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DAMTP-98-36

DSF/16-98

OUTP-98-42P

hep-th/9806099

June 1998

Revised April 1999

String Geometry and the Noncommutative Torus

[8mm]

Giovanni Landi111On leave from Dipartimento di Scienze Matematiche, Università di Trieste, P.le Europa 1, I-34127, Trieste, Italy. Also INFN, Sezione di Napoli, Napoli, Italy.
E-mail:

[3mm] Department of Applied Mathematics and Theoretical Physics

University of Cambridge, Silver Street, Cambridge CB3 9EW, U.K.

[6mm] Fedele Lizzi222E-mail:

[3mm] Dipartimento di Scienze Fisiche, Università di Napoli Federico II

and INFN, Sezione di Napoli, Mostra d’Oltremare Pad. 20, 80125 Napoli, Italy

[6mm] Richard J. Szabo333Present address: The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark.
E-mail:

[3mm] Department of Physics – Theoretical Physics, University of Oxford

1 Keble Road, Oxford OX1 3NP, U.K.

[10mm]

Abstract

We construct a new gauge theory on a pair of -dimensional noncommutative tori. The latter comes from an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra and the noncommutative torus. We show that the tachyon algebra of is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple proofs of the Morita equivalences between -dimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly duality-symmetric. The duality-invariant gauge theory is manifestly covariant but contains highly non-local interactions. We show that it also admits a new sort of particle-antiparticle duality which enables the construction of instanton field configurations in any dimension. The duality non-symmetric on-shell projection of the field theory is shown to coincide with the standard non-abelian Yang-Mills gauge theory minimally coupled to massive Dirac fermion fields.

1. Introduction

The noncommutative torus  is one of the basic examples of a noncommutative geometry [5, 6] which captures features of the difference between an ordinary manifold and a ‘noncommutative space’. Recent interest in such geometries has occurred in the physics literature in the context of their relation to -Theory . As shown in the seminal paper , the most general solutions to the quotient conditions for toroidal compactifications of Matrix Theory satisfy the algebraic relations of gauge connections on noncommutative tori. This has led to, among other things, new physical insights into the structure of the supergravity sector of -Theory by relating geometrical parameters of the noncommutative torus to physical parameters of the Matrix Theory gauge theories. In this paper we shall discuss the role played by the noncommutative torus in the short-distance structure of spacetime. In particular we shall construct a new duality-symmetric gauge theory on a pair of noncommutative tori.

One way to describe the noncommutative torus is to promote the ordinary coordinates , , of a -torus to non-commuting operators acting on an infinite-dimensional Hilbert space and obeying the commutation relations

 [^xi,^xj]=2πiℓij (1.1)

where are real numbers. Defining the basic plane waves

 Ui=expi^xi (1.2)

it follows from the Baker-Campbell-Hausdorff formula that these operators generate the algebra

 UiUj=e2πiℓijUjUi (1.3)

and these are the basic defining relations of the noncommutative torus, when viewed as an algebra of functions with generalized Fourier series expansions in terms of the plane waves (1.2). In the limit , one recovers the ordinary coordinates and the plane waves (1.2) become the usual ones of the torus. The antisymmetric matrix can therefore be thought of as defining the Planck scale of a compactified spacetime. The noncommutative torus then realizes old ideas of quantum gravity that at distance scales below the Planck length the nature of spacetime geometry is modified. At large distances the usual (commutative) spacetime is recovered.

