July 1998

UCB-PTH-98/38

LBNL-42104


Super Yang-Mills on the Noncomutative Torus **To appear in the Arnowitt Festschrift Volume “Relativity, Particle Physics, and Cosmology”, Texas A&M University, April 1998, published by World Scientific This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-95-14797

Bogdan Morariu email address: bmorariu@lbl.govand Bruno Zumino §§email address: zumino@thsrv.lbl.gov

Department of Physics

University of California

and

Theoretical Physics Group

Lawrence Berkeley National Laboratory

University of California

Berkeley, California 94720


After a brief review of matrix theory compactification leading to noncommutative supersymmetric Yang-Mills gauge theory, we present solutions for the fundamental and adjoint sections on a two-dimensional twisted quantum torus in two different gauges. We also give explicit transformations connecting different representations which have appeared in the literature. Finally we discuss the more mathematical concept of Morita equivalence of C*superscript𝐶C^{*} -algebras as it applies to our specific case.


Disclaimer


This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial products process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, or The Regents of the University of California.


Lawrence Berkeley National Laboratory is an equal opportunity employer.

1 Introduction

It was conjectured in [1]that the infinite momentum frame description of M-theory is given by the large n𝑛n limit of supersymmetric quantum mechanics (SQM) [2, 3, 4], obtained as the dimensional reduction of the 9+1919+1 dimensional U(n)𝑈𝑛U(n) supersymmetric Yang-Mills (SYM) gauge field theory. Shortly afterwards Susskind took this a step further [5], conjecturing that the discrete light cone quantization (DLCQ) of M-theory is equivalent to the finite n𝑛n matrix theory.

Toroidal compactification of M-theory can then be obtained by first considering matrix theory on the covering space and then imposing a periodicity condition on the matrix variable [1, 6, 7], also known as the quotient condition. The result is a SYM field theory on a dual torus.

If we consider the DLCQ of M-theory and compactify on a torus Tdsuperscript𝑇𝑑T^{d} for d2𝑑2d\geq 2 there are additional moduli coming from the three-form of 11-dimensional supergravity. For example, if we compactify on T2superscript𝑇2T^{2} along X1subscript𝑋1X_{1} and X2subscript𝑋2X_{2} then C-12subscript𝐶12C_{-12} cannot be gauged away, and is a modulus of the compactification. It was conjectured in [8]that turning on this modulus corresponds to deforming the SYM theory on the dual torus to a noncommutative SYM on a quantum torus [9]with deformation parameter θ𝜃\theta given by

θ=C-12𝑑X-𝑑X1𝑑X2.𝜃subscript𝐶12differential-dsubscript𝑋differential-dsubscript𝑋1differential-dsubscript𝑋2\theta=\int~{}C_{-12}\,dX_{-}dX_{1}dX_{2}.

Evidence for this conjecture comes from comparison of the BPS mass spectra of the two theories and of their duality groups. Further evidence and discussions of this conjecture followed in [10, 11, 12, 13, 14, 16, 15, 17, 18, 19].

In Section 2we present a review of matrix theory compactification leading to noncommutative SYM gauge theory on trivial quantum bundles. We follow the elementary treatment of [17]with an emphasis on giving explicit formulae that closely resemble the commutative case. We further present an explicit realization of the algebra of the quantum torus 𝒜(Tθ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{\theta}) in terms of quantum plane coordinates.

In Section 3we introduce non-trivial quantum bundles as in [17]corresponding to compactified DLCQ of M-theory in the presence of transversely wrapped membranes. We also explain in some detail how to solve the boundary conditions for sections in the fundamental and adjoint quantum bundle. Finally using the special form of the transition functions in the given gauge we find an equivalent but simpler form of the general solution for fundamental sections.

In Section 4we discuss the more abstract language of projective modules, as presented in [8]and references therein, and we then give the explicit map between this formulation and the more elementary formulation in [17]. We also explain the notion of Morita equivalence [20, 8, 18, 19, 25]applied to our specific case. For an expanded coverage of noncommutative geometry see [24]and for a brief description see [26].

Finally in Section 5we discuss the general theory of gauge transformations on the noncommutative torus and find an explicit gauge transformation that trivializes one of the transition functions. With trivial transition functions T-duality transformations take the standard form, allowing us to interpret the gauge field as D-strings on the dual torus.

2 Review of Matrix Compactification

In this section we present a review of matrix theory compactification closely following the description given by Ho in [17]. The P-=n/Rsubscript𝑃𝑛𝑅P_{-}=n/R sector of the DLCQ of uncompactified M-theory is given by the U(n)𝑈𝑛U(n) SQM [2, 3, 4]whose action in the temporal gauge is given by

S=12R𝑑tTr(X˙μX˙μ+μ>ν[Xμ,Xν]2+iΘTΘ˙-ΘTΓμ[Xμ,Θ])𝑆12𝑅differential-d𝑡Trsuperscript˙𝑋𝜇subscript˙𝑋𝜇subscript𝜇𝜈superscriptsubscript𝑋𝜇subscript𝑋𝜈2𝑖superscriptΘ𝑇˙ΘsuperscriptΘ𝑇subscriptΓ𝜇superscript𝑋𝜇ΘS=\frac{1}{2R}\int dt\,{\rm Tr}\left(\dot{X}^{\mu}\dot{X}_{\mu}+\sum_{\mu>\nu}% [X_{\mu},X_{\nu}]^{2}+i\Theta^{T}\dot{\Theta}-\Theta^{T}\Gamma_{\mu}[X^{\mu},% \Theta]\right) (1)

where μ,ν=0,,9formulae-sequence𝜇𝜈09\mu,\nu=0,\ldots,9 . We will compactify matrix theory on a rectangular 2-torus of radii R1subscript𝑅1R_{1} and R2subscript𝑅2R_{2} . First let us consider matrix variables on the covering space and impose the quotient condition

Ui-1XjUisuperscriptsubscript𝑈𝑖1subscript𝑋𝑗subscript𝑈𝑖\displaystyle U_{i}^{-1}X_{j}U_{i} =\displaystyle= Xj+2πRjδij,subscript𝑋𝑗2𝜋subscript𝑅𝑗subscript𝛿𝑖𝑗\displaystyle X_{j}+2\pi R_{j}\delta_{ij}, (2)
Ui-1XaUisuperscriptsubscript𝑈𝑖1subscript𝑋𝑎subscript𝑈𝑖\displaystyle U_{i}^{-1}X_{a}U_{i} =\displaystyle= Xa,subscript𝑋𝑎\displaystyle X_{a},
Ui-1ΘUisuperscriptsubscript𝑈𝑖1Θsubscript𝑈𝑖\displaystyle U_{i}^{-1}\Theta~{}U_{i} =\displaystyle= Θ,i,j=1,2a=3,,9.formulae-sequenceΘ𝑖𝑗12𝑎39\displaystyle\Theta,~{}~{}~{}i,j=1,2~{}~{}~{}a=3,\ldots,9.

Here the Uisubscript𝑈𝑖U_{i} are unitary operators. The consistency of these equation requires

U1U2=e2πiθU2U1.subscript𝑈1subscript𝑈2superscript𝑒2𝜋𝑖𝜃subscript𝑈2subscript𝑈1U_{1}U_{2}=e^{2\pi i\theta}U_{2}U_{1}.

