arXiv subjects

Siye Wu

Publications and source records attributed to Siye Wu.

At least 37 records · Page 2Linked to original sources

arXiv2018

Conditions of smoothness of moduli spaces of flat connections and of character varieties

We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give a complete proof of the slice theorem for the action of the group of gauge transformations on the space of flat connections. Consequently, the slice is smooth if the second cohomology of the manifold with coefficients in the semisimple part of the adjoint bundle vanishes. On the other hand, we find that the smoothness of the slice for the character variety of a finitely generated and presented group depends not only on the second group cohomology but also on the relation module of the presentation. However, when there is a single relator or if there is no relation among the relators in the presentation, our condition reduces to the minimality of the second group cohomology. This is also verified using Fox calculus. Finally, we compare the conditions of smoothness in the two approaches.

math.DG↗

arXiv2012

Topology and Flux of T-Dual Manifolds with Circle Actions

We present an explicit formula for the topology and H-flux of the T-dual of a general type II compactification, significantly generalizing earlier results. Our results apply to T-dualities with respect to any circle action on spacetime. As before, T-duality exchanges type IIA and type IIB string theories. A new consequence is that the T-dual spacetime is a singular space when the fixed point set is non-empty; the singularities correspond to Kaluza-Klein monopoles. We propose that the Ramond-Ramond charges of type II string theories on the singular dual are classified by twisted equivariant cohomology groups. We also include the K-theory approach.

hep-th↗

arXiv2011

Bundle gerbes and moduli spaces

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analytically constructed index bundle gerbe. We apply these constructions to certain moduli spaces associated to compact Riemann surfaces, constructing on these moduli spaces, natural bundle gerbes with connection and curving, whose 3-curvature represent Dixmier-Douady classes that are generators of the third de Rham cohomology groups of these moduli spaces.

math.DG↗

arXiv2011

Analytic torsion for twisted de Rham complexes

We define analytic torsion for the twisted de Rham complex, consisting of the spaces of differential forms on a compact oriented Riemannian manifold X valued in a flat vector bundle E, with a differential given by a flat connection on E plus an odd-degree closed differential form H on X. The difficulty lies in the fact that the twisted de Rham complex is only Z_2-graded, and so the definition of analytic torsion in this case uses pseudo-differential operators and residue traces. We show that when dim X is odd, then the twisted analytic torsion is independent of the choice of metrics on X and E and of the representative H in the cohomology class of H. We define twisted analytic torsion in the context of generalized geometry and show that when H is a 3-form, the deformation H -> H - dB, where B is a 2-form on X, is equivalent to deforming a usual metric g to a generalized metric (g,B). We establish some basic functorial properties. When H is a top-degree form, we compute the torsion, define its simplicial counterpart and prove an analogue of the Cheeger-Muller Theorem. We also study the relationship of the analytic torsion for T-dual circle bundles with integral 3-form fluxes.

math.DG↗

arXiv2010

Projective flatness in the quantisation of bosons and fermions

We compare the quantisation of linear systems of bosons and fermions. We recall the appearance of projectively flat connection and results on parallel transport in the quantisation of bosons. We then discuss pre-quantisation and quantisation of fermions using the calculus of fermionic variables. We then define a natural connection on the bundle of Hilbert spaces and show that it is projectively flat. This identifies, up to a phase, equivalent spinor representations constructed by various polarisations. We introduce the concept of metaplectic correction for fermions and show that the bundle of corrected Hilbert spaces is naturally flat. We then show that the parallel transport in the bundle of Hilbert spaces along a geodesic is the rescaled projection or the Bogoliubov transformation provided that the geodesic lies within the complement of a cut locus. Finally, we study the bundle of Hilbert spaces when there is a symmetry.

math.SG↗

arXiv2010

Analytic Torsion of Z_2-graded Elliptic Complexes

We define analytic torsion of Z_2-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analytic and twisted holomorphic torsions, etc. The definition uses pseudo-differential operators and residue traces. We also study properties of analytic torsion for Z_2-graded elliptic complexes, including the behavior under variation of the metric. For compact odd dimensional manifolds, the analytic torsion is independent of the metric, whereas for even dimensional manifolds, a relative version of the analytic torsion is independent of the metric. Finally, the relation to topological field theories is studied.

math.DG↗

arXiv2010

Twisted Analytic Torsion

We review the Reidemeister torsion, Ray-Singer's analytic torsion and the Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differential forms on the total space of a principal circle bundle twisted by an invariant flux form. We show that when the dimension is even, such a torsion is invariant under certain deformation of the metric and the flux form. Under T-duality which exchanges the topology of the bundle and the flux form and the radius of the circular fiber with its inverse, the twisted torsions of invariant forms are inverse to each other for any dimension.

math.DG↗

arXiv2009

Massey Product and Twisted Cohomology of A-infinity Algebras

We study the twisted cohomology groups of $A_\infty$-algebras defined by twisting elements and their behavior under morphisms and homotopies using the bar construction. We define higher Massey products on the cohomology groups of general $A_\infty$-algebras and establish the naturality under morphisms and their dependency on defining systems. The above constructions are also considered for $C_\infty$-algebras. We construct a spectral sequence converging to the twisted cohomology groups an show that the higher differentials are given by the $A_\infty$-algebraic Massey products.

