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

Подождите немного. Документ загружается.

6. E

(Relative entropy of entanglement

[V, M, A



]) This measure (Vedral et al. 1997)

is motivated geometrically: it is simply the

relative entropy distance of  to the separable

subset: E

() = inf



S(k), where  ranges over

all separable states. E

is an upper bound to E

However, it can be improved by taking the

distance to the PPT states rather than the

separable states, and by combining with E

,in

the following way:

7. E

(The Rains bound [V, M, A



, C]) Following

Rains (2001),weset

ðÞ¼inf



SðkÞþE

ðÞðÞ

where the infimum is over all states . This is

still an upper bound to E

, although clearly

smaller than both E

(take only separable )

and E

(take  = ). No example of E

() <

() is known, but any bound entangled non-

PPT state would be such an example.

8. E

(Squashed entanglement [V, M, A



, A

þþ

C, L]) This measure, introduced by Christandl

and Winter (2004), amazingly has all the good

properties, but is as difficult to compute as any

of the other measures. E

(

) is the infimum

over the entropy combination

Sð

ÞþSð

ÞSð

ABC

ÞSð

over all extensions 

ABC

of the given state 

a system enlarged by a part C, where the density

operators in the above expression are the

restrictions of 

ABC

to the subsystems indicated.

9. E

(Key rate [V, M, A



]) The bit rate at which

secret key can be generated is certainly larger

than E

, since distillation is one way to do it. It

is, in general, strictly larg er, sinc e there are

undistillable states with positive key rate.

10. E

(Concurrence [ V]) This measure was

originally only defined for qubit pairs, as a

step in Wootter’s (1998 ) formula for E

in this

case. It has an extension to arbitrary dimensions

(Rungta et al. 2001), namely the convex hull of

the function c(j ih j) =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2(1  tr(

))

, where

 = j ih j

is the reduced density operator.

Both upper and lower bounds exist in the

literature. The main interest in this measure

stems from the fact that it has interesting

extensions to the multipartite case.

To conclude, we would like to point out that

many of the themes discussed in this article were set

by Bennett et al. (1996); their article is worth

reading even today. Good review articles covering

entanglement measures, with more complete refer-

ences, are Plenio and Virmani (2005) , Bruß (2002),

and Donald et al. (2002).

See also: Entanglement; Entropy and Quantitative

Transversality; Quantum Channels: Classical Capacity.

Further Reading

Bennett CH, DiVincenzo DP, Smolin JA, and Woottes WK (1996)

Mixed-state entanglement and quantum error correction.

Physical Review A 54: 3824–3851.

Bruß D (2002) Characterizing entanglement. Journal of Mathe-

matical Physics 43: 4237.

Christandl M and Winter A (2004) Squashed entanglement – an

additive entanglement measure. Journal of Mathematical

Physics 45: 829–840.

Donald MJ, Horodecki M, and Rudolph O (2002) The unique-

ness theorem for entanglement measures. Journal of Mathe-

matical Physics 43: 4252–4272 and quant-ph/0105017.

Horodecki M, Horodecki P, and Horodecki R (2000) Unified

approach to quantum capacities: towards quantum noisy

coding theorem. Physical Review Letters 85: 433.

Nielsen M (1999) Conditions for a class of entanglement

transformations. Physical Review Letters 83: 436–439.

Plenio MB and Virmani S (2005) An introduction to entanglement

measures, quant-ph/0504163.

Rains E (2001) IEEE Trans. Inf. Theory 47: 2921–2933.

Rungta P et al. (2001) Universal state inversion and concurrence

in arbitrary dimensions. Physical Review A 64: 042315.

Vedral V, Plenio MB, Rippin MA, and Knight PL (1997)

Quantifying entanglement. Physical Review Letters 78: 2275.

Vollbrecht KGH and Werner RF (2001) Entanglement measures

under symmetry. Physical Review A 64: 062307.

Wootters WK (1998) Entanglement of formation of an

arbitrary state of two qubits. Physical Review Letters

80: 2245–2248.

236 Entanglement Measures

Entropy and Quantitative Transversality

G Comte, Universite

de Nice Sophia Antipolis,

Nice, France

Introduction

A mathematical law for a physical phenomenon,

describing the variation of a value y(2 R)interms

of parameters x

2 R, i 2 {1, ..., n}, is usually

given:

1. in the simplest cases (and hence in exceptional

cases), by an explicit functional equation

y = F(x

, ..., x

), or

2. by an implicit equation G(y, x

, ..., x

) = 0, or

3. more generally, by a partial differentiable

equation,

Hy;

j

@x



; ...;

j

@x



; x

; ...; x

¼ 0 þ initial values

In the first case, the exact equation y = F(x

, ..., x

)

fully describes the behavior of y as (x

, ..., x

)

vary, but in practice this information is too

substantive: using the Taylor formula, knowledge

of the value y

at some point (x

, ..., x

) and of the

value of

;...;x

ðÞ

; ...;