On the other hand, a viable model for Planck scale physics is string theory, and its various extensions to -brane field theory  and Matrix Theory . In these latter extensions, the short-distance noncommutativity of spacetime coordinates is represented by viewing them as matrices. The noncommutativity of spacetime in this picture is the result of massless quantum excitations yielding bound states of -branes with broken supersymmetry , or alternatively of the quantum fluctuations in topology of the worldsheets of the -branes . However, it is also possible to view the spacetime described by ordinary string theory more directly as a noncommutative geometry . The main idea in this context is to substitute the (commutative) algebra of continuous complex-valued functions on spacetime with the (noncommutative) vertex operator algebra of the underlying conformal field theory. Ordinary spacetime can be recovered by noticing that the algebra of continuous functions can be thought of a subalgebra of the vertex operator algebra. The fruitfulness of this approach is that the full vertex operator algebra is naturally invariant under target space duality transformations of the string theory. The simplest such mapping relates large and small radius circles to one another and leads directly to a fundamental length scale, usually the finite size of the string. Although the duality is a symmetry of the full noncommutative string spacetime, different classical spacetimes are identified under the transformation yielding a natural geometrical origin for these quantum symmetries of compactified string theory. This point of view therefore also describes physics at the Planck scale.

In this paper we will merge these two descriptions of the short-distance structure of spacetime as described by noncommutative geometry. We will show that a particular algebra obtained naturally from the tachyon algebra, which in turn is obtained by projecting out the string oscillatory modes, defines a twisted projective module over the noncommutative torus. This algebra is, in this sense, the smallest quantum deformation of ordinary, classical spacetime, and it represents the structure of spacetime at the Planck length. Thus strings compactified on a torus have a geometry which is already noncommutative at short distances. The remainder of the full vertex operator algebra acts to yield the non-trivial gauge transformations (including duality) of spacetime. This fact yields yet another interpretation for the noncommutativity of Planck scale spacetime. In string theory spacetime is a set of fields defined on a surface, and at short distances the interactions of the strings (described by the vertex operator algebra) causes the spacetime to become noncommutative. In the case of toroidally compactified string theory, spacetime at the the Planck scale is a noncommutative torus. The results of this paper in this way merge the distinct noncommutative geometry formalisms for strings by connecting them all to the geometry of the noncommutative torus. At hand is therefore a unified setting for string theory in terms of (target space) -brane field theory, Matrix Theory compactifications, and (worldsheet) vertex operator algebras.

Aside from these physically interesting consequences, we will show that the modules we obtain from the vertex operator algebra also bear a number of interesting mathematical characteristics. Most notably, the duality symmetries of the vertex operator algebra lead to a simple proof of the Morita equivalences of noncommutative tori with deformation parameters which are related by the natural action of the discrete group on the space of real-valued antisymmetric matrices. These Morita equivalences have been established recently using more direct mathematical constructions in . In  it was shown that Matrix Theories compactified on Morita equivalent tori are physically equivalent to one another, in that the BPS spectra of states are the same and the associated field theories can be considered to be duals of each other. Here we shall find a direct manifestation of this duality equivalence in terms of the basic worldsheet theory itself. The relationship between vertex operator algebras, duality and the noncommutative torus was originally pointed out in  and discussed further in . In a sense, this relationship shows that the duality properties of Morita equivalences in -Theory are controlled by the stringy sector of the dynamics. Morita equivalent noncommutative tori have also been constructed in  under the name discrete Heisenberg-Weyl groups and via the action of the modular group. There it was also suggested that this is the base for the duality principles which appear in string theory and conformal field theory.

We will also show that the fairly complete classification of the automorphism group of the vertex operator algebra, given in , can be used to characterize the symmetries of the twisted modules that we find. This immediately leads us to the construction of an action functional for this particular noncommutative geometry which is naturally invariant under the automorphism group. Generally, the action functional in noncommutative geometry can be used to construct invariants of modules of the given algebra and it presents a natural geometrical origin for many physical theories, such as the standard model [24, 25] and superstring theory . The spectral action principle of noncommutative geometry naturally couples gravitational and particle interactions from a very simple geometric perspective [26, 27]. In the following we shall construct both fermionic and bosonic actions for the twisted modules which possess the same properties as dictated by the spectral action principle. However, since we shall be neither concerned with coupling to gravity nor in renormalization effects, we shall use a somewhat simpler definition than that proposed in . A consequence of the invariance of the action under automorphisms of the algebra is that the action is explicitly duality-symmetric. The construction of explicitly electric-magnetic duality symmetric action functionals has been of particular interest over the years [28, 29] and they have had applications to the physics of black holes  and of -branes [31, 32]. These actions are of special interest now because of the deep relevance of duality symmetries to the spacetime structure of superstring theory within the unified framework of -Theory.

