow E (1996) Mirror symmetry

is T-duality. Nuclear Physics B 479: 243–259, hep-th/

9606040.

402 Calibrated Geometry and Special Lagrangian Submanifolds

Calogero–Moser–Sutherland Systems of Nonrelativistic

and Relativistic Type

S N M Ruijsenaars, Centre for Mathematics and

Computer Science, Amsterdam, The Netherlands

Introduction

Systems of Calogero–Moser–Sutherland (CMS) type

form a class of finite-di mensional dynamical systems

that are integrable both at the classical and at the

quantum level. The CMS systems describe N point

particles moving on a line or on a ring, interacting

via pair potentials that are specific functions of four

types, namely rational (I), hyperbolic (II), trigono-

metric (III), and elliptic (IV). They occur not only in

a nonrelativistic (Galilei-invariant), but also in a

relativistic (Poincare´-invariant) setting. Thus, one

can distinguish a hierarchy of 16 physically distinct

versions (classical/quantum, nonrelativistic/relativis-

tic, type I–IV), the most general one being the

quantum relativistic type IV system.

The nonrelativistic systems date back to pioneer-

ing work by Calogero, Sutherland, and Moser in the

early 1970s. The pair potential structure of the

interaction can be encoded in the root system A

N1

and there also exist integrable versions for all of the

remaining root systems. The classical systems are

given by N Poisson commuting Hamiltonians with a

polynomial dependence on the particle momenta

, ..., p

. Accordingly, the quantum versions are

described by N commuting Hamiltonians that are

partial differential operato rs.

The relativistic systems were intr oduced in the

mid-1980s, at the classical level by Ruijsenaars and

Schneider, and at the quantum level by Ruijsenaars.

They con verge to the nonrelativistic systems in the

limit c !1, where c is the speed of light. Again, the

systems can be related to the root system A

N1

, and

they admit integrable versions for other root

systems. All of the commuting classical Hamilto-

nians depend exponentially on generalized momenta

, ..., p

. Hence, the associ ated commuting quan-

tum Hamiltonians are analytic difference operators.

The above integrable systems can be further

generalized by allowing supersymmetry or internal

degrees of freed om (‘‘spins’’), coupled in quite

special ways to retain integrability. In this article,

however, the focus is on the 16 versions of the

N1

-symmetric CMS systems without internal

degrees of freedom. The primary aim is to acquaint

the reader with their definition and integrability,

and with their most prominent features and inter-

relationships. Second, we intend to give a rough

sketch of the state of the art concerning explicit

solutions for the various versions. This involves a

concretization of the action-angle maps and eigen-

function transforms that simultaneously diagonal ize

the commuting dynamics, paying special attention to

their remarkable duality prop erties.

It is beyond the scope of this article to review the

hundreds of papers specifically dealing with CMS

type systems, let alone the much larger literature

where they play some role. Indeed, the systems have

been encountered in a great many different contexts

and they are related to a host of other integrable

systems in various ways. Accordingly, they can be

studied from the perspective of various subfields of

mathematics and theoretical physics. First some of

these perspectives and relations to seemingly quite

different topics will be mentioned before embarking

on the far more focused survey.

Staying first within the confines of the CMS type

systems, some nonobvious limits yielding other

familiar finite-dimensional integrable systems will

be mentioned. To begin with, all of the A

N1

type

systems give rise to systems with a Toda type

(exponential ‘‘nearest neighbor’’) interaction via a

suitable limiting transition (basically a strong-

coupling limit). This leads to integrable N-partic le

systems with a classical/quantum, nonrelativistic/

relativistic, nonpe riodic/periodic version; starting

from the quantum relativistic periodic Toda system,

the remaining seven versions can be obtained by

suitable limits.

Next, we recall that the quantum system of N

nonrelativistic bosons on the line or ring interacting

via a pair potential of -function type is soluble via a

Bethe ansatz, with the ‘‘line version’’ exhibiting

quantum soliton behavior (factorized scattering). It

has been shown that there exist scaling limits of

eigenfunctions for suitable CMS systems that give

rise to the latter Bethe type eigenfunctions for N = 2,

while convergence for N > 2 is plausibl e, but has

not been demonstrated thus far.

Via suitable analytic continuations preserving

reality/formal self-adjointness, one can arrive at

CMS systems with more than one species of particle

(particles and ‘‘antiparticles’’). Likewise, analytic

continuations and appropriate limits of CMS sys-

tems associated with root sytems other than A

N1

lead to a further proliferation of N-dimensional

integrable systems. Typically, such limits refer either

Calogero–Moser–Sutherland Systems of Nonrelativistic and Relativistic Type 403

to the commuting Hamiltonians (the Toda limit

being a case in point) or to the joint eigenfunctions

(as exemplified by the -function system limit); it

seems difficult to control both sets of quantities at

once.

