In this paper we study the moduli space of standard holomorphic structures on a noncommutative complex two torus. It will be shown that the moduli space is naturally identified with the moduli space of stable bundles on an elliptic curve. We also propose that the mirror reflection of the noncommutative complex torus is the mirror reflection of the elliptic curve together with a linear foliation. From this we identify the moduli space of super cycles on the mirror reflection with the moduli space of standard holomorphic bundles on a noncommutative complex torus.
In this paper we study the category of standard holomorphic vector bundles on a noncommutative two-torus. We construct a functor from the derived category of such bundles to the derived category of coherent sheaves on an elliptic curve and prove that it induces an equivalence with the subcategory of stable objects. By the homological mirror symmetry for elliptic curves this implies an equivalence between the derived category of holomorphic bundles on a noncommutative two-torus and the Fukaya category of the corresponding symplectic (commutative) torus.
We construct explicitly noncommutative deformations of categories of holomorphic line bundles over higher dimensional tori. Our basic tools are Heisenberg modules over noncommutative tori and complex/holomorphic structures on them introduced by A. Schwarz. We obtain differential graded (DG) categories as full subcategories of curved DG categories of Heisenberg modules over the complex noncommutative tori. Also, we present the explicit composition formula of morphisms, which in fact depends on the noncommutativity.
We construct the so-called theta vectors on noncommutative T^4, which correspond to the theta functions on commutative tori with complex structures. Following the method of Dieng and Schwarz, we first construct holomorphic connections and then find the functions satisfying the holomorphic conditions, the theta vectors. The holomorphic structure in the noncommutative T^4 case is given by a 2x2 complex matrix, and the consistency requires its off-diagonal elements to be the same. We also construct the tensor product of these functions satisfying the consistency requirement.
We prove that every holomorphic vector bundle on a noncommutative two-torus $T$ can be obtained by successive extensions from standard holomorphic bundles considered in math.QA/0211262. This implies that the category of holomorphic bundles on $T$ is equivalent to the heart of certain $t$-structure on the derived category of coherent sheaves on an elliptic curve.
Ordinary theta-functions can be considered as holomorphic sections of line bundles over tori. We show that one can define generalized theta-functions as holomorphic elements of projective modules over noncommutative tori (theta-vectors). The theory of these new objects is not only more general, but also much simpler than the theory of ordinary theta-functions. It seems that the theory of theta-vectors should be closely related to Manin's theory of quantized theta-functions, but we don't analyze this relation.
A one parameter set of noncommutative complex algebras is given. These may be considered deformation quantisation algebras. The commutative limit of these algebras correspond to the algebra of polynomial functions over a manifold or variety. The topology of the manifold or variety depends on the parameter, varying from nothing, to a point, a sphere, a certain variety and finally a torus. The irreducible adjoint preserving representations of the noncommutative algebras are studied. As well as typical noncommutative sphere type representations and noncommutative torus type representations, a new…
In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative definition of noncommutative non-principal torus bundles.
This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In \cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect category of $\A^\bullet$, which corresponded to the category of coherent sheaves on a complex manifold. In this paper we enlarge this category to include objects which correspond to quasi-coherent sheaves. We then apply this framework to proving an equivalence of categories between derived categories on the noncommutative complex torus and on a holomorphic gerbe on t…
We give an interpretation of the Bargman transform as a correspondence between state spaces that is analogous to commonly considered intertwiners in representation theory of finite groups. We observe that the non-commutative torus is nothing else that the range of the star-exponential for the Heisenberg group within the Kirillov's orbit method context. We deduce from this a realization of the non-commutative torus as acting on a Fock space of entire functions.
We compute the Wodzicki residue of the inverse of a conformally rescaled Laplace operator over a 4-dimensional noncommutative torus. We show that the straightforward generalization of the Laplace-Beltrami operator to the noncommutative case is not the minimal operator.
We classify spin structures on the noncommutative torus, and find that the noncommutative n-torus has 2^n spin structures, corresponding to isospectral deformations of spin structures on the commutative n-torus. For n>3 the classification depends on Connes' spin manifold theorem. In addition, we study unitary equivalences of these spin structures.
We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein module of a cylinder over the torus.
