Let $M$ be a compact Riemannian manifold endowed with an isometric action of a compact Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on $M$. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.
Atiyah's classical work on circular symmetry and stationary phase shows how the $\hat{A}$-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable ''antiholomorphic sector'' in the $\mathbb{C}/Λ$-equivariant cohomology of the conformal double loop space $\mathrm{Maps}(\mathbb{C}/Λ,X)$ of a rationally string manifold $X$ produces the Witten genus of $X$. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation o…
We use localization formulas in the theory of equivariant cohomology to rederive the wall crossing formulas of Li-Liu and Okonek-Teleman for Seiberg-Witten invariants.
For a rational homology 3-sphere $Y$ with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological invariants. We establish exact sequences relating them, we show that they satisfy a duality under orientation reversal, and we explain their relation to ou previous construction of equivariant Seiberg-Witten Floer (co)homologies. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduce…
We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance.
We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed.
We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first playing the most important role. These versions are shown to be equivalent to each other. A few typos are removed.
Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of $A_\infty$ structures. We extend and prove Fukaya's conjecture relating this Witten's deformed equivariant de Rham complexes, to a new Morse theoretical $A_\infty$ complexes defined by counting gradient trees with jumping which are closely related to the $\mathbb{S}^1$ equivariant symplectic cohomology proposed by S…
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.
We establish the family rigidity and vanishing theorems on the equivariant $K$-theory level for the Witten type operators on String$^c$ manifolds introduced by Chen-Han-Zhang.
Motivated by the Landau-Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function $f$ near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of Thom-Smale complex of $f$ as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large $T$. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomol…
We consider a compact, oriented, smooth Riemannian manifold $M$ (with or without boundary) and we suppose $G$ is a torus acting by isometries on $M$. Given $X$ in the Lie algebra and corresponding vector field $X_M$ on $M$, one defines Witten's inhomogeneous coboundary operator $d_{X_M} = d+ι_{X_M}: Ω_G^\pm \toΩ_G^\mp$ (even/odd invariant forms on $M$) and its adjoint $δ_{X_M}$. In the 1980s Witten showed that the resulting cohomology classes have $X_M$-harmonic representatives (forms in the null space of $Δ_{X_M} = (d_{X_M}+δ_{X_M})^2$), and the cohomology groups are isomorphic to the ordinar…
Let $(F,J,ω)$ be an almost Kähler manifold, $α$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $ω$ invariant. We introduce gauge theoretical invariants for such triples $(F,α,K)$. The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface. We give explicite descriptions of the moduli spaces associated with the triple $(\Hom(\C^r,\C^{r_0}), α_{\rm can},U(r))$, where $α_{\rm can}$ denotes the canonical action of $\hat K=U(r)\times U(r_0)$ on $\Hom(\C^r,\C^{r_0})$.…
Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In particular, we identify a candidate target for the elliptic Bismut-Chern character.
The goal of this article is to generalise the Witten deformation to even dimensional conic manifolds and a class of functions called admissible Morse functions.
We express Witten's deformation of Morse functions using deformation to the normal cone and $C^*$-modules. This allows us to obtain asymptotics of the `large eigenvalues'. Our methods extend to Morse functions along a foliation. We construct the Witten deformation using any generic function on an arbitrary foliation on a compact manifold and establish the compactness of its resolvent. When the foliation has a holonomy invariant transverse measure we show that our result implies Morse inequalities obtained by Connes and Fack in a slightly more general situation.
In this undergraduate thesis, we present an analytical proof of the Morse inequalities for closed smooth $n$-manifolds following Witten's approach. Using techniques from PDE theory, the proof is reduced to study the eigenspaces and eigenvalues of harmonic oscillators on $\mathbb{R}^n$.
In this paper, we give a brute-force proof of the Kastler-Kalau-Walze type theorem for $7$-dimensional manifolds with boundary about Witten deformation, and give a theoritic explaination of the gravitational action for $7$ dimensional manifolds with boundary.