Before solving the quotient condition ( 2), it is convenient to introduce two more unitary operators U~i,i=1,2formulae-sequencesubscript~𝑈𝑖𝑖12\widetilde{U}_{i},~{}~{}i=1,2 which commute with the Uisubscript𝑈𝑖U_{i} ’s and satisfy the relation

U~1U~2=e-2πiθU~2U~1.subscript~𝑈1subscript~𝑈2superscript𝑒2𝜋𝑖𝜃subscript~𝑈2subscript~𝑈1\widetilde{U}_{1}\widetilde{U}_{2}=e^{-2\pi i\theta}\widetilde{U}_{2}% \widetilde{U}_{1}. (3)

One way to realize this algebra is by using canonical variables σisubscript𝜎𝑖\sigma_{i} satisfying

[σ1,σ2]=2πiθ.subscript𝜎1subscript𝜎22𝜋𝑖𝜃\left[\sigma_{1},\sigma_{2}\right]=2\pi i\theta. (4)

Then U~i=defeiσisuperscriptdefsubscript~𝑈𝑖superscripteisubscript𝜎i\widetilde{U}_{i}\stackrel{\rm def}{=}e^{i\sigma_{i}} satisfy ( 3). The variables σisubscript𝜎𝑖\sigma_{i} are noncommutative coordinates on the quantum plane which is the covering space of the quantum torus. The algebra of functions on the quantum torus denoted 𝒜(T-θ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{-\theta}) is generated by U~isubscript~𝑈𝑖\widetilde{U}_{i} . Similarly the Uisubscript𝑈𝑖U_{i} operators generate the algebra denoted 𝒜(Tθ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{\theta}) . To realize them we introduce partial derivative operators on the quantum plane, satisfying the following algebra

[i,σj]=δij,subscript𝑖subscript𝜎𝑗subscript𝛿𝑖𝑗\left[\partial_{i},\sigma_{j}\right]=\delta_{ij}, (5)
[i,j]=0.subscript𝑖subscript𝑗0\left[\partial_{i},\partial_{j}\right]=0.

Then, we realize Uisubscript𝑈𝑖U_{i} as

U1=eiσ1e2πθ2,U2=eiσ2e-2πθ1.formulae-sequencesubscript𝑈1superscript𝑒𝑖subscript𝜎1superscript𝑒2𝜋𝜃subscript2subscript𝑈2superscript𝑒𝑖subscript𝜎2superscript𝑒2𝜋𝜃subscript1U_{1}=e^{i\sigma_{1}}e^{2\pi\theta\,\partial_{2}},~{}~{}U_{2}=e^{i\sigma_{2}}e% ^{-2\pi\theta\,\partial_{1}}.

For θ=0𝜃0\theta=0 we have Ui=U~i=eiσisubscript𝑈𝑖subscript~𝑈𝑖superscript𝑒𝑖subscript𝜎𝑖U_{i}=\widetilde{U}_{i}=e^{i\sigma_{i}} , all generators commute allowing us to use either Uisubscript𝑈𝑖U_{i} ’s or U~isubscript~𝑈𝑖\widetilde{U}_{i} ’s to generate the algebra of functions on the classical torus. It is then easy to check that

Ui-11ijUi=1ij+δij.superscriptsubscript𝑈𝑖11𝑖subscript𝑗subscript𝑈𝑖1𝑖subscript𝑗subscript𝛿𝑖𝑗U_{i}^{-1}~{}\frac{1}{i}\partial_{j}~{}U_{i}=\frac{1}{i}\partial_{j}+\delta_{% ij}. (6)

This and many other formulae in this paper can be proven using the Campbell-Baker-Hausdorff formula which can be written in closed form since commutators like ( 4) and ( 5) are c𝑐c -numbers. Equation ( 6) is very similar to the quotient condition ( 2) so one can write a solution as a sum of the partial derivative and a fluctuating part that commutes with the Uisubscript𝑈𝑖U_{i} ’s. However this is just the definition of the covariant derivative

Xisubscript𝑋𝑖\displaystyle X_{i} =\displaystyle= -2πiRjδijj+Ai(U~i),2𝜋𝑖subscript𝑅𝑗subscript𝛿𝑖𝑗subscript𝑗subscript𝐴𝑖subscript~𝑈𝑖\displaystyle-2\pi iR_{j}\delta_{ij}\partial_{j}+A_{i}(\widetilde{U}_{i}), (7)
Xasubscript𝑋𝑎\displaystyle X_{a} =\displaystyle= Xa(U~i),subscript𝑋𝑎subscript~𝑈𝑖\displaystyle X_{a}(\widetilde{U}_{i}),
ΘΘ\displaystyle\Theta =\displaystyle= Θ(U~i),Θsubscript~𝑈𝑖\displaystyle\Theta(\widetilde{U}_{i}),

where Aisubscript𝐴𝑖A_{i} , Xasubscript𝑋𝑎X_{a} and each spinorial component of ΘΘ\Theta are n×n𝑛𝑛n\times n hermitian matrices with operator valued entries. Note that since the partial derivatives already satisfy the cocycle condition, the gauge fields Aisubscript𝐴𝑖A_{i} and the scalar fields Xasubscript𝑋𝑎X_{a} must satisfy a homogeneous quotient condition like the second relation in ( 2). Hence Aisubscript𝐴𝑖A_{i} and Xasubscript𝑋𝑎X_{a} must depend only on U~isubscript~𝑈𝑖\widetilde{U}_{i} . Hidden in this dependence is the fact that we are working on a trivial bundle over the quantum torus.

If one inserts ( 7) into the SQM action ( 1) the result is a noncommutative SYM gauge field theory in 2+1212+1 dimensions, with the space part given by the above quantum torus and a commutative time. For the commutative case, matrix compactification on Tdsuperscript𝑇𝑑T^{d} results in a SYM gauge theory in d+1𝑑1d+1 dimensions on the dual torus. In the limit when the size of the original torus vanishes the dual torus becomes 𝐑dsuperscript𝐑𝑑{\bf R}^{d} , therefore we obtain the opposite of dimensional reduction. If one starts from a Euclidean 10-dimensional SYM and dimensionally reduces in all directions including the Euclidean time one obtains the IKKT [27]functional. Matrix compactification of one direction in the IKKT functional results in the finite temperature action of the original theory ( 1).

3 Twisted Quantum Bundles

We can consider more general solutions of the quotient condition ( 2) which are connections on twisted bundles. They correspond to compactification of the DLCQ of M-theory in the presence of transversely wrapped membranes. Again the solution is a sum of two terms, a constant curvature connection Disubscript𝐷𝑖D_{i} and a fluctuating part

Xisubscript𝑋𝑖\displaystyle X_{i} =\displaystyle= -2πiRjδijDj+Ai(Zj),2𝜋𝑖subscript𝑅𝑗subscript𝛿𝑖𝑗subscript𝐷𝑗subscript𝐴𝑖subscript𝑍𝑗\displaystyle-2\pi iR_{j}\delta_{ij}D_{j}+A_{i}(Z_{j}), (8)
Xasubscript𝑋𝑎\displaystyle X_{a} =\displaystyle= Xa(Zi),subscript𝑋𝑎subscript𝑍𝑖\displaystyle X_{a}(Z_{i}),
ΘΘ\displaystyle\Theta =\displaystyle=  Θ(Zi). Θsubscript𝑍𝑖\displaystyle~{}\Theta(Z_{i}).

Here the Zisubscript𝑍𝑖Z_{i} ’s are n×n𝑛𝑛n\times n matrices with operator entries and, just like the U~isubscript~𝑈𝑖\widetilde{U}_{i} ’s for the trivial bundle, commute with the Uisubscript𝑈𝑖U_{i} ’s, but now are sections of the twisted bundle whose exact form will be discussed shortly. However, while for the trivial bundle Aisubscript𝐴𝑖A_{i} , Xasubscript𝑋𝑎X_{a} and the spinorial components of ΘΘ\Theta are n×n𝑛𝑛n\times n matrix functions, in ( 8) Aisubscript𝐴𝑖A_{i} , Xasubscript𝑋𝑎X_{a} and the components of ΘΘ\Theta are one-dimensional functions but with matrix arguments. Later, this will allow us to establish a relationship between a SYM on a twisted U(n)𝑈𝑛U(n) bundle and one on a U(1)𝑈1U(1) bundle.

Following [17], up to a gauge transformation the constant curvature connection can be written as

D1=1,D2=2-ifσ1,formulae-sequencesubscript𝐷1subscript1subscript𝐷2subscript2𝑖𝑓subscript𝜎1D_{1}=\partial_{1},~{}D_{2}=\partial_{2}-if\sigma_{1}, (9)

where f𝑓f is the constant field strength

[D1,D2]=-if.subscript𝐷1subscript𝐷2𝑖𝑓[D_{1},D_{2}]=-if.

