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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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}.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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 .
Franz Luef Yang-Mills theory over noncommutative tori
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 .
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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 .
Franz Luef Yang-Mills theory over noncommutative tori
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).
Franz Luef Yang-Mills theory over noncommutative tori
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〉.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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 .
Franz Luef Yang-Mills theory over noncommutative tori
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 .
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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〉
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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〉.
Franz Luef Yang-Mills theory over noncommutative tori
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).
Franz Luef Yang-Mills theory over noncommutative tori
Action functional on the set of projections
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.
Franz Luef Yang-Mills theory over noncommutative tori
Action functional on the set of projections
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))].
Franz Luef Yang-Mills theory over noncommutative tori
Action functional on the set of projections
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.
Franz Luef Yang-Mills theory over noncommutative tori
Action functional on the set of projections
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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 .
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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.
Franz Luef Yang-Mills theory over noncommutative tori
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)
Franz Luef Yang-Mills theory over noncommutative tori