Questions tagged [index-theory]
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173
questions
3
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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 ...
10
votes
1
answer
583
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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 ...
2
votes
0
answers
268
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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,
$$...
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5
votes
1
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308
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Fredholm theory of non elliptic operators
In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in ...
12
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0
answers
540
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Elementary-ish geometric proof of Hirzebruch signature theorem for Riemannian 4-manifolds?
The Hirzebruch signature theorem tells us that for a smooth compact oriented 4-manifold, the signature $\sigma(M)$ is proportional to the first Pontryagin number of $M$:
$$
3\sigma(M)= p_1(M) = k \...
- 33.9k
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0
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61
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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 $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
- 312
3
votes
1
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162
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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 ...
- 113
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votes
1
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98
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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 ...
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6
votes
1
answer
178
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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}(...
- 1,871
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0
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329
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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 ...
- 515
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2
answers
420
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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....
- 186
4
votes
1
answer
260
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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 ...
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votes
1
answer
625
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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^\...
- 179
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46
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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
6
votes
1
answer
275
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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 ...
- 2,525
2
votes
1
answer
139
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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 ...
- 2,525
2
votes
0
answers
90
views
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 ...
- 312
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votes
1
answer
270
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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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109
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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$ ...
- 767
3
votes
0
answers
81
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Some questions on defining the analytic index
The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the ...
- 1,087
13
votes
0
answers
1k
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Eta-Invariant and Atiyah-Patodi-Singer Index Theorem
In Quantum Field Theory and Jones Polynomial (equation 2.16), Witten used a formula relating the APS eta-invariant to the Chern-Simons action. Witten claimed that it is derived from the Atiyah-Patodi-...
- 615
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209
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Variation Formula of APS $\eta$-Invariant and Chern-Simons Theory
In Perturbative Expansion of Chern-Simons Theory with Noncompact Gauge Group, the author proved the variation formula of the APS $\eta$-invariant. I have a few questions about their proof. I will ...
- 615
3
votes
0
answers
136
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eta invariant and spectral flow
We know that for a family of first-order self adjoint elliptic (Fredholm) operator $A_t$, for $t\in [0,1]$ we have the formula
$$\eta(A_1)-\eta(A_0)=spfl(A_t)_{t\in[0,1]}+\int^1_0 \omega(s)ds,$$
where ...
- 1,905
9
votes
0
answers
218
views
Torsion in Atiyah Singer index formula
In the papers of Atiyah and Singer, they first show their index theorem and then derive some index formulas.
For the Fredholm index living in the integers, they use the fact that on spheres the Chern ...
- 659
2
votes
1
answer
163
views
Dirac operator on manifold with periodic end
Let $\tilde W$ be a spin closed oriented manifold, $Y$ is a codimension $1$ closed oriented submanifold of $\tilde W$, and denote the $W$ the cobordism from $Y$ to
itself obtained from cutting $\...
- 1,905
5
votes
0
answers
159
views
Integrand in equivariant Atiyah-Patodi-Singer index theorem
I am trying to determine the integrand in the equivariant Atiyah-Patodi-Singer index theorem by Donnelli ("Eta invariants for G-spaces") for spin manifolds if one only has isolated fixed points. In "...
- 291
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0
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331
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Beyond smoothness-the clear picture about the notion of a differential form
In this paper N.Teleman constructs the signature operator on an arbitrary (closed, oriented) Lipschitz manifold with coefficients in a vector bundle $\xi$. In particular the notion of a differential $...
- 9,150
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0
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202
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About mod 2 Index of Dirac Operators in 3D on Non-Orientable Manifold
I was reading Witten's paper "Fermion Path Integrals and Topological Phases" (https://arxiv.org/abs/1508.04715). He claimed that
Indeed, on
an orientable 3-manifold, the eigenvalues of the Dirac ...
- 101
9
votes
1
answer
463
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Functoriality for wrong way maps
In the K-theory formulation of the index theorem one defines the topological index in terms of the so called wrong way maps. Those maps are defined for embeddings of compact manifolds $i:X \to Y$: see ...
- 9,150
1
vote
1
answer
68
views
Example of a certain partitioned manifold
I'm looking for an example of a non-compact spin manifold $M$ and a compact subset $K\subseteq M$ such that $\partial K$ is a compact hypersurface in $M$ with $\hat{A}(\partial K)\neq 0$.
(At first I ...
- 1,871
5
votes
2
answers
237
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Elements of graded algebra associated with the algebra of differential operators as smooth sections
Let $M$ be a compact manifold and $E$ a complex vector bundle. We will consider differential operators $P$ acting between $\Gamma^{\infty}(M,E)$. Let $\mathcal{P}$ be the algebra of all differential ...
- 9,150
7
votes
0
answers
163
views
Vanishing of K-theoretic index and positive scalar curvature
I'm confused about a seemingly basic point about a classical result on positive scalar curvature and would appreciate it if an expert could help me out.
Let $M^n$ be a closed spin manifold with ...
