Gabor frames and Yang-Mills theory for noncommutative tori · Gabor frames and Yang-Mills theory for noncommutative tori Franz Luef Strobl June 5, 2014 Franz Luef Yang-Mills theory

Gabor frames and Yang-Mills theory fornoncommutative tori

Franz Luef

Strobl

June 5, 2014

Franz Luef Yang-Mills theory over noncommutative tori

Starting point

A Gabor frame is a line bundle over a noncommutativetorus.

A multi-window Gabor frame is a vector bundle over anoncommutative torus.

Basic idea

Covariant derivatives and connections on these line/vector bundlesallows one to do calculus on Gabor frames in a way that iscompatible with the structure of Gabor frames.


Setting

C ∗-algebra A.

Integration on A is given by a trace on A, i.e. a linearfunctional trA on A such that trA(AB) = trA(BA) for allA,B ∈ A.

Differential structure is given by a derivation δ on A such thatδ(AB) = δ(A)B + Aδ(B).

A spectrally invariant subalgebra A0 of A defines a smoothstructure on A.

Noncommutative torus: VU = e2πiθUV for two unitaryoperators U,V and a real number θ.


Noncommutative torus

Let c be a 2-cocyle on αZ× βZ given byc((αk , βl), (αm, βn)) = e−2πim(l−n)αβ. Then we consider thetwisted group algebra `1(αZ× βZ, c) with respect totwisted convolution

a\b(k , l) =∑m,n

a(m, n)b(k −m, l − n)e−2πim(l−n)αβ

and twisted involution of a given by

a∗(k , l) = e2πiklαβa(−k ,−l).

The envelopping C ∗-algebra of `1(αZ× βZ, c) is thenoncommutative torus A(αZ× βZ, c) for . Another way todefine A(αZ× βZ, c) is as the norm closure of{π(αk , βl) : k , l ∈ Z2}.


Noncommutative torus

smooth structure given by A∞(αZ× βZ, c) = {A : A =∑k,l aklπ(αk , βl) (akl) ∈ S(Z2)}.

Trace tr on A(αZ× βZ, c):

tr(∑k,l

aklπ(αk , βl)) = a00.

Derivations ∂1 and ∂2 on A∞(αZ× βZ, c):

∂1(A) = 2πiα∑k,l

kaklπ(αk, βl)

∂2(A) = 2πiβ∑k,l

laklπ(αk , βl)

δ1 and δ2 are commuting derivations.


Moyal plane–quantum plane

Let c be a 2-cocyle on R2 given byc((x , ω), (y , η)) = e−2πiy(ω−η). Then we consider the twistedgroup algebra L1(R2, c) and its enveloping C ∗-algebra orequivalently the norm closure of {π(z) : z ∈ R2}. We denoteit by A(R2, c).

A(R2, c) is the continuous version of a noncommutative torus.

smooth structure given byA∞(R2, c) = {A : A =

∫∫a(z)π(z) a ∈ S(R2)}.

Trace tr on A(R2, c):

tr(

∫∫a(z)π(z)) = a(0).

Derivations ∂1 and ∂2 on A∞(R2, c):

∂1(A) = 2πi

∫∫xa(x , ω)π(x , ω)dxdω

∂2(A) = 2πi

∫∫ωa(x , ω)π(x , ω)dxdω

δ1 and δ2 are commuting derivations.


Hilbert C ∗-module

Let V be a vector space.

left action of A on V : (A, g) 7→ A · gA-valued pairing A〈., .〉A〈A · f , g〉 = AA〈f , g〉 for all A ∈ AA〈f , g〉 = A〈g , f 〉∗

A〈f , f 〉 ≥ 0

V is complete with respect to the norm A‖f ‖ := ‖A〈f , f 〉‖1/2A .

Noncommutative vector bundles over A are finitely generatedprojective Hilbert C ∗-modules over A.

AV = PAn

for a P ∈ Mn(A) with P = P∗ = P2.


Frames for Hilbert C ∗-modules

Let A be a unital C ∗-algebra. A sequence {gj : j = 1, ..., n} ina (left) Hilbert A-module AV is called a standard moduleframe if there are positive reals C ,D such that

C A〈f , f 〉 ≤n∑

j=1

A〈f , gj〉A〈gj , f 〉 ≤ D A〈f , f 〉

for each f ∈ AV .

Let A be a unital C ∗-algebra. A sequence {gj : j = 1, ..., n} ina (left) Hilbert A-module AV is a standard module frame ifthe reconstruction formula

f =n∑

j=1

A〈f , gj〉 · gj for all f ∈ AV .


