Questions tagged [index-theory]
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154
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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
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0
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83
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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 ...
- 443
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97
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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,...
- 51
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383
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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–...
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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 ...
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0
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171
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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
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56
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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 ...
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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 ...
- 443
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108
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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
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131
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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 ...
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129
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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 ...
9
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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:...
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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 ...
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35
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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 ...
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171
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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 ...
- 2,047
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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 ...
- 253
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114
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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 ...
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1
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203
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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 $C_m(\tilde X)$ ...
- 1,940
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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:
$$\eta(s) = \...
- 41
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How to define an equivariant Kasparov's KK-theory map?
I'm looking for some references about how to construct an equivariant Kasparov's KK-theory map $$ \psi \ : \ KK^{G_{1}} ( A,B ) \to KK^{G_{2}} ( C,D ) $$, where, $ G_1 $ and $ G_2 $ are two distinct ...
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A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov
Let $ G $ be a locally compact group.
Let $ H $ and $ K $ be two normal subgroups of $ G $.
In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$
where, $$ F(G/H,G/K) = KK^{G/H} ( ...
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128
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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 ...
- 2,016
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199
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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 ...
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283
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Atiyah-Singer theorem in heat kernels and Dirac operators
I'm reading "Heat kernels and Dirac operators" by Berline, Getzler and Vergne.
I have some trouble to understand a identity on the bottom of page 146 which is essential for the proof of the ...
5
votes
1
answer
312
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McKean-Singer formula in Heat Kernels and Dirac Operators book
I'm reading "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne. The setting is: $E \to M$ is a $\mathbb{Z}_2$-graded vector bundle on a compact Riemannian manifold $M$
and $D :...
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Spectral flow of Dirac operator twisted by instanton
Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a ...
- 171
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Splitting formulas for spectral flows
I'm asking if there are splitting formulas for equivariant spectral flow and higher spectral flow (of Dai-Zhang) for paths of Dirac operators, concerning gluing together two smooth compact Spinc ...
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127
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Is being a deg 0 vector field equivalent to being locally non-surjective?
Let $X:\mathbb R^n\to\mathbb R^n$ be a $C^1$ (or smooth)-vector field, such that $X(0)=0$ is an isolated zero. So we can talk about the mapping of $0$ for $X$.
For convenience assume $0$ be the only ...
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Dixmier trace, Wodzicki residue and topological index
There are a well-known facts about Dixmier trace and Wodzicki residue. Let $P$ be an elliptic pseudodifferential operator of degree $−n$ on a compact Riemannian manifold $(M,g)$, than its Dixmier ...
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47
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About the orientation of index formula on orbifold
Let $X$ be a closed oriented orbifold with singularity $\Sigma X$. The singularity is defined as
$$ \Sigma X=\{(x)|~x\in X,~G_x\neq1\}, $$
where $G_x$ is the isotropy group.
For $u\in K_v(TX)$, as ...
- 1,855
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215
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Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$
The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by
$$
\eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
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$\mathbb{Z}_2$-grading by Hodge star operator (for signature theorem)
This question may be a bit low level for MO but I have not received any attention from the SE post.
Consider the algebra of exterior forms $\bigwedge T^*M$ on an even dimensional $n$-manifold $M$. We ...
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Index of Dirac operator and Chern character of symmetric product twisting bundle
I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text
We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an ...
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Twisting holomorphic vector bundles and Euler characteristics
Given a holomorphic vector bundle $\mathcal{V}$ over a compact complex manifold $M$, it seems that even if $\mathcal{V}$ is non-trivial, then it can still have trivial Euler characteristic, that it,
$$...
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Fredholm theory of non elliptic operators
In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in ...
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461
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Elementary-ish geometric proof of Hirzebruch signature theorem for Riemannian 4-manifolds?
The Hirzebruch signature theorem tells us that for a smooth compact oriented 4-manifold, the signature $\sigma(M)$ is proportional to the first Pontryagin number of $M$:
$$
3\sigma(M)= p_1(M) = k \...
- 31k
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56
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A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact parallelizable manifold
Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
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Is there a fixed-point index theorem that treats the fixed points on the boundary?
Suppose $f$ is a continuous function from a unit cube $[0,1]^n$ to itself, then $f$ has at least a fixed point. Further suppose $f$ is smooth, $0$ is a regular value of $f(x)-x$, and the fixed points ...
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An adjoint characterization of (unbounded) Fredholm operators
Let $\mathcal{H}_i$, for $i=1,2$, be Hilbert spaces, and $T:{\frak Dom}(T) \subseteq \mathcal{H}_1 \to \mathcal{H}_2$ a densely-defined closed operator. If the kernel of $T$, and the kernel of its ...
- 903
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Coarse index of Dirac operator on $\mathbb{R}$
Let $D=i\frac{d}{dx}$ be the Dirac operator on $\mathbb{R}$, acting on the spinor bundle $\mathbb{R}\times\mathbb{C}$. The bounded operator $F=\frac{D}{\sqrt{D^2+1}}$ has a coarse index
$$\text{Ind}(...
- 1,779
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Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?
I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields.
For example, the first Chern class of a complex line ...
- 565
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Matrix of cosecants appearing in equivariant index computations
In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8....
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Producing $K$-homology cycles from $KK$-cycles
For two unital (separable) $C^*$-algebras $A$ and $B$, let $(H,\rho,F)$ be a $KK$-cycle in the sense of Kasparov, or in the sense of Wikipedia :)
I wonder if there us a natural way to "forget" the ...
- 903
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471
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An integral of the Hodge-Neumann Laplacian on a Riemannian manifold
Background
Let $M$ be a compact, oriented Riemannian manifold with boundary, with $\text{vol}$ the Riemannian volume form. Let $\nu$ be the outwards unit normal vector field on $\partial M$ and $\nu^\...
- 151
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The Morse Index of a $T$- periodic geodesics is a integer number?
It is well known that compact Riemannian manifolds $(M, g)$ with
periodic geodesic flows have ( Besse Book) exceptional spectral properties: the spectrum
of $ \sqrt{ - \Delta}$, the square root of ...
- 245
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How to compute the eta invariant of torus
I was wondering how to compute the eta invariant $\eta(T^3)$ of a flat torus $T^3$, with respect to the signature operator.
In general, how can we compute the $\eta(T^3/\Gamma)$ of a finite quotient ...
- 2,465
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votes
1
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Perturbation of the adiabatic limit
Let $(M,g_M)$ be a closed oriented Riemannian manifold that has a fibration structure
$$
Y \rightarrow M \overset{\pi}{\rightarrow} B
$$
where $(Y,g_Y)$ and $(B,g_B)$ are closed Riemannian manifolds ...
- 2,465
2
votes
0
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88
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A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)
Let $M$ be a Riemannian manifold with boundary $\partial M$.
Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of ...
- 205
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Noncommutative Fredholm operators
Let $A$ be a unital $C^*$-algebra and $F:H_A\rightarrow H_A$ a Fredholm operator on the standard Hilbert $A$-module $H_A:=l^2(A)$. Is it true that $\mbox{ker}(F)$ and $\mbox{coker}(F)$ are finitely ...
- 99
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Complexifed Gauge action on determinant line bundle and change of metric
Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ ...
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