; ...; x



is enough to predict, with controlled accuracy, by

linear approximation, the behavior of y for para-

meters (x

, ..., x

) close to (x

, ..., x

In the case (2), both the parameters (x

, ..., x

) and

the value y belong to the set M = {( y , x

, ..., x

) 2

nþ1

; G(y, x

, ..., x

) = 0}, and we would like to

know whether or not this set may be (at least locally

around one of its point (y

, x

, ..., x

)) a graph of

some function (x

, ..., x

) 7!y = F(x

, ..., x

), as in

the case (1). Using the implicit function theorem, we

may try to reduce our equation to the explicit

equation of (1), and then perform a linear approx-

imation involving rF

,..., x

)

. Assuming that a priori

we know a value y

such that for

, ..., x

), ( y

, x

, ..., x

) 2 M, this reduction is

possible, locally around ( y

, x

, ..., x

), under the

condition that

; x

; ...; x



6¼ 0

In this situation

;...; x

ðÞ

¼

; ...;



 y

; x

; ...; x



; x

; ...; x



Now, as it is normally the case, when they come

from observation, the variables x

, ..., x

are known

with an estimate and one sees that the larger



; ...;



; x

; ...; x



; x

; ...; x





is, the worse the estimate on y near y

Furthermore, assuming that M is locally a graph

of a function (x

, ..., x

) 7!y = F(x

, ..., x

), for a

given (x

, ..., x

), the exact expression of

y = F(x

, ..., x

) and consequently the exact value

of r F

,..., x

)

is not possible to obtain; we have to

approach it using an algorithm (classically the

Newton algorithm), and closer

; x

; ...; x



is to 0, the more such an algorithm is unstable.

Finally, in the case (3), skipping technical details,

we encounter the same type of difficulties: we have

to avoid small values for some gradient functions at

a given point, in order to obtain, locally at some

point (x

, ..., x

), in a stable way, reliable informa-

tion on y in terms of (x

, ..., x

To sum up, the prediction of a physical phenom-

enon by a mathematical law greatly depends not

only on the noncancellation of some gradient

functions, but, as we deal wi th approximations and

algorithms, on how different those gradient func-

tions are from zero.

This principle, of course, extends directly to

applied problems (see the last of our examples in

the final section): being close to singular values

essentially means that the control (e.g., of the

positions of some device by a manipulator) is poor.

The geometric counterpart of this analytic phe-

nomenmon is called ‘‘transversality,’’ the condition

for some function G to have a nonzero partial

derivative

; x

; ...; x



is equivalent to the condition

; x

;...;x

ðÞ

 Ox

 x

¼ R

nþ1

Entropy and Quantitative Transversality 237

or to the condition

;...; x

ðÞ

M  Oy ¼ R

nþ1

where T

M is the tangent space of M at a 2 M.

We say that rG

, x

,..., x

)

is transverse to the

space of parameters Ox

x

at (y

, x

, ..., x

), or

that M is transverse to Oy at (y

, x

, ..., x

For some quantity >0, the condition that



; ...;



; x

; ...; x



; x

; ...; x







means that the angle  = (rG

, x

,..., x

)

,Ox

x

)

or the angle  = (T

, x

..., x

)

M, Oy) is smaller than 

(see Figure 1).

Our purpose in the sequel is to indicate how we

can quantify the situations described above (the

defect of transversality), in order to generically or

almost generically avoid them with quantified

accuracy.

Quantifying Transversality

Given two submanifolds M and N of the Euclidean

space R

, we can measure the transversality defect

of (M, N)atx 2 R

with a differential criterion,

both analytical and geometric.

Let us first introduce some notations. For a given

linear map L : R

! R

, the image by L of the unit

ball of R

is an r-dimensional ellipsoid in R

with

semi-axes denoted as l

(L) l

(L), where r is

the rank of L. For r < p, we denote l

rþ1

(L) =

0, ..., l

(L) = 0.

Now, let x 2 M \ N;let : R

! T

be the

projection onto the orthogonal space of T

N, p = n 

dim (N)and

the restriction of  to M.

Definitions

We say that (M, N) is transverse at x, and we denote

it by M \j

N, if and only if 

is a submersion at x,

that is, D

jM(x)

: T

M ! T

is onto.

For a given =(

, ...,

), 



,wesay

that (M, N)is-nontransverse at x, and we denote it

by M \j



N, if and only if l

(D

jM(x)

) 



,8 i 2 {1, ..., p}.

With these notations, we have: M \j

N (i:e:,(M, N)

nontransverse at x)ifandonlyifx 62 M \ N or M \j



for some  with 

= 0, and the more (M, N)is-

nontransverse, with  close to (

, ..., 

p1

, 0), the less

the manifolds M and N seem transverse at x 2 M \ N

(see Figure 2).

The final step in our formalism to give a convenient

quantitative approach of transversality is the following:

let X, Y be two (real) Riemannian manifolds, f : X ! Y

a (smooth) mapping, N  Y a submanifold of Y with

codimension p in N, y 2 N,and : O!R

submersion, where O is an open neighborhood of x in

Y, such that 

1

({0}) = N \O. Then we say that (f , N)

is transverse at x, and we denote it by f \j

N, if and only

if f   is submersive in x.