math.AT↗

arXiv2008

Miniscule representations, Gauss sum and modular invariance

After explaining the concepts of Langlands dual and miniscule representations, we define an analog of the Gauss sum for any compact, simple Lie group with a simply laced Lie algebra. We then show a reciprocity property when a Lie group is exchanged with its Langlands dual. We also explore the relation with theta functions and modular transformations. In the non-simply laced case, we construct a unitary representation of the Hecke group which involves interesting new phase factors.

math.RT↗

arXiv2008

$S$-duality in Vafa-Witten theory for non-simply laced gauge groups

Vafa-Witten theory is a twisted N=4 supersymmetric gauge theory whose partition functions are the generating functions of the Euler number of instanton moduli spaces. In this paper, we recall quantum gauge theory with discrete electric and magnetic fluxes and review the main results of Vafa-Witten theory when the gauge group is simply laced. Based on the transformations of theta functions and their appearance in the blow-up formulae, we propose explicit transformations of the partition functions under the Hecke group when the gauge group is non-simply laced. We provide various evidences and consistency checks.

hep-th↗

arXiv2005

Mathai-Quillen Formalism

We begin with the construction of Mathai-Quillen's Thom form. We also study the case with group actions, with a review of equivariant cohomology and then Mathai-Quillen's construction in this setting. Next, we show that much of the above can be formulated as a "field theory" on a superspace of one fermionic dimension. Finally, we present the interpretation of topological field theories using the Mathai-Quillen formalism.

hep-th↗

arXiv2004

Geometric Quantization, Parallel Transport and the Fourier Transform

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle H over the space J of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle H -> J is a rescaled orthogonal projection or Bogoliubov transformation. We then construct the kernel for the integral parallel transport operator. Finally, by extending geodesics to the boundary (for which the metaplectic correction is essential), we obtain the Segal-Bargmann and Fourier transforms as parallel transport in suitable limits.

math.SG↗

arXiv1998

A Note on Higher Cohomology Groups of Kähler Quotients

Consider a holomorphic torus action on a possibly non-compact Kähler manifold. We show that the higher cohomology groups appearing in the geometric quantization of the symplectic quotient are isomorphic to the invariant parts of the corresponding cohomology groups of the original manifold. For non-Abelian group actions on compact Kähler manifolds, this result was proved recently by Teleman and by Braverman. Our approach is applying the holomorphic instanton complex to the prequantum line bundles over the symplectic cuts. We also settle a conjecture of Zhang and the present author on the exact sequence of higher cohomology groups in the context of symplectic cutting.

math.SG↗

arXiv1998

Equivariant Holomorphic Morse Inequalities II: Torus and Non-Abelian Group Actions

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying directions in the Lie algebra of the torus. We apply the results to invariant line bundles over toric manifolds. If the group is non-Abelian, there is in addition an action of the Weyl group on the fixed-point set of its maximal torus. The sum over the fixed points can be rearranged into sums over the Weyl group (having incorporated the character of the isotropy representation on the fiber) and over its orbits.

dg-ga↗

arXiv1998

On the Instanton Complex of Holomorphic Morse Theory

Consider a holomorphic torus action on vector bundles over a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixed-point set are partially ordered, we construct, using sheaf-theoretical techniques, two spectral sequences that converges to the twisted Dolbeault cohomology groups and those with compact support, respectively. These spectral sequences are the holomorphic counterparts of the instanton complex in standard Morse theory. The results proved imply holomorphic Morse inequalities and fixed-point formulas on a possibly non-compact manifold. Finally, a number of examples and applications are given.

math.AG↗

arXiv1997

Equivariant Holomorphic Morse Inequalities III: Non-Isolated Fixed Points

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kahler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the fixed-point set. We apply the inequalities to obtain relations of Hodge numbers of the connected components of the fixed-point set and the whole manifold. We also investigate the consequences in geometric quantization, especially in the context of symplectic cutting.

dg-ga↗

arXiv1996

Equivariant Holomorphic Morse Inequalities I: A Heat Kernel Proof

Assume that the circle group acts holomorphically on a compact Kähler manifold with isolated fixed points and that the action can be lifted holomorphically to a holomorphic Hermitian vector bundle. We give a heat kernel proof of the equivariant holomorphic Morse inequalities. We use some techniques developed by Bismut and Lebeau. These inequalities, first obtained by Witten using a different argument, produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomologies in terms of the data of the fixed points.

dg-ga↗

arXiv1994

On the Mathai-Quillen Formalism of Topological Sigma Models

We present a Mathai-Quillen interpretation of topological sigma models. The key to the construction is a natural connection in a suitable infinite dimensional vector bundle over the space of maps from a Riemann surface (the world sheet) to an almost complex manifold (the target). We show that the covariant derivative of the section defined by the differential operator that appears in the equation for pseudo-holomorphic curves is precisely the linearization of the operator itself. We also discuss the Mathai-Quillen formalism of gauged topological sigma models.

hep-th↗