Such a gauge field can only exist in a non-trivial bundle. One can introduce transition functions ΩisubscriptΩ𝑖\Omega_{i} such that the sections of the fundamental bundle satisfy the twisted boundary conditions

Φ(σ1+2π,σ2)Φsubscript𝜎12𝜋subscript𝜎2\displaystyle\Phi(\sigma_{1}+2\pi,\sigma_{2}) =\displaystyle= Ω1(σ1,σ2)Φ(σ1,σ2),subscriptΩ1subscript𝜎1subscript𝜎2Φsubscript𝜎1subscript𝜎2\displaystyle\Omega_{1}(\sigma_{1},\sigma_{2})~{}\Phi(\sigma_{1},\sigma_{2}), (10)
Φ(σ1,σ2+2π)Φsubscript𝜎1subscript𝜎22𝜋\displaystyle\Phi(\sigma_{1},\sigma_{2}+2\pi) =\displaystyle= Ω2(σ1,σ2)Φ(σ1,σ2).subscriptΩ2subscript𝜎1subscript𝜎2Φsubscript𝜎1subscript𝜎2\displaystyle\Omega_{2}(\sigma_{1},\sigma_{2})~{}\Phi(\sigma_{1},\sigma_{2}).

Similarly the adjoint sections satisfy

Ψ(σ1+2π,σ2)=Ω1(σ1,σ2)Ψ(σ1,σ2)Ω1(σ1,σ2)-1,Ψsubscript𝜎12𝜋subscript𝜎2subscriptΩ1subscript𝜎1subscript𝜎2Ψsubscript𝜎1subscript𝜎2subscriptΩ1superscriptsubscript𝜎1subscript𝜎21\displaystyle\Psi(\sigma_{1}+2\pi,\sigma_{2})=\Omega_{1}(\sigma_{1},\sigma_{2}% )~{}\Psi(\sigma_{1},\sigma_{2})~{}\Omega_{1}(\sigma_{1},\sigma_{2})^{-1}, (11)
Ψ(σ1,σ2+2π)=Ω2(σ1,σ2)Ψ(σ1,σ2)Ω2(σ1,σ2)-1.Ψsubscript𝜎1subscript𝜎22𝜋subscriptΩ2subscript𝜎1subscript𝜎2Ψsubscript𝜎1subscript𝜎2subscriptΩ2superscriptsubscript𝜎1subscript𝜎21\displaystyle\Psi(\sigma_{1},\sigma_{2}+2\pi)=\Omega_{2}(\sigma_{1},\sigma_{2}% )~{}\Psi(\sigma_{1},\sigma_{2})~{}\Omega_{2}(\sigma_{1},\sigma_{2})^{-1}.

Consistency of the transition functions of the bundle requires that

Ω1(σ1,σ2+2π)Ω2(σ1,σ2)=Ω2(σ1+2π,σ2)Ω1(σ1,σ2).subscriptΩ1subscript𝜎1subscript𝜎22𝜋subscriptΩ2subscript𝜎1subscript𝜎2subscriptΩ2subscript𝜎12𝜋subscript𝜎2subscriptΩ1subscript𝜎1subscript𝜎2\Omega_{1}(\sigma_{1},\sigma_{2}+2\pi)~{}\Omega_{2}(\sigma_{1},\sigma_{2})=% \Omega_{2}(\sigma_{1}+2\pi,\sigma_{2})~{}\Omega_{1}(\sigma_{1},\sigma_{2}). (12)

This relation is known in the mathematical literature as the cocycle condition. The covariant derivatives transform just as the adjoint sections

Di(σ1+2π,σ2)=Ω1(σ1,σ2)Di(σ1,σ2)Ω1(σ1,σ2)-1,subscript𝐷𝑖subscript𝜎12𝜋subscript𝜎2subscriptΩ1subscript𝜎1subscript𝜎2subscript𝐷𝑖subscript𝜎1subscript𝜎2subscriptΩ1superscriptsubscript𝜎1subscript𝜎21D_{i}(\sigma_{1}+2\pi,\sigma_{2})=\Omega_{1}(\sigma_{1},\sigma_{2})~{}D_{i}(% \sigma_{1},\sigma_{2})~{}\Omega_{1}(\sigma_{1},\sigma_{2})^{-1},
Di(σ1,σ2+2π)=Ω2(σ1,σ2)Di(σ1,σ2)Ω2(σ1,σ2)-1.subscript𝐷𝑖subscript𝜎1subscript𝜎22𝜋subscriptΩ2subscript𝜎1subscript𝜎2subscript𝐷𝑖subscript𝜎1subscript𝜎2subscriptΩ2superscriptsubscript𝜎1subscript𝜎21D_{i}(\sigma_{1},\sigma_{2}+2\pi)=\Omega_{2}(\sigma_{1},\sigma_{2})~{}D_{i}(% \sigma_{1},\sigma_{2})~{}\Omega_{2}(\sigma_{1},\sigma_{2})^{-1}.

A particular solution for the transition functions compatible with the constant curvature connection ( 9) and satisfying the cocycle condition is given by

Ω1=eimσ2/nU,Ω2=V,formulae-sequencesubscriptΩ1superscript𝑒𝑖𝑚subscript𝜎2𝑛𝑈subscriptΩ2𝑉\Omega_{1}=e^{im\sigma_{2}/n}U,~{}\Omega_{2}=V, (13)

where U,V𝑈𝑉U,~{}V are n×n𝑛𝑛n\times n unitary matrices satisfying

UV=e-2πim/nVU𝑈𝑉superscript𝑒2𝜋𝑖𝑚𝑛𝑉𝑈UV=e^{-2\pi im/n}VU

and m𝑚m is an integer. For simplicity, here we will only consider the case when n𝑛n and m𝑚m are relatively prime. For the general case see [8, 14, 31]. Using the representation given in [17]one has

Ukl=e2πikm/nδk,l,Vkl=δk+1,l,formulae-sequencesubscript𝑈𝑘𝑙superscript𝑒2𝜋𝑖𝑘𝑚𝑛subscript𝛿𝑘𝑙subscript𝑉𝑘𝑙subscript𝛿𝑘1𝑙U_{kl}=e^{2\pi ikm/n}\delta_{k,l},~{}~{}V_{kl}=\delta_{k+1,l},

where the subscripts are identified with period n𝑛n .

We can express the above matrices in terms of the standard ’t Hooft matrices [28, 29]denoted here by Usuperscript𝑈U^{\prime} and Vsuperscript𝑉V^{\prime} and satisfying

UV=e-2πi/nVU,Un=Vn=1.formulae-sequencesuperscript𝑈superscript𝑉superscript𝑒2𝜋𝑖𝑛superscript𝑉superscript𝑈superscript𝑈𝑛superscript𝑉𝑛1U^{\prime}V^{\prime}=e^{-2\pi i/n}V^{\prime}U^{\prime},~{}~{}U^{\prime n}=V^{% \prime n}=1.

The relation is given by

U=e2πim/nUm,V=V.formulae-sequence𝑈superscript𝑒2𝜋𝑖𝑚𝑛superscript𝑈𝑚𝑉superscript𝑉U=e^{2\pi im/n}U^{\prime m},~{}~{}V=V^{\prime}. (14)

The phase in ( 14) is due to the nonstandard definition of U𝑈U used in [17]. This has certain advantages but similar phases will appear when comparing the results of [17]with similar results where the standard ’t Hooft matrices were used. We also introduce a unitary matrix K𝐾K which changes the representation so that Vsuperscript𝑉V^{\prime} is diagonal, and satisfies

KUK-1=V-1,KVK-1=U.formulae-sequence𝐾superscript𝑈superscript𝐾1superscript𝑉1𝐾superscript𝑉superscript𝐾1superscript𝑈KU^{\prime}K^{-1}=V^{\prime-1},~{}~{}KV^{\prime}K^{-1}=U^{\prime}. (15)

Note that n𝑛n is quantized since we are considering a U(n)𝑈𝑛U(n) gauge theory and m𝑚m is quantized since the magnetic flux f𝑓f through T2subscript𝑇2T_{2} is quantized

2πf=mn-mθ.2𝜋𝑓𝑚𝑛𝑚𝜃2\pi f=\frac{m}{n-m\theta}.