We define the notion of a holomorphic bundle on the noncommutative toric orbifold $T_θ/G$ associated with an action of a finite cyclic group $G$ on an irrational rotation algebra. We prove that the category of such holomorphic bundles is abelian and its derived category is equivalent to the derived category of modules over a finite-dimensional algebras $Λ$. As an application we finish the computation of $K_0$-groups of the crossed product algebras describing the above orbifolds initiated by Kumjian, Walters and Buck. Also, we describe a torsion pair in the category of $Λ$-modules, such that th…
We find and classify possible equivariant spin structures with Dirac operators on the noncommutative torus, proving that similarly as in the classical case the spectrum of the Dirac operator depends on the spin structure.
We compute the scalar curvature of a curved noncommutative 3-torus. To perturb the flat metric, the standard volume form on the noncommutative 3-torus is conformally perturbed and the corresponding perturbed Laplacian is analyzed. Using Connes' pseudodifferential calculus for the noncommutative 3-torus, we explicitly compute the first three terms of the small time heat kernel expansion for the perturbed Laplacian. The third term of the expansion gives a local formula for the scalar curvature. Finally, we show that in the classical limit when the deformation parameters vanish, our formula coinc…
The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p+1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation, which yields a recursive relation for computing all colored Jones polynomials of the knot.
We describe how and to what extent the noncommutative two-torus can be approximated by a tower of finite-dimensional matrix geometries. The approximation is carried out for both irrational and rational deformation parameters by embedding the algebra of the noncommutative torus into an approximately finite algebra. The construction is a rigorous derivation of the recent discretizations of noncommutative gauge theories using finite dimensional matrix models, and it shows precisely how the continuum limits of these models must be taken. We clarify various aspects of Morita equivalence using this …
We introduce a family of spectral triples that describe the curved noncommutative two-torus. The relevant family of new Dirac operators is given by rescaling one of two terms in the flat Dirac operator. We compute the dressed scalar curvature and show that the Gauss-Bonnet theorem holds (which is not covered by the general result of Connes and Moscovici).
Spectral triples over noncommutative principal $\T^n$-bundles are studied, extending recent results about the noncommutative geometry of principal U(1)-bundles. We relate the noncommutative geometry of the total space of the bundle with the geometry of the base space. Moreover, strong connections are used to build new Dirac operators. We discuss as a particular case the commutative case, the noncommutative tori and theta deformed manifolds.
It is known that the physics of open strings on a D2-brane on a two-torus is realized from the viewpoint of deformation quantization in the Seiberg-Witten limit. We study its T-dual theory, i.e. D1-brane physics on two-tori. Such theory is described by Kronecker foliation. The algebra of open strings on the D1-brane is then identified with the crossed product representation of a noncommutative two-torus. The Morita equivalence of noncommutative two-tori is also realized geometrically along this line. As an application, Heisenberg modules and the tensor product between them are discussed from t…
This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of as category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considere…
In this paper we show that the value at zero of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent of the choice of the volume element (Weyl factor) given by a (non-unimodular) state. We had obtained, in the late eighties, in an unpublished computation, a general formula for this value at zero involving modified logarithms of the modular operator of the state. We give here the detailed computation and prove that the result is independent of the Weyl factor as in the classical case, thus proving the analogue of the…
As a noncommutative generalization of the addition formula of theta functions, we construct a class of theta functions which are closed with respect to the Moyal star product of a fixed noncommutative parameter. These theta functions can be regarded as bases of the space of holomorphic homomorphisms between holomorphic line bundles over noncommutative complex tori.
We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using an analogue of Kontsevich-Vishik canonical trace, defined on Connes' algebra of classical pseudodifferential symbols for the noncommutative two torus, we compute the curvature form of the determinant line bundle by computing the second variation $δ_{w}δ_{\bar{w}}\log\det(Δ)$.
The twined almost commutative structure of the standard spectral triple on the noncommutative torus with rational parameter is exhibited, by showing isomorphisms with a spectral triple on the algebra of sections of certain bundle of algebras, and a spectral triple on a certain invariant subalgebra of the product algebra. These isomorphisms intertwine also the grading and real structure. This holds for all four inequivalent spin structures, which are explicitly constructed in terms of double coverings of the noncommutative torus (with arbitrary real parameter). These results are extended also t…
We construct explicitly the quantization of classical linear maps of $SL(2, R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of $SL(2, Z)$ to the subgroup $SL(2, Z)/Γ_l$, $Γ_l$ being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, $O_{l}(2)\subset SL(2,l)$, for arbitrary values of l are determin…
We study perturbations of the flat geometry of the noncommutative two-dimensional torus T^2_θ(with irrational θ). They are described by spectral triples (A_θ, \H, D), with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra A_θof T_θ. We show, up to the second order in perturbation, that the zeta-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We also calculate first two terms of the perturbative expansion of the corresponding local scalar curvature.