Frames for Hilbert C ∗-modules and projective modules

Recently, Rieffel gave a very useful description of projectiveA-modules:

Rieffel

Suppose AV is a finitely generated projective A-module withan A-valued innerproduct. Then every projection p ∈ Mn(A)such that AV ∼= pAn is of the form p = (pjk) with

pjk = A〈gj , gk〉

for some standard module frame {g1, .., gn} for AV .

Suppose AV is singly generated. Then every projection p in Ais of the form

p = A〈g , g〉

for a g ∈ AV such that f = A〈f , g〉 · g .


Maps on Hilbert C ∗-modules

Suppose V is a Hilbert A-module. Then a module mappingT : V → V is adjointable, if there is a mapping T ∗ : V → V :

A〈Tf , g〉 = A〈f ,T ∗g〉 for all f , g ∈ V .

End(V ) denotes the space of all adjointable mappings on V .

rank one operators SAg ,hf = A〈f , g〉 · h are adjointableoperators.

Morita equivalence

Let A and B be C ∗-algebras. Then an A-B-equivalence bimodule

AV B is an A-B-bimodule such that:

〈A · f , g〉B = 〈f ,A∗ · g〉B and A〈f · B, g〉 = A〈f , g · B∗〉;A〈f , g〉 · h = f · 〈g , h〉B.

A and B are called Morita equivalent if there exists an equivalencebimodule.


Morita equivalence

If A and B are called Morita equivalent, then V is a finitelygenerated projective A-module and a finitely generatedprojective right B-module.

Suppose A and B are Morita equivalent and{gj : j = 1, ..., n} a standard module frame for AV . Then

f =n∑

j=1

A〈f , gj〉 · gj =n∑

j=1

f · 〈gj , gj〉B.

A and End(AV ) are Morita equivalent.


Hilbert C ∗-module over A(R2, c)

Define for f , g ∈ S(R)

R2〈f , g〉 =

∫∫R2

〈f , π(z)g〉π(z)dz

R2〈A · f , g〉 =

∫∫R2

(a\Vg f )(z)π(z)dz .

Then S(R) is a Hilbert C ∗-module wrt ‖g‖R2 = ‖R2〈g , g〉‖1/2.

S(R) is a singly generated A(R2, c)-module: There exists ag ∈ S(R) with ‖g‖2 = 1 such that S(R) = pA(R2, c) forp = R2〈g , g〉.Note that ‖g‖2

R2 = ‖R2〈g , g〉‖ ≤∫∫

R2 |Vgg(z)|dz .


Line bundles over Moyal plane

Proof idea

An operator T on S(R) is adjointable wrt R2〈., .〉 if

〈f , π(z)Tg〉 = 〈f ,Tπ(z)g〉,

hence π(z)T = Tπ(z) for all z ∈ R2. By Schur’s theorem wehave that T = cI for some c ∈ C.

Make S(R) into a C right Hilbert C ∗-module:

〈f , g〉C = 〈g , f 〉2

g · λ := gλ λ ∈ C

For f , g , h ∈ S (R) we have R2〈f , g〉 · h = f · 〈g , h〉C.

〈Vg f ,Vhk〉L2(R2) = 〈f , h〉L2(R)〈k , g〉L2(R).


Conclusion

A∞(R2, c) is Morita equivalent to C.

In other words, there exists a g ∈ S (R) with ‖g‖L2 = 1 suchthat

f =

∫∫R2

〈f , π(z)g〉π(z)gdz for all f ∈ S (R).

In other words, S (R) is a line bundle over A∞(R2, c).

R2〈f , g〉 · h = f · 〈g , h〉C is just a fancy way of expressing∫∫R2

〈f , π(z)g〉π(z)h = f 〈g , h〉.


Hilbert C ∗(αZ× βZ, c)-module

Left action of A∞(αZ× βZ, c) on S(R)

A · g =[ ∑k,l∈Z

a(k , l)π(αk, βl)]g for a ∈ S (Z2)

For f , g in S(R)

•〈f , g〉 =∑k,l∈Z〈f , π(αk, βl)g〉π(αk , βl)

The completion of S(R) with respect to ‖g‖• = ‖•〈g , g〉‖1/2op

is a Hilbert C ∗(αZ× βZ, c)-module.


Hilbert C ∗( 1βZ×

1αZ, c)-module

right action for A∞( 1βZ×

1αZ, c) on S (R):

g · B =∑k,l∈Z

π( kβ ,lα)∗gb( kβ ,

lα) b ∈ S (Z2).

