# Tagged Questions

The tag has no usage guidance.

9k views

### What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.) The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem. The ...
6k views

### Intuitive explanation for the Atiyah-Singer index theorem

My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem. I'm trying to learn the ...
541 views

### Atiyah-Singer theorem-a big picture

So far I made several attempts to really learn Atiyah-Singer theorem. In order to really understand this result rather broad background is required: you need to know analysis (pseudodifferential ...
365 views

### What is the “real” meaning of the $\hat A$ class (or the Todd class)?

In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...
854 views

### Applications of Atiyah-Singer using pseudodifferential operators

Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential ...
1k views

### How we do actually compute the topological index in Atiyah-Singer?

This is migrated by math.stackexchange as I did not receive an answer. I do not know if it is too naive for this site. I am taking a lectured class in Atiyah-Singer this semester. While the class is ...
742 views

310 views

### Why does the index of the Dirac operator on a manifold with boundary live inside the Pfaffian line of the boundary Dirac operator?

I am trying to understand a statement from page 72 of What is an elliptic object? by Stolz and Teichner. They have a spin Riemannian 2-manifold $\Sigma$, with boundary 1-manifold $Y$. The Dirac ...
203 views

### Index of a family of operators

In the usual setting of the Atiyah-Singer index theorem the situation is as follows: we have a closed smooth manifold $M$ without boundary and $D$ is some elliptic differential operator acting on ...
748 views

### Understanding the analytic index map of the Atiyah-Singer index theorem

Hi, I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn. I do not understand why the analytic index map ...
504 views

### What's the relation between the heat kernel proof of the index theorem and deformation quantization?

In the book "Heat kernels and Dirac operators", I found the slogan by Quillen in the Introduction: "Dirac operators are a quantization of the theory of connections, and the supertrace of the heat ...
237 views

### Index of Modified Dirac Operator

Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, ...
221 views

### Index theorems and orientability

Given a Dirac operator $D$ acting on some Clifford bundle $\mathcal{E}$ over a compact, even-dimensional, oriented manifold $M$, the Atiyah-Singer index theorem states that its index is given by ...
270 views

### Further directions of index theory

The Atiyah-Singer theorem is a major achievement of twentieth century mathematics. It has inspired a lot of work and people started to develop generalizations of this theorem. I would like to know the ...
370 views

