Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

We present several results on the geometry of the quantum projective plane. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles with explicit computation of the corresponding ‘classical’ characteristic classes (via Fredholm modules); complete diagonalization of gauged Laplacians on these line bundles; ‘quantum’ characteristic classes via equivariant K-theory and q-indices.     

Higher-Degree Analogs of the Determinant Line Bundle

 In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-type operators whose index lies in the subspace of the reduced K-theory of the parametrizing space, we construct a set of Deligne cohomology classes of degree i whose curvatures are the i-form component of the Atiyah-Singer families index theorem. Received: 27 September 2001 / Accepted: 5 April 2002 Published online: 22 August 2002     

Instantonic branes, Atiyah–Hirzebruch spectral sequence, and duality of SYM

We study the equivalence among orientifold three-planes in the context of the Atiyah–Hirzebruch spectral sequence. This equivalence refers to the fact that two different cohomology classes in the same cohomology group, which classify the orientifolds, are lifted to the same trivial class in K-theory. The physical interpretation of this mathematical algorithm is given by the role of D-brane instantons. By following some recent ideas, we extend the sequence to include a classification of NS–NS fluxes. We find that such equivalences, in the low energy limit of the dynamics on the worldvolume of type IIB D3-branes on top of the orientifolds, are interpreted as the duality in four-dimensional SYM theories.     

A higher twist in string theory

Considering the gauge field and its dual in heterotic string theory as a unified field, we show that the equations of motion at the rational level contain a twisted differential with a novel degree seven twist. This generalizes the usual degree three twist that lifts to twisted K-theory and raises the natural question of whether at the integral level the abelianized gauge fields belong to a generalized cohomology theory. Some remarks on possible such extension are given.     

Comparison Theory and Smooth Minimal C*-Dynamics

We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar’s Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C*-algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C*-algebras with the same K-theory and tracial state space, providing a C*-algebraic analogue of McDuff’s uncountable family of II₁ factors. We prove in passing that the range of the radius of comparison is exhausted by simple C*-algebras. This research was supported in part by an NSERC Discovery Grant.     

Differential twisted K-theory and applications

In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann–Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.     

SPINORS ON KAHLER–NORDEN MANIFOLDS

It is known that the complex spin group Spin(n, ?) is the universal covering group of complex orthogonal group SO(n, ?). In this work we construct a new kind of spinors on some classes of Kahler–Norden manifolds. The structure group of such a Kahler–Norden manifold is SO(n, ?) and has a lifting to Spin(n, ?). We prove that the Levi-Civita connection on M is an SO(n, ?)-connection. By using the spinor representation of the group Spin(n, ?), we define the spinor bundle S on M. Then we define covariant derivative operator ? on S and study some properties of ?. Lastly we define Dirac operator on S.     

Periodicity and the Determinant Bundle

The infinite matrix ‘Schwartz’ group G ^−∞ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on G ^−∞. We show that while the higher (even, Schwartz) loop groups of G ^−∞, again classifying for odd K-theory, do not carry multiplicative determinants generating the first Chern class, ‘dressed’ extensions, corresponding to a star product, do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the eta invariant. In so doing we relate two distinct extensions of the eta invariant to self-adjoint elliptic operators and to elliptic invertible suspended families and show that the corresponding τ invariant is a determinant in this sense. The first author acknowledges the support of the National Science Foundation under grant DMS0408993, the second author acknowledges support of the Fonds québécois sur la nature et les technologies and NSERC while part of this work was conducted.     

Families Index Theorem in Supersymmetric WZW Model and Twisted K-Theory: The <Emphasis Type="Italic">SU</Emphasis>(2) Case

The construction of twisted K-theory classes on a compact Lie group is reviewed using the supersymmetric Wess-Zumino-Witten model on a cylinder. The Quillen superconnection is introduced for a family of supercharges parametrized by a compact Lie group and the Chern character is explicitly computed in the case of SU(2). For large euclidean time, the character form is localized on a D-brane.     

