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

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 $C^{*}$ -algebras as it applies to our specific case.

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## 1 Introduction

It was conjectured in [1]that the infinite momentum frame description of M-theory is given by the large $n$ limit of supersymmetric quantum mechanics (SQM) [2, 3, 4], obtained as the dimensional reduction of the $9+1$ dimensional $U(n)$ 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 $n$ 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 $T^{d}$ for $d\geq 2$ there are additional moduli coming from the three-form of 11-dimensional supergravity. For example, if we compactify on $T^{2}$ along $X_{1}$ and $X_{2}$ then $C_{-12}$ 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 $\theta$ given by

 $\theta=\int~{}C_{-12}\,dX_{-}dX_{1}dX_{2}.$

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 ${\cal A}(T^{2}_{\theta})$ 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 $P_{-}=n/R$ sector of the DLCQ of uncompactified M-theory is given by the $U(n)$ SQM [2, 3, 4]whose action in the temporal gauge is given by

 $S=\frac{1}{2R}\int dt\,{\rm Tr}\left(\dot{X}^{\mu}\dot{X}_{\mu}+\sum_{\mu>\nu}% [X_{\mu},X_{\nu}]^{2}+i\Theta^{T}\dot{\Theta}-\Theta^{T}\Gamma_{\mu}[X^{\mu},% \Theta]\right)$ (1)

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

 $\displaystyle U_{i}^{-1}X_{j}U_{i}$ $\displaystyle=$ $\displaystyle X_{j}+2\pi R_{j}\delta_{ij},$ (2) $\displaystyle U_{i}^{-1}X_{a}U_{i}$ $\displaystyle=$ $\displaystyle X_{a},$ $\displaystyle U_{i}^{-1}\Theta~{}U_{i}$ $\displaystyle=$ $\displaystyle\Theta,~{}~{}~{}i,j=1,2~{}~{}~{}a=3,\ldots,9.$

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

 $U_{1}U_{2}=e^{2\pi i\theta}U_{2}U_{1}.$

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

 $\widetilde{U}_{1}\widetilde{U}_{2}=e^{-2\pi i\theta}\widetilde{U}_{2}% \widetilde{U}_{1}.$ (3)

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

 $\left[\sigma_{1},\sigma_{2}\right]=2\pi i\theta.$ (4)

Then $\widetilde{U}_{i}\stackrel{\rm def}{=}e^{i\sigma_{i}}$ satisfy ( 3). The variables $\sigma_{i}$ 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 ${\cal A}(T^{2}_{-\theta})$ is generated by $\widetilde{U}_{i}$ . Similarly the $U_{i}$ operators generate the algebra denoted ${\cal A}(T^{2}_{\theta})$ . To realize them we introduce partial derivative operators on the quantum plane, satisfying the following algebra

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

Then, we realize $U_{i}$ as

 $U_{1}=e^{i\sigma_{1}}e^{2\pi\theta\,\partial_{2}},~{}~{}U_{2}=e^{i\sigma_{2}}e% ^{-2\pi\theta\,\partial_{1}}.$

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

 $U_{i}^{-1}~{}\frac{1}{i}\partial_{j}~{}U_{i}=\frac{1}{i}\partial_{j}+\delta_{% ij}.$ (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 $c$ -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 $U_{i}$ ’s. However this is just the definition of the covariant derivative