In M-theory m𝑚m is the transversal membrane wrapping number.

One can solve the boundary conditions ( 10) for the fundamental sections as in [17]generalizing a previous result for m=1𝑚1m=1 in the commutative case presented in [7]. Using the ordered exponential explained below, the general solution has the form

Φk(σ1,σ2)=s𝐙j=1mE(mn(σ22π+k+ns)+j,iσ1)ϕ^j(σ22π+k+ns+njm).subscriptΦ𝑘subscript𝜎1subscript𝜎2subscript𝑠𝐙superscriptsubscript𝑗1𝑚𝐸𝑚𝑛subscript𝜎22𝜋𝑘𝑛𝑠𝑗𝑖subscript𝜎1subscript^italic-ϕ𝑗subscript𝜎22𝜋𝑘𝑛𝑠𝑛𝑗𝑚\Phi_{k}(\sigma_{1},\sigma_{2})=\sum_{s\in{\bf Z}}\sum_{j=1}^{m}E\left(\frac{m% }{n}\left(\frac{\sigma_{2}}{2\pi}+k+ns\right)+j,i\sigma_{1}\right)\widehat{% \phi}_{j}\left(\frac{\sigma_{2}}{2\pi}+k+ns+\frac{nj}{m}\right).

The ordered exponential [17]is defined for two variables whose commutator is a c𝑐c -number

E(A,B)=11-[A,B]l=01l!AlBl.𝐸𝐴𝐵11𝐴𝐵superscriptsubscript𝑙01𝑙superscript𝐴𝑙superscript𝐵𝑙E(A,B)=\frac{1}{1-[A,B]}~{}\sum_{l=0}^{\infty}\frac{1}{l!}A^{l}B^{l}.

The normalization is such that

E(-B,A)E(A,B)=1𝐸𝐵𝐴𝐸𝐴𝐵1E(-B,A)E(A,B)=1

and it has the following desirable properties similar to the usual exponential

E(A+c,B)=E(A,B)ecB,𝐸𝐴𝑐𝐵𝐸𝐴𝐵superscript𝑒𝑐𝐵\displaystyle E(A+c,B)=E(A,B)e^{cB}, (16)
E(A,B+c)=ecAE(A,B).𝐸𝐴𝐵𝑐superscript𝑒𝑐𝐴𝐸𝐴𝐵\displaystyle E(A,B+c)=e^{cA}E(A,B).

The ϕ^jsubscript^italic-ϕ𝑗\widehat{\phi}_{j} functions are defined on the whole real axis and are unrestricted except for the behavior at infinity. They should be considered as vectors in a Hilbert space on which all the elements of the algebra are represented.

Next we explain in some detail how to obtain this result. First we define

ϕ(σ1,σ2)=defΦk(σ1,σ2-2π(k-1)).superscriptdefitalic-ϕsubscript𝜎1subscript𝜎2subscriptΦksubscript𝜎1subscript𝜎22𝜋k1\phi(\sigma_{1},\sigma_{2})\stackrel{\rm def}{=}\Phi_{k}(\sigma_{1},\sigma_{2}% -2\pi(k-1)).

The second boundary condition ( 10) implies that the definition of ϕitalic-ϕ\phi is consistent, i.e. k𝑘k -independent. Using Vn=1superscript𝑉𝑛1V^{n}=1 we also find that ϕitalic-ϕ\phi is a periodic function in σ2subscript𝜎2\sigma_{2}

ϕ(σ1,σ2+2πn)=ϕ(σ1,σ2).italic-ϕsubscript𝜎1subscript𝜎22𝜋𝑛italic-ϕsubscript𝜎1subscript𝜎2\phi(\sigma_{1},\sigma_{2}+2\pi n)=\phi(\sigma_{1},\sigma_{2}).

The other boundary condition gives

ϕ(σ1+2π,σ2)=eim(σ2+2π)/nϕ(σ1,σ2).italic-ϕsubscript𝜎12𝜋subscript𝜎2superscript𝑒𝑖𝑚subscript𝜎22𝜋𝑛italic-ϕsubscript𝜎1subscript𝜎2\phi(\sigma_{1}+2\pi,\sigma_{2})=e^{im(\sigma_{2}+2\pi)/n}\phi(\sigma_{1},% \sigma_{2}).

It is convenient to separate out a factor to eliminate the above twist

ϕ(σ1,σ2)=f(σ1,σ2)ϕˇ(σ1,σ2)italic-ϕsubscript𝜎1subscript𝜎2𝑓subscript𝜎1subscript𝜎2ˇitalic-ϕsubscript𝜎1subscript𝜎2\phi(\sigma_{1},\sigma_{2})=f(\sigma_{1},\sigma_{2})\check{\phi}(\sigma_{1},% \sigma_{2})

and to require a simpler periodicity condition for ϕˇˇitalic-ϕ\check{\phi}

ϕˇ(σ1+2π,σ2)=ϕˇ(σ1,σ2).ˇitalic-ϕsubscript𝜎12𝜋subscript𝜎2ˇitalic-ϕsubscript𝜎1subscript𝜎2\check{\phi}(\sigma_{1}+2\pi,\sigma_{2})=\check{\phi}(\sigma_{1},\sigma_{2}).

Then the function f𝑓f must satisfy

f(σ1+2π,σ2)=eim(σ2+2π)/nf(σ1,σ2).𝑓subscript𝜎12𝜋subscript𝜎2superscript𝑒𝑖𝑚subscript𝜎22𝜋𝑛𝑓subscript𝜎1subscript𝜎2f(\sigma_{1}+2\pi,\sigma_{2})=e^{im(\sigma_{2}+2\pi)/n}f(\sigma_{1},\sigma_{2}).

This is satisfied exactly for

f(σ1,σ2)=E(mn(σ22π+1),iσ1),𝑓subscript𝜎1subscript𝜎2𝐸𝑚𝑛subscript𝜎22𝜋1𝑖subscript𝜎1f(\sigma_{1},\sigma_{2})=E\left(\frac{m}{n}\left(\frac{\sigma_{2}}{2\pi}+1% \right),i\sigma_{1}\right),

where in the right hand side we used the ordered exponential defined above. Now we can Fourier transform ϕˇˇitalic-ϕ\check{\phi} in σ1subscript𝜎1\sigma_{1}

ϕˇ(σ1,σ2)=p𝐙eipσ1ϕp(σ2)ˇitalic-ϕsubscript𝜎1subscript𝜎2subscript𝑝𝐙superscript𝑒𝑖𝑝subscript𝜎1subscriptitalic-ϕ𝑝subscript𝜎2\check{\phi}(\sigma_{1},\sigma_{2})=\sum_{p\in{\bf Z}}~{}e^{ip\sigma_{1}}~{}% \phi_{p}(\sigma_{2})

and using the property ( 16) of the ordered exponential we obtain

ϕ(σ1,σ2)=p𝐙E(mn(σ22π+1)+p,iσ1)ϕp(σ2).italic-ϕsubscript𝜎1subscript𝜎2subscript𝑝𝐙𝐸𝑚𝑛subscript𝜎22𝜋1𝑝𝑖subscript𝜎1subscriptitalic-ϕ𝑝subscript𝜎2\phi(\sigma_{1},\sigma_{2})=\sum_{p\in{\bf Z}}E\left(\frac{m}{n}\left(\frac{% \sigma_{2}}{2\pi}+1\right)+p,i\sigma_{1}\right)\phi_{p}(\sigma_{2}).