- 1,871
4
votes
1
answer
261
views
Index formula with nonisolated fixed points
Consider a compact Riemannian manifold of even dimension $n$ admitting a $U(1)$ action. If the fixed points of the action are isolated, then Witten [1; eq. 35] gives the character-valued index of the ...
- 103
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0
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59
views
Ellipticity of certain differential operator associated to a pair of vector field via curvature tensor
What is a precise example of the following situation:
A compact Riemanian manifold $M$ admits two vector field $X,Y$ such that the the operator $$Z\mapsto R(X,Y)Z$$
Would be an elliptic operator and ...
- 312
1
vote
1
answer
154
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The index of certain differential operator on tori
Assume that $J$ is an almost complex structure on torus $\mathbb{T}^2$. Let $X$ be a non vanishing vector field on the torus. Let $g$ be a Riemannian metric with corresponding $LC$ connection $...
- 312
4
votes
0
answers
112
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Index of glued operator
Suppose $X_1$ is a manifold which has a tubular end $\mathbb R^+\times Y$, and $X_2$ is a manifold which has a tubular end $\mathbb R^-\times Y$. Here, $X_1,X_2$ are orientable manifolds and $Y$ is a ...
- 1,721
3
votes
1
answer
103
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Injectivity of the $\alpha$-genus
My question regards the map defined in Atiyah,Bott,Shapiro "Clifford modules", which equals the index of the Clifford-linear Dirac operator: $$\alpha:\Omega^\mathrm{Spin}_\ast(M)\longrightarrow KO^{-n}...
8
votes
1
answer
524
views
KK-theoretical proof of Atiyah-Singer index theorem
Does anyone know of any detailed proof of the Atiyah-Singer Index Theorem using KK-theory/ Kasparov products? References to any papers and textbooks are greatly appreciated. Thanks!
- 81
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1
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589
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supersymmetry and the de Rham complex
In Alvarez-Gaume's paper "Supersymmetry and the index theorem" there is
given a certain supersymmetric Lagrangian whose quantization, apparently, leads to the de Rham Laplacian on the exterior ...
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1
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519
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preliminary reading recommendation before embarking on Connes non commutative geometry book?
I want to try to understand non commutative geometry by reading Connes's book
..and I am discovering it is a hard book to read :-) as I miss a lot of background specially in operator algebra and ...
2
votes
1
answer
131
views
Could an inverse of (weak) Morse inequality exists in some special case?
Could an inverse Morse inequality hold in some sense? More precisely I wish the following result to be true:
Problem
$M$ is a smooth simply connected compact manifold, $dim(M)=n$, $f$ is a morse ...
- 697
2
votes
0
answers
80
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Relative version of Hopf cyclic cohomology
In this paper Connes and Moscovici introduced the Hopf algebra of transverse differential operators in order to compute the index formula for the diffeomorphism invariant geometry. They developed the ...
- 9,150
2
votes
1
answer
231
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Why is index unchanged after applying functional calculus?
Suppose $D$ is the Dirac operator on a closed spin manifold $M$, with spinors $S$. One can take the functional calculus of $D$ with respect to the continuous function $f:\mathbb{R}\rightarrow\mathbb{R}...
- 1,871
3
votes
0
answers
67
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Pseudodifferential calculus for the Diffeomorphism Invariant Geometry
In the paper "Local Index Formula in Noncommutative Geometry" Connes and Moscovici build the spectral triple $(A,H,D)$ where $A=C^{\infty}_c(P) \rtimes \Gamma$ where $\Gamma$ is an arbitrary subgroup ...
- 9,150
10
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0
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172
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Baum Connes conjecture and abstract isomorphism
Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...
- 9,150
5
votes
1
answer
310
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Reference request: Higson compactification
It seems that the idea of the Higson compactification first arose in the context of non-compact manifolds in a 1992 preprint of Higson called "The relative $K$-homology of Baum and Douglas".
It seems ...
- 1,871
3
votes
0
answers
146
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Subordination process and Bismut's proof of asymptotic formula for the heat kernel
J. Bismut proved the asymptotic formula for the heat kernel of the Laplace-Beltrami operator $\Delta$ on a manifold $M$ in one of his well-known books. Later, in his paper on the index theorem, ...
- 219
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0
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138
views
Local family index theorem, but with Chern class?
Let $\pi:X\to B$ be a proper submersion with spin fibers, and $E\to X$ a Hermitian vector bundle with a unitary connection $\nabla$. Then the local family index theorem for spin Dirac operator twisted ...
- 1,087
4
votes
0
answers
142
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Definition of the $G$-equivariant index map
My question concerns a statement on page 12 of the following paper of Baum, Connes, and Higson:
http://www.mmas.univ-metz.fr/~gnc/bibliographie/BaumConnes/Baum-Connes-Higson.pdf
about the definition ...
- 1,871
27
votes
2
answers
2k
views
Atiyah-Singer style index theorem for elliptic cohomology?
In 1994, Mike Hopkins wrote a paper called Topological Modular Forms, the Witten Genus, and the Theorem of the Cube. As usual, the introduction was fantastic, explaining the power of various cobordism ...
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