This is a simple reading report of professor Weiping Zhang's lectures. In this article we will mainly introduce the basic ideas of Witten deformation, which were first introduced by Edward Witten on, and some applications of it. The first part of this article mainly focuses on deformation of Dirac operators and some important analytic facts about the deformed Dirac operators. In the second part of this article some applications of Witten deformation will be given, to be more specific, an analytic proof of Poincar$\acute{e}$-Hopf index theorem and Real Morse Inequilities will be given. Also we …
As is well-known, the Witten deformation of the De Rham complex computes the De Rham cohomology. In this paper we study the Witten deformation on a noncompact manifold and restrict it to differential forms which behave polynomially near infinity. Such polynomial differential forms naturally appear on manifolds with a cylindrical structure. We prove that the cohomology of the Witten deformation acting on the complex of the polynomially growing forms can be computed as the relative cohomology of the manifold with respect to the negative remote fiber of the function. We show that the assumptions …
We introduce a Witten-Novikov type perturbation $\bar\partial_{\barω}$ of the Dolbeault complex of any complex Kähler manifold, defined by a form $ω$ of type $(1,0)$ with $\partialω=0$. We give an explicit description of the associated index density which shows that it exhibits a nontrivial dependence on $ω$. The heat invariants of lower order are shown to be zero.
The Witten deformation associated to a Morse function on a closed Riemannian manifold, via Rellich-Kato theorem, relates analytically the spectral package of the Riemannian manifold (eigenvalues and eigenforms) to the Morse complex defined by the pair (Morse function, Riemannian metric) coupled with the "multivariable harmonic oscillators" associated to the critical points of the Morse function. We survey this relation and discuss some implications, including the finite subset of the spectral package referred to as the "virtually small spectral package" .
Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as $\mathbb{C}^n$ with (quasi-homogeneous) polynomials potentials, it's already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li-Yau's famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel for the Witten Laplacian with strong remainder estimate. When the deformation …
We investigate U(1)-equivariant deformations of C. LeBrun's self-dual metric with torus action. We explicitly determine all U(1)-subgroups of the torus for which one can obtain U(1)-equivariant deformation that do not preserve semi-free U(1)-action. This gives many new self-dual metrics with U(1)-action which are not conformally isometric to LeBrun metric. We also count the dimension of the moduli space of self-dual metrics with U(1)-action obtained in this way.
Associated with every quaternionic representation of a compact, connected Lie group there is a Seiberg-Witten equation in dimension three. The moduli spaces of solutions to these equations are typically non-compact. We construct Kuranishi models around boundary points of a partially compactified moduli space. The Haydys correspondence identifies such boundary points with Fueter sections - solutions of a non-linear Dirac equation - of the bundle of hyperkähler quotients associated with the quaternionic representation. We discuss when such a Fueter section can be deformed to a solution of the Se…
We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.
We prove an equivariant deformation result for Hamiltonian stationary Lagrangian submanifolds of a Kahler manifold, with respect to deformations of its metric and almost complex structure that are compatible with an isometric Hamiltonian group action. This yields existence of Hamiltonian stationary Lagrangian submanifolds in possibly non-Kahler symplectic manifolds whose metric is arbitrarily close to a Kahler metric.
It is well known that the cohomology groups of a closed manifold $M$ can be reconstructed using the gradient dynamical of a Morse-Smale function $f\colon M\to \R$. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of $f$ in term of Betti numbers of $M$. These inequalities can be deduced through a purely analytic method by studying the asymptotic behaviour of the deformed Laplacian operator. This method was introduced by E. Witten and has inspired a numbers of great achievements in Geometry and Topology in few past decades. I…
For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, and that the set of critical points of the square of the moment map is a countable discrete union of compact sets. We show that certain integrals of equivariant cohomology classes localizes as a sum of contributions from these compact critical sets, and we bound the contribution from each critical set. In the case (1) that the contribution from higher critical set…
Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian Gromov-Witten invariants of the target manifold.