Let p=ms+j𝑝𝑚𝑠𝑗p=ms+j with j=1,,m𝑗1𝑚j=1,\ldots,m and s𝑠s is an integer. Then the solution can be written as

ϕ(σ1,σ2)=s𝐙j=1mE(mn(σ22π+1)+ms+j,iσ1)ϕs,j(σ2),italic-ϕsubscript𝜎1subscript𝜎2subscript𝑠𝐙superscriptsubscript𝑗1𝑚𝐸𝑚𝑛subscript𝜎22𝜋1𝑚𝑠𝑗𝑖subscript𝜎1subscriptitalic-ϕ𝑠𝑗subscript𝜎2\phi(\sigma_{1},\sigma_{2})=\sum_{s\in{\bf Z}}\sum_{j=1}^{m}E\left(\frac{m}{n}% \left(\frac{\sigma_{2}}{2\pi}+1\right)+ms+j,i\sigma_{1}\right)\phi_{s,j}(% \sigma_{2}),

where ϕs,j=defϕms+jsuperscriptdefsubscriptitalic-ϕ𝑠𝑗subscriptitalic-ϕmsj\phi_{s,j}\stackrel{\rm def}{=}\phi_{ms+j} . Periodicity in σ2subscript𝜎2\sigma_{2} then implies ϕs-1,j(σ2+2πn)=ϕs,j(σ2)subscriptitalic-ϕ𝑠1𝑗subscript𝜎22𝜋𝑛subscriptitalic-ϕ𝑠𝑗subscript𝜎2\phi_{s-1,j}(\sigma_{2}+2\pi n)=\phi_{s,j}(\sigma_{2}) so that using this recursively we have ϕs,j(σ2)=ϕ0,j(σ2+2πns)subscriptitalic-ϕ𝑠𝑗subscript𝜎2subscriptitalic-ϕ0𝑗subscript𝜎22𝜋𝑛𝑠\phi_{s,j}(\sigma_{2})=\phi_{0,j}(\sigma_{2}+2\pi ns) . Finally after defining ϕ~j(x)=defϕ0,j(2π(x-1))superscriptdefsubscript~italic-ϕ𝑗𝑥subscriptitalic-ϕ0j2𝜋x1\widetilde{\phi}_{j}(x)\stackrel{\rm def}{=}\phi_{0,j}(2\pi(x-1)) we obtain

Φk(σ1,σ2)=s𝐙j=1mE(mn(σ22π+k+ns)+j,iσ1)ϕ~j(σ22π+k+ns).subscriptΦ𝑘subscript𝜎1subscript𝜎2subscript𝑠𝐙superscriptsubscript𝑗1𝑚𝐸𝑚𝑛subscript𝜎22𝜋𝑘𝑛𝑠𝑗𝑖subscript𝜎1subscript~italic-ϕ𝑗subscript𝜎22𝜋𝑘𝑛𝑠\Phi_{k}(\sigma_{1},\sigma_{2})=\sum_{s\in{\bf Z}}\sum_{j=1}^{m}E\left(\frac{m% }{n}\left(\frac{\sigma_{2}}{2\pi}+k+ns\right)+j,i\sigma_{1}\right)\widetilde{% \phi}_{j}\left(\frac{\sigma_{2}}{2\pi}+k+ns\right).

This is the result mentioned above up to another redefinition

ϕ~j(x)=ϕ^j(x+nmj).subscript~italic-ϕ𝑗𝑥subscript^italic-ϕ𝑗𝑥𝑛𝑚𝑗\widetilde{\phi}_{j}(x)=\widehat{\phi}_{j}(x+\frac{n}{m}j).

While the solutions for the sections of the fundamental bundle given in [17]are suitable for showing the equivalence to the projective modules of [8]as we will discuss in Section 4, the appearance of the ordered exponential is somewhat inconvenient. Using the special form of the transition functions we were able to rewrite the solution in an equivalent but simpler form. The transition functions in this gauge do not contain σ1subscript𝜎1\sigma_{1} and it is convenient to order all σ1subscript𝜎1\sigma_{1} to the right in the solution. Using Vn=1superscript𝑉𝑛1V^{n}=1 in the second condition ( 10) one can express all n𝑛n components of ΦΦ\Phi in terms of a single function with period 2πn2𝜋𝑛2\pi n in σ2subscript𝜎2\sigma_{2} . After Fourier transforming in σ2subscript𝜎2\sigma_{2} and imposing both boundary conditions ( 10) we obtain the general solution

Φk(σ1,σ2)=p𝐙e2πi(σ2/2π+k)p/ne2πi(σ1/2π-p/m)m/nψ^p(σ1/2π-p/m),subscriptΦ𝑘subscript𝜎1subscript𝜎2subscript𝑝𝐙superscript𝑒2𝜋𝑖subscript𝜎22𝜋𝑘𝑝𝑛superscript𝑒2𝜋𝑖subscript𝜎12𝜋𝑝𝑚𝑚𝑛subscript^𝜓𝑝subscript𝜎12𝜋𝑝𝑚\Phi_{k}(\sigma_{1},\sigma_{2})=\sum_{p\in{\bf Z}}~{}e^{2\pi i(\sigma_{2}/2\pi% +k)p/n}~{}e^{2\pi i(\sigma_{1}/2\pi-p/m)m/n}~{}\widehat{\psi}_{p}(\sigma_{1}/2% \pi-p/m),

where only m𝑚m of the ψ^psubscript^𝜓𝑝\widehat{\psi}_{p} functions are independent, since

ψ^p+m(x)=ψ^p(x).subscript^𝜓𝑝𝑚𝑥subscript^𝜓𝑝𝑥\widehat{\psi}_{p+m}(x)=\widehat{\psi}_{p}(x).

Using the same technique one can show that an arbitrary adjoint section has the following expansion

Ψ(σ1,σ2)=s,t𝐙cstZ1sZ2-t.Ψsubscript𝜎1subscript𝜎2subscript𝑠𝑡𝐙subscript𝑐𝑠𝑡superscriptsubscript𝑍1𝑠superscriptsubscript𝑍2𝑡\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}~{}c_{st}Z_{1}^{s}Z_{2}^{-t}. (17)

Here cs,tsubscript𝑐𝑠𝑡c_{s,t} are c𝑐c -numbers and

Z1=eiσ1/(n-mθ)Vb,Z2=eiσ2/nU-b,formulae-sequencesubscript𝑍1superscript𝑒𝑖subscript𝜎1𝑛𝑚𝜃superscript𝑉𝑏subscript𝑍2superscript𝑒𝑖subscript𝜎2𝑛superscript𝑈𝑏Z_{1}=e^{i\sigma_{1}/(n-m\theta)}V^{b},~{}~{}Z_{2}=e^{i\sigma_{2}/n}U^{-b},

where b𝑏b is an integer, such that we can find another integer a𝑎a satisfying an-bm=1𝑎𝑛𝑏𝑚1an-bm=1 . For n𝑛n and m𝑚m relatively prime one can always find integer solutions to this equation. Again, we emphasize that the Zisubscript𝑍𝑖Z_{i} ’s commute with the Uisubscript𝑈𝑖U_{i} ’s. They are generators of the algebra of functions on a new quantum torus

Z1Z2=e2πiθZ2Z1,subscript𝑍1subscript𝑍2superscript𝑒2𝜋𝑖superscript𝜃subscript𝑍2subscript𝑍1Z_{1}Z_{2}=e^{2\pi i\theta^{\prime}}Z_{2}Z_{1},

where θsuperscript𝜃\theta^{\prime} is obtained by an SL(2,𝐙)𝑆𝐿2𝐙SL(2,{\bf Z}) fractional transformation from -θ𝜃-\theta

θ=a(-θ)+bm(-θ)+n.superscript𝜃𝑎𝜃𝑏𝑚𝜃𝑛\theta^{\prime}=\frac{a(-\theta)+b}{m(-\theta)+n}.

Now we outline how to obtain this result. Note first that

Ψ(σ1+2πn,σ2)=Ω1nΨ(σ1,σ2)Ω1-n=Ψ(σ1+2πθm,σ2).Ψsubscript𝜎12𝜋𝑛subscript𝜎2superscriptsubscriptΩ1𝑛Ψsubscript𝜎1subscript𝜎2superscriptsubscriptΩ1𝑛Ψsubscript𝜎12𝜋𝜃𝑚subscript𝜎2\Psi(\sigma_{1}+2\pi n,\sigma_{2})=\Omega_{1}^{n}~{}\Psi(\sigma_{1},\sigma_{2}% )~{}\Omega_{1}^{-n}=\Psi(\sigma_{1}+2\pi\theta m,\sigma_{2}).