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory

We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings of D-branes which generalize previous examples. We propose a definition for groups of differential characters associated to equivariant K-theory. We derive a Dirac quantization rule for Ramond-Ramond fluxes, and study flat Ramond-Ramond potentials on orbifolds.     

Representation of Spin Group Spin（p, q）

The representation η（P, q） of spin group Spin（p, q） in any dimensional space is given by induction, and the relation between two representations, which are obtained in two kinds of inductions from Spin（p, q） to Spin（p ＋ 1, q ＋ 1） are studied.     

On the Landau-Ginzburg type theory of quark confinement

We present a detailed investigation of the interaction of “magnetically” charged quarks in the vacuum of type II superconductivity (represented in a relativistic theory by a Higgs field). The analysis of confining forces is given in details. Spin dependence, relationship to other confining mechanisms and first quantization are discussed.     

Fermion Representation of Quantum Toroidal Algebra

We construct a fermion analogue of the Fock representation of quantum toroidal algebra and construct the fermion representation of quantum toroidal algebra on the K-theory of Hilbert scheme.     

Non-Commutative Methods for the K-Theory of C*-Algebras of Aperiodic Patterns from Cut-and-Project Systems

We investigate the C*-algebras associated to aperiodic structures called model sets obtained by the cut-and-project method. These C*-algebras are Morita equivalent to crossed product C*-algebras obtained from dynamics on a disconnected version of the internal space. This construction may be made from more general data, which we call a hyperplane system. From a hyperplane system, others may be constructed by a process of reduction and we show how the C*-algebras involved are related to each other. In particular, there are natural elements in the Kasparov KK-groups for the C*-algebra of a hyperplane system and that of its reduction. The induced map on K-theory fits in a six-term exact sequence. This provides a new method of the computation of the K-theory of such C*-algebras which is done completely in the setting of non-commutative geometry.     

Twisted Orbifold K-Theory

 We use equivariant methods to define and study the orbifold K-theory of an orbifold X. Adapting techniques from equivariant K-theory, we construct a Chern character and exhibit a multiplicative decomposition for K ^* _orb (X)⊗ℚ, in particular showing that it is additively isomorphic to the orbifold cohomology of X. A number of examples are provided. We then use the theory of projective representations to define the notion of twisted orbifold K–theory in the presence of discrete torsion. An explicit expression for this is obtained in the case of a global quotient. Received: 21 August 2001 / Accepted: 27 January 2003 Published online: 13 May 2003 RID="*" ID="*" Both authors were partially supported by the NSF RID="*" ID="*" Both authors were partially supported by the NSF Communicated by R.H. Dijkgraaf     

Calibrations and Intersecting Branes

We investigate the solutions of Nambu–Goto-type actions associated with calibrations. We determine the supersymmetry preserved by these solutions using the contact set of the calibration and examine their bulk interpretation as intersecting branes. We show that the supersymmetry preserved by such solutions is closely related to the spinor singlets of the subgroup G of Spin (9,1) or Spin (10,1) that rotates the tangent spaces of the brane. We find that the supersymmetry projections of the worldvolume solutions are precisely those of the associated bulk configurations. We also investigate the supersymmetric solutions of a Born–Infeld action. We show that in some cases this problem again reduces to counting spinor singlets of a subgroup of Spin (9,1) acting on the associated spinor representations. We also find new worldvolume solutions which preserve 1/8 of the supersymmetry of the bulk and give their bulk interpretation. Received: 20 May 1998 / Accepted: 16 November 1998     

Remarks on 2–q-bit states

We distinguish six classes of families of locally equivalent states in a straightforward scheme for classifying all 2–q-bit states; four of the classes consist of two subclasses each. The simple criteria that we stated recently for checking a given state’s positivity and separability are justified, and we discuss some important properties of Lewenstein–Sanpera decompositions. An upper bound is conjectured for the sum of the degree of separability of a 2–q-bit state and its concurrence. Received: 17 July 2000 / Published online: 6 December 2000