For a given =(

, ..., 

), 



, we say

that (f , N)is(, )-nontransverse at x, and we

denote it by f \j

(,)

N, if and only if l

(D[f  ]

(x)

)

 

,8 i 2 {1, ..., p}.

Clearly, we recognize the definition of transvers-

ality and of -nontransversality of two submani-

folds M, N of R

by letting f : M ! R

be the

inclusion and =

(for more details on transvers-

ality and stability, see, e.g., Golubitski and Guille-

min (1973)).

With the definitions and notations above, our

general problem may be posed as follow s:

For a C

-regular (k 2 N [ {1}) mapping f : R

! R

and a given =(

, ..., 

), how large is the set

(f , B

, ) = f ((f , B

, )), where (f , B

, ) = {x 2

 R

; l

(Df

(x)

)  

, 8i 2 {1, ..., p}} and B

is a ball

of radius r in R

∇G(a)

. . . x

Figure 1 Transversality of the manifold M and Oy.

⊥

(B )

Figure 2 Almost-nontransversality of M and N.

238 Entropy and Quantitative Transversality

The ‘‘bad’’ set (f , B

, ) is called the set of

-almost critical values of f (restricted to B

). Our

purpose is to show that one can control its size in

terms of k and . However, before explicitly stating

quantitative results, let us precise what we under-

stand by ‘‘big set’’ or by ‘‘size of a set.’’

Measure and Dimensions

We have a very natural way to measure a subset A

of a metric space. To do this, we consider   0a

real number and we denote



¼ðD

i2N

; A 

[

i2N

and jD

j

()

where jD

j is the diameter of D





ðAÞ¼inf

i2N



; ðD

i2N



()

and



ðAÞ¼lim

!0





ðAÞ2R \ f1g



(A) is called the -dimensional Hausdorff

measure of A. It appears that when H



(A) 6¼1,



(A) = 0 for 

>, and when H



(A) 6¼ 0,



(A) = 1 for 

<. This gives rise to the

following definition of the Hausdorff dimension

of A:

dim

ðAÞ¼inff; H



ðAÞ¼0g

¼ supf; H



ðAÞ¼1g

The Hausdorff dimension generalizes the classical

notions of dimension, for instance, when A is a

subset of R

, dim

(A)  n,ad-dimensional mani-

fold has Hausdorff dimension d, and H

(A) is the

same as the Lebesgue measure L

of A (for a very

large class of subset A, which we do not describe

here. For more details on geometric measure theory,

see Falconer (1986) and Federer (1969)).

Another convenient notion of dimensio n is the

(metric) entropy dimension. Let us brie fly define it.

For a bounded subset A in some metric space and a

real number >0, we denote M(, A) the minimal

number of closed balls of radius  , covering A.



(A) = log

(M(, A)) is called the -entropy of the

set A. This terminology was introduced in

Kolmogorov and Tihomirov (1961) and reflects the

fact that H



(A) is the amount of information needed

to digitally memorize A with accuracy . The

entropy dimension of A, dim

(A), is the order of

M(, A )as ! 0. Precisely,

dim

ðAÞ¼lim sup

!0

logðMð; AÞÞ

logð1=Þ

¼ inff; Mð; AÞð1=Þ



;

for sufficiently small g

We clearly have

dim

ðAÞdim

ðAÞ

For any bounded set A in R

, we can bound M(, A)

from above by a polynomial in 1= (see Ivanov

(1975) and Yomdin and Comte (2004)):

Mð; AÞcðnÞ

i¼0

ðAÞð1=Þ

where c(n) only depends on n and V

(A) (the ith

variation of the set A) is the mean value, with

respect to P (for a suitable measure), of the number

of connected components of A \ P, with P an affine

(n  i)-dimensional space of R

Since for A contained in a d-dimensional mani-

fold, V

(A) = 0 for i > d, we deduce from this

inequality that in this case M(, A) is bounded

from above by a polynomial of degree  d in 1=.

Our goal is to explain that we can be more precise

than this general inequality when A is a set of

critical or almost-critical values of a C

mapping.

Transversality Is a Generic Situation

The results in this section concern critical values,

and not almost-critical values. They show that a

‘‘generic’’ point of the target space is not a critical

value, and the more regular, the mapping the

smaller the set of critical values. Such theorems

relating the regularity of a mapping and the size of

its critical values are called Morse–Sard type

theorems (see Sard (1942, 1958, 1965)). The

simplest theorem in this direction is the following:

Theorem 1 (C

Morse–Sard theorem) (Morse

1939, Sard 1942, Holm 1987). Let f : R

! R

be a C

-regular mapping. Then H

((f , B

)) = 0,

where (f , B

) = f ( (f, B

)) and (f , B

) is the set of

points x 2 B

where rank(Df

(x)

) < p.

