Aim of the talk

Aim of the talk

• Explain “localization formulae".

A “localization formula" is a “compressed”

formula for sums over a large set

(integral points inside a convex polytope) or for some special integrals

on manifolds.

These special integrals are related to action of a symmetry group,

like rotation group on the manifold

or gauge transformations.

Method

• Use of equivariant cohomology as modified de Rham complex (Witten, Berline-Vergne, ∼ 1982) using the symmetry group.

Localization is

a generalization of the fundamental theorem

of calculus.

Inverse problem

• “Decompress" the short formulae

to obtain back the wanted information:

Witten formula for integrals over reduced spaces ∼ 1992.

Compute it by residue methods.

•

Relate discrete and continuous pictures.

Classical mechanics versus quantum mechanics.

Volume versus number of integral points

in convex polytopes.

Some Applications

Conjecture of Guillemin-Sternberg on Quantization of compact classical spaces

Meinrenken-Sjamaar, ∼ 1999,

and Generalization by Paradan ∼ 2003.

Conjecture on Geometric quantization

and transversally elliptic operators

Beautiful mathematical formulae 7→ algorithmic computations

First some examples of what we can compute related to convex polytopes.

Then : geometry,

equivariant cohomology, elliptic operators.

• But ideas occurred in the reverse order:

We: many collaborators

Welleda Baldoni, Matthias Beck, Nicole Berline, Michel

Brion, Charles Cochet, Michel Duflo, Shrawan Kumar,

Jesus de Loera, Paul-Emile Paradan, Andras Szenes.

A beautiful formula.

0 1 3 4 5 6 10000

10000

X

i=0

q

= 1

1 − q + q

1 − q

. - depends only of the end points.

N

X

i=0

1 = (N + 1).

Left hand side exponentially larger than the right hand side .

(see Barvinok’s talk at this ICM)

Atiyah-Bott fixed point formula.

A

X

i=−A

q

= q

1 − q + q

1 − q

Simple case of Atiyah-Bott fixed point formula.

x

p⁻ p⁺

Inverse problem: Decompression

Given a short expression for a sum,

compute an individual term of the sum.

Understand its geometric meaning, if it comes from geometry.

In the preceding example,

knowing left hand side, want right hand side q

1 − q + q

1 − q

= X

i

c

q

that is, find the coefficients of q

.

• Answer c

:= 1 , if −A ≤ i ≤ A :

geometric meaning from Atiyah-Bott theorem:

non zero if and only if

The continuous version: Integrals Continuous version of sums; P

i=A

q

=

+

Z

A

e

dx

the fundamental theorem of calculus

gives a compressed expression for the integral

Z

A

e

dx = e

t + e

−t e

=

(e

/t) :

apply fundamental theorem of calculus.

Relation between continuous and discrete versions Local Euler Maclaurin formula (Berline-Vergne 2005) P ⊂ R

convex polytope with rational vertices,

F face of P

D

local differential operators of infinite order with constant coefficients, such that

h a polynomial function:

X

a∈P∩Z^d

h(a) = Z

P

h + X

F;dim(F)<d

Z

F

D

h

D

computed in polynomial time (d, degree h) fixed.

N

X 1 = N + 1

+ 1

.

Work in progress: implement this formula Example: sums of polynomial functions over points in Z

inside polygons in R

;

with parameters: Sum of x

x

on T ∩ Z

dilated by the factor N .

N = 11

; result with 604 digits (expected number of digits).

Proof: prove on basic examples

Cone decompositions in convex geometry

B

X

i=A

q

= q

1 − q + q

1 − q

= − q

1 − q

− q

1 − q

χ([A, B ]) = χ([A, ∞[) + χ(] − ∞, B ]) − χ( R )

= χ( R ) − χ(] − ∞, A[) − χ(]B, ∞[).

A B

=

⁻

P

n=−∞

q

= 0

Cone decompositions

A

C

B

=

ΓA A

C B

+

ΓB A

C

B

+

ΓC A

C

B

−

ΓBC A C B

−

ΓAC A

C B

−

ΓAB A

C B

+

ΓABC A

C B

Use of Brion theorem and of Barvinok algorithm

Brion: Prove formulae for cones and exponentials.