In the last equality we used the fact that Un=1superscript𝑈𝑛1U^{n}=1 , and we also used the exponential formula to shift σ1subscript𝜎1\sigma_{1} . Using both boundary conditions we have

Ψ(σ1+2π(n-mθ),σ2)=Ψ(σ1,σ2),Ψsubscript𝜎12𝜋𝑛𝑚𝜃subscript𝜎2Ψsubscript𝜎1subscript𝜎2\Psi(\sigma_{1}+2\pi(n-m\theta),\sigma_{2})=\Psi(\sigma_{1},\sigma_{2}),
Ψ(σ1,σ2+2πn)=Ψ(σ1,σ2).Ψsubscript𝜎1subscript𝜎22𝜋𝑛Ψsubscript𝜎1subscript𝜎2\Psi(\sigma_{1},\sigma_{2}+2\pi n)=\Psi(\sigma_{1},\sigma_{2}).

We can expand the section as

Ψ(σ1,σ2)=s,t𝐙eisσ1/(n-mθ)e-itσ2/nΨs,t,Ψsubscript𝜎1subscript𝜎2subscript𝑠𝑡𝐙superscript𝑒𝑖𝑠subscript𝜎1𝑛𝑚𝜃superscript𝑒𝑖𝑡subscript𝜎2𝑛subscriptΨ𝑠𝑡\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}e^{is\sigma_{1}/(n-m\theta)}e^% {-it\sigma_{2}/n}\Psi_{s,t},

where Ψs,tsubscriptΨ𝑠𝑡\Psi_{s,t} is a n×n𝑛𝑛n\times n matrix and can be expanded as

Ψs,t=i=i0n+i0j=j0n+j0cs,t,i,jViUj.subscriptΨ𝑠𝑡superscriptsubscript𝑖subscript𝑖0𝑛subscript𝑖0superscriptsubscript𝑗subscript𝑗0𝑛subscript𝑗0subscript𝑐𝑠𝑡𝑖𝑗superscript𝑉𝑖superscript𝑈𝑗\Psi_{s,t}=\sum_{i=i_{0}}^{n+i_{0}}\sum_{j=j_{0}}^{n+j_{0}}c_{s,t,i,j}V^{% \prime i}U^{\prime j}. (18)

Here i0,j0subscript𝑖0subscript𝑗0i_{0},j_{0} are two arbitrary integers, allowing us to freely shift the summation limits assuming that cs,t,i+n,j=cs,t,i,j+n=cs,t,i,jsubscript𝑐𝑠𝑡𝑖𝑛𝑗subscript𝑐𝑠𝑡𝑖𝑗𝑛subscript𝑐𝑠𝑡𝑖𝑗c_{s,t,i+n,j}=c_{s,t,i,j+n}=c_{s,t,i,j} . Then one can obtain further restrictions on the cs,t,i,jsubscript𝑐𝑠𝑡𝑖𝑗c_{s,t,i,j} coefficients using the boundary conditions ( 11). For example using the first equation ( 11) and comparing like coefficients in the Fourier expansion we have

cs,t,i,je2πis/(n-mθ)=cs,t,i,je-2πimi/ne2πismθ/[n(n-,θ)].subscript𝑐𝑠𝑡𝑖𝑗superscript𝑒2𝜋𝑖𝑠𝑛𝑚𝜃subscript𝑐𝑠𝑡𝑖𝑗superscript𝑒2𝜋𝑖𝑚𝑖𝑛superscript𝑒2𝜋𝑖𝑠𝑚𝜃delimited-[]𝑛limit-from𝑛𝜃c_{s,t,i,j}e^{2\pi is/(n-m\theta)}=c_{s,t,i,j}e^{-2\pi imi/n}e^{2\pi ism\theta% /[n(n-,\theta)]}.

From this and the similar relation obtained by imposing the second equation ( 11) we have that cs,t,i,jsubscript𝑐𝑠𝑡𝑖𝑗c_{s,t,i,j} vanish unless (s+mi)/n=k𝑠𝑚𝑖𝑛𝑘(s+mi)/n=k and (t+j)/n=s𝑡𝑗𝑛𝑠(t+j)/n=s for k𝑘k and s𝑠s two integers. These equations have multiple solutions. However, if (i,j)𝑖𝑗(i,j) and (i,j)superscript𝑖superscript𝑗(i^{\prime},j^{\prime}) are two solutions then i-in𝐙𝑖superscript𝑖𝑛𝐙i-i^{\prime}\in n{\bf Z} and j-jn𝐙𝑗superscript𝑗𝑛𝐙j-j^{\prime}\in n{\bf Z} . This ensures that only one term survives in the sum ( 18) over i𝑖i and j𝑗j . Choosing for later convenience i0=sbsubscript𝑖0𝑠𝑏i_{0}=sb and j0=mbtsubscript𝑗0𝑚𝑏𝑡j_{0}=mbt we have

Ψ(σ1,σ2)=s,t𝐙eisσ1/(n-mθ)e-itσ2/ni=sbn+sbj=mbtn+mbtcs,t,i,jViUj.Ψsubscript𝜎1subscript𝜎2subscript𝑠𝑡𝐙superscript𝑒𝑖𝑠subscript𝜎1𝑛𝑚𝜃superscript𝑒𝑖𝑡subscript𝜎2𝑛superscriptsubscript𝑖𝑠𝑏𝑛𝑠𝑏superscriptsubscript𝑗𝑚𝑏𝑡𝑛𝑚𝑏𝑡subscript𝑐𝑠𝑡𝑖𝑗superscript𝑉𝑖superscript𝑈𝑗\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}~{}e^{is\sigma_{1}/(n-m\theta)% }e^{-it\sigma_{2}/n}\sum_{i=sb}^{n+sb}~{}\sum_{j=mbt}^{n+mbt}c_{s,t,i,j}~{}V^{% \prime i}U^{\prime j}.

Since n𝑛n and m𝑚m are relatively prime let a,b𝐙𝑎𝑏𝐙a,b\in{\bf Z} such that an-bm=1𝑎𝑛𝑏𝑚1an-bm=1 . Then

k=as,l=at,i=bs,j=mbtformulae-sequence𝑘𝑎𝑠formulae-sequence𝑙𝑎𝑡formulae-sequence𝑖𝑏𝑠𝑗𝑚𝑏𝑡k=as,~{}l=at,~{}i=bs,~{}j=mbt

is an integer solution inside the i𝑖i and j𝑗j summation range. Dropping the i,j𝑖𝑗i,j indices since they are determined by s𝑠s and t𝑡t we have

Ψ(σ1,σ2)=s,t𝐙cs,t(eiσ1/(n-mθ)Vb)s(eiσ2/nU-mb)-t,Ψsubscript𝜎1subscript𝜎2subscript𝑠𝑡𝐙subscript𝑐𝑠𝑡superscriptsuperscript𝑒𝑖subscript𝜎1𝑛𝑚𝜃superscript𝑉𝑏𝑠superscriptsuperscript𝑒𝑖subscript𝜎2𝑛superscript𝑈𝑚𝑏𝑡\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}c_{s,t}\left(e^{i\sigma_{1}/(n% -m\theta)}V^{\prime b}\right)^{s}\left(e^{i\sigma_{2}/n}U^{\prime-mb}\right)^{% -t},

which is just ( 17) after an additional phase redefinition of cs,tsubscript𝑐𝑠𝑡c_{s,t} to accommodate the phase difference between U𝑈U and Umsuperscript𝑈𝑚U^{\prime m} .