The set (f , B

) is the image, under f, of the points

of the ball B

in the source space at which f is not

submersive, that is, the set of critical values of f.

Consequently, the Morse–Sard theorem ensures that

for almost all points y in the target space, f

1

({y}) is

either empty or a smooth submanifold of the source

space of dimension n  p.

Entropy and Quantitative Transversality 239

Note that (f , B

) =(f , B

, ) for some conve-

nient =(

, ..., 

) with 

= 0, because

x 7!l

(Df

(x)

) is bounded on B

, for all i 2 {1, ..., p}.

Now, we can concentrate our attention on more

singular points than the critical ones, those at which

the rank  of f is prescribed. Let us denote such

points by 



(f , B

), for <p. By definition,





(f , B

) = f ( 



(f , B

)), where 



(f , B

) = {x 2 B



; rank(Df

(x)

)  }. With these notations, the result

for rank-r critical values is the following:

Theorem 2 (C

Morse–Sard theorem for rank-r

critical values) (Federer 1969). Let f : R

! R

a C

-regular mapping. Then H

þ(n)=k

(



(f , B

)) = 0. In particular,

dim

ð



ðf ; B

ÞÞ   þ

n  

One can produce examples sh owing that the

bound of Theo rem 2 is the sharpest one (see

Comte (1996), Whitney (1935), Grinberg (1985),

and Yomdin and Comte (2004)).

We note that Theorem 1 is a corollary of Theorem 2

(just replace k by 1 and  by p  1inTheorem 2).

This result tells nothing about the entropy dimen-

sion of 



(f , B

); in the next section, we will

bound the growth of entropy of almost-critical

values.

Almost-Transversality Is Almost Generic

In this section, f : R

! R

is a C

mapping. We

denote by K a Lipschitz consta nt of D

k1

f on B

and

by R

(f ) the quantity (K=(k  1)!)  r

. We have:

Theorem 3 (C

quantitative Morse–Sard theorem)

(Yomdin 1983 Yomdin and Comte 2004). Let

f : R

! R

be a C

mapping, =(

, ..., 





, and let us denote 

= 1. We have

(for   R

(f )):

Mð; ðf ; B

; ÞÞ

 C 

i¼0









ðf Þ





ðniÞ=k

where C is a constant depending only on n, p, and k.

As a corollary, one can bound the entropy

dimension of 



(f , B

)by þ (n  )=k, and hence

its Hausdorff dimension, again finding Theorem 2:

we just have to put 

þ1

= 0 and 

, ..., 



large

enough, that is, 

 

(Df

(x)

), for all x 2 B

,in

Theorem 3, to obtain:

Theorem 4 (C

entropy Morse–Sard theorem)

(Yomdin 1983 Yomdin and Comte 2004). Let

f : R

! R

be a C

mapping, let us denote 

= 1

and 

= sup {

(Df

(x)

); x 2 B

}, for i 2 {1, ..., }. We

have (for   R

(f )):

Mð; 



ðf ; B

ÞÞ  C 



i¼0









ðf Þ





ðniÞ=k

where C is a constant depending only on n, p, and k.

In particular,

dim

ð



ðf ; B

ÞÞ  dim

ð



ðf ; B

ÞÞ   þ

n  

Again we have examples showing that this bound

is sharp (see Yomdin and Comte 2004).

Furthermore, the mapping f in Theorems 2–4 may

be of real differentiability class (Ho¨ lder smoothness

class C

), with the same conclusions in these

theorems. That is, k may be a real number written

as k = p þ  with  2 [0, 1], p 2 Nn{0}, and f is C

means that f is p times differentiable and there exists

a constant C > 0 such that for all x, y 2 B

,kD

(x)



(y)

kC kx  yk



(see Yomdin and Comte

(2004)).

Examples

Let us denote by A the set of real polynomial

mappings of degree d and of the following type:

x 7!Qða; xÞ¼1 þ

j¼1

with a = (a

, ..., a

) and kak1 (where kk is the

Euclidean norm of R

). We identify the set A with

(0, 1) = { a 2 R

; kak1}.

We want to bound the -entropy of the set of

such polynomials for which the real roots are

multiple or almost multiple.

We denote by V the set V = {(a, x) 2 R

dþ1

;

Q(a, x) = 0}. At points (a, x)ofV with rQ

(a, x)

6¼ 0,

V is a C

manifold of codimension 1 of R

dþ1

We denote by V

reg

= {(a, x) 2 V;rQ

(a,x)

6¼ 0}

and by V

sing

= {(a, x) 2 V; rQ

(a, x)

= 0} = V n V

reg

By Whitney (1957), V

sing

is a union of smooth

manifolds of dimension  d  1.