 $\displaystyle X_{i}$ $\displaystyle=$ $\displaystyle-2\pi iR_{j}\delta_{ij}\partial_{j}+A_{i}(\widetilde{U}_{i}),$ (7) $\displaystyle X_{a}$ $\displaystyle=$ $\displaystyle X_{a}(\widetilde{U}_{i}),$ $\displaystyle\Theta$ $\displaystyle=$ $\displaystyle\Theta(\widetilde{U}_{i}),$

where $A_{i}$ , $X_{a}$ and each spinorial component of $\Theta$ are $n\times n$ hermitian matrices with operator valued entries. Note that since the partial derivatives already satisfy the cocycle condition, the gauge fields $A_{i}$ and the scalar fields $X_{a}$ must satisfy a homogeneous quotient condition like the second relation in ( 2). Hence $A_{i}$ and $X_{a}$ must depend only on $\widetilde{U}_{i}$ . 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 $2+1$ dimensions, with the space part given by the above quantum torus and a commutative time. For the commutative case, matrix compactification on $T^{d}$ results in a SYM gauge theory in $d+1$ dimensions on the dual torus. In the limit when the size of the original torus vanishes the dual torus becomes ${\bf R}^{d}$ , 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 $D_{i}$ and a fluctuating part

 $\displaystyle X_{i}$ $\displaystyle=$ $\displaystyle-2\pi iR_{j}\delta_{ij}D_{j}+A_{i}(Z_{j}),$ (8) $\displaystyle X_{a}$ $\displaystyle=$ $\displaystyle X_{a}(Z_{i}),$ $\displaystyle\Theta$ $\displaystyle=$ $\displaystyle~{}\Theta(Z_{i}).$

Here the $Z_{i}$ ’s are $n\times n$ matrices with operator entries and, just like the $\widetilde{U}_{i}$ ’s for the trivial bundle, commute with the $U_{i}$ ’s, but now are sections of the twisted bundle whose exact form will be discussed shortly. However, while for the trivial bundle $A_{i}$ , $X_{a}$ and the spinorial components of $\Theta$ are $n\times n$ matrix functions, in ( 8) $A_{i}$ , $X_{a}$ and the components of $\Theta$ are one-dimensional functions but with matrix arguments. Later, this will allow us to establish a relationship between a SYM on a twisted $U(n)$ bundle and one on a $U(1)$ bundle.

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

 $D_{1}=\partial_{1},~{}D_{2}=\partial_{2}-if\sigma_{1},$ (9)

where $f$ is the constant field strength

 $[D_{1},D_{2}]=-if.$

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

 $\displaystyle\Phi(\sigma_{1}+2\pi,\sigma_{2})$ $\displaystyle=$ $\displaystyle\Omega_{1}(\sigma_{1},\sigma_{2})~{}\Phi(\sigma_{1},\sigma_{2}),$ (10) $\displaystyle\Phi(\sigma_{1},\sigma_{2}+2\pi)$ $\displaystyle=$ $\displaystyle\Omega_{2}(\sigma_{1},\sigma_{2})~{}\Phi(\sigma_{1},\sigma_{2}).$

 $\displaystyle\Psi(\sigma_{1}+2\pi,\sigma_{2})=\Omega_{1}(\sigma_{1},\sigma_{2}% )~{}\Psi(\sigma_{1},\sigma_{2})~{}\Omega_{1}(\sigma_{1},\sigma_{2})^{-1},$ (11) $\displaystyle\Psi(\sigma_{1},\sigma_{2}+2\pi)=\Omega_{2}(\sigma_{1},\sigma_{2}% )~{}\Psi(\sigma_{1},\sigma_{2})~{}\Omega_{2}(\sigma_{1},\sigma_{2})^{-1}.$

Consistency of the transition functions of the bundle requires that

 $\Omega_{1}(\sigma_{1},\sigma_{2}+2\pi)~{}\Omega_{2}(\sigma_{1},\sigma_{2})=% \Omega_{2}(\sigma_{1}+2\pi,\sigma_{2})~{}\Omega_{1}(\sigma_{1},\sigma_{2}).$ (12)

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

 $D_{i}(\sigma_{1}+2\pi,\sigma_{2})=\Omega_{1}(\sigma_{1},\sigma_{2})~{}D_{i}(% \sigma_{1},\sigma_{2})~{}\Omega_{1}(\sigma_{1},\sigma_{2})^{-1},$
 $D_{i}(\sigma_{1},\sigma_{2}+2\pi)=\Omega_{2}(\sigma_{1},\sigma_{2})~{}D_{i}(% \sigma_{1},\sigma_{2})~{}\Omega_{2}(\sigma_{1},\sigma_{2})^{-1}.$

