Bai-Ling Wang

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.

GEOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY

We study twisted Spin c -manifolds over a paracompact Hausdorff space X with a twisting � : X ! K(Z, 3). We introduce the topological index and the analytical index on the bordism group of �-twisted Spin c -manifolds over (X,�), taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this paper is to establish the equality between the topological index and the analyt- ical index for smooth manifolds. We also define a notion of geo metric twisted K-homology, whose cycles are geometric cycles of (X,�) analogous to Baum-Douglass geometric cy- cles. As an application of our twisted index theorem, we discuss the twisted longitudinal index theorem for a foliated manifold (X,F ) with a twisting � : X ! K(Z, 3), which generalizes the Connes-Skandalis index theorem for foliations and the Atiyah-Singer fami- lies index theorem to twisted cases.

Higher bundle gerbes and cohomology classes in gauge theories

Abstract The notion of a higher bundle gerbe is introduced to give a geometric realisation of the highder degree integral cohomology of certain manifolds. We consider examples using the infinite-dimensional spaces arising in gauge theories.

Seiberg--Witten Monopoles in Three Dimensions

A topological quantum field theory is introduced which reproduces the Seiberg-Witten invariants of four-manifolds. Dimensional reduction of this topological field theory leads to a new one in three dimensions. Its partition function yields a three-manifold invariant, which can be regarded as the Seiberg-Witten version of Cassons invariant. A Geometrical interpretation of the three dimensional quantum field theory is also given.Dimensional reduction of the Seiberg--Witten equations leads to the equations of motion of a U(1) Chern--Simons theory coupled to a massless spinorial field. A topological quantum field theory is constructed for the moduli space of gauge equivalence classes of solutions of these equations. The Euler characteristic of the moduli space is obtained as the partition function which yields an analogue of Cassons invariant.A mathematically rigorous definition of the invariant isdeveloped for homology spheres using the theory of spectral flow ofself-adjoint Fredholm operators.

WEAK UCP AND PERTURBED MONOPOLE EQUATIONS

We give a simple proof of weak Unique Continuation Property for perturbed Dirac operators, using the Carleman inequality. We apply the result to a class of perturbations of the Seiberg–Witten monopole equations that arise in Floer theory.

Fusion of Symmetric D-Branes and Verlinde Rings

We explain how multiplicative bundle gerbes over a compact, connected and simple Lie group G lead to a certain fusion category of equivariant bundle gerbe modules given by pre-quantizable Hamiltonian LG-manifolds arising from Alekseev-Malkin-Meinrenken’s quasi-Hamiltonian G-spaces. The motivation comes from string theory namely, by generalising the notion of D-branes in G to allow subsets of G that are the image of a G-valued moment map we can define a ‘fusion of D-branes’ and a map to the Verlinde ring of the loop group of G which preserves the product structure. The idea is suggested by the theorem of Freed-Hopkins-Teleman. The case where G is not simply connected is studied carefully in terms of equivariant bundle gerbe modules for multiplicative bundle gerbes.

Delocalized Chern character for stringy orbifold K-theory

In this paper, we define a stringy product on K � orb(X) C, the orbifold K-theory of any almost complex presentable orbifold X. We establish that under this stringy product, the delocali zed Chern character chdeloc : K �(X) C ! H � CR(X), after a canonical modification, is a ring isomorphism. Here H � CR(X) is the Chen-Ruan cohomology of X. The proof relies on an intrinsic description of the obstruc tion bundles in the construction of the Chen- Ruan product. As an application, we investigate this stringy product on the equivariant K-theory K � G(G) of a finite group G with the conjugation action. It turns out that the stringy pr oduct is different from the Pontryagin product (the latter is also called the fusion pro duct in string theory).

Seiberg–Witten and Casson–Walker Invariants for Rational Homology 3-Spheres

AbstractWe consider a modified version of the Seiberg–Witten invariants for rational homology 3-spheres, obtained by adding to the original invariants a correction term which is a combination of η-invariants. We show that these modified invariants are topological invariants. We prove that an averaged version of these modified invariants equals the Casson–Walker invariant. In particular, this result proves an averaged version of a conjecture of Ozsváth and Szabó on the equivalence between their

Gluing Principle for Orbifold Stratified Spaces

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