Because the derivation of the duality-symmetric action involves quite a bit of mathematics, it is worthwhile to summarize briefly the final result here. We shall show that a general gauge theory on the twisted module leads naturally to a target space Lagrangian of the form

 L=(F+⋆F)ij(F+⋆F)ij−i¯¯¯¯¯¯ψ∗γi(∂i+i↔Ai)ψ−i¯¯¯¯ψγ∗i(∂i∗+i↔Ai∗)ψ∗ (1.4)

where

 Fij=∂iAj−∂jAi+i[Ai,Aj]−gikgjl(∂k∗Al∗−∂l∗Ak∗+i[Ak∗,Al∗]) (1.5)

and . Here the field theory is defined on a Lorentzian spacetime with coordinates , and is the (flat) metric of while the metric on the total spacetime is . We have defined and . The fields and are a dual pair of gauge fields, and are dual spinor fields, and and are Dirac matrices on and , respectively. The field strength is a certain “dual” to the field strength with respect to the Lorentzian metric of the spacetime. The bars on the fermion fields denote their “adjoints” and the double arrow on the gauge potentials in the fermionic part of the action denotes their left-right symmetric action on the fermion and anti-fermion fields (in a sense which we shall define more precisely in what follows). The commutators and actions of gauge potentials on fermion fields are defined using the noncommutative tachyon algebra structure, so that the action associated with (1.4) describes a certain nonabelian gauge theory coupled to fermions in a nontrivial representation of a gauge group (again this gauge group will be described more precisely in the following).

The action corresponding to (1.4) is explicitly invariant under the interchange of starred and un-starred quantities. This symmetry incorporates the -duality transformation which inverts the metric , and it moreover contains a particle-antiparticle duality transformation that represents a certain topological instanton symmetry of the field theory in any dimension . However, by its very construction, it is manifestly invariant under a much larger symmetry group, including the gauge group, which we shall describe in this paper. In this sense, we will see that the action functional corresponding to (1.4) measures the amount of duality symmetry as well as the strength of the string interactions present in the given spacetime theory. Like the usual formulations of electric-magnetic symmetric actions  (see also ), (1.4) involves an doublet of vector potentials . The crucial difference between (1.4) and the usual actions is that it is also manifestly covariant, without the need of introducing auxiliary fields . This general covariance follows from the fact that the diffeomorphism symmetries of the spacetime are encoded in the tachyon algebra as internal gauge symmetries, so that the gauge invariance of the action automatically makes it covariant. As a consequence of this feature, the on-shell condition for the field strengths is different than those in the usual formulations. Here it corresponds to a dimensional reduction in which the field theory becomes ordinary Yang-Mills theory minimally coupled to massive Dirac fermions. This reduction thus yields a geometrical origin for colour degrees of freedom and fermion mass generation. The field theory (1.4) can thereby be thought of as a first stringy extension of many physical models, such as the standard model and Matrix Theory. As a conventional field theory, however, the Lagrangian (1.4) is highly non-local because it contains infinitely many orders of derivative interactions through the definition of the commutators (derived from the algebra (1.3)). This non-locality, and its origin as an associative noncommutative product, reflects the nature of the string interactions.