A root x of a polynomial Q(a,  ) is multiple if and

only if

Qða; xÞ¼

ða; xÞ¼0

Consequently, the set A



of polynomials of A

with multiple roots is (V

sing

) [ (

reg

), where

 : R

dþ1

! R

is the standard projection (a, x) = a,

and (

reg

) is the set {(a, x) 2 V

reg

; Ox  T

(a,x)

reg

}

of critical values of 

reg

. By Sard’s theore m

240 Entropy and Quantitative Transversality

(Theorem 2), dim

((

reg

))  d  1. Since

dim

((V

sing

))  d  1, we obtain: dim



)  d 

1: thus, having distinct roots is a generic property.

Let, as above, =(

, ..., 

) with 



and 

= 1. A root x of a polynomial Q(a, ) 2 A is

said to be -almost multiple if and only if

Q(a, x) = 0 and V \j



Ox, that is, (a, x) 2 V

sing

sin (T

(a, x)

reg

,Ox)  

. This condition only con-

cerns 

and we can take 

=  = 

d1

= 1. We

denote A

,

to be the set of polynomials of A with

(at least) a -almost multiple root. By Theorem 3,

Mð; A

;

nðV

sing

ÞÞ  C 

d1

i¼0





þ





But (V

sing

) being a finite union of manifolds of

dimension at most d  1, we finally obtain

Mð; A

;

ÞC



d1

i¼0





þ





Thus, having no -almost multiple root is -almost

a generic property. In Figure 3, we represent V for

d = 3 and a

= 1,

W ¼ða ; xÞ2R



dþ1

;

ða; xÞ¼0



The next example comes from robotics: let us

consider a planar robotic manipulator consisting of

two jointed bars of length a and b, as presented in

Figure 4. We may parametrize the positions of the

endpoint P of this device by the angles  and (see

Figure 4). Now the distance r from the origin to P is

= kPk

= a

þ b

þ 2ab cos ( ). The critical points

of r are given by

ð Þ¼2ab sinð Þ¼0

and correspond to the circle = 0. The critical value

of r is a þ b. Near these critical positions, the control

of r with respect to is poor; we would like to avoid

those near-critical values. Given >0, the condition

ð Þ



 

implies j jarcsin(=2ab), and the -near-critical

values of r are

max

 r

 2ab½1  cosðarcsinð=2abÞÞ

where r

max

is a þ b; thus, they are contained in an

interval of length  c  

=(4ab  r

max

), and

M(,(r, ))  c  

=(4ab  r

max

 )(Theorem 3

gives M(, (r, ))  C(1 þ =).

See also: Entanglement; Entanglement Measures;

Quantum Entropy; Singularity and Bifurcation Theory.

Further Reading

Comte G (1996) Sur les hypothe` ses du the´ore` me de Sard pour le

lieu critique de rang 0. C. R. Academy of Science Paris

323(I): 143–146.

de Araujo Moreira CGT (2001) Hausdorff measures and

the Morse–Sard theorem. Publication Mathema` tiques

45(1): 149–162.

Σ,Λ

A\ A

Σ,Λ

Figure 3 The space of polynomials of type 1 þ a

x þ a

x þ x

with almost-multiple roots.

max

Figure 4 Almost-critical points of the distance function of P to

the origin.

Entropy and Quantitative Transversality 241

Falconer KJ (1986) The Geometry of Fractal Sets, Cambridge Tracts

in Mathematics, vol. 85. Cambridge: Cambridge University Press.

Federer H (1969) Geometric Measure Theory, Grundlehren

Math. Wiss., vol. 153. Springer.

Golubitski V and Guillemin V (1973) Stable Mappings and Their

Singularities, Graduate Texts in Mathematics, vol. 14.

Springer.

Grinberg EL (1985) On the smoothness hypothesis in Sard’s

theorem. American Mathematical Monthly 92(10): 733–734.

Holm P (1987) The theorem of Brown and Sard. L’enseignement

Mathe´matiques 33: 199–202.

Ivanov LD (1975) In: Vituskin AG (ed.) Variazii mnozˆhestv i funktsii

(Russian) (Variations of Sets and Functions). Moscow: Nauka.

Kolmogorov AN and Tihomirov VM (1961) "-Entropy and

"-capacity of sets in functional space. American Mathematical

Society Translations 17: 277–364.

Morse AP (1939) The behavior of a function on its critical set.

Annals of Mathematics 40(1): 62–70.

Sard A (1942) The measure of the critical values of differentiable

maps. Bulletin of the American Mathematical Society

48: 883–890.

Sard A (1958) Images of critical sets. Annals of Mathematics

68: 247–259.

Sard A (1965) Hausdorff measure of critical images on Banach

manifolds. American Journal of Mathematics 87: 158–174.

Whitney H (1935) A function not constant on a connected set of

critical points. Duke Mathematical Journal 1: 514–517.

Whitney H (1957) Elementary structure of real algebraic varieties.

Annals of Mathematics 66: 545–556.

Yomdin Y (1983) The geometry of critical and near-critical values

of differentiable mappings. Mathematische Annalen

264(4): 495–515.

Yomdin Y and Comte G (2004) Tame Geometry with Applica-

tion in Smooth Analysis, Lecture Notes in Mathematics,

vol. 1834. Berlin: Springer.