For f , g in S(R):

〈f , g〉• =∑

π( kβ ,lα)∗〈g , π( kβ ,

lα)f 〉.

The completion of S (R) with respect to ‖g‖• = ‖〈g , g〉•‖1/2op

is a Hilbert C ∗( 1βZ×

1αZ, c)-module.


Morita equivalence of noncommutative tori

S(R) is an equivalence bimodule between A∞(αZ× βZ, c)and A∞( 1

βZ×1αZ, c), i.e. there exist g1, ..., gn ∈ S (R) such

that

f =n∑

i=1

•〈f , gi 〉 · gi =n∑

i=1

f · 〈gi , gi 〉•.

In other words, {g1, ..., gn} are a standard module frame forthe equivalence bimodule S (R).

•〈f , g〉 · h = f · 〈g , h〉• is another way to express the Janssenrepresentation:∑k,l

〈f , π(αk, βl)g〉π(αk , βl)h = (αβ)−1∑k,l

〈h, π( kβ ,lα)g〉π( kβ ,

lα)f .


Differential geometry

Covariant derivation is a mapping on V that satisfies aLeibniz rule:

∇(A · g) = (∂A) · g + A · ∇g .

CC (V ) is the set of all compatible connections on V .

Gauge connection V is given by two covariant derivatives∇1,∇2 that are compatible with the Hermitian structure

A〈., .〉 and two derivatives ∂1, ∂2:

∂1(A〈f , g〉) = A〈∇1f , g〉+ A〈f ,∇1g〉∂2(A〈f , g〉) = A〈∇2f , g〉+ A〈f ,∇2g〉.

Curvature of a gauge connection is defined byF12 := [∇1,∇2]−∇[∂1,∂2].

Constant curvature connection is a connection withF12 = c IV for some constant c .


Complex geometry

Suppose ∂1, ∂2 are derivations on A. Then

∂(i) = 12 (∂1 − i∂2) and ∂(i) = 1

2 (∂1 + i∂2)

define a complex structure on A.

Laplacian ∆ = ∂21 + ∂2

2 has a factorization ∆ = 4∂(i)∂(i)

To the complex structure on A corresponds a complexstructure on the Hilbert C ∗-module V given by ∇1,∇2:

∇(i) = 12 (∇1 − i∇2) and ∇(i) =

1

2(∇1 + i∇2)

If ∇(i)g = 0 for a g ∈ V , then g is called a holomorphicvector.


Constant curvature connection – Moyal plane

Covariant derivatives ∇1 and ∇2 on S(R):

(∇1g)(t) = 2πitg(t), and (∇2g)(t) = g ′(t).

∇i (∫∫

a(z)π(z)dz) = (δiA) · g +∫∫

a(z)π(z)∇igdz

δi (R2〈f , g〉) = R2〈∇i f , g〉+ R2〈f ,∇ig〉∇2∇1 −∇1∇2 = 2πiI

∇1,∇2 define a constant curvature gauge connection onS(R).

The case of A = π(x , ω) and its higher-order variants areuseful in the description of the Schwartz space andGelfand-Shilov spaces: ‖∇p

1∇q2g‖∞ ≤ RCp+q(p!)α(q!)α.

Schwartz space and Gelfand-Shilov spaces are function spaceson the line bundle over the Moyal plane defined by growthconditions on the gauge connection.


Complex structure on Moyal plane

Facts

∂(i) = ∂1 − i∂2 and ∂i = ∂1 + i∂2

∇(i) = ∇1 − i∇2 and ∇i = ∇1 + i∇2

∇(i)g(t) = 2πitg(t)− ig ′(t) and ∇(i)g(t) = 2πitg(t) + ig ′(t)

∇(i) is the creation operator and ∇(i) the annihilationoperator

H = −(∇21 +∇2

2) Hamiltonian of quantum harmonicoscillator, 4∇(i)∇(i) = 2πI +∇2

1 +∇22

holomorphic vector for Moyal plane: Gaussians ϕ(t) = e−πt2


Complex structure on Moyal plane

Facts

(2πI +∇21 +∇2

2)(R2〈f , ϕ〉 · ϕ) =(2πI + δ2

1 + δ22)(R2〈f , ϕ〉) · ϕ+ R2〈f , ϕ〉 · (2πI +∇2

1 +∇22)ϕ

(2πI + δ21 + δ2

2)(R2〈f , ϕ〉) is up to some renormalization∫∫(1 + |z |2)〈f , π(z)ϕ〉π(z)ϕdz .