Thus, the natural action functional associated with the noncommutative geometry of string theory not only yields an explicitly duality-symmetric (non-local) field theory, but it also suggests a sort of noncommutative Kaluza-Klein mechanism for the origin of nonabelian gauge degrees of freedom and particle masses. The explicit invariance of (1.4) under the duality group of the spacetime thereby yields a physical interpretation of the mathematical notion of Morita equivalence. This gauge theory on the noncommutative torus is different from those discussed in the context of Matrix Theory  but it shares many of their duality properties. The formalism of the present paper may thus be considered as a step towards the formulation of Matrix Theory in terms of the framework of spectral triples in noncommutative geometry . It is a remarkable feature that such target space dynamics can be induced so naturally at the level of a worldsheet formalism.

The structure of the remainder of this paper is as follows. All ideas and results of noncommutative geometry which we use are briefly explained throughout the paper. In section 2 we will briefly define the vertex operator algebra associated with toroidally compactified string theory. In section 3 we study the tachyon algebra of the lattice vertex operator algebra and show that it defines a particular twisted module over the noncommutative torus. In section 4 we introduce a set of spectral data appropriate to the noncommutative geometry of string theory. In section 5 we exploit the duality symmetries of this noncommutative geometry to study some basic properties of the twisted module, including its Morita equivalence classes and its group of automorphisms. In section 6 the most general gauge theory on the module is constructed, and in section 7 that gauge theory is used to derive the duality-symmetric action functional. In section 8 we describe some heuristic, physical aspects of these twisted modules along the lines described in this section. For completeness, an appendix at the end of the paper gives a brief overview of the definition and relevant mathematical significance of Morita equivalence in noncommutative geometry.

2. Lattice Vertex Operator Algebras

In this section we will briefly review, mainly to introduce notation, the definition of a lattice vertex operator algebra (see [18, 33] and references therein for more details). Let be a free infinite discrete abelian group of rank with -bilinear form which is symmetric and nondegenerate. Given a basis of , the symmetric nondegenerate tensor

 gij=⟨ei,ej⟩L (2.1)

defines a Euclidean metric on the flat -dimensional torus . We can extend the inner product to the complexification by -linearity. The dual lattice to is then

 L∗={p∈Lc | ⟨p,w⟩L∈\mybb Z  ∀w∈L} (2.2)

which is also a Euclidean lattice of rank with bilinear form inverse to (2.1) that defines a metric on the dual torus .

Given the lattice and its dual (2.2), we can form the free abelian group

 Λ=L∗⊕L (2.3)

If is a basis of dual to a basis of , then the chiral basis of is , where

 ei±=1√2(ei±gijej) (2.4)

with an implicit sum over repeated indices always understood. Given and , we write the corresponding elements of with respect to the basis (2.4) as with components

 p±i=1√2(pi±⟨ei,w⟩L) (2.5)

Then with , we can define a -bilinear form by

 ⟨p,q⟩Λ≡p+igijq+j−p−igijq−j=⟨p,v⟩L+⟨q,w⟩L (2.6)

and this makes an integral even self-dual Lorentzian lattice of rank and signature which is called the Narain lattice. Note that, in the chiral basis (2.4), the corresponding metric tensor is

 (2.7)

The commutator map

 cΛ(p+,p−;q+,q−)=eiπ⟨q,w⟩L (2.8)

on is a two-cocycle of the group algebra generated by the vector space , the complexification of . This two-cocycle corresponds to a central extension of ,

 1 → \mybb Z2 → ˜Λc → Λc → 1 (2.9)

where as a set and the multiplication is given by

 (ρ;q+,q−)⋅(σ;r+,r−)=(cΛ(q+,q−;r+,r−)ρσ;q++r+,q−+r−) (2.10)

for and . This can be used to define the twisted group algebra associated with the double cover of . A realization of in terms of closed string modes is given as follows. Viewing and as abelian Lie algebras of dimension , we can consider the corresponding affine Lie algebras and , and also the affinization . In the basis (2.4), we then have

 ˆΛc≅ˆu(1)d+⊕ˆu(1)d− (2.11)

where the generators , , of satisfy the Heisenberg algebra

 [α(±)in,α(±)jm]=ngijδn+m,0 (2.12)