Equivariant Cohomology and the Cartan Model

E Meinrenken, University of Toronto, Toronto, ON,

Canada

Introduction

If a compact Lie group G acts on a manifold M, the

space M/G of orbits of the action is usually a singular

space. Nonetheless, it is often possible to develop a

‘‘differential geometry’’ of the orbit space in terms of

appropriately defined equivariant objects on M. This

article is mostly concerned with ‘‘differential forms

on M/G.’’ A first idea would be to work with the

complex of ‘‘basic’’ forms on M, but for many

purposes this complex turns out to be too small.

A much more useful complex of equivariant differ-

ential forms on M was introduced by Cartan (1950).

In retrospect, Cartan’s approach presented a differ-

ential form model for the equivariant cohomology of

M, as defined by A Borel (1960). Borel’s construction

replaces the quotient M/G by a better-behaved (but

usually infinite-dimensional) homotopy quotient M

and Cartan’s complex shou ld be viewed as a model

for forms on M

One of the features of equivariant cohomology are

the localization formulas for the integrals of equivar-

iant cocycles. The first instance of such an integration

formula was the ‘‘exact stationary phase formula,’’

discovered by Duistermaat and Heckman. This

formula was quickly recognized by Berline and

Vergne (1983) and Atiyah and Bott (1984),asa

localization principle in equivariant cohomology.

Today, equivariant localization is a basic tool in

mathematical physics, with numerous applications.

This articl e begins with Borel’s topological defini-

tion of equivariant cohomology, then proceeds to

describe H Cartan’s more algebraic approach, and

concludes with a discussion of localization principles.

As additional references for the material covered

here, we particularly recommend books by Berline,

Getzler, and Vergne (1992) and Guillemin and

Sternberg (1999).

Borel’s Model of H

(M)

Let G be a topological group. A G-space is a

topological space M on which G acts by transforma-

tions g 7!a

, in such a way that the action map

a : G  M ! M ½1

is continuous. An important special case of G-spaces

are principal G-bundles E ! B, that is, G-spaces

locally isomorphic to products U  G.

Definition 1 A classifying bundle for G is a

principal G-bu ndle EG ! BG, with the following

universal property: for any principal G-bundle

E ! B, there is a map f : B ! BG, unique up to

homotopy, such that E is isomorphic to the pullback

bundle f



EG. The map f is known as a ‘‘classifying

map’’ of the principal bundle.

To be precise, the base spaces of the principal

bundles considered here must satisfy some technical

condition. For a careful discussion, see Husemoller

(1994). Classifying bundles exist for all G (by a

construction due to Milnor (1956)), and are unique

up to G-homotopy equivalence.

It is a basic fact that principal G-bundles with

contractible total space are classifying bundles.

242 Equivariant Cohomology and the Cartan Model

Examples 2

(i) The bundle R ! R=Z = S

is a classifying

bundle for G = Z.

(ii) Let H be a separable complex Hilbert space,

dim H= 1. It is known that unit sphere S(H)is

contractible. It is thus a classifying U(1)-bundle,

with the projective space P(H) as base. More

generally, the Stiefel manifold St(k, H) of unitary

k-frames is a classifying U(k)-bundle, with base

the Grassmann manifold Gr(k , H)ofk-planes.

(iii) Any compact Lie group G arises as a closed

subgroup of U(k), for k sufficiently large.

Hence, the Stiefel manifold St(k, H) also serves

as a model for EG.

(iv) The based loop group G = L

K of a connected Lie

group K acts by gauge transformations on the

space of connections A (S

) =

, k). This is a

classifying bundle for L

K,withbaseK.The

quotient map takes a connection to its holonomy.

For any commutative ring R (e.g., Z, R, Z

), let

H( ; R) denote the (singular) cohomology with

coefficients in R. Recall that H( ; R) is a graded

commutative ring under cup product.

Definition 3 The equivariant cohomology H

(M) =

(M; R)ofaG-space M is the cohomology ring of

its homotopy quotient M

= EG 

ðM; RÞ¼HðM

; RÞ½2

Equivariant cohomology is a contravariant func-

tor from the category of G-spaces to the category of

R-modules. The G-map M ! pt induces an algebra

homomorphism from H

(pt) = H(BG)toH

(M). In

this way, H

(M) is a module over the ring H(BG).

Example 4 (Principal G-bundles). Suppose E ! B is

aprincipalG-bundle. The homotopy quotient E

may

be viewed as a bundle E 

EG over B. Since the fiber

is contractible, there is a homotopy equivalence

’ B ½3

and therefore H

(E) = H(B).

Example 5 (Homogeneous spaces). If K is a closed

subgroup of a Lie group G, the space EG may be

viewed as a model for EK, with BK = EG=K = EG 

(G=K). Hence,

ðG=KÞ¼HðBKÞ½4

Let us briefly describe two of the main techniques

for computing H

(M).