Shubin class Qs : ‖(2πI + δ21 + δ2

2)s/2(2R〈f , ϕ〉)‖2.


Connections on A∞(αZ× βZ, c)

∇1g(t) = 2πiαtg(t) and ∇2g(t) = βg ′(t)

∇2∇1 −∇1∇2 = 2παβiI.

∇1,∇2 is a constant curvature connection onA∞(αZ× βZ, c).

Connections on A∞(αZ× βZ, c)

∇◦1g(t) = 2πiβ−1tg(t) and ∇◦2g(t) = α−1g ′(t)

∇2∇1 −∇1∇2 = 2π(αβ)−1iI.


Balian-Low

Let G(g , αZ× βZ) be a Riesz basis for its closed span H inL2(R) and αβ = 1. Then ∇ig or ∇ih is not in H, where hdenotes the canonical dual Gabor atom h = S−1

g ,g .

There does not exist a line bundle over A∞(αZ× βZ) forαβ = 1 due to the existence of a constant curvatureconnection on S (R).

Proof is based on an observation of G. Battle, which uses the leftLeibniz property for A = π(αk , βl) implies:

〈∇1g , π(αk , βl)h〉 = 〈π(−αk ,−βl)g ,∇ih〉


Balian-Low theorem

Suppose ∇ig and ∇ih are in H for i = 1, 2. Then using the frameexpansion we have that

〈∇1g ,∇2h〉 = 〈∑k,l

〈∇1g , π(αk , βl)h〉π(αk, βl)g ,∇2h〉

=∑k,l

〈π(−αk ,−βl)g ,∇1h〉〈∇2g , π(−αk ,−βl)h〉

= 〈∇2g ,∑k,l

〈∇1h, π(αk , βl)g〉π(αk , βl)h〉

= 〈∇2g ,∇1h〉

However, ∇1∇2 −∇2∇1 = 2πiI , constant curvature connection,gives

1 = 〈g , h〉 = 〈∇2g ,∇1h〉 − 〈∇1g ,∇2h〉 = 0.


Projections in noncommuative tori

There is a link between tight Gabor frames and projections innoncommutative tori.

Suppose G(g , αZ× βZ) is a tight Gabor frame for g ∈ S (R).Then •〈g , g〉 is a projection in A(αZ× βZ, c).

Wexler-Raz: G(g , αZ× βZ) is a tight Gabor frame if and onlyif 〈g , g〉• = I .

Suppose p is a projection in A(αZ× βZ, c). Then there existsa g ∈ S (R) such that p = •〈g , g〉.


Action functional on the set of projections

Action functional

Define an action functional on the set of projections P of A:

S(p) =2

πtr[∂(i)(p)∂(i)(p)] =

1

πtr[p(∂2

1p + ∂22p)]

From the properties of a trace we get that S(p) is anon-negative real number.

Consider the tangent space to P at the point p.

An element δp of the tangent space at p must be hermitian,(δp)∗ = δp, and (p + δp)2 = p + δp + O(δp).



Action functional

δS(p) = 0 gives equation of motions for the action functionalS(p):

0 = δS(p) = − 12π tr[(p∆(p))z + ((1− p)∆(p)p)z∗]

Since z is arbitrary we get the following field equations:

p∆(p) = 0 and (1− p)∆(p)p,

which is equivalent to:

p∆(p)−∆(p)p = 0.

Therefore, the field equation is non-linear and of second order.



Action functional

The absolute minimum of S(p) fullfill first-order equations.

For a projection p in A the first Connes-Chern number isdefined by

Ψ1(p) = i2π tr[p(∂1(p)∂2(p)− ∂2(p)∂1(p))]

Ψ1(p) is integer-valued and is given by evaluating the cyclic2-cocycle at a0 = a1 = a2 = p:

Ψ(a0, a1, a2) = i2π tr[a0(∂1(a1)∂2(a2)− ∂2(a1)∂1(a2))].



Action functional

Crucial Fact: S(p) ≥ 2|Ψ1(p)|Equality in the preceding inequality occurs when theprojection p satisfies:

self duality equations:

[(∂1 ± i∂2)p]p = 0

anti-self duality equations:

[(∂1 ∓ i∂2)p]p = 0

self duality equations: ∂(i)(p)p = 0 and/or p∂(i)p = 0.

anti-self duality equations: ∂(i)(p)p = 0 and/or p∂(i)p = 0.