4 Projective Modules and Morita Equivalence

A classic mathematical result of Gel’fand states that compact topological spaces are in one to one correspondence with commutative C*superscript𝐶C^{*} -algebras. In one direction, to a topological space X𝑋X we associate the algebra of continuous functions C(X)𝐶𝑋C(X) . Conversely and rather nontrivially, the spectrum of a commutative C*superscript𝐶C^{*} -algebra is equivalent to a compact topological space. This important result allows for a dual description of topological spaces and brings powerful algebraic methods into the realm of topology. On the other hand, if we drop the commutativity requirement, a C*superscript𝐶C^{*} -algebra 𝒜𝒜{\cal A} describe what is called by correspondence a quantum space. To illustrate, consider the algebra of the quantum torus 𝒜(Tθ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{\theta}) generated by the Uisubscript𝑈𝑖U_{i} ’s. An arbitrary element a𝑎a has the form

a=k,l𝐙ak,lU1kU2l,𝑎subscript𝑘𝑙𝐙subscript𝑎𝑘𝑙superscriptsubscript𝑈1𝑘superscriptsubscript𝑈2𝑙a=\sum_{k,l\in{\bf Z}}a_{k,l}U_{1}^{k}U_{2}^{l}, (19)

where some restrictions (which we do not discuss here) are imposed on the c𝑐c -number coefficients ak,lsubscript𝑎𝑘𝑙a_{k,l} . For θ=0𝜃0\theta=0 , formula ( 19) reduces to the Fourier expansion of functions on T2superscript𝑇2T^{2} . Thus we can read the compact space from the commutative algebra.

Using the same strategy one can describe other spaces of classical geometry in commutative algebraic terms and then remove the commutativity requirement. A quantum vector bundle is a projective 𝒜𝒜{\cal A} -module {\cal E} . First consider the classical commutative picture. The set {\cal E} of global sections of a vector bundle over a base space X𝑋X has the structure of a projective module over the algebra C(X)𝐶𝑋C(X) . Having a module essentially means that we can add sections and can multiply them by functions. Not all modules over a commutative algebra are vector bundles. For example the set of sections on a space consisting of a collection of fibers of different dimensions over a base space also form a module. However, projective modules over the algebra of functions on a topological space are in one to one correspondence with vector bundles over that space. By definition a projective module is a direct summand in a free module. A free module 0subscript0{\cal E}_{0} over an algebra 𝒜𝒜{\cal A} is a module isomorphic to a direct sum of a finite number of copies of the algebra

0=𝒜𝒜.subscript0direct-sum𝒜𝒜{\cal E}_{0}={\cal A}\oplus\ldots\oplus{\cal A}.

Trivial bundles correspond to free modules since the description of their sections in terms of components is global, and each component is an element of C(X)𝐶𝑋C(X) . For every vector bundle we can find another one such that their direct sum is a trivial bundle. In dual language this implies that the module of sections {\cal E} is projective

0=.subscript0direct-sumsuperscript{\cal E}_{0}={\cal E}\oplus{\cal E}^{\prime}.

Again it is nontrivial to show the converse, that every projective module is isomorphic to the set of sections of some vector bundle. Finally projective modules over noncommutative algebras are the quantum version of vector bundles.

In the noncommutative case we distinguish between left and right projective modules. Multiplying fundamental sections from the right with elements of 𝒜(T-θ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{-\theta}) preserves the boundary conditions ( 10) while multiplication on the left gives something that no longer is a global section. Thus the set of sections of the fundamental bundle form a right projective module over the 𝒜(T-θ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{-\theta}) algebra which we denote n,mθsuperscriptsubscript𝑛𝑚𝜃{\cal F}_{n,m}^{\theta} . This is no longer true for the adjoint sections since in ( 11) the transition functions multiply from both the left and right. However one can check that the fundamental and the adjoint are both left and right projective modules over the 𝒜(Tθ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{\theta}) algebra. This is because the exponents of the Uisubscript𝑈𝑖U_{i} ’s satisfy

[iσ1+2πθ2,σi]=0,[iσ2-2πθ1,σi]=0;formulae-sequence𝑖subscript𝜎12𝜋𝜃subscript2subscript𝜎𝑖0𝑖subscript𝜎22𝜋𝜃subscript1subscript𝜎𝑖0[i\sigma_{1}+2\pi\theta\partial_{2},\sigma_{i}]=0,~{}~{}[i\sigma_{2}-2\pi% \theta\partial_{1},\sigma_{i}]=0; (20)

thus the Uisubscript𝑈𝑖U_{i} ’s can be commuted over the transition functions in ( 10) and ( 11). Additionally, the fact that n,mθsuperscriptsubscript𝑛𝑚𝜃{\cal F}_{n,m}^{\theta} is both a left 𝒜(Tθ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{\theta}) -module and a right 𝒜(T-θ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{-\theta}) -module can be understood as follows. Since [Ui,σj]=0subscript𝑈𝑖subscript𝜎𝑗0[U_{i},\sigma_{j}]=0 we have

UiΦ(σ1,σ2)=Φ(σ1,σ2)U~i,subscript𝑈𝑖Φsubscript𝜎1subscript𝜎2Φsubscript𝜎1subscript𝜎2subscript~𝑈𝑖U_{i}\Phi(\sigma_{1},\sigma_{2})=\Phi(\sigma_{1},\sigma_{2})\widetilde{U}_{i},

where we dropped the derivatives when there was nothing to their right. Thus multiplying on the left with a𝑎a is equivalent to multiplying on the right with a~~𝑎\widetilde{a}

aΦ=Φa~,𝑎ΦΦ~𝑎a\Phi=\Phi\widetilde{a}, (21)

where a~=k,l𝐙ak,lU~2lU~1k~𝑎subscript𝑘𝑙𝐙subscript𝑎𝑘𝑙superscriptsubscript~𝑈2𝑙superscriptsubscript~𝑈1𝑘\widetilde{a}=\sum_{k,l\in{\bf Z}}a_{k,l}\widetilde{U}_{2}^{l}\widetilde{U}_{1% }^{k} is the same function as a𝑎a but with U~isubscript~𝑈𝑖\widetilde{U}_{i} ’s as arguments and with all the factors written in reversed order.

As mentioned in [17]the construction in Section 2is equivalent to the projective modules discussed in [8]. By solving the boundary conditions we went from a local to a global description. Here we present explicit formulae for this equivalence. First one has to express the left actions on the fundamental sections as actions on the Hilbert space [17]. For example the action of the Zisubscript𝑍𝑖Z_{i} generators is given by

(Z1ϕ^)j(x)=ϕ^j-a(x-1m),(Z2ϕ^)j(x)=e-2πij/me2πix/(n-mθ)ϕ^j(x).formulae-sequencesubscriptsubscript𝑍1^italic-ϕ𝑗𝑥subscript^italic-ϕ𝑗𝑎𝑥1𝑚subscriptsubscript𝑍2^italic-ϕ𝑗𝑥superscript𝑒2𝜋𝑖𝑗𝑚superscript𝑒2𝜋𝑖𝑥𝑛𝑚𝜃subscript^italic-ϕ𝑗𝑥(Z_{1}\widehat{\phi})_{j}(x)=\widehat{\phi}_{j-a}(x-\frac{1}{m}),~{}~{}(Z_{2}% \widehat{\phi})_{j}(x)=e^{-2\pi ij/m}e^{2\pi ix/(n-m\theta)}\widehat{\phi}_{j}% (x).

This can be written as

Z1=W1aV1,Z2=W2V2,formulae-sequencesubscript𝑍1superscriptsubscript𝑊1𝑎subscript𝑉1subscript𝑍2subscript𝑊2subscript𝑉2Z_{1}=W_{1}^{a}V_{1},~{}~{}Z_{2}=W_{2}V_{2},

where Visubscript𝑉𝑖V_{i} and Wisubscript𝑊𝑖W_{i} are operators acting on the Hilbert space as

(V1ϕ^)j(x)=ϕ^j(x-1m),(V2ϕ^)j(x)=e2πix/(n-mθ)ϕ^j(x),formulae-sequencesubscriptsubscript𝑉1^italic-ϕ𝑗𝑥subscript^italic-ϕ𝑗𝑥1𝑚subscriptsubscript𝑉2^italic-ϕ𝑗𝑥superscript𝑒2𝜋𝑖𝑥𝑛𝑚𝜃subscript^italic-ϕ𝑗𝑥(V_{1}\widehat{\phi})_{j}(x)=\widehat{\phi}_{j}(x-\frac{1}{m}),~{}~{}(V_{2}% \widehat{\phi})_{j}(x)=e^{2\pi ix/(n-m\theta)}\widehat{\phi}_{j}(x),
(W1ϕ^)j(x)=ϕ^j-1(x),(W2ϕ^)j(x)=e-2πij/mϕ^j(x).formulae-sequencesubscriptsubscript𝑊1^italic-ϕ𝑗𝑥subscript^italic-ϕ𝑗1𝑥subscriptsubscript𝑊2^italic-ϕ𝑗𝑥superscript𝑒2𝜋𝑖𝑗𝑚subscript^italic-ϕ𝑗𝑥(W_{1}\widehat{\phi})_{j}(x)=\widehat{\phi}_{j-1}(x),~{}~{}(W_{2}\widehat{\phi% })_{j}(x)=e^{-2\pi ij/m}\widehat{\phi}_{j}(x).