The basic operators of interest to us are the chiral Fubini-Veneziano fields

 Xi±(z±)=xi±+igijp±jlogz±+∑n≠01inα(±)inz−n± (2.13)

and the chiral Heisenberg fields

 αi±(z±)=−i∂z±Xi±(z±)=∞∑n=−∞α(±)inz−n−1±      ;      α(±)i0≡gijp±j (2.14)

where . Classically, the fields (2.13) are maps from the Riemann sphere into the torus , with . When they are interpreted as classical string embedding functions from a cylindrical worldsheet into a toroidal target space, the first two terms in (2.13) represent the center of mass (zero mode) motion of a closed string while the Laurent series represents its oscillatory (vibrational) modes. The in (2.5) represent the winding modes of the string about each of the cycles of , while the fields (2.14) are the conserved currents which generate infinitesimal reparametrizations of the torus.

Upon canonical quantization, the oscillatory modes satisfy (2.12) and the zero modes form a canonically conjugate pair,

 [xiα,pβj]=iδβαδji (2.15)

where . Then the multiplication operators

 εq+q−(p+,p−)≡e−iq+ixi+−iq−ixi− cΛ(p+,p−;q+,q−) (2.16)

generate the twisted group algebra . The fields now become quantum operators which act formally on the infinite-dimensional Hilbert space

 h=L2(Td×T∗d,∏di=1dxidx∗i(2π)2)⊗F+⊗F− (2.17)

where and define, respectively, local coordinates on the torus and on the dual torus . The space in (2.17) is generated by the canonical pairs (2.15) of zero modes and is subject to the various natural isomorphisms

 L2(Td×T∗d,∏di=1dxidx∗i(2π)2)≅L2(Td,∏di=1dxi2π)⊗\mybbs CL2(T∗d,∏di=1dx∗i2π)≅⨁w∈LL2(Td,∏di=1dxi2π)≅⨁p∈L∗L2(T∗d,∏di=1dx∗i2π) (2.18)

The Hilbert space (2.17) is thus a module for the group algebras and , and also for . The dense subspace of smooth complex-valued functions on is a unital -algebra which is spanned by the eigenstates of the operators and on , where and . The spaces are bosonic Fock spaces generated by the oscillatory modes which act as annihilation operators for and as creation operators for on some vacuum states .

The basic single-valued quantum fields which act on the Hilbert space (2.17) are the chiral tachyon operators

 Vq±(z±)=∘∘e−iq±iXi±(z±)∘∘ (2.19)

where and denotes the Wick normal ordering defined by reordering the operators (if necessary) so that all , , and occur to the left of all , , and . The twisted dual of the Hilbert space (2.17) is spanned by operators of the form

 Ψ=εq+q−(p+,p−)⊗∏kα(+)ik−nk⊗∏lα(−)jl−ml (2.20)

To (2.20) we associate the vertex operator

 V(Ψ;z+,z−)=∘∘Vq+q−(z+,z−) ∏k1(nk−1)!∂nk−1z+αik+(z+) ∏l1(ml−1)!∂ml−1z−αjl−(z−)∘∘ (2.21)

where

 Vq+q−(z+,z−)≡V(εq+q−(p+,p−)⊗\mybb I;z+,z−)=cΛ(p+,p−;q+,q−)∘∘Vq+(z+)Vq−(z−)∘∘=eiπ⟨q,w⟩L∘∘e−iq+iXi+(z+)−iq−iXi−(z−)∘∘ (2.22)

are the basic tachyon vertex operators. This gives a well-defined linear map on , the latter being the space of endomorphism-valued Laurent polynomials in the variables . This mapping is known as the operator-state correspondence.