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

 $\Omega_{1}=e^{im\sigma_{2}/n}U,~{}\Omega_{2}=V,$ (13)

where $U,~{}V$ are $n\times n$ unitary matrices satisfying

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

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

 $U_{kl}=e^{2\pi ikm/n}\delta_{k,l},~{}~{}V_{kl}=\delta_{k+1,l},$

where the subscripts are identified with period $n$ .

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

 $U^{\prime}V^{\prime}=e^{-2\pi i/n}V^{\prime}U^{\prime},~{}~{}U^{\prime n}=V^{% \prime n}=1.$

The relation is given by

 $U=e^{2\pi im/n}U^{\prime m},~{}~{}V=V^{\prime}.$ (14)

The phase in ( 14) is due to the nonstandard definition of $U$ 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 $K$ which changes the representation so that $V^{\prime}$ is diagonal, and satisfies

 $KU^{\prime}K^{-1}=V^{\prime-1},~{}~{}KV^{\prime}K^{-1}=U^{\prime}.$ (15)

Note that $n$ is quantized since we are considering a $U(n)$ gauge theory and $m$ is quantized since the magnetic flux $f$ through $T_{2}$ is quantized

 $2\pi f=\frac{m}{n-m\theta}.$

In M-theory $m$ 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 $m=1$ in the commutative case presented in [7]. Using the ordered exponential explained below, the general solution has the form

 $\Phi_{k}(\sigma_{1},\sigma_{2})=\sum_{s\in{\bf Z}}\sum_{j=1}^{m}E\left(\frac{m% }{n}\left(\frac{\sigma_{2}}{2\pi}+k+ns\right)+j,i\sigma_{1}\right)\widehat{% \phi}_{j}\left(\frac{\sigma_{2}}{2\pi}+k+ns+\frac{nj}{m}\right).$

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

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

The normalization is such that

 $E(-B,A)E(A,B)=1$

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

 $\displaystyle E(A+c,B)=E(A,B)e^{cB},$ (16) $\displaystyle E(A,B+c)=e^{cA}E(A,B).$

The $\widehat{\phi}_{j}$ 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

 $\phi(\sigma_{1},\sigma_{2})\stackrel{\rm def}{=}\Phi_{k}(\sigma_{1},\sigma_{2}% -2\pi(k-1)).$

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

 $\phi(\sigma_{1},\sigma_{2}+2\pi n)=\phi(\sigma_{1},\sigma_{2}).$

The other boundary condition gives

 $\phi(\sigma_{1}+2\pi,\sigma_{2})=e^{im(\sigma_{2}+2\pi)/n}\phi(\sigma_{1},% \sigma_{2}).$

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

 $\phi(\sigma_{1},\sigma_{2})=f(\sigma_{1},\sigma_{2})\check{\phi}(\sigma_{1},% \sigma_{2})$

and to require a simpler periodicity condition for $\check{\phi}$

 $\check{\phi}(\sigma_{1}+2\pi,\sigma_{2})=\check{\phi}(\sigma_{1},\sigma_{2}).$

Then the function $f$ must satisfy

 $f(\sigma_{1}+2\pi,\sigma_{2})=e^{im(\sigma_{2}+2\pi)/n}f(\sigma_{1},\sigma_{2}).$

This is satisfied exactly for

 $f(\sigma_{1},\sigma_{2})=E\left(\frac{m}{n}\left(\frac{\sigma_{2}}{2\pi}+1% \right),i\sigma_{1}\right),$

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

 $\check{\phi}(\sigma_{1},\sigma_{2})=\sum_{p\in{\bf Z}}~{}e^{ip\sigma_{1}}~{}% \phi_{p}(\sigma_{2})$