For quantum torus generated by unitaries $UV = e(θ)VU$ there exist nontrivial strong Morita autoequivalences in case when $θ$ is real quadratic irrationality. A.Polishchuk introduced and studied the graded ring of holomorphic sections of powers of the respective bimodule (depending on the choice of a complex structure). We consider a Segre square of this ring whose graded components are spanned by Rieffel scalar products of Polishchuk's holomorphic vectors. These graded components are linear spaces of quantum theta functions in sense of Yu.Manin.
We give a local expression for the {\it scalar curvature} of the noncommutative two torus $ A_θ = C(\mathbb{T}_θ^2)$ equipped with an arbitrary translation invariant complex structure and Weyl factor. This is achieved by evaluating the value of the (analytic continuation of the) {\it spectral zeta functional} $ζ_a(s): = \text{Trace}(a \triangle^{-s})$ at $s=0$ as a linear functional in $a \in C^{\infty}(\mathbb{T}_θ^2)$. A new, purely noncommutative, feature here is the appearance of the {\it modular automorphism group} from the theory of type III factors and quantum statistical mechanics in t…
We construct a new gauge theory on a pair of d-dimensional noncommutative tori. The latter comes from an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra A and the noncommutative torus. We show that the (truncated) tachyon subalgebra of A is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding even real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple pr…
We define holomorphic structures on canonical line bundles of the quantum projective space $\qp^{\ell}_q$ and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of $\qp^{\ell}_q$ is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of Riemann-Roch formula and Serre duality for $\qp^{1}_q$ and $\qp^{2}_q$.
We prove the analogue of the Riemann-Roch formula for the noncommutative two torus $ A_θ = C(\mathbb{T}_θ^2)$ equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive element $k\in C^{\infty}(\mathbb{T}_θ^2)$. We consider a topologically trivial line bundle equipped with a general holomorphic structure and the corresponding twisted Dolbeault Laplacians. We define an spectral triple ($A_θ, \mathcal{H}, D)$ that encodes the twisted Dolbeault complex of $ A_θ$ and whose index gives the left hand side of the Riemann-Roch formula. Using Connes' …
The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudo-differential operators on $\mathbb R^n$, one may derive a similar pseudo-differential calculus on noncommutative $n$ tori $\mathbb T_θ^n$, and with the development of this calculus came many results concerning the local differential geometry of noncommutative tori for $n=2,4$, as shown in the groundbreaking paper in which t…
We define a noncommutative residue for classical Euclidean pseudodifferential operators on a torus of arbitrary dimension. We prove that, up to multiplication by a constant, it is the unique trace on the algebra of classical pseudodifferential operators modulo infinitely smoothing operators. In the case of the two torus, we show that the noncommutative residue is the semiclassical limit of a noncommutative residue defined on classical pseudodifferential operators on noncommutative two tori.
We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive an explicit formula for the partition function of Yang-Mills theory defined on a projective module for arbitrary noncommutativity parameter θwhich is manifestly invariant under gauge Morita equivalence. The energy observables are shown to be smooth functions of θ. The construction of noncommutative …
We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which ar…
This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential graded algebra. This is compared to current ideas on noncommutative algebraic geometry.
The scalar curvature for the noncommutative four torus $\mathbb{T}_Θ^4$, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite differenc…
We construct the join of noncommutative Galois objects (quantum torsors) over a Hopf algebra H. To ensure that the join algebra enjoys the natural (diagonal) coaction of H, we braid the tensor product of the Galois objects. Then we show that this coaction is principal. Our examples are built from the noncommutative torus with the natural free action of the classical torus, and arbitrary anti-Drinfeld doubles of finite-dimensional Hopf algebras. The former yields a noncommutative deformation of a non-trivial torus bundle, and the latter a finite quantum covering.