With an appropriate regularization (see the next section), the vertex operators (2.21) yield well-defined and densely-defined operators acting on . They generate a noncommutative unital -algebra with the usual Hermitian conjugation and operator norm (defined on an appropriate dense domain of bounded operators in ). The various algebraic properties of can be found in [18, 33], for example. It has the formal mathematical structure of a vertex operator algebra. In this paper we shall not discuss these generic properties of , but will instead analyse in detail a particular ‘subalgebra’ of it which possesses some remarkable properties.

It is important to note that is the -twist of the algebra , where are the chiral algebras generated by the operators (2.19). The lattice itself yields the structure of a vertex operator algebra without any reference to its dual lattice or the Narain lattice with its chiral sectors. Indeed, given a basis of and two-cocycles corresponding to an covering group of the dual lattice, the operators

 ˜Vq(z)=eiπ⟨q,p⟩L∗∘∘e−iqiXi(z)∘∘ (2.23)

generate a vertex operator algebra, acting on the Hilbert space , in an analogous way that the operators (2.22) do . Similarly one defines a vertex operator algebra associated with the lattice . Modulo the twisting factors, the algebras and are isomorphic since the latter representation comes from changing basis on from the chiral one (2.4) to the canonical one . Physically, the difference between working with the full chirally symmetric algebra and only the chiral ones or , is that the former represents the algebra of observables of closed strings while the latter ones are each associated with open strings. Furthermore, the algebra is that which encodes the duality symmetries of spacetime and maps the various open string algebras among each other via duality transformations. Much of what is said in the following will therefore not only apply as symmetries of a closed string theory, but also as mappings among various open string theories. It is in this way that these results are applicable to -brane physics and -Theory.

3. Tachyon Algebras and Twisted Modules over the Noncommutative Torus

The operator product algebra of the chiral tachyon operators (2.19) can be evaluated using standard normal ordering properties to give 

 Vq±(z±)Vr±(z′±)=(z±−z′±)q±igijr±j∘∘Vq±(z±)Vr±(z′±)∘∘ (3.1)

which leads to the operator product

 (3.2)

of the full vertex operators (2.22). They are the defining characteristics of the vertex operator algebra which in the mathematics literature are collectively referred to as the Jacobi identity of [18, 33]. The product of a tachyon operator with its Hermitian conjugate is given by

 Vq±(z±)Vq±(z′±)† = (z±z′±)p±igijq±j(1−z′±z±)q±igijq±j∏n>0eq±iα(±)i−nn(zn±−z′±n) (3.3) ×∏n>0eq±iα(±)inn(z−n±−z′±−n)

The tachyon vertex operators thus form a -algebra which, according to the operator-state correspondence of the previous section, is associated with the subspace defined by

 h0 ≡ ⨂n>0d⨂i=1(kerα(+)in⊗kerα(−)in) ≅ L2(Td×T∗d,∏di=1dxidx∗i(2π)2) (3.4)

Algebraically, is the subspace of highest weight vectors for the representation of on the space, and it is spanned by the vectors . The irreducible (highest weight) representations of this current algebra are labelled by the charges , and carries a representation of given by the actions of . Then,

 h=⎛⎝⨁(q+,q−)∈ΛVq+q−⎞⎠⊗F+⊗F− (3.5)

where and are the irreducible modules for the representation of the Kac-Moody algebra on . In this sense, the tachyon operators generate the full vertex operator algebra , and the algebraic relations between any set of vertex operators (2.21) can be deduced from the operator product formula (3.2) . In the following we will therefore focus on the tachyon sector of the vertex operator algebra.