Starting from the spin type CMS systems, another

kind of limit can be taken. Specifically, by ‘‘freez-

ing’’ the pa rticles at equilibrium positions, it is

possible to arrive at integrable spin chains of

Haldane–Shastry and Inozemtsev type.

At this point, it is expedient to insert a brief

remark on finite-dimensional integrable systems. As

the term suggests, one may expect that, with due

effort, such systems can be ‘‘integrated,’’ or, equiva-

lently, ‘‘solved.’’ But it should be noted that the

latter terms (let alone the qualifier ‘‘due effort’’)

have no unambiguous mathematical meaning. Cer-

tainly, ‘‘solving’’ involves obtaining explicit infor-

mation on the action-angle map and joint

eigenfunction transform at the classical and quan-

tum level, resp., but a priori it is not at all clear how

far one can proceed.

Focusing again on the CMS systems and their

relatives, it should be stressed that, in many cases,

one is still far rem oved from a complete solution,

especially for the elliptic CMS systems. In this

regard the previous remark serves not only as a

caveat, but also to make clear why the various

vantage points provided by different subfields in

mathematics and physics are crucial: typically, they

yield complementary insights and distinct represen-

tations for solutions, serving different purposes.

To be sure, in first approximation the mathe-

matics involved at the classical and quantum level is

symplectic geometry and Hilbert space theory, resp.

In point of fact, however, far more ingredients have

turned out to be quite natural and useful. On the

classical level, these include the theory of groups, Lie

algebras and symmetric spaces, lin ear algebra and

spectral theory, Riemann surface theory, and more

generally algebraic geome try.

On the quantum level, the viewpoint of harmonic

analysis on symmetric spaces is particularly natural

and fruitful for the nonrelativistic CMS systems and

their arbitrary root-system versions, whereas quan-

tum groups/algebras/symmetric spaces can be tied in

with the relativistic systems and their versions for

other root systems. (The c !1limit amoun ts to the

q ! 1 limit in the quantum group picture.) As a

matter of fact, the whole area of special functions

and their q-analogs is intimately related to the

quantum CMS type systems (cf. also the last section

of this article). Finally, the occurrence of commut-

ing analytic difference operators in the relativistic

(q 6¼ 1) systems leads to largely uncharted territory

in the intersection of the theory of Hilbert space

eigenfunction expansions and the theory of linear

analytic difference equations.

The study of the thermodynamics (N !1limit

with temperature 0 and density 0 fixed) asso-

ciated with the trigonometric and elliptic CMS

systems and their spin cousins yields its own circle

of problems. It was initiated by Sutherland three

decades ago, and even though a host of results on

partition functions, correlation functions, fractional

statistics, strong–weak coupling duality, relations to

Yangians, etc., have meanwhile been obtained,

many questions are still open. This area also has

links with random-matrix theory, but the input from

this field is thus far limited to certain discrete

couplings.

The above N-dimensional integrable systems are

related to a great many infinite-dimensional integr-

able systems, both at the classical and at the

quantum level. On the one hand, there are structural

analogs that have been used to advantage in the

study of CMS systems, including Lax pair and R-

matrix formulations, zero-curvature representations,

bi-Hamiltonian formalism, Ba¨cklund transforma-

tions, time discretizations, and tools such as Baker–

Akhiezer funct ions, Bethe ansatz, separation of

variables, and Baxter-type Q-operators.

On the other hand, there are striking physical

similarities between various soliton field theories

(a prominent one being the sine-Gordon field

theory) and infinite soliton lattices (in particular

several Toda type lattices), and the CMS systems for

special parameter values. Particularly conspicuous

are the ties between the classical CMS systems and

the KP and two-dimensional Toda hierarchies. The

latter relations actually extend beyond the solitons,

including rational and theta function solutions.

CMS systems are relevant in various other

contexts not yet mentioned. A prominent one

among these is a class of supersymmetric gauge

field theories. In this quantum context, the classical

CMS systems have surfaced in the description

of moduli spaces encoding the vacuum structure

(Seiberg–Witten theory). Equally surprising, certain

classical CMS systems (with internal degrees

of freedom) have found a second application in a

quantum context, namely in the description of

quantum chaos (level repulsion).

We conclude this introduction by listing addi-

tional disparate subjects where connections with

CMS type systems have been found. These include

the theory of Sklyanin, affine Hecke, Kac–Moody,

Virasoro an d W-algebras, equations of Knizhnik–

Zamolodchikov, Yang–Baxter, Witten–Dijkgraaf–

Verlinde–Verlinde, and Painleve´ type, Gaudin,

404 Calogero–Moser–Sutherland Systems of Nonrelativistic and Relativistic Type