Suppose V is an equivalence bimodule between A and B. Letg ∈ V be such that 〈g , g〉B = I . Then pg = 〈g , g〉A is aprojection in A.

If there exists an element λ ∈ B such that g is a solution of

∇g − gλ = 0,

then pg is a solution of the self-duality equation:

∂(i)(pg )pg = 0.

If pg is a solution of the self-duality equation, then g satisfiesthis equation for

λ = (〈g , g〉A)−1〈g ,∇g〉A.


Connes-Chern number of line bundles over Moyal plane

Since every projetion in A∞(R2, c) is given by some pg = R2〈g , g〉for some g ∈ S (R) with ‖g‖L2 = 1, we have that theConnes-Chern character of a line bundle over A∞(R2, c) is given by

Ψ1(pg ) = − 12πi tr[pg (∂1(pg )∂2(pg )− ∂2(pg )∂1(pg ))] = 1.

The constant curvature connection ∇1,∇2 is crucial for this result.


Self-duality equations – Moyal plane

Let g ∈ S (R) be such that ‖g‖L2 = 1. Then pg = R2〈g , g〉 is aprojection in A∞(R2, c). Let ∇(i) = ∇1 + i∇2 be theanti-holomorphic connection on S (R). Then pg is a solution ofthe self duality equations:

∂(i)(p)p = 0

if and only if g satisfies

∇g − λg = 0

for some λ ∈ C.


Self-duality equations – Moyal plane

∇g − λg = 0

for some λ ∈ C. The solutions to this equations are Gaussiansg(t) = Ce−π(t2+2iλt).

Proposition

If pg is a solution of the self-duality equation: ∂(i)(p)p = 0, then gsolves the equation

∇g − λg = 0

for λ = 〈g ,∇g〉−1L2 .


Self-duality – noncommutative torus

Let g ∈ S (R) be such that 〈g , g〉Λ◦ = I. Then pg = •〈g , g〉is a projection in A∞(αZ× βZ, c).

Let ∇(i) = ∇1 + i∇2 be the anti-holomorphic connection onS (R). Then pg is a solution of the self duality equations:

∂(i)(p)p = 0

if and only if g satisfies

∇g − λg = 0

for some λ ∈ A∞( 1βZ×

1αZ, c).

Connes-Chern number Ψ1(pg ) = αβ.


Yang-Mills theory

In differential geometry Yang-Mills functional measures “thestrength” of the curavture of a connection. Main object ofstudy are the critical values of this functional, which leads viasome variational calculus to Euler-Lagrange equations. Theseare notoriously complicated pde’s and extremely hard tounderstand.

These investigations are also related to the study of harmoincmaps between Riemannian manifolds, as initiated by J. Eels.

Atiyah, Bott, Donaldson, Drinfield, Hitchin, Manin, Taubesand many more have made fundamental contributions toYang-Mills theory in three and fourdimensional Riemannianmanifolds.


Yang-Mills functional

The Yang-Mills functional is defined on the the set of allcomaptible connections on CC (V )

YM(∇1,∇2) = −(αβ)−1tr(F12,F12)

F12 = [∇1,∇2]−∇[∂1,∂2]

Yang-Mills problem: Determine the nature of the set ofconnections where YM attains its minimum or the set ofcritical points.

The Yang-Mills equations are the Euler-Lagrange equationsfor the associated variational problem.


Connes-Rieffel

The set of compatible connections where YM attains its minimum,consists exactly of all compatible connections with constantcurvature and all of these have the same curvatures. In the case ofthe noncommutative torus these are of the form ∇1 + iλ and∇2 + iλ for real λ.


Quantum field theory on Gabor frames

action functional for a g ∈ S (R):

Sλ,µ0,θ[g , g ] :=

∫R

(− d2

dx2 + 4π2

θ2 x2)g(x)g(x)dx + µ20‖g‖2

2

+λ2

∑k,l∈Z

∫Rdxg(x + mα + nx

β )g(x + nxβ )g(x + mα).

Langman-Szabo duality: Sλ,µ0,θ[g , g ] = 1θ2Sθλ,θ2µ0,1/θ[g , g ]

classical field equations: δS[g ,g ]δg = 0

(− d2

dx2 + 4π2

θ2 x2 + µ20)g(x) =

−λ∑k,l∈Z

∫Rdxg(x + mα + nx

β )g(x + nxβ )g(x + mα)


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Gabor frames and Yang-Mills theory for noncommutative tori · Gabor frames and Yang-Mills theory for noncommutative tori Franz Luef Strobl June 5, 2014 Franz Luef Yang-Mills theory