If is the orthogonal projection onto the subspace (3.4), then the low energy tachyon algebra is defined to be

 A0=P0AP0 (3.6)

If we consider only the zero mode part of the tachyon operators (by projecting out the oscillatory modes) then the corresponding algebra generators are given by

 P0Vq+q−(z+,z−)P0=K(z+,z−) eiπ⟨q,w⟩L e−iqixi−ivix∗i=K(z+,z−)εq+q−(p+,p−) (3.7)

with a worldsheet dependent normalization equal to unity at . The generators of coincide with the generators (2.16) of . From the multiplication property

 εq+q−(p+,p−)εr+r−(p+,p−)=cΛ(q+,q−;r+,r−) ε(q++r+)(q−+r−)(p+,p−) (3.8)

we find the clock algebra

 εq+q−(p+,p−)εr+r−(p+,p−)=eiπγΛ(q+,q−;r+,r−) εr+r−(p+,p−)εq+q−(p+,p−) (3.9)

where

 γΛ(q+,q−;r+,r−)=⟨r,v⟩L−⟨q,w⟩L (3.10)

is a two-cocycle of . This sector represents the extreme low energy of the string theory, in which the oscillator modes are all turned off. In this limit the geometry is a “commutative” one and the physical theory is that of a point particle theory. All stringy effects have been eliminated. In the following we will attempt to go beyond this limit.

The expressions (3.1)–(3.3) are singular at and are therefore to be understood only as formal relationships valid away from coinciding positions of the operators on . Moreover the vertex operators, seen as operators on the Hilbert space, are in general not bounded, and to define a -algebra one usually ‘smears’ them111For issues related to the boundedness of vertex operators see .. Nevertheless the ultraviolet divergence related to the product of the operators at coinciding points can be cured by introducing a cutoff on the worldsheet, a common practice in conformal field theory. We will implement this cutoff by considering a “truncated” algebra obtained by considering only a finite number of oscillators. With this device we solve the unboundedness problem as well. We define

 Vq±(z±)=∏n≥0W±n(z±) (3.11)

where contains the zero modes and , while the ’s for involve only the oscillator modes and . The truncated vertex operator is then defined as

 VNq±(z±)=N±NN∏n=0W±n(z±) (3.12)

The quantity is a normalization constant which we choose to be . With this normalization the operators (3.12) are unitary. There is now no problem in multiplying operators at coinciding points, but the relation (3.1) will change and be valid only in the limit .

The operator product formula (3.1) (and its modification in the finite case) imply a cocycle relation among the operators . Interchanging the order of the two operators and using standard tricks of operator product expansions ), it follows that the truncated algebra is generated by the elements subject to the relations

 VNei±(z±i)VNej∓(z∓j) = VNej∓(z∓j)VNei±(z±i) (3.13) VNei±(z±i)VNei±(z±i)† = VNei±(z±i)†VNei±(z±i) = \mybb I (3.14) VNei±(z±i)VNej±(z±j) = e2πiωNij± VNej±(z±j)VNei±(z±i)    ,    i≠j (3.15)

where is the -involution on , the are distinct points, and

 ωNij±=±gij⎡⎣log(z±iz±j)−N∑n=11n⎛⎝(z±jz±i)n−(z±iz±j)n⎞⎠⎤⎦  . (3.16)

The mutually commuting pair of (identical) algebras (3.14),(3.15) consists of two copies of the algebra of the noncommutative -torus . Choosing an appropriate closure, the algebra can be identified with the abstract unital -algebra generated by elements , subject to the cocycle relations (3.14),(3.15),

 UiU†i = U†iUi = \mybb I (3.17) UiUj = e2πiωijN UjUi    ,    i≠j (3.18)

The presentation of a generic ‘smooth’ element of in terms of the is

 f=∑p∈L∗fp Up11Up22⋯Updd      ,      fp∈S(L∗) (3.19)

where is a Schwartz space of rapidly decreasing sequences. When , one can identify the operator with the multiplication by on the ordinary torus , so that , the algebra of smooth complex-valued functions on . In this case the expansion (3.19) reduces to the usual Fourier series expansion .