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

 $\phi(\sigma_{1},\sigma_{2})=\sum_{p\in{\bf Z}}E\left(\frac{m}{n}\left(\frac{% \sigma_{2}}{2\pi}+1\right)+p,i\sigma_{1}\right)\phi_{p}(\sigma_{2}).$

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

 $\phi(\sigma_{1},\sigma_{2})=\sum_{s\in{\bf Z}}\sum_{j=1}^{m}E\left(\frac{m}{n}% \left(\frac{\sigma_{2}}{2\pi}+1\right)+ms+j,i\sigma_{1}\right)\phi_{s,j}(% \sigma_{2}),$

where $\phi_{s,j}\stackrel{\rm def}{=}\phi_{ms+j}$ . Periodicity in $\sigma_{2}$ then implies $\phi_{s-1,j}(\sigma_{2}+2\pi n)=\phi_{s,j}(\sigma_{2})$ so that using this recursively we have $\phi_{s,j}(\sigma_{2})=\phi_{0,j}(\sigma_{2}+2\pi ns)$ . Finally after defining $\widetilde{\phi}_{j}(x)\stackrel{\rm def}{=}\phi_{0,j}(2\pi(x-1))$ we obtain

 $\Phi_{k}(\sigma_{1},\sigma_{2})=\sum_{s\in{\bf Z}}\sum_{j=1}^{m}E\left(\frac{m% }{n}\left(\frac{\sigma_{2}}{2\pi}+k+ns\right)+j,i\sigma_{1}\right)\widetilde{% \phi}_{j}\left(\frac{\sigma_{2}}{2\pi}+k+ns\right).$

This is the result mentioned above up to another redefinition

 $\widetilde{\phi}_{j}(x)=\widehat{\phi}_{j}(x+\frac{n}{m}j).$

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 $\sigma_{1}$ and it is convenient to order all $\sigma_{1}$ to the right in the solution. Using $V^{n}=1$ in the second condition ( 10) one can express all $n$ components of $\Phi$ in terms of a single function with period $2\pi n$ in $\sigma_{2}$ . After Fourier transforming in $\sigma_{2}$ and imposing both boundary conditions ( 10) we obtain the general solution

 $\Phi_{k}(\sigma_{1},\sigma_{2})=\sum_{p\in{\bf Z}}~{}e^{2\pi i(\sigma_{2}/2\pi% +k)p/n}~{}e^{2\pi i(\sigma_{1}/2\pi-p/m)m/n}~{}\widehat{\psi}_{p}(\sigma_{1}/2% \pi-p/m),$

where only $m$ of the $\widehat{\psi}_{p}$ functions are independent, since

 $\widehat{\psi}_{p+m}(x)=\widehat{\psi}_{p}(x).$

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

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

Here $c_{s,t}$ are $c$ -numbers and

 $Z_{1}=e^{i\sigma_{1}/(n-m\theta)}V^{b},~{}~{}Z_{2}=e^{i\sigma_{2}/n}U^{-b},$

where $b$ is an integer, such that we can find another integer $a$ satisfying $an-bm=1$ . For $n$ and $m$ relatively prime one can always find integer solutions to this equation. Again, we emphasize that the $Z_{i}$ ’s commute with the $U_{i}$ ’s. They are generators of the algebra of functions on a new quantum torus

 $Z_{1}Z_{2}=e^{2\pi i\theta^{\prime}}Z_{2}Z_{1},$

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

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

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

 $\Psi(\sigma_{1}+2\pi n,\sigma_{2})=\Omega_{1}^{n}~{}\Psi(\sigma_{1},\sigma_{2}% )~{}\Omega_{1}^{-n}=\Psi(\sigma_{1}+2\pi\theta m,\sigma_{2}).$

