Questions tagged [index-theory]
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181 questions
1 vote
1 answer
166 views
Endomorphisms of Clifford modules
Assume $n>0$ is even. Let ${\mathbb C}\text{l}_n=\text{Cl}_n\otimes{\mathbb C}$, where $\text{Cl}_n$ is the (real) Clifford algebra associated with $V={\mathbb R}^n$. Let $S$ be (the) irreducible ...
- 649
6 votes
1 answer
246 views
Leading symbols and index theorem for arbitrary pseudo-differential operators
Let $M$ be a compact smooth manifold. If $E,F$ are smooth complex bundles over $M$ and $P:C^{\infty}(M;E)\to C^{\infty}(M;F)$ is a sufficiently nice $\Psi$DO (pseudo-differential operator) of degree $...
4 votes
1 answer
273 views
"Heat Kernels and Dirac Operators" existence and uniqueness proof
I'm trying to better understand the heat equation for vector bundles over Riemannian manifolds, so I've been reading "Heat Kernels and Dirac Operators" by Berlin, Getzler, and Vergne. I'm ...
- 195
0 votes
0 answers
175 views
Calculi of pseudodifferential operators and K-theory
I am reading the thesis of Chris Kottke (https://dspace.mit.edu/bitstream/handle/1721.1/60193/681923895-MIT.pdf) and I would need some help to try to understand intuitively why he makes the choice of ...
- 339
2 votes
0 answers
165 views
Is the fixed point index bounded?
I am working with the notion of fixed point index presented in the book "The Lefschetz fixed point theorem" of Robert Brown (MR283793, Zbl 0216.19601) and I would like to know if given any ...
- 21
5 votes
0 answers
155 views
Fredholm index of degenerate elliptic PDE
We consider the following degenerate elliptic PDE on the unit ball $B\subset \mathbb{R}^n$: $$ L(u):= -\operatorname{div}(a \, \nabla u) = 0, $$ where $a\in C^\infty(B;[0,+\infty))$ satisfies $a(0)=0$...
- 625
2 votes
0 answers
138 views
Quillen bundles and 2D CFTs
Roughly speaking, a (mathematical) genus-$0$ conformal field theory (CFT) is a projective symmetric monoidal functor $Z$ from $C$ to $GrVec$ [1], where $GrVec$ is the category of graded complex vector ...
- 5,748
9 votes
1 answer
253 views
Is there a relationship between fusion and S^1-equivariance for spinors on loop space?
A while ago (officially in 1987), Witten conjectured that string structures on a manifold $M$ correspond to "$S^1$-equivariant" (or more precisely $\mathrm{Diff}^+(S^1)$-equivariant) spin ...
- 2,255
14 votes
1 answer
773 views
Different proof techniques of the Atiyah-Singer index theorem
I am aware of the usual K-theoretical (cobordism, operator algebras) and heat kernel proofs of the index theorem, as answered in other questions in this site, e.g. here. However, I recently read this ...
0 votes
0 answers
219 views
Dirac operator on a manifold with boundary
I have been trying to understand construction of Dirac operator on a manifold with boundary. I came across the following paper: Hijazi, O., Montiel, S. & Roldán, A. Eigenvalue Boundary Problems ...
- 65
1 vote
0 answers
111 views
Does the local family index theorem hold for compact manifolds with corners?
Let $\pi:X\to B$ be a submersion with closed, oriented and spin fibers of even dimension. Suppose $X$ and $B$ are compact, and let $E\to X$ be a complex vector bundle over with a Hermitian metric $g^E$...
- 1,273
1 vote
0 answers
181 views
Witten's QFT Jones polynomial work on Atiyah Patodi Singer theorem and $\hat A$ genus over Chern character
In Witten's 1989 QFT and Jones polynomial paper, he wrote in eq.2.22 that Atiyah Patodi Singer theorem says that the combination: $$ \frac{1}{2} \eta_{grav} + \frac{1}{12}\frac{I(g)}{2 \pi} $$ is a ...
- 467
4 votes
0 answers
118 views
Pfaffian elements and anomalies
If $X$ is a compact even dimensional spin manifold, then we have a family of chiral Dirac operators parametrized by $Met(X)$, the (infinite dimensional) manifold of Riemannian metrics on $X$. This is ...
- 6,897
0 votes
1 answer
235 views
Can any Clifford module bundle be extended to a Dirac bundle?
I assume that the question in the title is clear, so let me talk about its relevance: According to theorem 4.3 in Heat Kernels and Dirac Operators the index theorem \begin{equation}\tag{1} \mathrm{ind}...
