Modules Over the Noncommutative Torus and Elliptic Curves

Using the Weil–Brezin–Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely generated projective modules over the algebra A _θ of the noncommutative torus. We show that such A _θ -modules have a natural interpretation as Moyal deformations of vector bundles over an elliptic curve E _τ , under the condition that the deformation parameter θ and the modular parameter τ satisfy a non-trivial relation.     

A Fourier–Mukai transform for real torus bundles

We construct a Fourier–Mukai transform for smooth complex vector bundles E over a torus bundle π:M→B, the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles E with a flat partial unitary connection, that is families or deformations of flat vector bundles (or unitary local systems) on the torus T. This leads to a correspondence between such objects on M and relative skyscraper sheaves supported on a spectral covering , where is the flat dual fiber bundle. Additional structures on (E,) (flatness, anti-self-duality) will be reflected by corresponding data on the transform . Several variations of this construction will be presented, emphasizing the aspects of foliation theory which enter into this picture.     

Nonassociative Strict Deformation Quantization of C*-Algebras and Nonassociative Torus Bundles

In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with H-flux. In particular, the Octonions fit nicely into our theory.     

Quantum Torus Symmetry of the KP,KdV and BKP Hierarchies

In this paper, we construct the quantum torus symmetry of the KP hierarchy and further derive the quantum torus constraint on the tau function of the KP hierarchy. That means we give a nice representation of the quantum torus Lie algebra in the KP system by acting on its tau function. Comparing to the W _∞ symmetry, this quantum torus symmetry has a nice algebraic structure with double indices. Further by reduction, we also construct the quantum torus symmetries of the KdV and BKP hierarchies and further derive the quantum torus constraints on their tau functions. These quantum torus constraints might have applications in the quantum field theory, supersymmetric gauge theory and so on.     

Categories of Holomorphic Vector Bundles on Noncommutative Two-Tori

 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. Received: 24 November 2002 / Accepted: 25 November 2002 Published online: 28 February 2003 RID="⋆" ID="⋆" The work of both authors was partially supported by NSF grants. Communicated by A. Connes     

Equivariant gerbes on complex tori

We explore a new direction in representation theory which comes from holomorphic gerbes on complex tori. The analogue of the theta group of a holomorphic line bundle on a (compact) complex torus is developed for gerbes in place of line bundles. The theta group of symmetries of the gerbe has the structure of a Picard groupoid. We calculate it explicitly as a central extension of the group of symmetries of the gerbe by the Picard groupoid of the underlying complex torus. We discuss obstruction to equivariance and give an example of a group of symmetries of a gerbe with respect to which the gerbe cannot be equivariant. We calculate the obstructions to invariant gerbes for some group of translations of a torus to be equivariant. We survey various types of representations of the group of symmetries of a gerbe on the stack of sheaves of modules on the gerbe and the associated abelian category of sheaves on the gerbe (twisted sheaves).     

Torus Knots and the Topological Vertex

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S ³. The key role is played by the \({SL(2, \mathbb{Z})}\) transformation, which generates a general torus knot from the unknot. Applying the topological vertex to the proposed A-branes, we rederive the colored HOMFLY polynomials for torus knots, in agreement with the Rosso and Jones formula. We show that our A-model construction is mirror symmetric to the B-model analysis of Brini, Eynard and Mariño. Compared to the recent proposal by Aganagic and Vafa for knots on S ³, we demonstrate that the disk amplitude of the A-brane associated with any knot is sufficient to reconstruct the entire B-model spectral curve. Finally, the construction of toric Lagrangian A-branes is generalized to other local toric Calabi–Yau geometries, which paves the road to study knots in other three-manifolds such as lens spaces.     

Mirror Symmetry in Two Steps: A–I–B

We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I–model. The equivalence between the A–model and the I–model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T–duality. On the other hand, the I–model is closely related to the twisted Landau-Ginzburg model (the B–model) that is mirror dual to the A–model. Thus, the mirror symmetry is realized in two steps, via the I–model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I–model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.Partially supported by the DARPA grant HR0011-04-1-0031 and the NSF grant DMS-0303529.Partially supported by the Federal Program 40.052.1.1.1112, by the Grants INTAS 03-51-6346, NSh-1999/2003.2 and RFFI-04-01- 00637.     