We study the homogeneous coordinate rings of real multiplication noncommutative tori as defined by A. Polishchuk. Our aim is to understand how these rings give rise to an arithmetic structure on the noncommutative torus. We start by giving an explicit presentation of these rings in terms of their natural generators. This presentation gives us information about the rationality of the rings and is the starting point for the analysis of the corresponding geometric data. Linear bases for these rings are also discussed. The basic role played by theta functions in these computations gives relations …
In this paper, we define NC complex spaces as complex spaces together with a structure sheaf of associative algebras in such a way that the abelization of the structure sheaf is the sheaf of holomorphic functions.
We introduce and study holomorphically finitely generated (HFG) Fréchet algebras, which are analytic counterparts of affine (i.e., finitely generated) $\mathbb C$-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some standard constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions…
We prove that the Shimizu L-function of a real quadratic field is obtained from a (Lorentzian) spectral triple on a noncommutative torus with real multiplication, as an adiabatic limit of the Dirac operator on a 3-dimensional solvmanifold. The Dirac operator on this 3-dimensional geometry gives, via the Connes-Landi isospectral deformations, a spectral triple for the noncommutative tori obtained by deforming the fiber tori to noncommutative spaces. The 3-dimensional solvmanifold is the homotopy quotient in the sense of Baum--Connes of the noncommutative space obtained as the crossed product of…
We compute the Picard group $ Pic(A_q) $ of the noncommutative algebraic 2-torus $A_q$, describe its action on the space $ R(A_q) $ of isomorphism classes of rk 1 projective modules and classify the algebras Morita equivalent to $ A_q $. Our computations are based on a quantum version of the Calogero-Moser correspondence relating projective $A_q$-modules to irreducible representations of the double affine Hecke algebras (DAHA) $ H_{t, q^{-1/2}}(S_n) $ at $ t = 1 $. We show that, under this correspondence, the action of $ Pic(A_q) $ on $ R(A_q) $ agrees with the action of $ SL_2(Z) $ on $ H_{t,…
We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules.We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors generalizes the theory of theta-functions. The paper is self-contained; it can be used also as an introduction to the theory of noncommutative spaces with simplest space of this kind thoroughly analyzed as a basic example.
The spectral eta function for certain families of Dirac operators on noncommutative $3$-torus is considered and the regularity at zero is proved. By using variational techniques, we show that $η_{D}(0)$ is a conformal invariant. By studying the Laurent expansion at zero of $\text{TR} (|D|^{-z})$, the conformal invariance of $ζ'_{|D|}(0)$ for noncommutative $3$-torus is proved. Finally, for the coupled Dirac operator, a local formula for the variation $\partial_Aη_{D+A}(0)$ is derived which is the analogue of the so called induced Chern-Simons term in quantum field theory literature.
We recall a construction of non-commutative algebras related to a one-parameter family of (deformed) spheres and tori, and show that in the case of tori, the *-algebras can be completed into C*-algebras isomorphic to the standard non-commutative torus. As the former was constructed in the context of matrix (or fuzzy) geometries, it provides an important link to the framework of non-commutative geometry, and opens up for a concrete way to study deformations of non-commutative tori. Furthermore, we show how the well-known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of f…
We prove that, for the moduli space of flat SU(2)-connections on the torus, the Weyl quantization and the quantization using the quantum group of SL(2,C) are unitarily equivalent. This is done by comparing the matrices of the operators associated by the two quantization to cosine functions. We also discuss the *-product of the Weyl quantization and show that it satisfies the product-to-sum formula for noncommutative cosines on the noncommutative torus.
We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the colored Jones polynomials of the knot.
Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended in following two directions. First, bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized. Second, bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated, though in this case the base space of the bundles is best viewed as a topological stack. Some methods developed for the con…
We compute the Ricci curvature of a curved noncommutative three torus. The computation is done both for conformal and non-conformal perturbations of the flat metric. To perturb the flat metric, the standard volume form on the noncommutative three torus is perturbed and the corresponding perturbed Laplacian is analyzed. Using Connes' pseudodifferential calculus for the noncommutative tori, we explicitly compute the second term of the short time heat kernel expansion for the perturbed Laplacians on functions and on 1-forms. The Ricci curvature is defined by localizing heat traces suitably. Equiv…