1. Leray spectral sequences.IfR is a field, the

equivariant cohomology may be computed as the

term of the spectral sequence for the fibration

! BG.IfBG is simply connected (as is the

case for all compact connected Lie groups), the

-term of the spectral sequence reads

p;q

¼ H

ðBGÞH

ðMÞ½5

2. Mayer–Vietoris sequences.IfM = U

[ U

is a

union of two G-invariant open subsets, there is a

long exact sequence

!H

ðMÞ!H

ðU

ÞH

ðU

Þ!

! H

ðU

\ U

Þ!H

kþ1

ðMÞ!

More generally, associated to any G-invariant open

cover, there is a spectral sequence converging to

(M).

Example 6 Consider the standard U(1)-action on

by rotations. Cover S

by two open sets U



, given

as the complement of the south pole and north pole,

respectively. Since U

\ U



retracts onto the equa-

torial circle, on which U(1) acts freely, its equivar-

iant cohomology vanishes except in degree 0. On the

other hand, U



retract onto the poles p



. Hence, by

the Mayer–Vietoris sequence the map H

U(1)

) ﬃ

U(1)

)  H

U(1)



) given by pullback to the fixed

points is an isomorphism for k > 0. Since the

pullback map is a ring homomorphism, we conclude

that H

U(1)

; R) is the commutative ring generated

by two elements x



of degree 2, subject to a single

relation x



= 0.

g-Differential Algebras

Let G be a Lie group, with Lie algebra g.A

G-manifold is a manifold M together with a

G-action such that the action map [1] is smooth.

We would like to introduce the concept of equivar-

iant differential forms on M. This complex should

play the role of differential forms on the infinite-

dimensional space M

. In Cartan’s approach, the

starting point is an algebraic model for the differ-

ential forms on the classifying bundle EG.

The algebraic machinery will only depend on the

infinitesimal action of G. It is therefore convenient

to introduce the following concept.

Definition 7 Let g be a finite-dimensional Lie

algebra. A g-manifold is a manifold M, together with

a Lie algebra homomorphism a : g ! X(M),  7!a



into the Lie algebra of vector fields on M,suchthat

the map g  M ! TM,(, m) 7!a



(m)issmooth.

Any G-manifold M becomes a g-manifold by

taking a



to be the generating vector field



ðmÞ :¼



t¼0

expðtÞ

ðmÞ½6

Equivariant Cohomology and the Cartan Model 243

Conversely, if G is simply connected, and M is a

g-manifold for which all of the vector fields a



are

complete, the g-action integrates uniquely to an

action of the group G.

The de Rham algebra ((M), d) of differential

forms on a g-manifold M carries graded derivations



= L(a



) (Lie derivatives, degree 0) and 



= (a



)

(contractions, degree 1). One has the following

graded commutation relations:

½d; d¼0; ½L



; d¼0; ½



; d¼L



½7

½



;



¼0; ½L



; L



¼L

½;

; ½L



;



¼

½;

½8

More generally, the following definitions are

introduced.

Definition 8 A g-differential algebra (g-da) is a

commutative graded algeb ra A =

n¼0

, equipped

with graded derivations d, L



, 



of degrees 1, 0, 1

(where L



, 



depend linearly on  2 g), satisfying the

graded commutation relations [7] and [8].

Definition 9 For any g-da A, one defines the

horizontal subalgebra A

hor



ker(



), the invar-

iant subalgebra A



ker(L



), and the basic

subalgebra A

basic

= A

hor

Note that the basic subalgebra is a differential

subcomplex of A.

Definition 10 A connection on a g-da is an

invariant element  2 A

 g, with the property





 = . The curvature of a connection is the element



2 A

 g given as F



= d þ (1/2)[,  ]

g-da’s A admitting connections are the algebraic

counterparts of (smoo th) principal bundles, with

basic

playing the role of the base of the principal

bundle.

Weil Algebra

The Weil algebra Wg is the algebraic analog to the

classifying bundle EG. Similar to EG, it may be

characterized by a universal property:

Theorem 11 There exists a g-da Wg with

connection 

, having the following universal

property: if A is a g-da with connection , there

is a unique algebra homomorphism c : Wg !A

taking 

to .

Clearly, the universal property character izes Wg

up to a unique isomorphism. To get an explicit

construction, choose a basis {e

}ofg, with dual

basis {e

}ofg



.Lety



be the corresponding

generators of the exterior algebra, and v

2 S



the

generators of the symmetric algebra. Let

g ¼

2iþj¼n



^



½9

carry the differential

¼ v

½10

¼f

½11

where f

= he

,[e

, e

]

i are the structure constants

of g. Define the contractions 

= 



¼ 

;

¼ 0 ½12

and let L

= [d, 

]. Then L

are the generators for

the adjoint action on Wg. The element 

= y



2 W

g  g is a connection on Wg. Notice that

we could also use y

and dy

as generators of Wg.

This identifies Wg with the Koszul algebra, and

implies:

Theorem 12 Wg is acyclic, that is, the inclusion

R ! Wg is a homotopy equivalence.

Acyclicity of Wg corresponds to the contractibil-

ity of the total space of EG.