- 339
4 votes
1 answer
537 views
"The index is independent of the Dirac operator"
Fix a Clifford module bundle $E$ on a compact Riemannian manifold $M$ and let $D_0$ and $D_1$ be two Dirac operators (compatible with the Clifford action). The proof of the Atiyah-Singer index theorem ...
- 339
2 votes
0 answers
277 views
Heat kernels and Dirac operators - Why are half densities invoked in the definition of heat kernels?
The authors of Heat kernels and Dirac operators chose to consider a generalized Laplacian $H$ on a bundle $E\otimes|\Lambda|^{1/2}$ and heat kernels of $H$ are defined to be sections of $(E\otimes|\...
- 339
3 votes
1 answer
358 views
Determinant line of Fredholm operators and composition of morphisms
Let $P$ be a polarization of a Hilbert space $\mathcal{H}$, i.e. a bounded idempotent: consider a group $G=GL_{res}(\mathcal{H}):=\{g \in GL(\mathcal{H}): [g,P] \in HS\}$ (where $HS$ is the set of all ...
- 9,634
4 votes
1 answer
733 views
Proof of the Hirzebruch-Riemann-Roch theorem using the Atiyah-Singer index theorem
I am trying to read the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), but I do not understand the few last steps (theorem 4.11, page ...
- 339
2 votes
0 answers
325 views
$\hat{A}$-genus of a complex manifold
I am trying to understand the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), and at the end they say that since $$TM \otimes \mathbb{C} =...
- 339
3 votes
1 answer
279 views
Explicit computations of Bismut-Cheeger eta form for $S^{2n}$ bundles
I'm interested in computing eta invariants of Dirac operators (on spinor bundles tensored with some vector bundles) on the total space of $S^{2n}$ bundles over odd-dimensional manifolds. I found the ...
- 6,143
6 votes
0 answers
155 views
K-homology fundamental class for singular varieties?
Given a smooth $\text{Spin}^c$ compact manifold without boundary $M$, a suitable normalization of the Dirac operator defines the fundamental class of $M$ in Kasparov's $KK(\mathbb{C}, C^0(M))$. This ...
- 945
3 votes
0 answers
189 views
A question about index of Dirac operator
Let $\Phi: M\to S^n$ be a map from an even-dimensional, $\dim M=n$, spin manifold $M$ with the boundary $\partial M$ to a unit sphere. And $\Phi$ is locally constant near $\partial M$. If we take a ...
- 563
3 votes
0 answers
101 views
A confusion about an assumption in the setting of the local family index theorem
Let $\pi:M\to B$ be a proper submersion with closed, oriented and spin fibers. Then one can state the local family index theorem as an equality of differential forms (ignoring the details here). I am ...
- 1,273
6 votes
1 answer
254 views
Size of Hilbert space in geometric quantization from index theorem
In these notes on geometric quantization by Nair, on page 24, the Bohr-Sommerfeld rule in quantum mechanics is interpreted in terms of the Atiyah-Singer index theorem. To be precise, the polarization ...
- 1,145
3 votes
1 answer
158 views
Index formula for elliptic operators acting on Sobolev sections vanishing on the boundary (say $D: H_0^k(\Omega) \to H_0^{k-1}(\Omega)$)
Given a first order elliptic operator $D:\Gamma(X; E)\to \Gamma(X; F)$ where $X$ is a closed manifold, and $E\to X, F\to X$ vector bundles, we know that $D$ induces a Fredholm operator between the ...
- 2,562
1 vote
0 answers
108 views
A proof that the analytic index for families is multiplicative
I am looking for a detailed proof of the analytic index for families (of geometrically defined Dirac operators is good enough and assuming the existence of the kernel bundle) is multiplicative. Any ...
- 1,273
7 votes
1 answer
319 views
What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?
H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin Geometry, (1989), p. xi: ...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic ...
- 619
5 votes
1 answer
282 views
Coefficient of the top Pontryagin class in $L$-genus
The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows: $$L_1=\frac{1}{3}p_1,$$ $$L_2=\frac{1}{45}(7p_2-p_1^2),$$ $$L_3=\frac{1}{945}(62p_3-...
- 707
1 vote
0 answers
140 views
Existence of a local spinor bundle
I am confused about the existence of a local spinor bundle. My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open ...
- 563
6 votes
0 answers
349 views
Was an index theorem for manifold with local boundary condition proven?
I would like to ask a question on the bibliography of the index theorems on manifold with boundary. Before my bibliographical research my understanding of the field was that for manifold with boundary,...
- 61
9 votes
1 answer
4k views
Is there a version of the Poincaré–Hopf theorem for manifold with corners?
As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–...
- 193
5 votes
0 answers
375 views
Atiyah–Singer Index theorem for the pedestrian / layperson
So I came across the so-called Atiyah–Singer Index Theorem (ASIT) and claims of it being an extremely powerful and versatile tool. Question. What is a truly simple application of the ASIT to obtain a ...