Barvinok: Signed decomposition of cones;

7→ Theoretical result: P

P∩Z^d

h

Polynomial time in the data of P , h when (d, order) fixed.

Cones:=Basic examples.

The geometric analogue will be

tubular neighborhoods of a critical set.

Move on Geometry: Stationary phase M a compact manifold of dimension n , f smooth function on M ,

dm smooth density.

F (t) :=

Z

M

e

dm

finite number of non-degenerate critical points:

When t → ∞

F (t) ∼ X

p∈C

e

∞

X

k≥0

a

t

. Asymptotically, the integral “localizes"

at a finite number of critical points p .

Example

M = {x

+ y

+ z

= A

}

x

−A A x

f := x two critical points projecting on A and −A .

Duistermaat-Heckman :Exact stationary phase

F (t) :=

Z

M

e

dm = Z

−A

e

dx = e

it + e

−it Here F (t) is exactly equal

to the first term of the local expression.

Duistermaat-Heckman theorem

M compact symplectic manifold, dimension n , with circle action, f Hamiltonian vector field

dm Liouville measure.

Z

M

e

dm = X

p∈C

e

a

t

.

Reason: equivariant cohomology.

Definitions

K compact Lie group acting on manifold M . k Lie algebra of K . φ ∈ k ,

vector field V φ : For φ ∈ k , V

φ := d

dǫ exp(−ǫφ) · x|

.

0

Equivariant complex

Let A(M ) be the algebra of differential forms on M .

d the exterior derivative.

ι(V ) the contraction by a vector field V . Equivariant form: α : k → A(M )

α commutes with action of K .

The equivariant de Rham operator D (Witten (82), Berline-Vergne (82))

D(α)(φ) := d(α(φ)) − ι(V φ)α(φ).

Then D

= 0 .

Equivariant cohomology

Equivariant cohomology algebra

H

( k , M ) = Ker(D)/Im(D ).

If α(φ) polynomial in φ , obtain H

( k , M ) : = topological equivariant cohomology (Henri Cartan, Atiyah-Bott)

But need to integrate e

for M non compact:

First application: Fourier transform of the measure on x

+ y

− z

= 1 .

(Harish-Chandra character formulae for

non compact reductive groups)

Equivariant integration R

M

α(φ) well defined if M compact oriented.

( integration top degree term α(φ)[dim M ] ).

This is a function on k .

may be defined as a generalized function on k , when M is not compact:

F (φ) a test function on k ; R

k

α(φ)F (φ)dφ :

differential form on M .

If this differential form is integrable on M , then R

M

α is defined h

Z

M

α, F dφi :=

Z

M

Z

k

α(φ)F (φ)dφ.

Hamiltonian spaces

Examples of equivariantly closed forms arise in Hamiltonian geometry.

M symplectic manifold, symplectic form Ω . Hamiltonian action of K on M :

Moment map µ : M → k

for every φ ∈ k ,

d(hφ, µi) = ι(V φ) · Ω.

Noether’s theorem: hφ, µi constant on the orbit of V φ .

Atiyah-Guillemin-Sternberg theorem:

K abelian, M compact.

The image of M by µ is a convex polytope.

(Kirwan polytope if K connected compact)

Example M := P

( C ) .

|z

|

+ |z

|

+ |z

|

= 1

[z

, z

, z

] ∼ [e

z

, e

z

, e

z

] Action (e

z

, e

z

, z

) .

Moment map (|z

|

, |z

|

) : Image

j

Homogeneous polynomials of degree n : z

z

z

Basis: integral points in t

≥ 0, t

≥ 0, t

+ t

≤ n .

Equivariant symplectic form M Hamiltonian manifold.

Ω(φ) := hφ, µi + Ω is a closed equivariant form.

1

(2iπ )

Z

M

e

Fourier transform of the pushforward

of the Liouville measure under the moment map.

Basic Example 1: R

with action of S

Coordinates (x, y ) .

Ω := dx ∧ dy.

V φ := φ(y∂

− x∂

).

µ := x

+ y

2

Equivariant symplectic form is

Ω(φ) = φ( x

+ y

2 ) + dx ∧ dy.

The moment map µ(x, y) =

Image of dxdy under the moment map µ :

0

More generally R

with diagonal action Then image of R

: Cone,

and image of measure a multivariate spline function.

Multivariate spline functions.