The basic subalgebra of Wg is equal to (Sg



)

, and

the differential restricts to zero on this subalgebra,

since d changes parity. Hence, if A is a g-da with

connection, the characteristic homomorphism

c : Wg !A induces an algebra homomorphism,

(Sg



)

! H(A

basic

). This homomorphis m is indepen-

dent of :

Theorem 13 Suppose 

, 

are two connections on

a g-da A. Then their characteristic homomorphisms

, c

: Wg !Aare g-homotopic. That is, there is a

chain homotopy intertwining contractions and Lie

derivatives.

Remark 14 One obtains other interesting exam-

ples of g-da’s if one drops the commutativity

assumption from the definition. For instance,

suppose g carries an invariant scalar product. Let

Cl(g) be the corresponding Clifford algebra, and

U(g) the enveloping alge bra. The noncommu-

tative Weil algebra (introd uced by Alekseev and

Meinrenken 2002)

Wg ¼ Ug  ClðgÞ½13

is a (noncommutative) g-da, with the derivations d,

, 

defined on generators by the same formulas as

for Wg.

244 Equivariant Cohomology and the Cartan Model

Equivariant Cohomology of g-da’s

In analogy to H

(M):= H(M

), we now declare:

Definition 15 The equivariant cohomology algebra

of a g-da A is the cohomolo gy of the differential

algebra A

:= (Wg A)

basic

ðAÞ :¼ HðA

Þ½14

The equivariant cohomolo gy H

(A) has functorial

properties parallel to those of H

(M). In particular,

(A) is a module over

ðf0gÞ ¼ HððWgÞ

basic

Þ¼ðSg



½15

Theorem 16 Suppose A is a g-da with connection

, and let c : Wg !A be the characteristic homo-

morphism. Then

Wg A!A; w  x 7! cð wÞx ½16

is a g-homotopy equivalence, with g-homotopy

inverse the inclusion

A!Wg A; x 7!1  x ½17

In particular, there is a canonical isomorphism

HðA

basic

ÞﬃH

ðAÞ ½18

Proof By Theorem 13, the automorphism w 

x 7!1  c(w) x of Wg A is g-homotopic to the

identity map.

The above definition of the complex A

is often

referred to as the Weil model of equivariant

cohomology, while the term Cartan model is reserved

for a slightly different description of A

. Identify

the space (Sg



A)

with the algebra of equivariant

A-valued polynomial functions  : g !A. Define a

differential d

on this space by setting

ðd

ÞðÞ¼d ððÞÞ  



ðÞ½19

Theorem 17 (H Cartan). The natural projection

Wg A!Sg



A restricts to an isomorphism of

differential algebras, A

ﬃ (Sg



A)

Suppose A carries a connection .Theg-homotopy

equivalence [16] induces a homotopy equivalence

basic

of the basic subcomplexes. By explicit

calculation, the corresponding map for the Cartan

model is given by

ðSg



AÞ

basic

;7! P



hor

ððF



ÞÞ ½20

Here (F



) 2A

is the result of substituting the

curvature of ,andP

hor

: A!A

hor

is horizontal

projection. On elements of (Sg



)

 (Sg



A)

the map [20] specializes to the Chern–Weil

homomorphism.

There is an algebraic counterpart of the Leray

spectral sequence: introduce a filtration

pþq

:¼

2ip

ðS



A

½21

Since second term in the equivariant differential

[19] raises the filtration degree by 2, it follows that

p;q

¼ðS

p=2



 H

ðAÞ ½22

for p even, E

p, q

= 0 for p odd. In fortunate cases, the

spectral sequence collapses at the E

-stage (see

below).

Equivariant de Rham Theory

We will now restrict ourselves to the case that

A=(M) is the algebra of differential forms on a

G-manifold, where G is compact and connected.

Theorem 18 (Equivariant de Rham theorem). Sup-

pose G is a compact, connected Lie group, and

that M is a G-manifold. Then there is a canonical

isomorphism

ðM; RÞﬃH

ððMÞÞ ½23

where the left-hand side is the equivariant cohomol-

ogy as defined by the Borel construction.

Motivated by this result, the nota tion can be

changed slightly; write



ðMÞ¼ðSg



 ðMÞÞ

½24

for the Cartan complex of equivariant differential

forms, and d

for the equivariant differential [19].

Remark 19 Theorem 18 fails, in general, for

noncompact Lie groups G . A differential form

model for the noncompact case was developed by

Getzler (1990).

Example 20 Let (M, !) be a symplectic manifold,

and a : G ! Diff(M) a Hamiltonian group action.

That is, a preserves the symplectic form, a



! = !,

and there exists an equivariant mom ent map

 : M ! g



such that 



! þ dh, i= 0. Then the

equivariant sympl ectic form !

():= ! þh, i is

equivariantly closed.

Example 21 Let G be a Lie group, and denote,

respectively, by



¼ g

1

dg and 

¼ dgg

1

½25

Equivariant Cohomology and the Cartan Model 245