Gerbes over Orbifolds and Twisted K-Theory

In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H³(X) via Chern-Weil theory. For an arbitrary gerbe , a twisting K_orb(X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], Atiyah-Segal [2] and Bowknegt et. al. [4] in the smooth case and by Adem-Ruan [1] for discrete torsion on an orbifold.The first author was partially supported by the National Science Foundation and Conacyt-México     

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     

Twisted K-Theory and Finite-Dimensional Approximation

We provide a finite-dimensional model of the twisted K-group twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta’s generalized vector bundle, and the other is a finite-dimensional approximation of Fredholm operators.     

Unoriented WZW Models and Holonomy of Bundle Gerbes

The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models. manche meinen lechts und rinks kann man nicht velwechsern werch ein illtum Ernst Jandl [Jan95] K.W. is supported with scholarships by the German Israeli Foundation (GIF) and by the Rudolf und Erika Koch–Stiftung.     

Chern-Weil Construction for Twisted K-Theory

We give a finite-dimensional and geometric construction of a Chern character for twisted K-theory, introducing a notion of connection on a twisted vectorial bundle which can be considered as a finite-dimensional approximation of a twisted family of Fredholm operators. Our construction is applicable to the case of any classes giving the twisting, and agrees with the Chern character of bundle gerbe modules in the case of torsion classes.     

Fractional Loop Group and Twisted K-Theory

We study the structure of abelian extensions of the group L _q G of q-differentiable loops (in the Sobolev sense), generalizing from the case of the central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of the supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on G is discussed.     

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H⁴(BG, ℤ) for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H⁴(BG, ℤ) to H³(G, ℤ). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.The authors acknowledge the support of the Australian Research Council. ALC thanks MPI für Mathematik in Bonn and ESI in Vienna and BLW thanks CMA of Australian National University for their hospitality during part of the writing of this paper.     

Higgs Fields,Bundle Gerbes and String Structures

We use bundle gerbes and their connections and curvings to obtain an explicit formula for a de Rham representative of the string class of a loop group bundle. This is related to earlier work on calorons.Both authors acknowledge the support of the Australian Research Council.     

Nonabelian Bundle Gerbes,Their Differential Geometry and Gauge Theory

Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential.Acknowledgement We have benefited from discussions with L. Breen, D. Husemoller, A. Alekseev, L. Castellani, J. Kalkkinen, J. Mickelsson, R. Minasian, D. Stevenson and R. Stora.     

Chern Character in Twisted K-Theory: Equivariant and Holomorphic Cases

 It was argued in [25, 5] that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are classified by twisted K-theory. In [4], it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary Hilbert bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective unitary bundle is in turn determined by the twist. This paper studies in detail the Chern-Weil representative of the Chern character of bundle gerbe K-theory that was introduced in [4], extending the construction to the equivariant and the holomorphic cases. Included is a discussion of interesting examples. Received: 10 January 2002 / Accepted: 9 December 2002 Published online: 25 February 2003 RID="⋆" ID="⋆" The authors acknowledge the support of the Australian Research Council Communicated by R.H. Dijkgraaf     

Gerbes and Duality

We describe a global approach to the study of duality transformations between antisymmetric fields with transitions and argue that the natural geometrical setting for the approach is that of gerbes; these objects are mathematical constructions generalizing U(1) bundles and are similarly classified by quantized charges. We address the duality maps in terms of the potentials rather than on their field strengths and show the quantum equivalence between dual theories which in turn allows a rigorous proof of a generalized Dirac quantization condition on the couplings. Our approach needs the introduction of an auxiliary form satisfying a global constraint which in the case of 1-form potentials coincides with the quantization of the magnetic flux. We apply our global approach to refine the proof of the duality equivalence between the d=11 supermembrane and d=10 IIA Dirichlet supermembrane.     

Equivariant Holonomy for Bundles and Abelian Gerbes

This paper generalizes Bismut’s equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local version of the higher Hochschild complex, resulting in differential forms given by iterated integrals. Connections to two dimensional topological field theories are indicated. Similarly, this local higher Hochschild complex is used to calculate the 2-holonomy of an abelian gerbe along any closed oriented surface, as well as the derivative of 2-holonomy, which in the case of a torus fits into a sequence of higher holonomies and their differentials.     

Gravitational Anomalies, Gerbes, and¶Hamiltonian Quantization

In ref. [1], Schwinger terms in hamiltonian quantization of chiral fermions coupled to vector potentials were computed, using some ideas from the theory of gerbes, with the help of the family index theorem for a manifold with boundary. Here, we generalize this method to include gravitational Schwinger terms. Received: 5 May 1999 / Accepted: 30 January 2000     

K-Theory, Reality, and Orientifolds

We use equivariant K-theory to classify charges of new (possibly non-supersymmetric) states localized on various orientifolds in Type II string theory. We also comment on the stringy construction of new D-branes and demonstrate the discrete electric-magnetic duality in Type I brane systems with p+q=7, as proposed by Witten.     

Index Theory, Gerbes, and Hamiltonian Quantization

We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms (Faddeev-Mickelsson cocycle) for the gauge group action. We relate the APS construction to the bundle gerbe approach discussed recently by Carey and Murray, including an explicit computation of the Dixmier-Douady class. An advantage of our method is that it can be applied whenever one has a form of the APS theorem at hand, as in the case of fermions in an external gravitational field. Received:     

Gerbes,Simplicial Forms and Invariants for Families of Foliated Bundles

The notion of smooth Deligne cohomology is conveniently reformulated in terms of the simplicial deRham complex. In particular the usual Chern-Weil and Chern-Simons theory is well adapted to this framework and rather easily gives rise to characteristic Deligne cohomology classes associated to families of bundles and connections. In turn this gives invariants for families of foliated bundles. The construction provides representing cocycles in the usual ech-deRham model for smooth Deligne cohomology called gerbes with connection as they generalize usual Hermitian line bundles with connection. A special case is the Quillen line bundle associated to families of flat SU(2)-bundles.Work supported in part by the Erwin Schrödinger International Institute of Mathematical Physics, Wien, Austria and by the Statens Naturvidenskabelige Forskningsråd, DenmarkSupported in part by the European Union Network EDGE.Supported in part by Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 14195 MATAcknowledgement The results of the paper go back a few years but the presentation follows a talk given by the first author in November 2002 during the program Aspects of Foliation Theory at the Erwin Schrödinger Institute in Vienna. Both authors gratefully acknowledge the hospitality and support of the Erwin Schrödinger Institute. The second author visited Å;rhus on several occasions during the preparation of this work and would like to thank the Department of Mathematics at Aarhus University for its hospitality and support. Finally we want to thank the referee for some very useful comments in particular on the terminology of gerbes and Deligne cohomology.     

Twisted parafermions

A new type of nonlocal currents (quasi-particles), which we call twisted parafermions, and its corresponding twisted Z-algebra are found. The system consists of one spin-1 bosonic field and six nonlocal fields of fractional spins. Jacobi-type identities for the twisted parafermions are derived, and a new conformal field theory is constructed from these currents. As an application, a parafermionic representation of the twisted affine current algebra A⁽²⁾₂ is given.     

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.