- 7,043
2 votes
0 answers
259 views
Riemann-Roch theorem for higher-dimensional complex manifolds
Does an analogue of the Riemann-Roch theorem hold for higher-dimensional complex manifolds? (Hirzebruch-Riemann-Roch theorem is for algebraic manifolds, but not for general complex manifolds, right?)
- 29
1 vote
0 answers
92 views
Relationship with between Clifford multiplication and pullback
Let $X$ be a smooth vector field on the even-dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
- 563
0 votes
1 answer
264 views
Question about Clifford multiplication
Let $X$ be a smooth vector field on the even dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...
- 563
2 votes
0 answers
183 views
Friedrichs Inequality
I'm a little confused with the following proof of Friedrichs inequality in Lawson's & Michelsohn's book Spin geometry, page 194, Theorem 5.4. I don't understand why the last inequality, i.e. $$ C(\...
- 21
4 votes
2 answers
576 views
Comments and reference-request on books for KK-theory
I am looking for a good reference to learn Kasparov's KK-theory, where my motivation is to understand (and hopefully can do something about) the Atiyah-Singer index theorem in terms of KK-theory. I ...
- 1,273
6 votes
0 answers
201 views
Elliptic operators with with same index but non homotopic symbols
Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$. Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold. In Atiyah-Singer "the index of ...
- 2,562
9 votes
0 answers
551 views
Why is the symbol map in Atiyah–Singer paper continuous?
I am reading "Index of elliptic operators: I", by Atiyah and Singer these days and I am trying to understand all the paper. I find it difficult to verify the following statement on page 512:...
4 votes
0 answers
76 views
Dependence of Roe algebra and coarse index on the Riemannian metric
Let $(M,g)$ be a spin Riemannian manifold. The coarse index of the Dirac operator $D$ lies in the $K$-theory of the Roe algebra, which I will denote by $C^*(M,g)$ since its construction uses $g$. I ...
- 1,943
1 vote
0 answers
54 views
Extending the Dirac operator on an open subset of a manifold and preserving positivity
Let $M$ be a spin manifold and $U\subseteq M$ an open ball. Let $D$ be the Dirac operator on $M$ with respect to some Riemannian metric $g$, acting on sections of the spinor bundle $S\to M$. Suppose ...
- 1,943
3 votes
0 answers
369 views
Discrete spectrum of Dirac operator
It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete. For example at least for $d=4$, this ...
- 2,187
15 votes
1 answer
1k views
Atiyah's proof of the moduli space of SD irreducible YM connections
In the paper "Self-duality in Four-dimensional Riemannian Geometry" (1978), Atiyah, Hitchin and Singer present a proof that the space of self-dual irreducible Yang-Mills connections is a ...
- 385
4 votes
0 answers
225 views
Push forward of Chern character and index theorem
I have some trouble understanding a proposition in Leung's paper "Symplectic Structures on Gauge Theory" published in Commun. Math. Phys. 193, 47 – 67 (1998). I expose here the setup for my ...
- 799
2 votes
1 answer
318 views
can the actions of fundamental groups annihilate homology?
Let $X$ be a path-connected manifold (or a CW complex). Let $\pi_1(X)$ be the fundamental group of $X$. Let $\pi: \tilde X\longrightarrow X$ be a covering map. For each $m\geq 0$, let $C_m(\tilde X)$ ...
- 2,000
3 votes
0 answers
85 views
Analyticity of the regularized $\eta$-invariant
The APS $\eta$- invariant of an operator $B$ with eigenvalues $\lambda$ is defined as $$\eta = \sum_\lambda sgn (\lambda)$$ which is a divergent sum and it can be regularized as follows: $$\eta(s) = \...
- 31
0 votes
1 answer
146 views
How to define an equivariant Kasparov's KK-theory map?
I'm looking for some references about how to construct an equivariant Kasparov's KK-theory map $$ \psi \ : \ KK^{G_{1}} ( A,B ) \to KK^{G_{2}} ( C,D ) $$, where, $ G_1 $ and $ G_2 $ are two distinct ...
- 325
1 vote
0 answers
85 views
A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov
Let $ G $ be a locally compact group. Let $ H $ and $ K $ be two normal subgroups of $ G $. In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$ where, $$ F(G/H,G/K) = KK^{G/H} ( ...
- 325
2 votes
0 answers
557 views
Spectrum of the Witten Laplacian on compact Riemannian manifolds
Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$ How generally is it true that this ${\rm ...
- 2,256
5 votes
1 answer
295 views
Equivalence of families indexes of Fredholm operators
Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space. For a continuous ...
- 153