Example:

(Z

, Z

, Z

) ∈ C

⊕ C

⊕ C

e

Z

, e

Z

, e

Z

Moment map (a = |Z

|

+ |Z

|

, b = |Z

|

+ |Z

|

) : G(a, b) = b

(7a

− 7ab + 2b

)

14 · 5!

b

G(b, a)

G(a, b)

Basic Example 2: The cotangent bundle T

S

Free action of S

M := T

S

= S

× R . Coordinates (θ, t) .

Ω := dt ∧ dθ.

V φ = −φ∂

.

µ = t.

The moment map of µ[t, θ] = t

Image of the measure dt ∧ dθ under µ :

0

1 2iπ

Z

T^∗S¹

e

= δ

(φ).

Fourier transform of the pushforward

of the Liouville measure under the moment map.

Example 3: The sphere

x

−A A x

Ω := dy ∧ dz

x , V φ := φ(y∂

− z∂

).

µ := x.

Relation with Duistermaat-Heckman

Ω(φ) = φ x + Ω.

1 2iπ

Z

M

e

= Z

M

e

dm.

1 2iπ

Z

M

e

= ( e

iφ + e

−iφ ).

Push-forward of Liouville measure K abelian:

Duistermaat-Heckman proved exact stationary phase formula by showing that if

M

(ξ ) = µ

(ξ )/K symplectic reduction, then

Z

M×k

e

e

dφ = vol(M

(ξ )).

Reduced spaces for R

and T

S

are points, so vol(M

(ξ )) = 1 or 0 ,

if ξ belong to µ(M ) or not.

G(a, b) symplectic volume of a Kahler manifold of complex dimension 7 .

Kahler cone of dimension 2 :

Two generalizations

Abelian and "non abelian" localization formulae.

Abelian localization

(Berline-Vergne, Witten, Atiyah-Bott ∼ 1982) S

acting on compact M

isolated fixed points.

α(φ) closed equivariant form:

(2π)

Z

M

α(φ) = X

p∈{fixed points}

i

α(φ)

p det

L

(φ) .

Will explain the reason later.

Witten "non abelian" localization formula ∼ 1992 K compact group, M Hamiltonian:

µ : M → k

.

M

= µ

(0)/K symplectic reduction with symplectic form Ω

.

α(φ) closed equivariant form with polynomial coefficients gives a cohomology class α

on M

Then

Z

M×k

e

α(φ)dφ = c ∗ Z

µ⁻¹(0)/K

e

α

If α = 1 , Duistermaat theorem.

Equivariant cohomology with generalized coefficients α(φ) distribution with value forms on M :

Z

k

α(φ)F (φ)dφ is a differential form on N .

The operator D is well defined: (Duflo-Kumar-Vergne (1993));

Parallel theory to equivariant index theorem

for Atiyah-Singer transversally elliptic operators

Equivariant index: a distribution on the group.

Localization or 1 = 0

Basic observation ( Paradan ∼ 2000) K compact Lie group acting on M : κ a K -invariant vector field

tangent to the orbits of K and never vanishing:

Theorem

1 = 0

in equivariant cohomology with generalized coefficients.

That is 1 = DB with

B defined with generalized coefficients.

Example

Example C

with coordinates [r, θ ] . 1 = D (iY

(φ)dθ)

0

Indeed:

D (iY

(φ)dθ)

= iY

(φ)d(dθ)) − iφY

(φ)ι(∂

)dθ = 1

Applications

M manifold with action of K : κ a K -invariant vector field tangent to the orbits of K :

C = ∪

C

the set of zeroes of κ ,

divided in connected components;

1 = 0 on M − C :

Reduces the computation on a neighborhood of the

zeroes of κ .

Abelian localization

For an action of S

, use κ = V φ :

Zeroes of κ : Fixed points of the action.

Integration of closed equivariant form reduced to the basic case R

(Tangent space to fixed points is a sum of R

).

Abelian localization formula

Witten non abelian localization M → k

Hamiltonian manifold.

k ∼ k

κ

= d

dǫ exp(ǫµ(m)) ∗ m|

Zeroes of κ : Critical set of |µ|

.

