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Sep 18, 2013
09/13

by
A. L. Carey; Bai-Ling Wang

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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...

Source: http://arxiv.org/abs/math-ph/0505040v2

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Sep 21, 2013
09/13

by
Alan Carey; Matilde Marcolli; Bai-Ling Wang

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We establish a surgery formula for 3-dimensional Seiberg-Witten monopoles under (+1) Dehn surgery on a knot in a homology 3-sphere. (substantial revision)

Source: http://arxiv.org/abs/math/9907065v3

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Sep 19, 2013
09/13

by
Alan L. Carey; Bai-Ling Wang

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We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable proper map $f: X\to Y$ (not necessarily K-oriented). The push-forward map generalizes the push-forward map in ordinary K-theory for any $K$-oriented differentiable proper map and the Atiyah-Singer index theorem of Dirac operators on Clifford modules. For $D$-branes satisfying Freed-Witten's anomaly cancellation condition in a manifold with...

Source: http://arxiv.org/abs/math/0507414v4

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Sep 19, 2013
09/13

by
Alan L. Carey; Bai-Ling Wang

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This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion $\spinc$ structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.

Source: http://arxiv.org/abs/math/0206047v1

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Sep 21, 2013
09/13

by
Alan L. Carey; Bai-Ling Wang

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The goal of this paper is to apply the universal gerbe of \cite{CMi1} and \cite{CMi2} to give an alternative, simple and more unified view of the relationship between index theory and gerbes. We discuss determinant bundle gerbes \cite{CMMi1} and the index gerbe of \cite{L} for the case of families of Dirac operators on odd dimensional closed manifolds. The method also works for a family of Dirac operators on odd dimensional manifolds with boundary, for a pair of Melrose-Piazza's...

Source: http://arxiv.org/abs/math/0407243v2

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Sep 19, 2013
09/13

by
Alan L. Carey; Bai-Ling Wang

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In this paper, we establish the Riemann-Roch theorem in twisted K-theory extending our earlier results. As an application, we prove a twisted index formula and show that D-brane charges in Type I and Type II string theory are classified by twisted KO-theory and twisted K-theory respectively in the presence of B-fields as proposed by Witten.

Source: http://arxiv.org/abs/0909.4848v1

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Sep 19, 2013
09/13

by
Alan L. Carey; Jouko Mickelsson; Bai-Ling Wang

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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.

Source: http://arxiv.org/abs/0708.3114v4

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Sep 20, 2013
09/13

by
Alan L. Carey; Stuart Johnson; Michael K. Murray; Danny Stevenson; Bai-Ling Wang

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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 $H^4(BG, \ZZ)$ 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...

Source: http://arxiv.org/abs/math/0410013v2

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Sep 22, 2013
09/13

by
Bai-Ling Wang

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We give the definition of the Seiberg-Witten-Floer homology group for a homology 3-sphere. Its Euler characteristic number is a Casson-type invariant. For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten invariant is defined taking values in the Seiberg-Witten-Floer homology group, these relative Seiberg-Witten invariants are applied to certain homology spheres bounding Stein surfaces.

Source: http://arxiv.org/abs/dg-ga/9602003v3

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Sep 23, 2013
09/13

by
Bai-Ling Wang

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We study twisted $Spin^c$-manifolds over a paracompact Hausdorff space $X$ with a twisting $\alpha: X \to K(\ZZ, 3)$. We introduce the topological index and the analytical index on the bordism group of $\alpha$-twisted $Spin^c$-manifolds over $(X, \alpha)$, 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 analytical index. We also define a notion of...

Source: http://arxiv.org/abs/0710.1625v2

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Sep 18, 2013
09/13

by
Bai-Ling Wang

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A systematic description of the Wess-Zumino-Witten model is presented. The symplectic method plays the major role in this paper and also gives the relationship between the WZW model and the Chern-Simons model. The quantum theory is obtained to give the projective representation of the Loop group. The Gauss constraints for the connection whose curvature is only focused on several fixed points are solved. The Kohno connection and the Knizhnik-Zamolodchikov equation are derived. The holonomy...

Source: http://arxiv.org/abs/dg-ga/9504001v1

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Jun 26, 2018
06/18

by
Bohui Chen; An-Min Li; Bai-Ling Wang

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In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold stratified spaces. We introduce a concept of good gluing structure to ensure a smooth structure on the stratified space. As an application, we provide an orbifold structure on the coarse moduli space $\bar{M}_{g, n}$ of stable genus $g$ curves with $n$-marked points....