### Index of a differential operator between trivial bundles.

Let $M$ be a closed parallelizable manifold and $D: \Gamma(E) \to \Gamma(F)$ an elliptic differential operator between trivial vector bundles $E,F \to M$. The Atiyah Singer index theorem implies that ...
Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $... 0answers 103 views ### Chern-Simons form and Rarita-Schwinger operator The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2. I was wondering if there exists any reference concerning the ... 0answers 212 views ### Relative index theorem for Clifford linear Dirac operators Dear community, there is relative index theorem due to Gromov and Lawson (Thm. 4.18 in POSITIVE SCALAR CURVATURE AND THE DIRAC OPERATOR ON COMPLETE RIEMANNIAN MANIFOLDS) which states that \begin{... 1answer 180 views ### Injectivity/Surjectivity of$F_A :=\frac{d}{dt} +A(t) $for a hyperbolic path of matrices$A(t)$on$H^1 $I am looking for a reference to the following problem: Given a hyperbolic (no purely imaginary eigenvalues), continuous path of matrices$A(t)$in$\mathbb{R}$with hyperolic limits at$\pm \infty $. ... 1answer 196 views ### Parametrized Atiyah-Singer index theorem Let$M$be any smooth manifold (could be unorientable - I think). Let$E,F \to M$be two complex vector bundles. Let$S$be any compact space, and let$D_s:E\to F,s\in S$be a continuous family ... 1answer 258 views ### Do we have a “topological assembly map” in the Baum-Connes conjecture? In the equivariant Atiyah-Singer index theorem, when$G$is a compact group acting on a manifold$M$and$R(G)$is the representation ring of$G$. We have the analytic index $$\text{a-ind}: K^*_G(TM)\... 0answers 259 views ### The space of Fredholm operators as a classifying space Is it true that the space of Fredholm operators on a separable Hilbert space is the classifying space for K-theory in the category of paracompact spaces? Everyone quotes the theorem of Atiyah-Janich ... 2answers 601 views ### Elliptic operators corresponds to non vanishing vector fields Let X be a non vanishing vector field on a compact manifold M. The only differential operator associated with X which I am aware of, is the derivational operator D(g)=X.g. Unfortunately ... 1answer 711 views ### The equivariant index of Dirac operator Let us consider the Dirac complex $$D_{\rm Dirac}:S^+\to S^-$$ where S^{\pm} are the chiral-spinor bundles on \mathbb{R}^4. Using the fact that the bundle S^+ is ... 1answer 105 views ### Lifting a differential operator Let D be a differential operator acting between the spaces of smooth sections of two vector bundles E,F over compact manifold M. If M is not simply connected one can construct the universal ... 1answer 232 views ### Fredholm subvector spaces of B(\mathcal{H}) Let \mathcal{H} be a separable Hilbert space with orthonormal base \{e_i\}\;\;i\in \mathbb{N}. Definition: We say a subvector space W\subset B(\mathcal{H}) is a Fredholm subspace if ... 0answers 130 views ### Submodules of a Hilbert space with finite Jones index with respect to a von Neumann algebra While studying some basic theory of Cartan subalgebras of von Neumann algebras I found the following fact that I couldn't prove: Let H be a Hilbert space, A and B trace von Neumann subalgebras ... 0answers 138 views ### How does the Atiyah-Singer index theorem in a relative setting related to “ringed spaces and pseudocoherent complexes of finite tor-dimension”? I come across the following paragraph from the article Reminiscences of Grothendieck and His School, here is from the part of the interview by Luc Illusie,: " I was indeed looking for an Atiyah-... 0answers 207 views ### The proof of the splitting principle in equivariant K-theory via flag manifolds In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely: Let j: T\... 1answer 113 views ### Local index formula for >ungraded< elliptic operators Let P\colon E \to F be an elliptic pseudodifferential operator over M. Assuming that P defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic ... 0answers 245 views ### Local Index Formula vs Atiyah Singer Index Theorem I have a question concerning the so called Local Index Formula by Connes in noncommutative geometry. First issue: why it is called Index Formula? I spoke to one person about this and he gave me the ... 2answers 334 views ### How to compute the index of such operator? Let M be a compact Riemannian manifold, with R nowhere-vanishing vector field on M(whose orbit may be closed/ not closed). E and F are two vector bundle (Edit: which are sub-bundles of \... 1answer 157 views ### derivative of the adiabatic limit of the eta invariant To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ... 1answer 333 views ### Normal coordinates near the boundary Let M be an Riemannian manifold with boundary \partial M and e_n be a unit normal vector on \partial M. With respect to e_n, around a point p on boundary, we have the usual normal ... 1answer 285 views ### The relation between the heat kernel on the principal bundle and the heat kernel on the base manifold This question is mainly about Section 5.2 of the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne. Let M be a compact Riemannian manifold without boundary and P\rightarrow M ... 1answer 333 views ### Kasparov's Dirac element and the index map In Kasparov's 1988 paper Equivariant KK-theory and the Novikov conjecture section 4 he defined the Dirac element for a (non-spin) G- Riemanian manifold X as an element in the K-homology K^0_G(... 1answer 320 views ### Index of elliptic operators III: H-structure invariant under a group G In the Atiyah-Singer paper mentioned above, they introduced on p.557 a concept called H-structure which is used to describe the Chern character of special elements of K(TX). It is roughly the ... 0answers 112 views ### Connecting homomorphism in generalized cohomology theory I have some compact manifold with boundary (M,\partial M), and there is a long exact sequence$$\cdots\to KO^{-1}(\partial M)\xrightarrow{\partial} KO^{0}(M,\partial M)\to KO^0(M)\to KO^0(\partial ... 2answers 184 views ### Proving the non-degeneracy of the critical points of the potential function for a certain vector field with$ n $point-singularities This question is an expansion of another question that I asked over at Math Stack Exchange. In what follows,$ \alpha \in \mathbb{R}_{> 1} $is a constant,$ n $a fixed integer$ \geq 2 $, and$ [...
A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...