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

 $\Psi(\sigma_{1}+2\pi(n-m\theta),\sigma_{2})=\Psi(\sigma_{1},\sigma_{2}),$
 $\Psi(\sigma_{1},\sigma_{2}+2\pi n)=\Psi(\sigma_{1},\sigma_{2}).$

We can expand the section as

 $\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}e^{is\sigma_{1}/(n-m\theta)}e^% {-it\sigma_{2}/n}\Psi_{s,t},$

where $\Psi_{s,t}$ is a $n\times n$ matrix and can be expanded as

 $\Psi_{s,t}=\sum_{i=i_{0}}^{n+i_{0}}\sum_{j=j_{0}}^{n+j_{0}}c_{s,t,i,j}V^{% \prime i}U^{\prime j}.$ (18)

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

 $c_{s,t,i,j}e^{2\pi is/(n-m\theta)}=c_{s,t,i,j}e^{-2\pi imi/n}e^{2\pi ism\theta% /[n(n-,\theta)]}.$

From this and the similar relation obtained by imposing the second equation ( 11) we have that $c_{s,t,i,j}$ vanish unless $(s+mi)/n=k$ and $(t+j)/n=s$ for $k$ and $s$ two integers. These equations have multiple solutions. However, if $(i,j)$ and $(i^{\prime},j^{\prime})$ are two solutions then $i-i^{\prime}\in n{\bf Z}$ and $j-j^{\prime}\in n{\bf Z}$ . This ensures that only one term survives in the sum ( 18) over $i$ and $j$ . Choosing for later convenience $i_{0}=sb$ and $j_{0}=mbt$ we have

 $\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}~{}e^{is\sigma_{1}/(n-m\theta)% }e^{-it\sigma_{2}/n}\sum_{i=sb}^{n+sb}~{}\sum_{j=mbt}^{n+mbt}c_{s,t,i,j}~{}V^{% \prime i}U^{\prime j}.$

Since $n$ and $m$ are relatively prime let $a,b\in{\bf Z}$ such that $an-bm=1$ . Then

 $k=as,~{}l=at,~{}i=bs,~{}j=mbt$

is an integer solution inside the $i$ and $j$ summation range. Dropping the $i,j$ indices since they are determined by $s$ and $t$ we have

 $\Psi(\sigma_{1},\sigma_{2})=\sum_{s,t\in{\bf Z}}c_{s,t}\left(e^{i\sigma_{1}/(n% -m\theta)}V^{\prime b}\right)^{s}\left(e^{i\sigma_{2}/n}U^{\prime-mb}\right)^{% -t},$

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

## 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 $C^{*}$ -algebras. In one direction, to a topological space $X$ we associate the algebra of continuous functions $C(X)$ . Conversely and rather nontrivially, the spectrum of a commutative $C^{*}$ -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 $C^{*}$ -algebra ${\cal A}$ describe what is called by correspondence a quantum space. To illustrate, consider the algebra of the quantum torus ${\cal A}(T^{2}_{\theta})$ generated by the $U_{i}$ ’s. An arbitrary element $a$ has the form

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

where some restrictions (which we do not discuss here) are imposed on the $c$ -number coefficients $a_{k,l}$ . For $\theta=0$ , formula ( 19) reduces to the Fourier expansion of functions on $T^{2}$ . 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 ${\cal A}$ -module ${\cal E}$ . First consider the classical commutative picture. The set ${\cal E}$ of global sections of a vector bundle over a base space $X$ has the structure of a projective module over the algebra $C(X)$ . 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 ${\cal E}_{0}$ over an algebra ${\cal A}$ is a module isomorphic to a direct sum of a finite number of copies of the algebra

 ${\cal E}_{0}={\cal A}\oplus\ldots\oplus{\cal A}.$

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 $C(X)$ . 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