We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact Kähler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves like a Toeplitz operator, and compute its first coefficient. We then use this result to establish a semi-classical trace formula for the induced quantization of symplectic maps, and give an application to Witten's asymptotic expansion conjecture for the quantum representations of t…
Let M be a spin manifold with a circular action. Given an elliptic curve E, we introduce, as in Grojnowski, elliptic bouquets of germs of holomorphic equivariant cohomology classes on M. Following Bott-Taubes and Rosu, we show that integration of an elliptic bouquet is well defined. In particular, this imply Witten's rigidity theorem. We emphasize the similarity between integration in K-theory and in elliptic cohomology.
There is a natural way to deform a Killing foliation with non-closed leaves, due to Ghys and Haefliger--Salem, into a closed foliation, i.e., a foliation whose leaves are all closed. Certain transverse geometric and topological properties are preserved under these deformations, as previously shown by the authors. For instance, the basic Euler characteristic is invariant. In this article we show that the equivariant basic cohomology ring structure is preserved under these deformations, which in turn leads to a sufficient algebraic condition (namely, equivariant formality) for the Betti numbers …
In this paper we give detailed construction of $G$-equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with $G$-action, for an arbitrary compact Lie group $G$. The proof is based on the deformation theory of {\it unstable} marked curves using the language of Lie groupoid (which is {\it not} necessary etale) and the Riemannnian center of mass technique. This proof is actually similar to [FOn,Sections 13 and 15] except the usage of the language of Lie groupoid makes the argument more transparent. This version correct some errors of the previous vers…
In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group $G$ which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a $G$-stratified space carries a system of $G$-equivariant control data. As an application, we show that if $A \subset X$ is a closed $G$-stratified subspace which is a union of strata of $X$, then the inclusion $i : A \hookrightarrow X$ is a $G$-equivariant cofibration. In particular, this theorem applies whenever $X$ is a $G$-invariant analytic subspace of an analytic $G$-manifold $M$ and …
Consider a flat vector bundle F over compact Riemannian manifold M and let f be a self-indexing Morse function on M. Let g be a smooth Euclidean metric on F. Set g_t=exp(-2tf)g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g_t. Assuming that the vector field $grad f$ satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log(ρ(t)) for t\to\infty of the form a_0+a_1t+b log(t)+o(1). We present explicit formulae for coefficients a_0,a_1 and b. In particular, we show that b is a half integer.
In this paper, we construct a smooth vector bundle over the deformation to the normal cone $\text{DNC}(V,M)$ through a rescaling of a vector bundle $E\to V$, which generalizes the construction of the spinor rescaled bundle over the tangent groupoid by Nigel Higson and Zelin Yi. We also provide an equivariant version of their construction. As the main application, we recover the Kirillov character formula for the equivariant index of Dirac-type operators. As another application, we get an equivariant generalization of the description of the Witten and the Novikov deformations of the de Rham-Dir…
The main theorem describes the behaviour of the stable cohomotopy invariant defined in the first article (joint with M. Furuta) in this series of two under the operation of taking connected sums of four-manifolds: The invariant of a connected sum is the smash product (in the sense of equivariant spectra) of the invariants of the respective summands. This provides nonvanishing results even in cases in which both the integer valued Seiberg-Witten invariants and Donaldson invariants have to vanish. As a consequence, one gets new results on diffeomorphism types of decomposable four-manifolds.
In this paper, a vanishing theorem is stated and proved. If a 4-manifold $M$ admits a smooth action by a cyclic group $\mathbb{Z}_r$, then given an $\mathbb{Z}_r$-equivariant $Spin^c$-structure $\mathcal{C}$ on $M$, the Seiberg-Witten invariant $SW\mathcal{(C)}$ is zero modulo $r$ under some slight assumptions. Here $r$ can be any positive integer. This theorem is a generalization of the mod p vanishing theorem when $\mathbb{Z}_p$ is prime order cyclic proved by F.Fang.