For a general non-vanishing anti-symmetric bilinear form , we shall think of the algebra as a quantum deformation of the algebra . While the -involution is the usual complex-conjugation , given their deformed product is defined to be

 (f⋆ωNg)(x)=exp(iπωijN∂∂xi∂∂x′j)f(x)g(x′)∣∣x′=x (3.20)

Furthermore, the unique normalized trace is represented by the classical average 

 τ(f)=∫Tdd∏i=1dxi2π f(x) (3.21)

which is equivalent to projecting onto the zero mode in the Fourier series expansion (3.19), . This trace will be used later on to construct a duality-invariant gauge theory.

Once we have established the connection (at finite ) between the Vertex Operator Algebra and the Noncommutative Torus we can then take the limit . Then, the cocycle defined in (3.16) converges to

 ωij±=±gij sgn(argz±i−argz±j)      ,      i≠j . (3.22)

With we obtain two-cocyles. We have chosen the branch on of the logarithm function for which the imaginary part of lies in the interval . The cocycles depend only on the relative orientations of the phases of the given tachyon operators. If we order these phases so that , then for . This choice is unique up to a permutation in of the coordinate directions of . Thus we obtain the noncommutative torus characterized by defined in (3.22).

We will indicate with the algebra obtained in this fashion and will consider this algebra to act on the tachyon Hilbert space, which is motivated by the fact that at up to Planckian energies only the tachyonic states are excited. Of course at higher energies also the oscillatory (Fock space) modes of the Hilbert space will have to be considered. The algebra that we are considering can also be seen as a deformation of defined in (3.6), in which the product of the elements has to be made in the full algebra, and then the result is to be projected on the tachyon Hilbert space. Namely, given , the operator product expansion with the oscillators, gives non-trivial relations among the tachyons which we identify with the usual relations of the noncommutative torus so that we can define a deformed product by

 V0∗W0=P0(VW)P0  . (3.23)

This is a very natural way to deform the tachyonic algebra which takes the presence of oscillator modes into account, the projection operators being positioned in such a way to ensure that all oscillatory contribute to the product. The -algebra therefore defines a module for the noncommutative torus which possesses some remarkable properties that distinguish it from the usual modules for . Although it is related to the projective regular representation of the twisted group algebra of the Narain lattice, this is not the algebraic feature which determines the cocycle relation (3.15). The clock algebra (3.9) is very different from the algebra defined by (3.15) which arises from the operator product (3.1). The latter algebra is actually associated with the twisted chiral operators , although their chiral-antichiral products do not yield the -twist of the tachyon vertex operators (3.7). Thus the product of the two algebras in (3.13)–(3.15) have deformation matrix associated with the bilinear form (2.7). However, the minus sign which appears in the antichiral sector is irrelevant since by a relabeling of the generators .

The algebra thus defines a -dimensional -twisted module over the noncommutative torus, where carries a double representation of . The non-trivial -twist of this module is important from the point of view of the noncommutative geometry of the string spacetime. Note that its algebra is defined by computing the operator product relations in (3.13)–(3.15) in the full vertex operator algebra , and then afterwards projecting onto the tachyon sector. Otherwise we arrive at the (trivial) clock algebra, so that within the definition of there is a particular ordering that must be carefully taken into account. This module is therefore quite different from the usual projective modules over the noncommutative torus [3, 10], and we shall exploit this fact dramatically in what follows. Note that if we described using instead the canonical basis of the Narain lattice , i.e. taking as generators for twisted products of operators of the form (2.23) and their duals, then we would have arrived at a twisted representation of associated with quantum deformations of the -torus and its dual. That will be a consequence of a set of Morita equivalences that we shall prove in section 5. In particular, this observation shows that for any vertex operator algebra associated with a positive-definite lattice of rank , there corresponds a module over the -dimensional noncommutative torus , with deformation matrix given as above in terms of the bilinear form of , which is determined by an -twisted projective regular representation of the group algebra . We can summarize these results in the following

Proposition 1. Let be a positive-definite lattice of rank with bilinear form , and let be the vertex operator algebra associated with . Then the algebra defines a unitary equivalence class of finitely-generated self-dual -twisted projective modules of the double noncommutative torus with generators
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