 ${\cal E}_{0}={\cal E}\oplus{\cal E}^{\prime}.$

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 ${\cal A}(T^{2}_{-\theta})$ 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 ${\cal A}(T^{2}_{-\theta})$ algebra which we denote ${\cal F}_{n,m}^{\theta}$ . 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 ${\cal A}(T^{2}_{\theta})$ algebra. This is because the exponents of the $U_{i}$ ’s satisfy

 $[i\sigma_{1}+2\pi\theta\partial_{2},\sigma_{i}]=0,~{}~{}[i\sigma_{2}-2\pi% \theta\partial_{1},\sigma_{i}]=0;$ (20)

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

 $U_{i}\Phi(\sigma_{1},\sigma_{2})=\Phi(\sigma_{1},\sigma_{2})\widetilde{U}_{i},$

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

 $a\Phi=\Phi\widetilde{a},$ (21)

where $\widetilde{a}=\sum_{k,l\in{\bf Z}}a_{k,l}\widetilde{U}_{2}^{l}\widetilde{U}_{1% }^{k}$ is the same function as $a$ but with $\widetilde{U}_{i}$ ’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 $Z_{i}$ generators is given by

 $(Z_{1}\widehat{\phi})_{j}(x)=\widehat{\phi}_{j-a}(x-\frac{1}{m}),~{}~{}(Z_{2}% \widehat{\phi})_{j}(x)=e^{-2\pi ij/m}e^{2\pi ix/(n-m\theta)}\widehat{\phi}_{j}% (x).$

This can be written as

 $Z_{1}=W_{1}^{a}V_{1},~{}~{}Z_{2}=W_{2}V_{2},$

where $V_{i}$ and $W_{i}$ are operators acting on the Hilbert space as

 $(V_{1}\widehat{\phi})_{j}(x)=\widehat{\phi}_{j}(x-\frac{1}{m}),~{}~{}(V_{2}% \widehat{\phi})_{j}(x)=e^{2\pi ix/(n-m\theta)}\widehat{\phi}_{j}(x),$
 $(W_{1}\widehat{\phi})_{j}(x)=\widehat{\phi}_{j-1}(x),~{}~{}(W_{2}\widehat{\phi% })_{j}(x)=e^{-2\pi ij/m}\widehat{\phi}_{j}(x).$

These operators satisfy the following relations

 $V_{1}V_{2}=e^{-2\pi i/[m(n-m\theta)]}V_{2}V_{1},~{}~{}W_{1}W_{2}=e^{2\pi i/m}W% _{2}W_{1},~{}~{}[V_{i},W_{j}]=0$

and can be used to express other operators acting in the Hilbert space. For example we have $U_{1}=W_{1}V_{1}^{n-m\theta}$ and $U_{2}=W_{2}^{n}V_{2}^{n-m\theta}$ .

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

 $p=n,~{}~{}q=-m,~{}~{}\theta_{\rm CDS}=-\theta,~{}~{}\theta^{\prime}_{\rm CDS}=% \theta^{\prime}.$

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

 $\widehat{\phi}_{k}(x)=\sum_{l=1}^{m}{\cal K}_{kl}~{}({\cal S}_{(-\frac{m}{n-m% \theta})}f)(x,l).$

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

 ${\cal S}_{\lambda}=\lambda E((\lambda-1)x,\partial_{x}).$

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

 $v_{0}=V_{2}^{n-m\theta},~{}~{}v_{1}=V_{1}^{n-m\theta},~{}~{}w_{0}=e^{2\pi in/m% }W_{2}^{n},~{}~{}w_{1}=e^{2\pi i/m}W_{1}$
 $z_{0}=e^{2\pi i/m}Z_{2},~{}~{}z_{1}=e^{-2\pi ia/m}Z_{1}^{-1},~{}~{}u_{0}=e^{2% \pi in/m}U_{2},~{}~{}u_{1}=e^{2\pi i/m}U_{1}.$

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

 $T(\Phi f)=T(\Phi)f,~{}~{}\Phi\in{\cal E},~{}~{}f\in{\cal A}.$

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