These operators satisfy the following relations

V1V2=e-2πi/[m(n-mθ)]V2V1,W1W2=e2πi/mW2W1,[Vi,Wj]=0formulae-sequencesubscript𝑉1subscript𝑉2superscript𝑒2𝜋𝑖delimited-[]𝑚𝑛𝑚𝜃subscript𝑉2subscript𝑉1formulae-sequencesubscript𝑊1subscript𝑊2superscript𝑒2𝜋𝑖𝑚subscript𝑊2subscript𝑊1subscript𝑉𝑖subscript𝑊𝑗0V_{1}V_{2}=e^{-2\pi i/[m(n-m\theta)]}V_{2}V_{1},~{}~{}W_{1}W_{2}=e^{2\pi i/m}W% _{2}W_{1},~{}~{}[V_{i},W_{j}]=0

and can be used to express other operators acting in the Hilbert space. For example we have U1=W1V1n-mθsubscript𝑈1subscript𝑊1superscriptsubscript𝑉1𝑛𝑚𝜃U_{1}=W_{1}V_{1}^{n-m\theta} and U2=W2nV2n-mθsubscript𝑈2superscriptsubscript𝑊2𝑛superscriptsubscript𝑉2𝑛𝑚𝜃U_{2}=W_{2}^{n}V_{2}^{n-m\theta} .

We can now present the correspondence between [8]and [17]. The two integers p𝑝p and q𝑞q and the angular variable θCDSsubscript𝜃CDS\theta_{\rm CDS} labeling the projective module p,qθCDSsuperscriptsubscript𝑝𝑞subscript𝜃CDS{\cal H}_{p,q}^{\theta_{\rm CDS}} of [8], and θCDSsubscriptsuperscript𝜃CDS\theta^{\prime}_{\rm CDS} can be expressed in terms of the quantities used in this paper or in [17]

p=n,q=-m,θCDS=-θ,θCDS=θ.formulae-sequence𝑝𝑛formulae-sequence𝑞𝑚formulae-sequencesubscript𝜃CDS𝜃subscriptsuperscript𝜃CDSsuperscript𝜃p=n,~{}~{}q=-m,~{}~{}\theta_{\rm CDS}=-\theta,~{}~{}\theta^{\prime}_{\rm CDS}=% \theta^{\prime}.

Then n,mθn,-m-θsuperscriptsubscript𝑛𝑚𝜃superscriptsubscript𝑛𝑚𝜃{\cal F}_{n,m}^{\theta}\cong{\cal H}_{n,-m}^{-\theta} . The Hilbert space representation of [8]written in terms of the function f(s,k)𝑓𝑠𝑘f(s,k) with s𝐑𝑠𝐑s\in{\bf R} and k𝐙q𝑘subscript𝐙𝑞k\in{\bf Z}_{q} is linearly related to the ϕ^k(x)subscript^italic-ϕ𝑘𝑥\widehat{\phi}_{k}(x) representation

ϕ^k(x)=l=1m𝒦kl(𝒮(-mn-mθ)f)(x,l).subscript^italic-ϕ𝑘𝑥superscriptsubscript𝑙1𝑚subscript𝒦𝑘𝑙subscript𝒮𝑚𝑛𝑚𝜃𝑓𝑥𝑙\widehat{\phi}_{k}(x)=\sum_{l=1}^{m}{\cal K}_{kl}~{}({\cal S}_{(-\frac{m}{n-m% \theta})}f)(x,l).

Here 𝒦𝒦{\cal K} is an m×m𝑚𝑚m\times m representation changing matrix defined as in ( 15) but for m𝑚m -dimensional ’t Hooft matrices, and 𝒮λsubscript𝒮𝜆{\cal S}_{\lambda} is the rescaling operator (𝒮λf)(x,k)=f(λx,k)subscript𝒮𝜆𝑓𝑥𝑘𝑓𝜆𝑥𝑘({\cal S}_{\lambda}f)(x,k)=f(\lambda x,k) which can be expressed using the ordered exponential

𝒮λ=λE((λ-1)x,x).subscript𝒮𝜆𝜆𝐸𝜆1𝑥subscript𝑥{\cal S}_{\lambda}=\lambda E((\lambda-1)x,\partial_{x}).

Also, using lower case to distinguish them from our current notation which follows [17], the operators in [8]represented in the ϕ^k(x)subscript^italic-ϕ𝑘𝑥\hat{\phi}_{k}(x) basis are given by

v0=V2n-mθ,v1=V1n-mθ,w0=e2πin/mW2n,w1=e2πi/mW1formulae-sequencesubscript𝑣0superscriptsubscript𝑉2𝑛𝑚𝜃formulae-sequencesubscript𝑣1superscriptsubscript𝑉1𝑛𝑚𝜃formulae-sequencesubscript𝑤0superscript𝑒2𝜋𝑖𝑛𝑚superscriptsubscript𝑊2𝑛subscript𝑤1superscript𝑒2𝜋𝑖𝑚subscript𝑊1v_{0}=V_{2}^{n-m\theta},~{}~{}v_{1}=V_{1}^{n-m\theta},~{}~{}w_{0}=e^{2\pi in/m% }W_{2}^{n},~{}~{}w_{1}=e^{2\pi i/m}W_{1}
z0=e2πi/mZ2,z1=e-2πia/mZ1-1,u0=e2πin/mU2,u1=e2πi/mU1.formulae-sequencesubscript𝑧0superscript𝑒2𝜋𝑖𝑚subscript𝑍2formulae-sequencesubscript𝑧1superscript𝑒2𝜋𝑖𝑎𝑚superscriptsubscript𝑍11formulae-sequencesubscript𝑢0superscript𝑒2𝜋𝑖𝑛𝑚subscript𝑈2subscript𝑢1superscript𝑒2𝜋𝑖𝑚subscript𝑈1z_{0}=e^{2\pi i/m}Z_{2},~{}~{}z_{1}=e^{-2\pi ia/m}Z_{1}^{-1},~{}~{}u_{0}=e^{2% \pi in/m}U_{2},~{}~{}u_{1}=e^{2\pi i/m}U_{1}.

Next we introduce the Morita equivalence of two algebras [20, 21, 22, 23, 18, 19], which can be used to describe a subgroup of the T-duality group of the M-theory compactification in the language of noncommutative SYM gauge theory.

Two C*superscript𝐶C^{*} -algebras 𝒜𝒜{\cal A} and 𝒜superscript𝒜{\cal A}^{\prime} are Morita equivalent if there exists a right 𝒜𝒜{\cal A} -module {\cal E} such that the algebra End𝒜𝐸𝑛subscript𝑑𝒜End_{\cal A}{\cal E} is isomorphic to 𝒜superscript𝒜{\cal A}^{\prime} . Here End𝒜𝐸𝑛subscript𝑑𝒜End_{\cal A}{\cal E} denotes the set of endomorphisms of the 𝒜𝒜{\cal A} -module {\cal E} . It consists of linear maps T𝑇T on {\cal E} where linearity is not only with respect to c𝑐c -numbers but also with respect to right multiplication by elements of 𝒜𝒜{\cal A}

T(Φf)=T(Φ)f,Φ,f𝒜.formulae-sequence𝑇Φ𝑓𝑇Φ𝑓formulae-sequenceΦ𝑓𝒜T(\Phi f)=T(\Phi)f,~{}~{}\Phi\in{\cal E},~{}~{}f\in{\cal A}.

An example of Morita equivalent algebras is 𝒜(T-θ2)𝒜subscriptsuperscript𝑇2𝜃{\cal A}(T^{2}_{-\theta}) and 𝒜(Tθ2)