Topics: Mathematics, Geometric Topology, Symplectic Geometry

Source: http://arxiv.org/abs/1502.05103

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Jun 30, 2018
06/18

by
Bohui Chen; Bai-Ling Wang

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Let $(X,\omega)$ be a compact symplectic manifold with a Hamiltonian action of a compact Lie group $G$ and $\mu: X\to \mathfrak g$ be its moment map. In this paper, we study the $L^2$-moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends. We studied a circle-valued action functional whose gradient flow equation corresponds to the symplectic vortex equations on a cylinder $S^1\times \mathbb R$. Assume that $0$ is a regular value of the moment map $\mu$, we show that the...

Topics: Symplectic Geometry, Mathematics

Source: http://arxiv.org/abs/1405.6387

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Sep 23, 2013
09/13

by
Jianxun Hu; Bai-Ling Wang

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In this paper, we define a stringy product on $K^*_{orb}(\XX) \otimes \C $, the orbifold K-theory of any almost complex presentable orbifold $\XX$. We establish that under this stringy product, the de-locaized Chern character ch_{deloc} : K^*_{orb}(\XX) \otimes \C \longrightarrow H^*_{CR}(\XX), after a canonical modification, is a ring isomorphism. Here $ H^*_{CR}(\XX)$ is the Chen-Ruan cohomology of $\XX$. The proof relies on an intrinsic description of the obstruction bundles in the...

Source: http://arxiv.org/abs/1110.0953v2

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Jul 20, 2013
07/13

by
Matilde Marcolli; Bai-Ling Wang

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This is the third part of the work on the exact triangles. We construct chain homomorphisms and show exactness of the resulting sequence.

Source: http://arxiv.org/abs/math/0009157v1

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Sep 21, 2013
09/13

by
Matilde Marcolli; Bai-Ling Wang

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This is the second part of the proof of the exact traiangles in Seiberg-Witten Floer theory. We analyse the splitting and gluing of flow lines of the Chern-Simons-Dirac functional when the underlying three-manifold splits along a torus. (two corrections added)

Source: http://arxiv.org/abs/math/9907080v2

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Sep 18, 2013
09/13

by
Matilde Marcolli; Bai-Ling Wang

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This paper circulated previously in a draft version. Now, upon general request, it is about time to distribute the more detailed (and much longer) version. The main technical issues revolve around the fine structure of the compactification of the moduli spaces of flow lines and the obstruction bundle technique, with related gluing theorems, needed in the proof of the topological invariance of the equivariant version of the Floer homology.

Source: http://arxiv.org/abs/dg-ga/9606003v3

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Jul 20, 2013
07/13

by
Matilde Marcolli; Bai-Ling Wang

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Some aspects of the construction of SW Floer homology for manifolds with non-trivial rational homology are analyzed. In particular, the case of manifolds that are obtained as zero-surgery on a knot in a homology sphere, and for torsion spinc structures. We discuss relative invariants in the case of torsion spinc structures.

Source: http://arxiv.org/abs/math/0009159v1

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Sep 18, 2013
09/13

by
Matilde Marcolli; Bai-Ling Wang

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We establish the exact triangle in Seiberg-Witten-Floer theory relating the monopoloe homologies of any two closed 3-manifolds which are obtained from each other by $\pm 1$-surgery. We also show that the sum of the modified version of the Seiberg-Witten invariants for any closed rational homology 3-sphere $Y$ over all $Spin^c$ structures equals to $\frac 12 |H_1(Y, \Z)| \lambda (Y)$ where $\lambda (Y)$ is the Casson-Walker invariant.

Source: http://arxiv.org/abs/math/0101127v1

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Jun 30, 2018
06/18

by
Paulo Carrillo Rouse; Bai-Ling Wang

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We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant $PU(H)-$principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the...

Topics: Mathematics, K-Theory and Homology, Operator Algebras, Geometric Topology

Source: http://arxiv.org/abs/1402.3456

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Jul 19, 2013
07/13

by
Paulo Carrillo Rouse; Bai-Ling Wang

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For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of Connes-Skandalis longitudinal index theorem. When the foliation is given by...

Source: http://arxiv.org/abs/1005.3842v1

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Sep 21, 2013
09/13

by
Vicente Muñoz; Bai-Ling Wang

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We determine the Seiberg-Witten-Floer homology groups of the three-manifold which is the product of a surface of genus $g \geq 1$ times the circle, together with its ring structure, for spin-c structures which are non-trivial on the three-manifold. We give applications to computing Seiberg-Witten invariants of four-manifolds which are connected sums along surfaces and also we reprove the higher type adjunction inequalities previously obtained by Oszv\'ath and Szab\'o.

Source: http://arxiv.org/abs/math/9905050v2