µ = 0

one connected component of the zeroes of κ . Witten idea: use Kirwan vector field:

7→ non abelian localization formula;

reduces essentially double integration on M × k

to integration on T

S

× R

by considering N neighborhood of µ = 0

Tubular neighborhoods of the critical set of kµk

x²

−A 0 A x

−A A

p⁻ p⁺

Computations

Interest: Use both localization formulae:

abelian+non abelian; 7→

to obtain computations of intersection pairings on some reduced spaces

(as toric varieties, moduli spaces of flat bundles);

Residue computation theoretical: Jeffrey-Kirwan,

Brion-Vergne, Szenes-Vergne, de Concini-Procesi;

Efficient algorithms for Computations

Volume V (a, b) of the reduced fiber µ : C

→ R

V (a, b) = res

res

e

e

dx

dx

(x

)

(x

)

(x

+ x

)

. That is integration on

|x

| = ǫ

, |x

| = ǫ

with ǫ

< ǫ

. Reverse the order if a ≤ b .

Efficient (exact) computation of number of points in Network polytopes (Baldoni-de Loera-Vergne)

of Kostant partition functions (Beck-Baldoni-Cochet-Vergne)

Multiplicities for tensor products for representations of

compact groups (Cochet).

Equivariant integration on non compact spaces K compact Lie group acting on M :

κ a K -invariant vector field tangent to the orbits of K .

assume C = zeroes of κ compact

: then as 1 = 0 on M − C , there exists a canonical class P

∈ H

( k , M )

with compact support (neighborhood of C ) and equal to 1 (without support condition).

Equivariant integration defined on H

( k , M ) by Z

M

α(φ)P

(φ).

The result is well defined as a generalized function:

(depending possibly on κ if M is non compact).

The equivariant volume and its quantized version Basic example: T

S

Equivariant volume:

1 2iπ

Z

T^∗S¹

e

= Z

R

e

dt = δ

(φ).

Quantized version of T

S

= L

(S

) .

L

(S

) = ⊕

C e

.

Tr

(e

) =

∞

X

n=−∞

e

= δ

(e

).

Basic example R

Equivariant volume:

1 2iπ

Z

R²

e

=

Z

0

e

dr == 1

−iφ . Quantization of R

:

Stone Von-Neumann theorem (1930) :

F ock( C ) = {f ∈ O( C );

Z

C

|f (z )|

e

dxdy < ∞}.

F ock = ⊕

C z

Tr

(e

) =

∞

X e

= 1

1 − e

.

Formula for the character of the quantized space M symplectic manifold : classical space.

H (M ) Hilbert space with a representation Still mysterious

( Kirillov for homogeneous Hamiltonian spaces:

The orbit method )

I reinterpreted Kirillov’s formula for the character as an equivariant cohomology formula.

Works for more general cases.

In general ?? I conjecture

an equivariant cohomology formula in some cases

Quantization and transversally elliptic operators Let M be a Hamiltonian manifold.

Symplectic form ω integral.

K compact acting on M in a Hamiltonian way.

µ : M → k

moment map.

κ Kirwan Vector field:

assume zeroes of κ compact.

Theorem: Paradan-Berline-Vergne

There exists a (virtual) Hilbert space H (M ) (the index of a transversally elliptic operator), and a representation of K in H (M )

Tr

(e

) = Z

M

e

T odd(φ, M )P

(φ)

T odd(φ, M ) is an equivariant cohomology

Quantization and reduction

When M is compact, H (M ) constructed as the index of an elliptic operator:

Guillemin-Sternberg conjecture.

Theorem: Meinrenken-Sjamaar:

The multiplicities of the representations of K in H (M ) are related to the moment map

( dim H (M )

= dim H (M

) ).

Qualitative statement:

The irreducible representations of K

are classified by integral orbits of K in k

(orbits of highest weight).

Any representation arising in H (M ) is associated to an orbit

contained in µ(M ) ⊂ k

.

Proper moment map.

M non compact: H (M ) computed as

the index of a transversally elliptic operator if zeroes of Kirwan vector field is compact.

Conjecture

The conjecture of Guillemin-Sternberg should be true;

That is H (M ) reflects well the geometry of M .

Levels of energy

True for M = T

K ,

M = V symplectic space.

0

The spectrum of iφ in H (M ) is

composed of the discrete values of the energy.

Discrete series

True for discrete series (Paradan);

λ