A note on symmetric connections

In this paper we analyze a reciprocal of the fundamental theorem of Riemannian geometry. We give a condition for a symmetric connection to be locally the Levi-Civita connection of a metric. We also construct a couple of natural examples of connections on the n-dimensional torus and investigate the global problem.     

Nonlinear science abstracts

The class of exactly integrable non-linear evolution equations (NLEE) related to the general first order n × n linear problem is studied. The set of independent scattering data $\text{J}$ is determined and trace identities are obtained. Next we use a special integro-differential operator Λ and obtain the expansions of the potential and its variation in the eigenfunctions of Λ. These enable us to consider the inverse scattering method as a generalized Fourier transform and to write down in a compact form the integrals of motion. We prove that there exist a hierarchy of Hamiltonian structures generated by Λ. At the end we calculate the symplectic form in terms of the scattering data variations.     

Torus Knot Polynomials and Susy Wilson Loops

We give, using an explicit expression obtained in (Jones V, Ann Math 126:335, 1987), a basic hypergeometric representation of the HOMFLY polynomial of (n, m) torus knots, and present a number of equivalent expressions, all related by Heine’s transformations. Using this result, the \({ (m, n) \leftrightarrow (n, m)}\) symmetry and the leading polynomial at large N are explicit. We show the latter to be the Wilson loop of 2d Yang–Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang–Mills theory, which is known to give averages of Wilson loops in \({\mathcal{N}}\) = 4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones–Rosso representation in terms of q-harmonic oscillators.     

On the geometrization of matter by exotic smoothness

In this paper we discuss the question how matter may emerge from space. For that purpose we consider the smoothness structure of spacetime as underlying structure for a geometrical model of matter. For a large class of compact 4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of Fintushel and Stern to change the smoothness structure. The influence of this surgery to the Einstein–Hilbert action is discussed. Using the Weierstrass representation, we are able to show that the knotted torus used in knot surgery is represented by a spinor fulfilling the Dirac equation and leading to a Dirac term in the Einstein–Hilbert action. For sufficient complicated links and knots, there are “connecting tubes” (graph manifolds, torus bundles) which introduce an action term of a gauge field. Both terms are genuinely geometrical and characterized by the mean curvature of the components. We also discuss the gauge group of the theory to be U(1) × SU(2) ×?SU(3).     

Dirac operators on Lagrangian submanifolds

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge–de Rham Laplacian provided the complex structure identifies the spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.     

Supersymmetric topological terms in D = 3 nonlinear sigma models

We discuss generalized Wess-Zumino-Witten terms in supersymmetric nonlinear sigma models on three-dimensional spacetime, and give a natural N = 2 supersymmetric term for any Kähler manifold. We show that the coefficient is not quantized if isometries of the manifold are not gauged. The relation between generalized Wess-Zumino-Witten terms in different spacetime dimensions is discussed, and some useful superspace methods are introduced.     

Geometry of Higgs and Toda fields on Riemann surfaces

We discuss geometrical aspects of Higgs systems and Toda field theory in the framework of the theory of vector bundles on Riemann surfaces of genus greater than one. We point out how Toda fields can be considered as equivalent to Higgs systems — a connection on a vector bundle E together with an End(E)-valued one form both in the standard and in the Conformal Affine case. We discuss how variations of Hodge structures can arise in such a framework and determine holomorphic embeddings of Riemann surfaces into locally homogeneous spaces, thus giving hints to possible realizations of W_n-geometries.     

Towards a finite N = 2 susy GUT

We consider finite, N = 2 supersymmetric GUTs based on gauge groups SU(n) and SO(n). As an example, we discuss a semirealistic model based on SO(12). We argue that in finite, N = 2 supersymmetric GUTs, gauge symmetry breaking should occur dynamically. We present a heuristic picture in which this is induced by soft, finiteness preserving SUSY breaking terms. The bound states formed cause a very rapid evolution of the SO(12) coupling constant and break SO(12) into SU(4)×SU(3)_C×U(1).     

Generalized Kähler Geometry

Generalized Kähler geometry is the natural analogue of Kähler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We prove that generalized Kähler geometry is equivalent to the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry. We also prove the existence of natural holomorphic Courant algebroids for each of the underlying complex structures, and that these split into a sum of transverse holomorphic Dirac structures. Finally, we explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.